Seminars
By Year
Title: Algebra & Combinatorics Seminar: Duadic codes over some finite rings and their applications
Speaker: Raj Kumar (Government College Hanumangarh, India)
Date: Fri, 07 Feb 2025
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, we will discuss about duadic codes (duadic group algebra codes) over some special class of finite rings. These codes are a family of abelian codes, which are themselves generalization of cyclic codes.
We will discuss about duadic codes of odd length over $\mathbb{Z}_4+u\mathbb{Z}_4, u^2=0$ and over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2, u^2=v^2=0, uv=vu$. We study these codes by considering them as a class of abelian codes and using the Fourier transform approach. In general, we will consider the algebraic structure of abelian codes over these rings. Some properties of the torsion and residue codes of abelian codes are studied. We will discuss about some results related to self-duality and self-orthogonality of duadic codes. Some conditions on the existence of self-dual augmented and extended codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ as well as over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ will be determined.
We will also discuss about a new Gray map over $\mathbb{Z}_4+u\mathbb{Z}_4$ under which an abelian code over $\mathbb{Z}_4+u\mathbb{Z}_4$ is an abelian code over $\mathbb{Z}_4$. We have obtained five new linear codes of length $18$ over $\mathbb{Z}_4$ from duadic codes of length $9$ over $\mathbb{Z}_4+u\mathbb{Z}_4$ as images of Gray map and a new map defined from $\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}_4^2$. The parameters of these codes are $[18, 4^42^{10}, 4], [18, 4^52^8, 4], [18, 4^42^5, 8], [18, 4^02^9, 8]$ and $[18, 4^22^5, 6]$. The code with parameters $[18, 4^02^9, 8]$ is self-orthogonal.
We will then discuss about abelian codes over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$, and their Gray images.
Title: Geometry & Topology Seminar: Einstein's Equations in Large Data Regime
Speaker: Puskar Mondal (CMSA, Harvard University)
Date: Fri, 31 Jan 2025
Time: 10 am
Venue: online on MS Teams
In this talk, our object of study is the Einstein-Vlasov system with a massless Vlasov matter field in topologically trivial spacetimes. Complementing various important works obtaining the stability of Minkowski spacetime as a solution to this system, we look at the large data regime, motivated in turn by the signature for decay rates of various Ricci coefficients, curvature and matter components, first introduced by X. An. Our work provides a semi-global existence result and a trapped surface formation result for the Einstein-Vlasov system in the absence of any symmetry and restriction on data size. Our proof is based on a double null gauge. Interestingly, we give a new way of obtaining estimates for the Vlasov matter, purely by commuting with various vector fields and without the need to use Jacobi fields. This is joint work with N. Athanasiou at Oxford.
Title: Algebra & Combinatorics Seminar: Frobenius and homological algebra
Speaker: Alapan Mukhopadhyay (EPFL, Lausanne, Switzerland)
Date: Fri, 31 Jan 2025
Time: 3 pm
Venue: LH-1, Mathematics Department
Frobenius or the $p$-th power map is crucial in defining singularity classes in characteristic $p > 0$, especially those appearing in the birational classification of algebraic varieties. On the other hand, the obstruction to smoothness is homological, according to a celebrated theorem of Serre. In this talk, we will show that Frobenius witnesses this homological obstruction to smoothness. This will explain the effectiveness of Frobenius in detecting singularities, from a homological point of view. The key will be to produce (explicit) generators of the bounded derived category of a variety in characteristic $p > 0$ from perfect complexes using the Frobenius pushforward functor. Our results recover earlier characterizations of smoothness using Frobenius, such as Kunz’s theorem. Part of the talk will report a joint work with Matthew Ballard, Patrick Lank, Srikanth Iyengar and Josh Pollitz.
Title: APRG Seminar: Slides A theorem of Strichartz for multipliers on homogeneous trees
Speaker: Sumit Kumar Rano (IISER Bhopal)
Date: Wed, 29 Jan 2025
Time: 3 pm
Venue: Microsoft Teams (online)
A theorem of Strichartz states that if a uniformly bounded bi-infinite sequence of functions on Euclidean spaces satisfies the property that the Laplacian acting on one function in the sequence yields the next, then every function in this sequence is an eigenfunction of the Laplacian. This result was later extended by replacing the standard Euclidean Laplacian with operators such as the d’Alembertian, the harmonic oscillator, and constant-coefficient linear partial differential operators on $\mathbb{R}^n$.
In this talk, we will explore several variants of this result for homogeneous trees, where the Euclidean Laplacian is replaced by the combinatorial Laplacian and the uniform boundedness condition is appropriately adjusted. We will then explore possible generalizations when the combinatorial Laplacian is substituted with multipliers on homogeneous trees. After presenting the result in this broader context, we will narrow our focus to specific cases, including key examples of multiplier operators such as the heat and Schrödinger operators, as well as ball and sphere averages of functions. The talk is based on a joint work with Rudra P. Sarkar.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Boomerang subalgebras of the group von Neumann algebra
Speaker: Tattwamasi Amrutam (IMPAN, Warsaw, Poland)
Date: Mon, 27 Jan 2025
Time: 4 pm
Venue: Microsoft Teams (online)
We will associate two particular objects with a countable group $\Gamma$. Consider its subgroup space $\text{Sub}(\Gamma)$, the collection of all subgroups of $\Gamma$. We can also associate with the group von Neumann algebra $L(\Gamma)$.
Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. They generalize the notion of normal subgroups. They strengthen the well-known Margulis’s normal subgroup Theorem, among many other remarkable results.
More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of subalgebras of $L(\Gamma)$.
Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of $L(\Gamma)$. In this talk, we shall connect these two very seemingly distant notions. If time permits, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We’ll also talk about its connection to understanding IRAs in such groups.
Title: Eigenfunctions Seminar: Modularity of elliptic curves, and beyond, and beneath
Speaker: Fred Diamond (King's College London, UK)
Date: Fri, 24 Jan 2025
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
In his celebrated work completed in 1995, Wiles, in part with Taylor, proved that every semistable elliptic curve over $\mathbb{Q}$ is modular, in the sense that its $L$-function is also that of a modular form. Their methods were subsequently extended by Breuil, Conrad, Taylor and myself to prove the modularity of all elliptic curves over $\mathbb{Q}$. The Modularity Theorem can be viewed as a special case of Langlands reciprocity conjectures, which continue to see exciting advances stemming from Wiles’ work in combination with further innovations. In the first half of the talk, I’ll give an overview of Wiles’ method and subsequent developments.
In addition to its most famous consequence, namely Fermat’s Last Theorem, modularity also underpins all major progress on the Birch–Swinnerton-Dyer Conjecture. Like the Modularity Theorem, the Birch–Swinnerton-Dyer Conjecture can also be viewed as an instance of a vast family of conjectures, in this case relating arithmetic invariants to special values of $L$-functions. In the second half of the talk, I’ll explain how the proof of the Modularity Theorem is itself related, by work of Hida, to another instance of these conjectures, namely for adjoint $L$-functions.
Title: APRG Seminar: Multivariate version of the Ghirlanda-Guerra identity and an application
Speaker: Aaradhya Pandey (Princeton University, USA)
Date: Wed, 22 Jan 2025
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we will introduce Ghirlanda-Guerra identities in the context of mean-field spin glasses, and prove a multivariate version. As an application, we will resolve some aspects of the problem of limiting spectra of the matrix of spin correlations coming from mean-field spin glasses.
Title: Number Theory Seminar: Reciprocity of central L-values
Speaker: Subhajit Jana (Queen Mary University of London, UK)
Date: Mon, 20 Jan 2025
Time: 4 pm
Venue: Online on MS-Teams
A reciprocity formula usually relates certain moments of two different families of $L$-functions which apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central $L$-values for $\mathrm{GL}(2)$. In this talk, we describe some instances of reciprocity formulas both in low and high-rank groups and give certain applications to subconvexity and non-vanishing of central $L$-values. These are based on the joint works with Nunes (https://arxiv.org/abs/2111.02297) and Blomer–Nelson (https://arxiv.org/abs/2404.10692).
Title: APRG Seminar: Mathematical analysis of the motion of a piston (point mass) in a fluid with density-dependent viscosity
Speaker: Abu Sufian (Universidad de Concepción, Chile)
Date: Mon, 20 Jan 2025
Time: 11:30 am
Venue: LH-1, Mathematics Department
In this talk, we examine a free boundary value problem that models the motion of a piston interacting with a viscous compressible fluid. The fluid dynamics are governed by the one-dimensional compressible Navier-Stokes equations, where the viscosity coefficient may be degenerate. The motion of the piston (point mass) is coupled with the fluid through Newton’s second law. We will discuss the existence and uniqueness of a global-in-time solution to this initial boundary value problem and explore the large-time behavior of the system.
Title: APRG Seminar: The Fourier transform on harmonic manifolds
Speaker: Kingshook Biswas (Indian Statistical Institute, Kolkata)
Date: Fri, 17 Jan 2025
Time: 3 pm
Venue: Microsoft Teams (online)
The classical Fourier inversion formula writes any nice function on Euclidean space as a superposition of “plane waves”, i.e. eigenfunctions of the Laplacian which are constant on a family of parallel hyperplanes. For the n-dimensional real hyperbolic space, and more generally for symmetric spaces of noncompact type, there is an analogous Fourier transform, called the Helgason Fourier transform, which similarly allows one to write functions as superpositions of “plane waves”, where in this case the plane waves are eigenfunctions of the Laplacian which are constant on a family of “horospheres” (these are given by the family of hypersurfaces normal to the set of geodesics meeting at a common boundary point on the boundary at infinity of the symmetric space).
We generalize the Fourier transform to the purely geometric setting of harmonic manifolds. These are Riemannian manifolds where harmonic functions satisfy the mean-value property with respect to the Riemannian surface measure on geodesic spheres. Examples of such manifolds include Euclidean space, the n-dimensional hyperbolic space, and more generally rank one symmetric spaces of noncompact type. We define a Fourier transform for negatively curved harmonic manifolds and prove a Fourier inversion formula in this setting.
In later joint work with Knieper and Peyerimhoff, this Fourier inversion formula was extended to the case of harmonic manifolds of purely exponential volume growth, a class which includes all known examples of simply connected, noncompact, nonflat harmonic manifolds.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: General purpose alternative finite difference WENO schemes (AFD-WENO) for hyperbolic conservative systems and systems with non-conservative products (with PCP and Div-preserving)
Speaker: Deepak Bhoriya (University of Notre Dame, USA)
Date: Wed, 15 Jan 2025
Time: 4 pm
Venue: Microsoft Teams (online)
Classical FD-WENO (and AFD-WENO) schemes have been available for conservation laws since the early papers by Shu and Osher [1988,1989]. Until very recently, all variants of the Finite Difference WENO scheme have indeed been restricted to treating only hyperbolic systems that are in the conservation form. The recent emergence of several classes of hyperbolic systems with non-conservative products exposes a dire need for a new class of finite difference WENO schemes that can handle such systems. To fulfill this need, we present an AFD-WENO (Alternative Finite Difference WENO) algorithm for hyperbolic systems with non-conservative products.
We present the methodology in a fluctuation form that is carefully engineered to retrieve the flux form when warranted and nevertheless extends to non-conservative products. The method is flexible because it allows any Riemann solver to be used. The formulation we arrive at is such that when non-conservative products are absent, it reverts exactly to the formulation in the citation above, which is in the exact flux conservation form. The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm. At all orders, AFD-WENO outperforms FD-WENO. We also show a very desirable result that higher-order variants of AFD-WENO schemes do not cost that much more than their lower-order variants. This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache-friendly. This should have great and very beneficial implications for the role of our AFD-WENO schemes in Peta- and Exascale computing.
We apply the method to several stringent test problems drawn from the Baer-Nunziato system, two-layer shallow water equations, and the multicomponent debris flow. The method meets its design accuracy for smooth flow and can handle stringent problems in one and multiple dimensions. Because of the pointwise nature of its update, AFD-WENO for hyperbolic systems with non-conservative products has also been shown to be a very efficient performer in solving problems with stiff source terms.
Title: Number Theory Seminar: Unipotent representations in the local Langlands correspondence
Speaker: Beth Romano (King's College London, UK)
Date: Wed, 15 Jan 2025
Time: 11 am
Venue: LH-1
The local Langlands correspondence (LLC) is a kaleidoscope of conjectures relating local Galois theory, complex Lie theory, and representations of $p$-adic groups. This talk will give an introduction to the part of the LLC involving unipotent representations. Reducing modulo $p$, we can move from representations of $p$-adic groups to representations of finite reductive groups, which have a rich structure developed by Deligne–Lusztig. I will talk about joint work with Anne-Marie Aubert and Dan Ciubotaru in which we lift some of this structure to $p$-adic groups. I will not assume previous familiarity with these topics; instead I’ll give an introduction to these ideas via examples.
Title: Number Theory Seminar: Rational points near manifolds and the Khintchine theorem
Speaker: Shreyasi Datta (University of York, UK)
Date: Mon, 13 Jan 2025
Time: 4 pm
Venue: Online on Microsoft Teams (Joint with the APRG Seminar)
We discuss a problem in Diophantine approximation which is related to counting rational points near a manifold. The proof uses tools from homogeneous dynamics and geometry of numbers. This is a joint work with Victor Beresnevich.
Title: Algebra & Combinatorics Seminar: Results in Extremal Combinatorics
Speaker: Debsoumya Chakraborti (Mathematics Institute, University of Warwick, UK)
Date: Fri, 10 Jan 2025
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
A key objective of extremal combinatorics is to investigate various conditions on combinatorial structures (such as graphs, set systems, and simplicial complexes) that guarantee the existence of specific substructures. In this talk, I will concentrate on two central topics within this theme of extremal combinatorics:
- Turán problems and
- Embedding spanning subgraphs.
I will begin with a gentle introduction to the first topic, highlighting a few fundamental questions in the field. In this context, I will introduce the Erdös–Sauer problem that asks for the maximum possible number of edges that an $n$-vertex graph can have without containing an $r$-regular subgraph. The problem had seen no progress since Pyber’s work in 1985 until recently when Janzer and Sudakov resolved this problem up to a multiplicative constant depending on $r$. We resolve the Erdös–Sauer problem up to an absolute constant factor (not depending on $r$) as follows. There exists an absolute constant $C$ such that the following holds. For each positive integer $r$, there exists some $n_0=n_0(r)$ such that if $n\geq n_0$, then every $n$-vertex graph with at least $Cr^2n\log \log n$ edges contains an $r$-regular subgraph. Moreover, we show this to be tight up to the value of $C$.
Next, I will transition to the second topic, starting with two classical results on embedding the Hamilton cycle (a cycle that visits every vertex exactly once):
- Dirac’s theorem, which establishes a sharp minimum degree condition on a graph to ensure the existence of a Hamilton cycle, and
- Theorems on various orientations of Hamilton cycles in tournaments.
In the last decade, extending subgraph embedding problems to the setting of transversals over a collection of graphs has sparked significant interest in the literature. I will introduce this concept and then discuss the transversal generalizations of (1) and (2). Some of these include results from my own work in various papers.
Title: APRG Seminar: Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions
Speaker: The Bui (Macquarie University, Sydney, Australia)
Date: Wed, 08 Jan 2025
Time: 3 pm
Venue: Microsoft Teams (online)
Let $\nu \in [-1/2, \infty)^n$, $n \geq 1$, and let $\mathcal{L}_\nu$ be a self-adjoint extension of the differential operator
\begin{equation} L_\nu := \sum_{i=1}^n \left[ - \frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2} (\nu_i^2 - \frac{1}{4}) \right] \end{equation}
on $C_c^\infty(\mathbb{R}^n_+)$ as the natural domain. In this talk, we will discuss the boundedness of the Riesz transforms associated with $\mathcal{L}_\nu$. In addition, we develop the theory of Hardy spaces and Campanato spaces associated with $\mathcal{L}_\nu$ and prove that the Riesz transforms are bounded on these Hardy spaces and Campanato spaces. This completes the description of the boundedness of the Riesz transform in the Laguerre expansion setting.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Quantum K-invariants of Grassmannian
Speaker: Shubham Sinha (ICTP Trieste, Italy)
Date: Mon, 06 Jan 2025
Time: 4 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
I will begin by reviewing some basic facts about vector bundles on the Grassmannian $Gr(r,n)$ and state the Borel–Weil–Bott theorem. The space of maps from a smooth projective curve $C$ to $Gr(r,n)$ is compactified by the Quot scheme. In this talk, we define $K$-theoretic invariants involving Euler characteristics of vector bundles over these Quot schemes. We show that these invariants naturally fit into a topological quantum field theory. Additionally, we demonstrate that the genus-zero invariants recover the quantum $K$-ring of $Gr(r,n)$, and provide a novel approach for deriving explicit formulas.
Title: Number Theory Seminar: Twisted triple product $p$-adic $L$-function for finite slope families and a $p$-adic Gross-Zagier formula
Speaker: Ananyo Kazi (UniDistance Suisse, Swetzerland)
Date: Mon, 06 Jan 2025
Time: 11 am
Venue: LH-1
In the theory of $p$-adic $L$-functions a $p$-adic Gross-Zagier formula gives interpretation to special values of $p$-adic $L$-functions outside the region of interpolation using $p$-adic integration. Seen as a first step towards “explicit reciprocity laws” they have important applications towards proving various instances of the Bloch-Kato conjecture, as in the work of Darmon–Rotger. We construct a $p$-adic twisted triple product $L$-function associated to finite slope families of Hilbert modular forms, assuming $p$ unramified in the totally real fields. We use techniques of $p$-adic iteration of the Gauss–Manin connection on sheaves of nearly overconvergent modular forms, as developed by Andreatta–Iovita. In joint work with Ting-Han Huang, we prove a $p$-adic Gross-Zagier formula for this $L$-function for a pair of an elliptic modular form and a quadratic Hilbert modular form. This generalises work of Blanco-Chacon and Fornea for the case of Hida families, and we overcome a technical assumption in their work of $p$ being split in the quadratic field.
Title: Number Theory Seminar: Poles of non-cuspidal Eisenstein series
Speaker: Devadatta Hegde (Brandeis University, USA)
Date: Tue, 31 Dec 2024
Time: 11 AM
Venue: LH-1
Let $G$ be a split semisimple group over $\mathbb{Q}$ and let $P = N \rtimes M$ be a maximal parabolic subgroup of $G$ defined over $\mathbb{Q}$. The Eisenstein series $E_P(s, \varphi)$ is an automorphic form on $G(\mathbb{Q})\G(\mathbb{A})$ built from a square-integrable automorphic form $\varphi$ on $M(\mathbb{Q})\M(\mathbb{A})^1$ and depends meromorphically on a complex spectral parameter $s \in \mathbb{C}$. The poles of these Eisenstein series in the region $\text{Re}(s) > 0$ play a central role in the spectral decomposition of automorphic forms.
In the 1960s, Langlands showed that when $\varphi$ is a cuspform, the poles of $E_P(s, \varphi)$ are determined via $L$-functions attached to $\varphi$ using
the adjoint representation of $\hat{M}$ on $\text{Lie}(\hat{N})$, where $\hat{\cdot}$ stands for the Langlands dual group. No such structural result was known about the poles of $E_P(s, \varphi)$ when $\varphi$ is not a cuspform.
It has been known since the 1980s, at least for several important examples, that there is an Arthur parameter $\mathrm{SL}_2(\mathbb{C}) \to \hat{M}$ attached to a non-cuspidal $\varphi$ on $M(\mathbb{Q})\M(\mathbb{A})^1$. The simplest example is when $\varphi = 1$, where the Arthur parameter is the principal homomorphism
$\mathrm{SL}_2(\mathbb{C}) \to \hat{M}$.
In this talk, we provide evidence that the poles of non-cuspidal Eisenstein series $E_P(s, \varphi)$ in the region $\text{Re}(s) > 0$ are related to the highest weights occurring in the decomposition of the $\mathrm{SL}_2(\mathbb{C})$ representation on $\text{Lie}(\hat{N})$ induced by the corresponding Arthur parameter, by determining the poles of the unramified degenerate Eisenstein series $E_P(s, \varphi = 1)$ using a straightforward global argument.
Title: PhD Thesis defence: Symmetric real-closed subsets of affine Kac-Moody Lie Algebras
Speaker: Dipnit Biswas (IISc Mathematics)
Date: Mon, 30 Dec 2024
Time: 11 AM
Venue: LH-1, Mathematics Department
In this thesis, we study some important classification problems related to affine Kac–Moody Lie algebras. First, we address the combinatorial problem of classifying symmetric real-closed subsets of affine root systems (which are roots of affine Kac–Moody Lie algebras). Secondly, we understand the correspondence between these symmetric real-closed subsets and the regular subalgebras generated by them, using the aforementioned classification. Motivation for this work comes from the celebrated work of Dynkin (1952), where he classified the semi-simple subalgebras of a given finite-dimensional semisimple Lie algebra $\mathfrak{g}$. He introduced the notion of regular subalgebras in order to achieve this classification. It is not hard to see that the regular subalgebras of $\mathfrak{g}$ correspond to symmetric closed subsets of roots of $\mathfrak{g}$, so the problem of classifying regular subalgebras comes down to the combinatorial problem of classifying these subsets. The analogous problem of studying regular subalgebras of affine Kac–Moody Lie algebras was initiated by Anna Felikson et al. in 2008 and continued by Roy–Venkatesh in 2019, where they addressed the part of the combinatorial problem, namely provided the classification of maximal real closed subroot systems of affine root systems.
In the finite case, it is well known that symmetric closed subsets are in fact closed subroot systems. It is not true in general for affine root systems. So, it is natural to ask the following questions:
-
When a given symmetric real-closed subset of affine root system is a closed subroot system?
-
Is it possible to classify all symmetric real-closed subsets of affine root systems?
We give affirmative answers to these questions in this thesis. In the untwisted setting, we prove that any symmetric real-closed subset is indeed a closed subroot system. Twisted types need more careful analysis, since the finite part of a symmetric real-closed subset has two possibilities in these types, namely closed and semi-closed. For semi-closed case, the behavior of symmetric real-closed subsets varies for each type. We prove that there are three types of irreducible symmetric real-closed subsets for reduced real affine root systems, one of which did not appear in Roy–Venkatesh’s work. We conclude our classification for the twisted case by determining explicitly when a symmetric closed subset is a closed subroot system, including the case when the ambient Lie algebra is the non-reduced affine Lie algebra $A_{2n}^{(2)}$.
In the second part of the thesis, we explore the correspondence between symmetric real-closed subsets and regular subalgebras generated by them. Roy–Venkatesh proved that the map between closed subroot systems and the regular subalgebras generated by them is injective. We observe that it is not true in general when we extend this map to symmetric real-closed subsets. Let $\psi$ be a real-closed subroot system. We describe the types of symmetric closed subsets that can appear in the fiber of the subalgebra generated by $\psi$. Moreover, we determine when these fibers are finite. In certain cases, we are able to describe very explicitly the defining parameters of the symmetric closed subsets appearing in the fiber.
Title: Eigenfunctions Seminar: Adjoint L-value and the Tate conjecture
Speaker: Haruzo Hida (University of California at Los Angeles, USA)
Date: Mon, 23 Dec 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
We sketch a strategy to prove the Tate conjecture on algebraic cycles for a good amount of quaternionic Shimura varieties. A key point is a twisted adjoint L-value formula relative to each quaternion algebra $D/F$ for a totally real field $F$ and its scalar extension $B=D\otimes_F E$ for a totally real quadratic extension $E_{/F}$. The theta base-change lift $\mathcal{F}$ of a Hilbert modular form $f$ to $B^\times$ has period integral over the Shimura subvariety $Sh_D\subset Sh_B$ given by $L(1,Ad(f)\otimes(\frac{E/F}{}))\ne0$; so, $Sh_D$ gives rise to a non-trivial Tate cycle in $H^{2r}(Sh_B,\mathbb{Q}_l(r))$ for $r=\dim Sh_D=\dim Sh_B/2$.
Title: Number Theory Seminar: Homogeneous dynamics and applications
Speaker: Mahbub Alam (Uppsala University, Sweden)
Date: Fri, 13 Dec 2024
Time: 2.30 pm
Venue: Online on MS-Teams
I will provide an overview of the quantitative and distributional aspects of homogeneous dynamics, focusing on Diophantine approximation and lattice point counting as key examples. In this talk, I will introduce a dynamical approach for counting the number of solutions to Diophantine inequalities over number fields. This method links the count of solutions to the problem of counting lattice points within specific regions, and to the moments of the lattice point counting function. I will discuss these moments in the broader framework of adeles. If time permits, I will also touch upon related topics, such as the sphere packing problem and the covering radius problem, emphasizing their connections to the aforementioned topics.
Title: Number Theory Seminar: Integral period relations for $\mathrm{GL}_n$, $n$ odd
Speaker: Jacques Tilouine (LAGA, Universite Paris 13, Paris, France)
Date: Thu, 05 Dec 2024
Time: 2 pm
Venue: LH-1
In a joint work in progress with B. Balasubramanyam (IISER Pune),
we generalize in two different directions, for higher odd dimensions,
an earlier work of T.-Urban on integral period relations for $\mathrm{GL}_2$.
Title: Geometry & Topology Seminar: Maps between Boundaries of Relatively Hyperbolic Groups
Speaker: Abhijit Pal (IIT Kanpur)
Date: Mon, 02 Dec 2024
Time: 04:00 pm
Venue: LH-1, Department of Mathematics
Quasi-isometric classification of groups is one of the central problems in Geometric group theory. In this talk, we will be focusing on quasi-isometries obtained from a homeomorphism between boundaries of hyperbolic groups and relatively hyperbolic groups. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalence, then those two hyperbolic groups are quasi-isometric. In this talk, we will extend Paulin’s results to relatively hyperbolic groups by introducing the notion of ‘relative quasi-Mobius maps’ between their Bowditch boundaries.
Title: Algebra & Combinatorics Seminar: A bijection between two branching models
Speaker: V. Sathish Kumar (Harish-Chandra Research Institute, Prayagraj)
Date: Fri, 29 Nov 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We prove a bijection between the branching models of Kwon and Sundaram, conjectured by Lenart–Lecouvey. To do so, we use a symmetry of the Littlewood–Richardson coefficients in terms of the hive model. Along the way, we introduce a new branching rule with flagged hives. This talk is based on a joint work with Dr. Jacinta Torres.
Title: Geometry & Topology Seminar: Tangent cones of Kahler-Ricci flow singularity models
Speaker: Maximilien Hallgren (Rutgers University)
Date: Mon, 25 Nov 2024
Time: 10:00 am
Venue: online, MS Teams
By the compactness theory of Bamler, any finite time singularity of the Kahler-Ricci flow is modeled on a singular Kahler-Ricci soliton, and such solitons are infinitesimally metric cones. In this talk, we will see that these cones are normal affine algebraic varieties, using a new method for proving Hormander-type L^2 estimates on singular shrinking solitons.
Title: Number Theory Seminar: p-adic L-functions for U(2,1) X U(1,1)
Speaker: Ming-Lun Hsieh (National Taiwan University, Taiwan)
Date: Fri, 22 Nov 2024
Time: 11 am
Venue: LH-1
In this talk, I plan to explain a construction of a five-variable $p$-adic $L$-function associated with Hida families on $U(2, 1) \times U(1, 1)$ and explain a possible connection with the p-adic Ichino-Ikeda formula.
This construction consists of two steps:
(i) the p-adic interpolation of the GGP period integrals for Hida families on $U(2, 1) \times U(1, 1)$;
(ii) the explicit evaluation of GGP period integrals via the the Ichino-Ikeda formula.
I will primarily focus on some key ingredients in the step (i), in particular, the use of $p$-integrality of the Taylor expansion of CM theta functions due to K. Bannai and S. Kobayashi. This is a joint work with M. Harris and S. Yamana.
Title: APRG Seminar: Hardy–Littlewood maximal operators on trees and Riemannian manifolds
Speaker: Stefano Meda (Università degli Studi di Milano Bicocca, Italy)
Date: Wed, 20 Nov 2024
Time: 3 pm
Venue: Microsoft Teams (online)
We consider the centred and the uncentred Hardy–Littlewood maximal operators on metric measure spaces with exponential volume growth and “locally bounded geometry”. We discuss the problem of finding the optimal range of $p$’s such that either the centred or the uncentred HL maximal operator is bounded on $L^p$.
The prototypes of the metric measure spaces we consider are the symmetric spaces of the noncompact type and the homogeneous trees, where sharp $L^p$ estimates on the HL operators are available in the literature.
We shall show that interesting phenomena arise when homogeneous trees are replaced by non-homegeoeus ones, and noncompact symmetric spaces by Riemannian manifolds with pinched negative sectional curvature.
Title: MS Thesis colloquium: λ-Decomposition of Hochschild Homology and Cyclic Homology
Speaker: Sahil Chavda Ramjibhai
Date: Wed, 20 Nov 2024
Time: 2 pm
Venue: LH-3, Department of Mathematics
Alain Connes developed Cyclic homology as a non-commutative analogue of de Rham cohomology, which can be viewed as a Lie analogue of algebraic $K$-theory. In this talk, we will begin by defining chain complexes, followed by an introduction to Hochschild homology and various perspectives on Cyclic homology. A key focus will be on the theorem stating that if $k$ contains $\mathbb{Q}$, the $k$-algebra $A$ is commutative, and the $A$-bimodule $M$ is symmetric, then the Hochschild complex $C_*(A, M)$ splits into a sum of sub-complexes $C_*^{(i)}$, $i \geq 0$, and the bicomplex $\mathcal{B}(A)$ similarly decomposes into sub-complexes $\mathcal{B}(A)^{(i)}$, $i \geq 0$. We will also discuss the Eulerian idempotents, the Eulerian decomposition of $S_n$, and Cyclic descents, which facilitate a $\lambda$-decomposition of Hochschild and Cyclic homology.
Title: APRG Seminar: Slides Quantum resonances and scattering poles
Speaker: Joachim Hilgert (Universität Paderborn, Germany)
Date: Wed, 13 Nov 2024
Time: 3 pm
Venue: Microsoft Teams (online)
For negatively curved symmetric spaces such as real hyperbolic spaces scattering matrices are defined via the standard intertwining operators for the spherical principal representations of the isometry group. They depend meromorphically on spectral parameters and it is known that their poles are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. In this talk I will explain these facts and discuss extensions to locally symmetric spaces of negative curvature.
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Slides Exceptional directions in hyperbolic FPP
Speaker: Mahan Mj (TIFR Mumbai)
Date: Fri, 08 Nov 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
First passage percolation (FPP) gives a well-known model of random geometry on a fixed background infinite graph. When we specialize to Cayley graphs of Gromov-hyperbolic groups $G$, random trees $T$ emerge naturally. The first part of the talk will dwell on setting up hyperbolic FPP and outlining its basic properties. This will have a probabilistic emphasis.
In the second part, we will specialize to the study of exceptional directions, i.e. distinct random geodesics in $T$ that converge asymptotically to the same point in the boundary $\partial G$ of $G$. This will have a geometric group theoretic emphasis (joint work with Riddhipratim Basu).
If time permits, we will describe how to reconstruct the bulk random hyperbolic geometry from the boundary. Random trees leave a trace on the boundary $\partial G$ of $G$ in the form of an evolving random partition. It turns out that there is an inverse construction, where the random metric can be reconstructed (up to bounded errors) from the data of an evolving random partition on the boundary.
Title: APRG Seminar: Slides How to represent a function in a quantum computer?
Speaker: Christoph Thiele (Universität Bonn, Germany)
Date: Wed, 06 Nov 2024
Time: 3 pm
Venue: Microsoft Teams (online)
Recently it was observed that an important algorithm in quantum signal processing (QSP) is the same as what is called elsewhere the nonlinear Fourier transform (NLFT). This has led to a new provably stable algorithm to compute the tuning parameters for this quantum algorithm. We will discuss QSP, NLFT, and the algorithm. This is joint work with Michel Alexis and Gevorg Mnatsakanyan and in part also with Lin Lin and Jiasu Wang.
The video of this talk is available on the IISc Math Department channel.
Title: Bangalore Probability Seminar: Probabilistic Solution of a Conjecture on Graphical Partitions
Speaker: Somabha Mukherjee (National University of Singapore)
Date: Mon, 04 Nov 2024
Time: 2 pm
Venue: LH-3, Department of Mathematics, Indian Institute of Science
An integer partition is called graphical if it is the degree sequence of a simple graph. In this talk, we will show that the probability that a uniformly chosen partition of size $n$ is graphical, decreases to 0 faster than $n^{-0.003}$, thereby addressing a conjecture of Boris Pittel. A lower bound of $n^{-0.5}$ for this rate was already proved by Paul Erdős and Bruce Richmond in 1993, and so this demonstrates that the probability actually decreases polynomially. Key to our argument is an asymptotic characterization of the joint distribution of the leading rows and columns of the Young diagram of a uniformly random partition, combined with a celebrated characterization of graphical partitions due to Erdős and Gallai. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.
Zoom link: https://us02web.zoom.us/j/88670406480 Meeting ID: 886 7040 6480
Title: Geometry & Topology Seminar: Visibility domains in complex manifolds
Speaker: Rumpa Masanta (IISc)
Date: Mon, 04 Nov 2024
Time: 04:00 pm
Venue: LH-1, Department of Mathematics
In this talk, we extend the notion of visibility with respect to the Kobayashi distance to domains in arbitrary complex manifolds. Visibility here is a weak notion of negative curvature and refers to a property resembling visibility in the sense of Eberlein–O’Neill for Riemannian manifolds. However, we do not assume Cauchy-completeness, with respect to the Kobayashi distance, of the domains in question. The visibility property of a domain $D$ can be used to deduce many properties of certain holomorphic mappings into $D$, ranging from their continuous extendibility to the iterative dynamics of such self-maps of $D$. Here, we present a few sufficient conditions for visibility in the above setting, and with these conditions, we see that the class of domains with the visibility property is very large. Time permitting, we will discuss an application of visibility and establish a far-reaching generalisation of the classical Wolff–Denjoy theorem.
Title: Bangalore Probability Seminar: Condensation in scale free geometric random graphs
Speaker: Neeladri Maitra (T.U. Eindhoven)
Date: Mon, 04 Nov 2024
Time: 3:15 pm
Venue: LH-3, Department of Mathematics, Indian Institute of Science
Geometric random graphs with a scale-free degree distribution has been the go-to model of random graphs in the network science community to model a large class of real-life networks. In this talk, we will focus on how condensation effects arise in such models. More specifically, we look at upper tail large deviations of the total number of edges in such graphs and show that the excess number of edges leading to the large deviation event, come from a condensation effect in the underlying degree distribution. This is in sharp contrast with the condensation effect in the `classical’ random geometric graph observed by Chatterjee and Harel (https://arxiv.org/abs/1401.7577), where the condensation instead takes place in the underlying space. This difference is due to the scale-free nature of our model. Here, the randomness coming from the highly variable degrees overpower the randomness of the underlying vertex locations, which gives rise to degree condensates - vertices with high degrees responsible for the excess number of edges in the large deviation event. We will also review the local structure and the degree distribution of the conditional graph. Based on joint work with Remco van der Hofstad, Pim van der Hoorn, Céline Kerriou and Peter Mörters.
Zoom link: https://us02web.zoom.us/j/88670406480 Meeting ID: 886 7040 6480
Title: APRG Seminar: Slides Norms on Harish-Chandra modules
Speaker: Pritam Ganguly (Universität Paderborn, Germany)
Date: Wed, 30 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, we plan to explore the natural question of how to compare different $G$-continuous norms on a Harish-Chandra module with a fixed growth rate. We demonstrate that the space of equivalence classes of these norms features both maximal and minimal elements, as well as a natural Sobolev distance. This leads to a new invariant, which we term the “Sobolev gap.” We will also discuss the natural question of estimating the Sobolev gap uniformly over all irreducible Harish-Chandra modules, offering a quantitative version of the Casselman-Wallach globalization theorem. Additionally, we provide explicit values for the Sobolev gap in the case of Harish-Chandra modules for $SL(2,\mathbb{R})$.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Geometry of Hyperconvex representations
Speaker: James Farre (Max Plank Institute, Leipzig)
Date: Mon, 28 Oct 2024
Time: 04:00 pm
Venue: online on MS Teams
A discrete group of conformal automorphisms of the Riemann sphere is called a Kleinian group. Anosov Kleinian surface groups are quasi-conformal deformations of Fuchsian surface groups; they preserve a Jordan curve cutting the sphere into two disks on which the action of the group is nice. Bers proved that quasi-Fuchsian surface group representations are determined uniquely by the pair of conformal structures at infinity, giving a natural parameterization by a product of Teichmüller spaces. We study the higher rank analogue: hyperconvex Anosov surface group representations into $\mathrm{PSL}(d,C)$, introduced by Pozzetti—Sambarino—Wienhard. We define a natural map into a product of Teichmüller spaces of Riemann surface foliations and prove the following analogue of a famous theorem of Bowen from the 70’s: The Hausdorff dimension of the limit set of a fully hyperconvex surface subgroup into $\mathrm{PSL}(d,C)$ is equal to 1 iff it is conjugated into $\mathrm{PSL}(d,R)$. This is joint work with Beatrice Pozzetti and Gabriele Viaggi.
Title: APRG Seminar: Symmetry for solutions of a Quasilinear equation on Hyperbolic Space
Speaker: Ramya Dutta (Institut Camille Jordan, Universite Claude Bernard Lyon 1, Villeurbanne, France)
Date: Fri, 25 Oct 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We will use the Alexandrov moving plane method to prove radial symmetry of positive finite energy solutions of a quasilinear elliptic equation arising from the Poincare Sobolev inequality in the Hyperbolic space. We will also discuss the existence and non-existence results in some particular cases.
Title: APRG Seminar: Slides Estimates for maximal Riesz transforms in terms of the Riesz transforms: dimension-free bounds on L^p
Speaker: Blazej Wrobel (Institute of Mathematics of the Polish Academy of Science and University of Wroclaw, Poland)
Date: Wed, 23 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In 1983 E. M. Stein proved that the vector of classical Riesz transforms has $L^p$ bounds on $\mathbb R^d$ which are independent of the dimension $d$. I will discuss an analogous result for the vector of maximal Riesz transforms. I will also mention generalizations to higher order Riesz transforms. A main principle of our approach is a comparison between a maximal Riesz transform and the corresponding Riesz transform. The talk is based on joint work with Maciej Kucharski and Jacek Zienkiewicz (Wrocław).
The video of this talk is available on the IISc Math Department channel.
Title: MS Thesis defence: On the study of Toeplitz operators on Hardy Space over a multiply connected domain
Speaker: Sushoban Das (IISc Mathematics)
Date: Tue, 22 Oct 2024
Time: 11 am
Venue: Microsoft Teams (online)
The classical theory of Toeplitz Operators on Hardy Space over the unit disk is a well-developed area in Operator Theory. If we substitute the domain disk $\Delta$ with a bounded multiply connected domain $D$, where $\partial D$ consists of finitely many smooth closed curves, what kinds of difficulties arise? This question motivates us to explore the Theory for Toeplitz Operators on Hardy Space over multiply connected domains $D$. In 1974, M.B. Abrahamse’s Ph.D. thesis made significant contributions in this topic, extending well-known results for the disk like characterizations of commutator ideals for the Banach Algebra generated by Toeplitz operators with continuous $\mathbb{C}(\partial D)$ or $H^{\infty+C}(\partial D)$ symbols, and the characterization of Fredholm operators with $H^{\infty+C}$ symbols to those for the multiply connected domain $D$. Also, he came up with a striking reduction theorem, which roughly says that modulo compact operators, the Toeplitz operator defined on the Hardy space over a multiply connected domain $D$, can be written as the direct sum of Toeplitz operators defined on the Hardy space over the unit disk.
In this talk, we will provide the definition of the Hardy Space $H^p$ over multiple connected domains $D$, where $1 \leq p \leq \infty$, and build some prerequisites to present the aforementioned characterization theorems obtained by Abrahamse for the case of multiple connected domains $D$. We will present the proofs of some of these theorems originally done by Abrahamse. Following that, we will examine the proof of the reduction theorem and explore some of its applications.
Title: Geometry & Topology Seminar: Positivity properties of the vector bundle Monge-Ampere equation
Speaker: Aashirwad Ballal (IISc)
Date: Mon, 21 Oct 2024
Time: 04:00 pm
Venue: LH-1, Mathematics Department
Monge-Ampere-positivity (MA-positivity) is a notion of positivity which arises naturally in the study of a generalization the complex Monge-Ampere equation to vector bundles. In particular, preservation of MA-positivity along a continuity path turns out to be crucial in proving the existence of solutions to the vector bundle Monge-Ampere (vbMA) equation. In this talk, we briefly introduce the vbMA equation and discuss some recent results such as the preservation of MA-positivity for rank-two holomorphic bundles over complex surfaces and the existence of counterexamples to an algebraic version of MA-positivity preservation for vector bundles of rank-three and higher over complex manifolds of dimension greater than one.
Title: Eigenfunctions Seminar: Phase transition for Elephant Random Walks on lattices
Speaker: Parthanil Roy (ISI, Bangalore)
Date: Fri, 18 Oct 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Random processes with strong memory and/or self-excitation properties arise naturally in various disciplines including physics, economics, biology, engineering, geology, etc. Many of these processes exhibit superdiffusive growth due to the effect of self-excitation. In order to model such situations, two statistical physicists Schütz and Trimper (2004) introduced a class of processes called Elephant Random Walks, in which the random walker (an elephant 🐘 with strong memory!) remembers the past steps and repeats them with some probability. In the first half of this talk, we shall discuss the basics of random walks and the elephant random walk in a lucid language. The second half will focus on a couple of models investigated by the speaker and his collaborators leading to phase transition results that verify a bunch of conjectures of Saha (2022).
Special care will be taken so that students can follow both halves of this talk. Most of the notions will be introduced and nothing more than basic probability will be prerequisite for this talk. (Based on collaborations with A. Haldar, K. Maulik, S. S. Manna and T. Sadhukhan.)
Title: Algebra & Combinatorics Seminar: Expansion, Group Homomorphism Testing, and Cohomology
Speaker: Bharatram Rangarajan (Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel)
Date: Thu, 17 Oct 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
Expansion in groups (or their Cayley graphs) is a valuable and well-studied notion in both mathematics and computer science, and describes a robust form of connectivity of graphs (a gap property of fixed points of representations of groups). It can also be interpreted as a graph on which connectivity is efficiently locally testable.
Group stability, on the other hand, is concerned with another robustness property – but of homomorphisms (or representations). Namely, is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This too can be interpreted as a local testability property of group homomorphisms in the right settings.
Expansion in groups (or property (T)) had been classically reformulated in the language of algebraic topology – in terms of the vanishing of the first cohomology of the group. In this talk we will see approaches in capturing group stability in terms of the vanishing of a second cohomology of the group, motivating higher-dimensional generalizations of expansion.
Based on joint (previous and ongoing) works with Monod, Glebsky, Lubotzky, Fournier-Facio, Dogon.
Title: APRG Seminar: Slides Quantitative characterizations of weights and parabolic boundary value problems
Speaker: Olli Saari (Universitat Politècnica de Catalunya, Barcelona, Spain)
Date: Wed, 16 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
Weights (non-negative locally integrable functions) satisfying a reverse H"older condition are important in the study of harmonic measure and boundary value problems for elliptic and parabolic partial differential equations. In this talk, I will discuss a quantitative version of a Carleson measure characterization of reverse H"older weights (originally found by Fefferman, Kenig and Pipher in the 90s) and its application to elliptic measure. In addition, I will explain extensions and modifications of the results needed in the analogous theory for parabolic measures. This is based on joint work (and work in progress) with Simon Bortz and Moritz Egert.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: The monopole-dimer model and eccentricities for Cartesian product of graphs
Speaker: Anita Arora (IISc Mathematics)
Date: Tue, 15 Oct 2024
Time: 4 PM
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
This thesis comprises two main parts. The details of the two parts are as follows:
The first part of the thesis deals with the monopole-dimer model. The dimer (resp. monomer-dimer) model deals with weighted enumeration of perfect matchings (resp. matchings). The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. A more general model called the loop-vertex model has also been defined for an oriented graph and the partition function in this case can also be written as a determinant. However, this model depends on the orientation of the graph. The monopole-dimer model interprets the loop-vertex model independent of the orientation for planar graphs with Pfaffian orientation.
The first part of the thesis focuses on the extension of the monopole-dimer model for planar graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof. We show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs so long as they are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs a la Kasteleyn and Temperley–Fischer, which turns out to be fourth power of a polynomial when all grid lengths are even. Further, we generalise this product formula to higher dimensions, again obtaining an explicit product formula. We also discuss about the asymptotic formulas for the free energy and monopole densities.
Lu and Wu (Physics Letters A, 1999) evaluated the partition function of the dimer model on two-dimensional grids embedded on a M"obius strip and a Klein bottle. We first prove a product formula for the partition function of the monopole-dimer model for the higher dimensional grid graphs with cylindrical and toroidal boundary conditions. We then consider the monopole-dimer model on high-dimensional M"obius and Klein grids, and evaluate the partition function for three-dimensional M"obius and Klein grids. Further, we show that the formula does not generalise for the higher dimensions in any natural way. Finally, we present a relation between the product formulas for three-dimensional grids with cylindrical and M"obius boundary conditions, generalising a result of Lu and Wu.
Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max{d(v,w): w\in V(G)}$. The eccentric graph of $G$, denoted $Ec(G)$, is a graph defined on the vertices of $G$ in which two vertices are adjacent if one is eccentric to the other. In the second part of the thesis, we find the structure and the girth of the eccentric graph of trees, and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a complete classification of when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and Cartesian product of two cycles. Finally, we determine the conditions under which the eccentricity matrix of Cartesian product of trees becomes invertible.
Title: Bangalore Probability Seminar: Optimization and dynamics on large graphs
Speaker: Raghavendra Tripathi (UW Seattle/NYU Abu Dhabi)
Date: Mon, 14 Oct 2024
Time: 2 pm to 3:45 pm
Venue: LH-1, Mathematics Department
The problem of optimizing the edge weights of large networks arises in many contexts, for example, machine learning, extremal graph theory, and large deviations of exponential random graphs. Finding the exact minimizers in these problems is often very difficult. In practice, various dynamic optimization schemes such as gradient descent or stochastic gradient descent are used to find the approximate minimizers. From a practical viewpoint, it is often important to study these algorithms or optimization problems in large dimensions. These algorithms yield a process of graphs/matrices and it is natural to investigate the limits of these processes as the dimension goes to infinity. This is done via taking the graphon limit of these processes.
In the first part of this talk, we give several motivating examples where such optimization problems arise and the algorithms that are used in practice. We give a brief introduction to the theory of graphons and recast these problems in the framework of graphons. In the second part of the talk, we introduce the notion of gradient flow on graphons and argue the existence and uniqueness of gradient flow for a rich class of functions. We then show that a large class of algorithms on large graphs does have a graphon limit and the limit can be described as the gradient flow or regularization. Time permitting, we will discuss some applications and limitations of this approach and end with several problems that remain to be investigated.
Title: Algebra & Combinatorics Seminar: Plücker inequalities for weakly separated coordinates in the TNN Grassmannian
Speaker: Prateek Kumar Vishwakarma (IISc Mathematics)
Date: Wed, 09 Oct 2024
Time: 4:15 pm
Venue: LH-1, Mathematics Department
We show that the partial sums of the long Plücker relations for pairs of weakly separated Plücker coordinates oscillate around 0 on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher–Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat–Vishwakarma (2024). In fact we obtain a characterization of weak separability, by showing that no other pair of Plücker coordinates satisfies this property.
Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley–Lieb immanants, and Plücker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian. This is joint work with Daniel Soskin.
Title: APRG Seminar: Slides $L^p$ bounds of maximal operators given by Fourier multipliers with some dilation sets
Speaker: Jin Bong Lee (Seoul National University, South Korea)
Date: Wed, 09 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, the speaker considers two types of maximal operators given by Fourier multipliers, and suggests criteria for Fourier multipliers so that the associated maximal operators are bounded on $L^p$ for each $p$. In other words, we first consider maximal operators given by taking supremum over $t > 0$ where $t$ is a dilation factor of Fourier multipliers, $m(t\xi)$. The condition for $m$ may be understood as a vector-valued analogue of the Hörmander–Mikhlin multiplier theorem. For the second type of maximal operators, we take the supremum over $t \in E$ with $0 \leq \dim(E) < 1$. Together with the dimension of $E$, the condition for $m$ associated with the first maximal operators is still valid for the second maximal operators.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: A Shannon–Kneser–Poulsen Theorem
Speaker: Gautam Aishwarya (Technion, Haifa, Israel)
Date: Tue, 08 Oct 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
Consider the following questions.
Question 1: Does the volume of a union of balls decrease if their centres are brought pairwise closer?
Question 2: Does communication over an additive white Gaussian noise channel worsen if the transmitters are brought pairwise closer?
These questions appeal to our basic intuition about geometry and information transmission, which seems to suggest the answer to both of them is yes. The first question is open; the Kneser–Poulsen conjecture asserts that it has an affirmative answer. In this talk, based on well-known analogies between convex geometry and information theory, we will frame (and prove) the natural entropic formulation of the Kneser–Poulsen conjecture. As a corollary, an affirmative answer to the second question is obtained. This talk is based on joint work with Dongbin Li.
Title: Geometry & Topology Seminar: Dynamics of Fuchsian meromorphic connections with real period
Speaker: Marco Abate (Università di Pisa and Università degli Studi Guglielmo Marconi, Italy)
Date: Mon, 07 Oct 2024
Time: 04:00 pm
Venue: LH-1, Mathematics Department (Joint with the APRG Seminar)
An interesting class of examples of holomorphic maps tangent to the identity at a fixed point in several complex variables is given by the time-1 maps of homogeneous vector fields. It is known that the study of the dynamics of these maps can be reduced to the study of the dynamics of the geodesic field of meromorphic connections on Riemann surfaces. In this talk we shall describe some recent results, obtained in collaboration with Karim Rakhimov, on the dynamics of the geodesic field for Fuchsian meromorphic connections having real periods. The main tools used are: a generalization to general Fuchsian meromorphic connections of a classical formula proved by Teichmüller for quadratic differentials, and the relationship between Fuchsian meromorphic connections with real periods and singular flat Hermitian metrics. In particular, we obtain a description of the possible $\omega$-limit sets of simple geodesics that extends and makes more precise results known for the particular case of Riemann surfaces endowed with a meromorphic $k$-differential.
Title: PhD Thesis defence: The Momentum Construction Method for Higher Extremal Kähler and Conical Higher cscK Metrics
Speaker: Rajas Sandeep Sompurkar (IISc Mathematics)
Date: Fri, 04 Oct 2024
Time: 3:30 pm
Venue: Hybrid - Google Meet (online) and LH-3, Mathematics Department
This Ph.D. thesis consists of two parts. In both the parts we study two new notions of canonical Kahler metrics introduced by Pingali viz. ‘higher extremal Kähler metric’ and ‘higher constant scalar curvature Kähler (higher cscK) metric’ both of whose definitions are analogous to the definitions of extremal Kähler metric and constant scalar curvature Kähler (cscK) metric respectively. On a compact Kähler manifold a higher extremal Kähler metric is a Kähler metric whose corresponding top Chern form equals its corresponding volume form multiplied by a smooth real-valued function whose gradient is a holomorphic vector field, while a higher cscK metric is a Kahler metric whose top Chern form is a real constant multiple of its volume form or equivalently whose top Chern form is harmonic. In both the parts we consider a special family of minimal ruled complex surfaces called as ‘pseudo-Hirzebruch surfaces’ which are the projective completions of holomorphic line bundles of non-zero degrees over compact Riemann surfaces of genera greater than or equal to two. These surfaces have got some nice symmetries in terms of their fibres and the zero and infinity divisors which enable the use of the momentum construction method of Hwang-Singer (a refinement of the Calabi ansatz procedure) for finding explicit examples of various kinds of canonical metrics on them.
In the first part of this Ph.D. thesis we will prove by using the momentum construction method that on a pseudo-Hirzebruch surface every Kahler class admits a higher extremal Kahler metric which is not a higher cscK metric. The construction of the required metric boils down to solving an ODE depending on a parameter on a closed and bounded interval with some boundary conditions, but the ODE is not directly integrable and requires a very delicate analysis for getting the existence of a solution satisfying all the boundary conditions. Then by doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude from this that higher cscK metrics (momentum-constructed or otherwise) do not exist in any Kahler class on this Kahler surface. We will briefly see the analogy of this problem with the related problem of constructing extremal Kähler metrics which are not cscK metrics on a pseudo-Hirzebruch surface which has been previously studied by Tønnesen-Friedman and Apostolov et al..
In the second part of this Ph.D. thesis we will see that if we relax the smoothness condition on our metrics a bit and allow our metrics to develop ‘conical singularities’ along at least one of the zero and infinity divisors of a pseudo-Hirzebruch surface then we do get ‘conical higher cscK metrics’ in each Kahler class of the Kahler surface by the momentum construction method. Even in this case the construction of the required metric boils down to solving a very similar ODE on the same interval but with different parameters and slightly different boundary conditions. We will show that our constructed metrics are conical Kahler metrics satisfying the strongest condition for conical metrics viz. the ‘polyhomogeneous condition’ of Jeffres-Mazzeo-Rubinstein, and we will interpret the conical higher cscK equation globally on the surface in terms of currents by using Bedford-Taylor theory. We will then employ the top ‘log Bando-Futaki invariant’ to obtain the linear relationship between the cone angles of the conical singularities of the metrics at the zero and infinity divisors of the surface.
Title: PhD Thesis defence: Holomorphic mappings and Kobayashi geometry of domains
Speaker: Annapurna Banik (IISc Mathematics)
Date: Fri, 04 Oct 2024
Time: 10 am
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
In this presentation, we shall study certain aspects of the geometry of the Kobayashi (pseudo)distance and the Kobayashi (pseudo)metric for domains in $\mathbb{C}^n$. We will focus on the following themes: on the interaction between Kobayashi geometry and the extension of holomorphic mappings, and on certain negative-curvature-type properties of Kobayashi hyperbolic domains equipped with their Kobayashi distances.
In the initial part of this talk, we shall present a couple of results on local continuous extension of proper holomorphic mappings $F:D \to \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $bD$ and $b\Omega$. These results are motivated by a well-known work by Forstneric–Rosay. However, our results allow us to have much lower regularity, for the patches of $bD, b\Omega$ that are relevant, than in earlier results in the literature. Moreover, our assumptions allow $b\Omega$ to contain boundary points of infinite type.
We will also discuss another type of extension phenomenon for holomorphic mappings, namely, Picard-type extension theorems. Well-known works by Kobayashi, Kiernan, and Joseph–Kwack have showed that Picard-type extension results hold true when the target spaces of the relevant holomorphic mappings belong to a class of Kobayashi hyperbolic complex manifolds – viewed as complex submanifolds embedded in some ambient complex manifold – with certain analytical properties. Beyond some classical examples, identifying such a target manifold by its geometric properties is, in general, hard. Restricting to $\mathbb{C}^n$ as the ambient space, we provide some geometric conditions on $b\Omega$, for any unbounded domain $\Omega \varsubsetneq \mathbb{C}^n$, for a Picard-type extension to hold true for holomorphic mappings into $\Omega$. These conditions are suggested, in part, by an explicit lower bound for the Kobayashi metric of a certain class of bounded domains. We establish the latter estimates using the regularity theory for the complex Monge–Ampere equation. The notion that allows us to connect these estimates with Picard-type extension theorems is called “visibility”.
In the concluding part of this presentation, we will explore the notion of visibility for its own sake. For a Kobayashi hyperbolic domain $\Omega \varsubsetneq \mathbb{C}^n$, $\Omega$ being a visibility domain is a notion of negative curvature of $\Omega$ as a metric space equipped with the Kobayashi distance $K_{\Omega}$ and encodes a specific way in which $(\Omega, K_{\Omega})$ resembles the Poincare disc model of the hyperbolic plane. The earliest examples of visibility domains, given by Bharali–Zimmer, are pseudoconvex. In fact, all examples of visibility domains in the literature are, or are conjectured to be, pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.
Title: APRG Seminar: Slides Bilinear Bochner–Riesz operator for convex domains in the plane
Speaker: Surjeet Singh Choudhary (IISER Bhopal)
Date: Thu, 03 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
For $\alpha\geq0$, the bilinear Bochner–Riesz operator of order $\alpha$ in $\mathbb{R}^n$ is defined as
\begin{equation} \mathcal{B}^\alpha_R(f,g)(x)=\int_{\mathbb{R}^n\times \mathbb{R}^n} \left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)^\alpha_+ \hat{f}(\xi) \hat{g}(\eta) e^{2\pi ix\cdot(\xi+\eta)} d\xi d\eta. \end{equation}
For $\alpha>n-\frac{1}{2}$, $\mathcal{B}^{\alpha}_R$ maps $L^{p_1}(\mathbb{R}^n)\times L^{p_2}(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ whenever $p_1,p_2\geq1$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. Thus, $\alpha=n-\frac{1}{2}$ is commonly referred as the critical index for the bilinear Bochner–Riesz operator. Recently, there have been some results on $L^p$-boundedness of the bilinear Bochner–Riesz operator $\mathcal{B}^{\alpha}_R$ when $\alpha\leq n-\frac{1}{2}$.
In this talk, we extend the bilinear Bochner–Riesz operator to convex domains in the plane and discuss some $L^p$-boundedness results.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: Invariants Associated with Complete Nevanlinna-Pick Spaces
Speaker: Abhay Jindal (IISc Mathematics)
Date: Tue, 01 Oct 2024
Time: 11:15 am
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we shall talk about two invariants associated with complete Nevanlinna-Pick (CNP) spaces. One of the invariants is an operator-valued multiplier of a given CNP space, and another invariant is a positive real number. These two invariants are called characteristic function and curvature invariant, respectively. The origin of these concepts can be traced back to the classical theory of contractions by Sz.-Nagy and Foias.
We extend the theory of Sz.-Nagy and Foias about the characteristic function of a contraction to a commuting tuple $(T_{1}, \dots, T_{d})$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. Surprisingly, there is a converse, which roughly says that if a kernel $k$ admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain. So, what can be said if $(T_{1}, \dots,T_{d})$ is $1/k$-contractive when $k$ is an irreducible unitarily invariant kernel, but does not have the complete Nevanlinna-Pick property? We shall see that if $k$ has a complete Nevanlinna-Pick factor $s$, then much can be retrieved.
We associate with a $1/k$-contraction its curvature invariant. The instrument that makes this possible is the characteristic function. We present an asymptotic formula for the curvature invariant. In the special case when the $1/k$-contraction is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of the $1/k$-contraction specifically when its characteristic function is a polynomial.
Title: Geometry & Topology Seminar: Stability, moment maps and holomorphic submersions
Speaker: Annamaria Ortu (University of Gothenburg)
Date: Mon, 30 Sep 2024
Time: 04:00 pm
Venue: online, MS Teams
A guiding problem in Kähler geometry is to find an equivalence between the existence of a given special metric and an algebro-geometric stability condition. In this context, a natural step towards proving the equivalence is to show that, given a manifold that admits a special metric, stable deformations of the manifold still admit a special metric. We propose a new technique to address this type of problem that relies on restricting to a finite-dimensional problem and applying the theory of moment maps and the moment map flow. We will show how to apply it to constant scalar curvature metrics and, if time permits, to holomorphic submersions.
Title: APRG Seminar: Slides Directional singular integrals in codimension one
Speaker: Ioannis Parissis (University of the Basque Country, Leioa, Spain)
Date: Wed, 25 Sep 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk I will give a brief overview of the theory of maximal directional averages and singular integrals in 2 and higher dimensions. We will see the main obstructions to the boundedness of these objects which will naturally lead us to the discussion of the connections with the Kakeya conjecture and the Stein and Zygmund conjectures. Finally, I will present a sharp estimate for maximal directional singular integrals in codimension one and general ambient dimension. This reports on joint work with O. Bakas (U. of Patras), F. Di Plinio (Napoli, Federico II) and L. Roncal (BCAM).
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Slides Algebraic positivity versus analytic positivity
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Fri, 20 Sep 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Elementary root separation arguments imply that a non-negative polynomial on the real line is a sum of two squares of polynomials. A non-constructive and ingenious enumerative geometry observation of Hilbert shows that in two or more variables, non-negative polynomials are not always sums of squares. This led to Hilbert’s 17-th problem, asking whether such a decomposition is possible in the field of rational functions. The answer is yes, due to Emil Artin. The gap between polynomials which are positive on semi-algebraic sets and corresponding weighted sums of squares was elucidated by Tarski’s elimination of quantifiers principle. The first part of the lecture will contain accessible details and historical notes on these topics, now part of Real Algebra and Real Algebraic Geometry.
In the second part of the lecture, I will show how sums of squares decompositions led F. Riesz to the definitive form of the spectral theorem for self-adjoint transforms of a Hilbert space. Implying for instance novel positivity results of harmonic analysis. Reversing the historical arrow, I will show how operator theory has put some essential marks on purely Real Algebra chapters. We will also touch positivity in non-commutative *-algebras and Lie-algebras.
The third part of the lecture will contain applications of relatively recent positivity certificates to global, non-convex optimization, stability of dynamical systems and construction of wavelet frames.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: Maps between non-compact surfaces
Speaker: Sumanta Das (IISc Mathematics)
Date: Thu, 19 Sep 2024
Time: 11:30 am
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
In this talk, we discuss proper maps between two non-compact surfaces, with a particular emphasis on facts stemming from two fundamental questions in topology: whether every homotopy equivalence between two $n$-manifolds is homotopic to a homeomorphism, and whether every degree-one self-map of an oriented manifold is a homotopy equivalence.
Topological rigidity is the property that every homotopy equivalence between two closed $n$-manifolds is homotopic to a homeomorphism. This property refines the notion of homotopy equivalence, implying homeomorphism for a particular class of spaces. According to Nielsen’s results from the 1920s, compact surfaces exhibit topological rigidity. However, topological rigidity fails in dimensions three and above, as well as for compact bordered surfaces.
We prove that all non-compact, orientable surfaces are properly rigid. In fact, we prove a stronger result: if a homotopy equivalence between any two noncompact, orientable surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided that the surfaces are neither the plane nor the punctured plane. As an application, we also prove that any $\pi_1$-injective proper map between two non-compact surfaces is properly homotopic to a finite-sheeted covering map, given that the surfaces are neither the plane nor the punctured plane.
An oriented manifold $M$ is said to be Hopfian if every self-map $f : M \to M$ of degree one is a homotopy equivalence. This is the natural topological analog of Hopfian groups. H. Hopf posed the question of whether every closed, oriented manifold is Hopfian. We prove that every oriented infinite-type surface is non-Hopfian. Consequently, an oriented surface $S$ is of finite type if and only if every proper self-map of $S$ of degree one is homotopic to a homeomorphism.
Title: Algebra & Combinatorics Seminar: Positivity preservers over finite fields
Speaker: Prateek Kumar Vishwakarma (IISc Mathematics)
Date: Wed, 18 Sep 2024
Time: 4:15 pm
Venue: LH-1, Mathematics Department
We discuss an algebraic version of Schoenberg’s celebrated theorem [Duke Math. J., 1942] characterizing entrywise matrix transforms that preserve positive definiteness. Compared to the classical real and complex settings, we consider matrices with entries in a finite field and obtain a complete characterization of such preservers for matrices of a fixed dimension. When the dimension of the matrices is at least 3, we prove that, surprisingly, the positivity preservers are precisely the positive multiples of the field’s automorphisms. Our work makes crucial use of the well-known character-sum bound due to Weil, and of a result of Carlitz [Proc. Amer. Math. Soc., 1960] that leads to characterizing the automorphisms of Paley graphs. This is joint with Dominique Guillot and Himanshu Gupta.
Title: APRG Seminar: Slides Hyponormal quantization
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Wed, 18 Sep 2024
Time: 10 am – 12:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
After a review of the Markov–Krein transform in the case of a one variable, and the Hilbert space interpretation of it (the phase shift), we will specialize the Markov–Krein transform to 2D. This will bring us to a relaxation of the Heisenberg commutation relation, this time filled by bounded linear transforms. The spectral invariant of this class of so called hyponormal operators is called the principal function. It is a measurable function of compact support, carrying a degree of shade. We will sketch the main specific results pertaining to hyponormal operators.
For the rest of the lecture we will link the resulting inverse spectral problem to image processing, potential theory, Hele–Shaw flows, integrable systems, and the regularity of free boundaries. Current advances with precise open questions will be detailed.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Slides Fourier uncertainty and bandlimited functions
Speaker: Emanuel Carneiro (ICTP Trieste, Italy)
Date: Wed, 18 Sep 2024
Time: 3 pm
Venue: Microsoft Teams (online)
This will be a broad discussion on a few different types of Fourier uncertainty principles for bandlimited functions (i.e. functions that have Fourier transforms compactly supported), and their connections to multiplication operators in certain Hilbert spaces of entire functions. Some of these problems are related to applications in analysis, PDEs and number theory, and I might describe a few of these if time permits. The talk should be accessible to a broad audience.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Slides The Fantappie transform
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Fri, 13 Sep 2024
Time: 3 – 5:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
There are several analogs of Cauchy’s integral transform in the theory of functions of several complex variables. Fantappie’s transform is one of them, possibly the simplest and oldest, with some remarkable properties we will discuss in detail. First, it is immediately connected to Radon transform, much exploited today in inverse problems. The characterization of Fantappie transforms of positive measures resonates with Bernstein’s theorem linking Laplace transforms and completely monotonic functions. One of the spectacular applications of Fantappie’s transform is the multivariate analog of Koethe–Grothendieck duality of spaces of analytic functions. The multiplicative structure of Fantappie’s transform brings us to the classical Markov–Krein correspondence, much investigated these days by probabilists.
An application to mathematical economics will be sketched.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Computing the zeta function of varieties over finite fields
Speaker: Madhavan Venkatesh (IIT Kanpur)
Date: Fri, 13 Sep 2024
Time: 4 pm
Venue: LH-2
The focus of this talk will be on computing the point counts
of algebraic varieties, i.e., number of solutions of a system of
polynomial equations over finite fields. The zeta function encodes the
point counts over an infinite tower of finite field extensions and enjoys
the property of being a rational function over $\mathbb{Q}$. Further, the zeta
function can be recovered from certain invariants of the variety in
question, using an appropriate cohomology theory. I will review the state of the art on efficient algorithms to compute the
zeta function of varieties, including the dimension one case of curves
(covering the works of Schoof, and Pila) and report on our generalisations
for the first cohomology (joint work with Diptajit Roy and Nitin
Saxena) and ongoing work on the second cohomology, which addresses a question
of Edixhoven.
Title: APRG Seminar: Slides Strichartz estimates in the Heisenberg group and generalizations
Speaker: Davide Barilari (Università degli Studi di Padova, Italy)
Date: Wed, 11 Sep 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk I will discuss Strichartz estimates on the Heisenberg group for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on the Heisenberg group is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available.
Our approach, inspired by the Fourier transform restriction method initiated by Tomas and Stein, is based on Fourier restriction theorems, using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.
Title: Bangalore Probability Seminar: Classical random matrix ensembles and planar random growth models
Speaker: Riddhipratim Basu (ICTS Bangalore)
Date: Mon, 09 Sep 2024
Time: 3:15 pm
Venue: LH-1, Mathematics Department
There are some remarkable bijections that connect the spectrum of classical random matrix ensembles (such as Gaussian and Laguerre unitary ensembles) to certain planar random growth models. I shall review some of these connections and discuss a few results on either side that uses the interplay.
Based on joint works with Jnaneshwar Baslingker, Sudeshna Bhattacharjee and Manjunath Krishnapur.
Title: APRG Seminar: Slides Moment indeterminateness
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Mon, 09 Sep 2024
Time: 10 am – 12:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Contrary to Fourier, Laplace, Cauchy, or Poisson transforms, the power moments of a positive measure, fast decaying at infinity on the real line, do not determine the original measure. The non-uniqueness phenomenon was analyzed in depth by Stieltjes, via continued fraction expansions of the formal generating series of moments. I will sketch the main ideas of Stieltjes celebrated memoir, to continue with an account of a not less foundational contribution put by Carleman in book format. Then we will touch on Marcel Riesz’s pioneering work on extensions of positive linear functionals, to return from another perspective to the Christoffel-Darboux kernel. All in 1D.
Some unfinished parallel studies in $n$D, marred by pitfalls and open problems will be discussed.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Ray structures on Teichmuller spaces
Speaker: Huiping Pan (South China University of Technology)
Date: Mon, 09 Sep 2024
Time: 06:30 pm
Venue: online, MS Teams
Teichmuller space admits several ray structures, such as the Teichmuller geodesic ray, Thurston stretch ray, harmonic map (dual) ray, grafting ray, etc. In the first part of this talk, we will depict harmonic map ray structures on Teichmuller space as a geometric transition between Teichmuller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces, the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to a Teichmuller geodesic. In the second part, we will discuss applications to the Thurston metric. This is a joint work with Michael Wolf.
Title: Bangalore Probability Seminar: Interchangeability theorems for interacting particle systems
Speaker: Arvind Ayyer (IISc Mathematics)
Date: Mon, 09 Sep 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
I will focus on two interacting particle systems with inhomogeneous rates on the finite ring with site-dependent rates. The first is the Totally Asymmetric Zero Range Process (TAZRP) and the second is the PushTASEP, where TASEP stands for the Totally Asymmetric Simple Exclusion Process. In both cases, I will present results which say that the distribution of the path of the process restricted to certain intervals is unchanged under permutations of the rates. Our proofs build on Weber’s theorem for exponential queues. I will aim to keep the talk self-contained.
This is joint work with O. Mandelshtam and J. Martin (arXiv:2209.09859, Math. Z., to appear) and with J. Martin (arXiv:2310.09740).
Title: Eigenfunctions Seminar: Slides Two perspectives in number theory: explicit and probabilistic
Speaker: Kaneenika Sinha (IISER Pune)
Date: Fri, 06 Sep 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
“Explicit” number theory is the name given to the study of what are called zero-free regions of the Riemann zeta function and other L-functions. An explicit determination of such regions often reveals deep arithmetic properties of the underlying object attached to the concerned L-function. More generally, it could refer to the use of “explicit”, often technical methods to understand an arithmetic object. On the other hand, “probabilistic” number theory attempts to investigate arithmetic properties of an object by treating the object as one in a family of many, and exploring these families of varying sizes through the viewpoint of probability. In this talk, we will explore both perspectives and compare the wealth of information each perspective presents to us. In particular, we will address the study of Fourier coefficients of certain modular forms (called the Hecke eigenforms) through both the above viewpoints.
Title: Algebra & Combinatorics Seminar: Graphical models, causality and algebraic perspectives
Speaker: Prateek Misra (Technical University of Munich, Germany)
Date: Thu, 05 Sep 2024
Time: 4 pm
Venue: LH-3, Mathematics Department
Algebraic Statistics is a relatively new field of research where tools from Algebraic Geometry, Combinatorics and Commutative Algebra are used to solve statistical problems. A key area of research in this field is the Gaussian graphical models, where the dependence structure between jointly normal random variables is determined by a graph. In this talk, I will explain the algebraic perspectives on Gaussian graphical models and present some of my key results on understanding the defining equations of these models. In the end, I will talk about the problem of structural identifiability and causal discovery and how algebraic techniques can be implemented to tackle them.
Title: APRG Seminar: Slides The Christoffel–Darboux Kernel
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Thu, 05 Sep 2024
Time: 2 – 4:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Given a positive measure in Euclidean space, rapidly decaying at infinity, a point $a$, and a filtration of the polynomial ring by the degree, the optimal bounds for point evaluations at $a$ in Lebesgue space $L^2$ are provided by a reproducing kernel known as the Christoffel–Darboux kernel. For a century and a half, this object continues to intrigue by surprising new turns.
We will touch the asymptotics of orthogonal polynomials in the complex plane, with emphasis on Szego’s Limit Theorem. Then turn to spectral analysis on the line, culminating in Weyl’s circle phenomenon.
Recent applications to dynamical systems (via Koopman’s operator formalism) and the statistics of geometric data will be presented together with some numerical experiments.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Algebras of functions and heat kernel estimates
Speaker: Tommaso Bruno (University of Genova, Italy)
Date: Wed, 04 Sep 2024
Time: 4 pm
Venue: Microsoft Teams (online)
I will discuss a theory of function spaces defined in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. This includes inhomogeneous and homogeneous Sobolev, Besov and Triebel–Lizorkin spaces on Lie groups and Grushin settings.
Title: Number Theory Seminar: A finer Sato-Tate conjecture and Bias conjecture for certain families of elliptic curves
Speaker: Sudhir Pujahari (NISER Bhubaneshwar)
Date: Fri, 30 Aug 2024
Time: 2 pm
Venue: LH-1
In this talk, we will study moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. In conclusion, we will obtain the Sato-Tate distribution for the trace of certain families of Elliptic curves in arithmetic progressions. As a special case we will recover a result of Birch proving Sato-Tate distribution for certain families of elliptic curves. Moreover, we will see that these results follow from asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. Finally, if time permits, we will discuss the bias conjecture in the finite field setting. These are joint works with Kathrin Bringmann, Ben Kane, and Zichen Yang.
Title: APRG Seminar: Slides Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on hyperbolic spaces
Speaker: Effie Papageorgiou (Universität Paderborn, Germany)
Date: Wed, 28 Aug 2024
Time: 3 pm
Venue: Microsoft Teams (online)
On a Riemannian manifold, consider the Laplace-Beltrami operator $-\Delta$, and the associated extension problem
\begin{equation} \Delta v+\frac{(1-2\sigma)}{t}\frac{\partial v}{\partial t}+\frac{\partial^2 v}{\partial t^2}=0, \quad 0 <\sigma < 1,\quad t>0, \end{equation}
introduced by Caffarelli and Silvestre on Euclidean space to recover the fractional Laplacian $(-\Delta)^{\sigma}$, as $t$ approaches zero.
On hyperbolic spaces this gives rise to a family of convolution operators, including the Poisson operator $e^{-t\sqrt{-\Delta}}$, $t>0$; moreover,
the kernels of these operators are subordinated to the heat kernel.
Motivated by Euclidean results for the Poisson semigroup, but also by results on the heat semigroup on Riemannian manifolds and the influence of underlying geometry, we study the long-time asymptotic behavior of solutions to the extension problem for $L^1$ initial data. If the initial datum is radial, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence can break down in the non-radial case. The results extend to all noncompact symmetric spaces of arbitrary rank.
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Old and new results on bounds for codes and sphere packings: An overview
Speaker: Alexander Barg (University of Maryland, College Park, USA)
Date: Fri, 23 Aug 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
The problem of maximum packing density of the $n$-dimensional real space with spheres is a classic one. Exact values of the density are known only in a few cases ($n=1,2,3,8,24$), and there have been several recent improvements of the bounds for other small dimensions. In a parallel development, researchers have studied the maximum size of packings of the $n$-dimensional Hamming space, known as error-correcting codes. While existence bounds in both cases are found by random choice, the best known impossibility results are obtained by an application of a general method commonly known as Delsarte’s linear programming. The best known upper bound on the maximum size of a code with a given minimum distance for large $n$ was obtained in 1977, and it has proved surprisingly resistant to various improvement attempts, including the semidefinite programming extension of LP.
In the first part of the talk we will introduce the general problem and give an overview of the known results on upper bounds on codes and related problems such as equiangular lines and families of finite sets with restricted intersections. In the second part, we will delve into the details of the proofs for the case of codes and highlight some obstacles for further improvements.
Title: PhD Thesis defence: On two complex Hessian equations and convergence of corresponding flows
Speaker: Ramesh Mete (IISc Mathematics)
Date: Thu, 22 Aug 2024
Time: 11 am
Venue: Hybrid - LH-3, Department of Mathematics and Google meet (online)
This thesis consists of three parts. Two important complex Hessian equations are studied on certain compact Kahler manifolds from different perspectives. The first one is the J-equation introduced independently by S.K. Donaldson and X.X. Chen from different point of view. The second one is the deformed Hermitian Yang Mills (dHYM) equation which has connection to the mirror symmetry in string theory.
There is a notion of (global) slopes for both equations. It is known that they admit smooth solutions with the global slopes if and only if certain Nakai-Moishezon (NM) type criterion holds. In the first part, our aim is to find some appropriate singular solutions of the equations when the NM-type criterion fails–this is the so-called unstable case. An algebro-geometric characterization of the slopes is formulated – which we call the minimal J-slope for the J-equation and the maximal dHYM-slope for the dHYM equation. There is a natural weak (i.e. singular) version of the equations replacing the standard wedge product with a more generalized product, called the non-pluripolar product. We settle the existence and uniqueness problem for the singular J and dHYM equation on compact Kahler surfaces. More precisely, for the J-equation we show that there exists a unique closed $(1,1)$-Kahler current solving the singular J-equation on a compact K"ahler surface with the minimal J-slope. Analogous result is established for the singular dHYM equation on compact Kahler surfaces with the maximal dHYM-slope. Furthermore, we conjecture analogous existence and uniqueness result for higher dimensions.
In the second part, we study the convergence behaviour of the J-flow, which is the parabolic version of the J-equation, on certain generalized projective bundles using the Calabi Symmetry in the J-unstable case. An invariant version of the minimal J-slope is introduced for these bundles. Furthermore, we show that the flow converges to some unique limit in the weak sense of currents, and the limiting current solves the singular J-equation with the invariant minimal J-slope. This result resolves the invariant version of our conjecture for the J-equation on these examples with symmetry.
In the third part, we study the convergence behaviour of a flow, called the cotangent flow, for the dHYM equation in the dHYM-unstable case on the blowup of $\mathbb{C}\mathbb{P}^2$ or $\mathbb{C}\mathbb{P}^3$. Analogous to our results in the second part, we show that the flow converges to some unique limit, and the limiting current solves the singular dHYM equation with the (invariant) maximal dHYM-slope.
Title: APRG Seminar: Slides Smooth barycentric decompositions of Weyl chambers - a helpful tool to handle inverse spherical Fourier transforms on noncompact symmetric spaces of higher rank Fourier and Schur idempotents
Speaker: Jean-Philippe Anker (Institut Denis Poisson, Université d'Orléans, France)
Date: Wed, 21 Aug 2024
Time: 4 pm
Venue: Microsoft Teams (online)
The study of invariant dispersive PDE on noncompact symmetric spaces, such as the wave equation or the Schrödinger equation, requires to analyze oscillating integrals arising from the inverse spherical Fourier transform. While this can be achieved by classical though nontrivial tools in rank one, a major problem in higher rank lies in the fact that the Plancherel density is not a differentiable symbol in general, and thus integration by parts produces no additional global decay at infinity. In this talk, we will explain a way to overcome this problem by introducing a smooth barycentric decomposition of Weyl chambers, which leads eventually to the same dispersive and Strichartz estimates as in rank one. This work started 15 years ago as a joint project with S. Meda, V. Pierfelice, M. Vallarino and was finally achieved in collaboration with H.-W. Zhang.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Slides Hypoelliptic damped wave equations on graded lie groups with initial data from negative order Sobolev spaces
Speaker: Vishvesh Kumar (Ghent University, Ghent, Belgium)
Date: Wed, 14 Aug 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, I will discuss the semilinear hypoelliptic damped wave equation with power-type nonlinearity associated with a Rockland operator on graded Lie groups. Specifically, we will concentrate on the case when the initial data belongs to Sobolev spaces of negative order. We show the global-in-time existence of small data Sobolev solutions of lower regularity for the supercritical range and a finite-time blow-up of weak solutions for the subcritical range. For the particular settings of the Heisenberg group and Euclidean space, we will prove that the critical exponent belongs to the blow-up case. Furthermore, to precisely characterize the blow-up time, we derive sharp upper and lower bound estimates for the lifespan in the subcritical cases.
This talk is based on my joint research with Aparajita Dasgupta (IIT Delhi), Shyam Swarup Mondal (ISI Kolkata), Michael Ruzhansky (Ghent University), and Berikbol Torebek (Ghent University).
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: p and hp Spectral Element Methods for Elliptic Boundary Layer Problems
Speaker: Akhlaq Husain (Jamia Millia Islamia, Delhi)
Date: Tue, 13 Aug 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
In this talk, we consider p and h-p least-squares spectral element methods for elliptic boundary layer problems in one dimension. We derive stability estimates and design a numerical scheme based on minimizing the residuals in the sense of least-squares in appropriate Sobolev norms. We prove parameter robust uniform error estimates i.e. error in the approximation is independent of the boundary layer parameter for the p and hp-version. Numerical results are presented for a number of model elliptic boundary layer problems confirming the theoretical estimates and uniform convergence results.
Title: Geometry & Topology Seminar: Ribbon concordance is a partial order (after Ian Agol)
Speaker: Siddhartha Gadgil (IISc)
Date: Mon, 12 Aug 2024
Time: 11:00 am
Venue: LH-1, Mathematics Department
A couple of years ago Ian Agol proved a long-standing conjecture showing that a relation called “Ribbon concordance” on knots is a partial order. The proof was a six page paper involving a blend of topology, combinatorial group theory and real algebraic geometry, with ribbon concordance implying a relation between representation varieties. In this talk I describe Agol’s proof and some background results.
Title: Geometry & Topology Seminar: A Moduli Space of Marked Hyperbolic Structures for Big Surfaces.
Speaker: Chaitanya Tappu (Cornell University)
Date: Mon, 12 Aug 2024
Time: 04:00 pm
Venue: LH-1, Mathematics Department
We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space. This marked moduli space reduces to the usual Teichm"uller space for finite type surfaces. Since a big mapping class group is a topological group, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative.
Title: APRG Seminar: Dentability and related concepts in the unit ball of Banach spaces
Speaker: Sudeshna Basu (Loyola University, Baltimore, USA)
Date: Fri, 09 Aug 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
Let $X$ be a Banach space. Let $C$ be a subset of $X$. Let $x^*$ be a functional on $X$. Then $S(C, x^*, \alpha) := \{ x \in C : x^*(x) > \sup x^*(C) - \alpha \}$, $\alpha > 0$, is called the open slice of $C$ determined by $x^*$ and $\alpha$. $X$ has Radon Nikodym Property if and only all closed bounded convex sets admit slices of arbitrarily small diameter i.e. these sets are dentable. The geometry of Banach space is an area of research which characterizes the topological and measure theoretic concepts in Banach spaces in terms of geometric structure of the space. The related concepts were initiated developed and extensively studied in the context of Radon Nikodym Property and Krein Milman Property by Ghoussoub, Godefroy, Maurey, and Scachermayer [Memoirs AMS 1987]. In this work, we look at Banach spaces where the unit ball admits slices of arbitrarily small diameter. We look at some related properties as well. We prove that all these properties are stable under $l_p$ sum for $1 \leq p \leq ∞$, sum and Lebesgue Bochner spaces. We show that these are three space properties under certain conditions on the quotient space. We also study these properties in ideals of Banach spaces. This is based on two papers jointly written with my graduate student, Susmita Seal in [J. Math. Anal. Appl. 2022] and [J. Convex Anal. 2023]. The only prerequisite for this talk is the statement of the Hahn Banach Theorem.
Title: APRG Seminar: On the phase diagram of the polymer model
Speaker: Arjun Krishnan (University of Rochester, USA)
Date: Thu, 08 Aug 2024
Time: 3:40 pm
Venue: LH-1, Mathematics Department
In dimension 1, the directed polymer model is in the celebrated KPZ universality class, and for all positive temperatures, a typical polymer path shows non-Brownian KPZ scaling behavior. In dimensions 3 or larger, it is a classical fact that the polymer has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can induce the phase transition in the directed polymer model.
Title: APRG Seminar: Ruin in double halve gambling
Speaker: Aditya Guha Roy (Indiana University, Bloomington, USA)
Date: Thu, 08 Aug 2024
Time: 2:30 pm
Venue: Hybrid - LH-1, Mathematics Department and Microsoft Teams (online)
We study the ruin probability of a gambler in a scheme where the bet size is doubled after every win in a round and halved after every loss; we show some paradoxical results such as that the ruin probability is one if and only if the probability of winning in any round is at least half. We also verify some conjectures about the behaviour of the ruin probability as a function of the initial fortune and the probability of winning.
Title: APRG Seminar: Slides Fourier and Schur idempotents
Speaker: Javier Parcet (Instituto de Ciencias Matemáticas, Madrid, Spain)
Date: Wed, 07 Aug 2024
Time: 2:30 pm
Venue: Microsoft Teams (online)
What happens to an $L_p$ function when one truncates its Fourier transform to a domain? This is in the root of foundational problems in harmonic analysis. Fefferman’s celebrated theorem for the ball (1971) imposes that, to preserve $L_p$-integrability, the boundary of such domain must be flat. What if we truncate on a curved space like a Lie group? And if we truncate the entries of a given matrix? What happens with the singular numbers of it or with its Schatten $p$-norm? We fully characterize the local geometry of such $L_p$-preserving truncations for these (apparently unrelated) problems, in terms of a surprisingly lax notion of boundary flatness. The matrix ones are all diffeomorphic variations of a fundamental example: the triangular projection. The Lie group ones are all modeled on one of three fundamental examples: the classical Hilbert transform, and two new examples of Hilbert transforms that we call affine and projective. This vastly generalizes Fefferman’s theorem to nontrigonometric and noncommutative scenarios. It confirms the intuition that Schur multipliers share profound similarities with Euclidean Fourier multipliers – even in the lack of a Fourier transform connection – and complete, for Lie groups, a longstanding search of Fourier $L_p$-idempotents. Joint work with M. de la Salle and E. Tablate.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis colloquium: Symmetric real-closed subsets of affine Kac-Moody Lie Algebras
Speaker: Dipnit Biswas (IISc Mathematics)
Date: Mon, 29 Jul 2024
Time: 11 am
Venue: LH-1, Mathematics Department
In this thesis, we study some important classification problems related to affine Kac–Moody Lie algebras. First, we address the combinatorial problem of classifying symmetric real-closed subsets of affine root systems (which are roots of affine Kac–Moody Lie algebras). Secondly, we understand the correspondence between these symmetric real-closed subsets and the regular subalgebras generated by them, using the aforementioned classification. Motivation for this work comes from the celebrated work of Dynkin (1952), where he classified the semi-simple subalgebras of a given finite-dimensional semisimple Lie algebra 𝔤. He introduced the notion of regular subalgebras in order to achieve this classification. It is not hard to see that the regular subalgebras of 𝔤 correspond to symmetric closed subsets of roots of 𝔤, so the problem of classifying regular subalgebras comes down to the combinatorial problem of classifying these subsets. The analogous problem of studying regular subalgebras of affine Kac–Moody Lie algebras was initiated by Anna Felikson et al. in 2008 and continued by Roy–Venkatesh in 2019, where they addressed the part of the combinatorial problem, namely provided the classification of maximal real closed subroot systems of affine root systems.
In the finite case, it is well known that symmetric closed subsets are in fact closed subroot systems. It is not true in general for affine root systems. So, it is natural to ask the following questions:
- When a given symmetric real-closed subset of affine root system is a closed subroot system?
- Is it possible to classify all symmetric real-closed subsets of affine root systems?
We give affirmative answers to these questions in this thesis. In the untwisted setting, we prove that any symmetric real-closed subset is indeed a closed subroot system. Twisted types need more careful analysis since the finite part of a symmetric real-closed subset has two possibilities in these types, namely closed and semi-closed. For semi-closed cases, the behavior of symmetric real-closed subsets varies for each type. We prove that there are three types of irreducible symmetric real-closed subsets for reduced real affine root systems, one of which did not appear in Roy–Venkatesh’s work. We conclude our classification for the twisted case by determining explicitly when a symmetric closed subset is a closed subroot system, including the case when the ambient Lie algebra is the non-reduced affine Lie algebra $A_{2n}^{(2)}$.
In the second part of the thesis, we explore the correspondence between symmetric real-closed subsets and regular subalgebras generated by them. Roy-Venkatesh proved that the map between closed subroot systems and the regular subalgebras generated by them is injective. We observe that it is not true in general when we extend this map to symmetric real-closed subsets. Let ψ be a real-closed subroot system. We describe the types of symmetric closed subsets that can appear in the fiber of the subalgebra generated by ψ. Moreover, we determine when these fibers are finite. In certain cases, we are able to describe very explicitly the defining parameters of the symmetric closed subsets appearing in the fiber.
Title: Algebra & Combinatorics Seminar: Milnor–Witt cycle modules over excellent DVR
Speaker: Rakesh Pawar (UMPA, École Normale Supériure de Lyon, France)
Date: Wed, 24 Jul 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
I will briefly recall Milnor cycle modules over a field as defined by Rost (1996) and their significance and properties. Recently, ‘modules’ over Milnor–Witt K-theory or alternatively Milnor–Witt cycle modules over fields have been formalized by N. Feld (2020).
I will talk about recent joint work with Chetan Balwe and Amit Hogadi, where we considered the Milnor–Witt cycle modules over excellent DVR and studied a subclass of these that satisfy certain lifting conditions on residue maps associated with horizontal valuations. As an important example, Milnor–Witt K-theory of fields belongs to this subclass. Moreover, this condition is sufficient to deduce the local acyclicity property and $A^1$-homotopy invariance of the associated Gersten complex.
Title: Geometry & Topology Seminar: Simplicial cell decompositions of CP^n
Speaker: Basudeb Datta (IAI, TCG Crest, Kolkata)
Date: Thu, 18 Jul 2024
Time: 11 am
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
According to a well-known result in geometric topology, we have $(S^2)^n/Sym(n) = \mathbb{CP}^n$, where $Sym(n)$ acts on $(S^2)^n$ by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $\mathbb{CP}^n$ for each $n > 1$. In more detail, we take the standard two triangle crystallisation $S^2_3$ of the 2-sphere $S^2$, in its $n$-fold Cartesian product. We then simplicially subdivide, and prove that naively taking the $Sym(n)$ quotient yields a simplicial cell decomposition of $\mathbb{CP}^n$. Taking the first derived subdivision of this cell complex produces a triangulation of $\mathbb{CP}^n$. To the best of our knowledge, this is the first explicit description of triangulations of $\mathbb{CP}^n$ for $n > 3$. This is a joint work with Jonathan Spreer, University of Sydney.
Title: MS Thesis defence: Higher categories of n-bordisms
Speaker: Arghan Dutta (IISc Mathematics)
Date: Tue, 16 Jul 2024
Time: 11 am
Venue: Hybrid - LH-3, Department of Mathematics and Microsoft Teams (online)
In this talk, we will discuss the notion of a complete Segal space – a model of an infinity category, and then study the infinity category of $n$-bordisms.
Higher category theory is the generalization of category theory to a context where there are not only morphisms between objects, but generally $k$-morphisms between $(k−1)$-morphisms, for all $k \in \N$. The theory of higher categories or $(\infty, 1)$-categories, as it is sometimes called, however, can be very intractable at times. That is why there are now several models which allow us to understand what a higher category should be. Among these models is the theory of quasi-categories, introduced by Bordman and Vogt, and much studied by Joyal and Lurie. There are also other very prominent models such as simplicial categories (Dwyer and Kan), relative categories (Dwyer and Kan), and Segal categories (Hirschowitz and Simpson). One of those models, complete Segal spaces, was introduced by Charles Rezk in his seminal paper “A model for the homotopy theory of homotopy theory”. Later they were shown to be a model for $(\infty, 1)$-categories.
One major application of higher category theory and one of the driving forces in developing it has been extended topological quantum field theory. This has recently led to what may become one of the central theorems of higher category theory, the proof of the cobordism hypothesis, conjectured by Baez and Dolan. Lurie suggested passing to $(\infty, n)$-categories for a proof of the Cobordism Hypothesis in arbitrary dimension $n$. However, finding an explicit model for such a higher category poses one of the difficulties in rigorously defining these $n$-dimensional TFTs, which are called “fully extended”. Our focus will be on the $(\infty, 1)$-category $\mathrm{Bord}_n^{(n -1)}$, a variant of the fully extended $\mathrm{Bord}_{n}$. Our goal is to sketch a detailed construction of the $(\infty, 1)$-category of $n$-bordisms as a complete Segal space.
Title: Algebra & Combinatorics Seminar: When do the natural embeddings of classical invariant rings split?
Speaker: Vaibhav Pandey (Purdue University, West Lafayette, USA)
Date: Fri, 12 Jul 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field. Objects of broad interest in commutative algebra, representation theory, and algebraic geometry like generic determinantal rings, Plücker coordinate rings of Grassmannians, symmetric determinantal rings, rings defined by Pfaffians of alternating matrices etc. arise as the invariant rings $S^G$ of such group actions.
In characteristic zero, reductive groups are linearly reductive and therefore the embedding of the invariant ring $S^G$ in the ambient polynomial ring $S$ splits. This explains a number of good algebro-geometric properties of the invariant ring in characteristic zero. In positive characteristic, reductive groups are typically no longer linearly reductive. We determine, for the natural actions of the classical groups, precisely when $S^G$ splits from $S$ in positive characteristic.
This is joint work with Melvin Hochster, Jack Jeffries, and Anurag K. Singh.
Title: PhD Thesis defence: On regular subalgebras of Kac-Moody algebras and related combinatorics
Speaker: Md. Irfan Habib (IISc Mathematics)
Date: Fri, 12 Jul 2024
Time: 11:30 am
Venue: LH-1, Department of Mathematics
In this thesis, we study two aspects of Kac-Moody algebras. One is to understand the subalgebras that can be embedded inside a Kac-Moody algebra as subalgebras generated by real root vectors. The other one is to explicitly classify the regular subalgebras and the maximal regular subalgebras of an untwisted affine Kac-Moody algebra.
Dynkin classified the semisimple regular subalgebras of a finite-dimensional semisimple Lie algebra back in $1949.$ One of the key tools he used for the classification is $\pi$-systems. For non-finite Kac-Moody algebras, $\pi$-system became an integral part of und erstanding the embedding of different types of algebras in a Kac-Moody algebra. Till now all the articles existing in the literature, which study $\pi$-systems, assume that the $\pi$-systems are either linearly independent or finite. It seems that our work is the first one to address infinite $\pi$ systems in the context of the embedding problem. This paves a way for us to understand the infinite (linearly independent) $\pi$-systems for Borcherds Kac-Moody algebras and understand the embedding problem in that setting. We used Deodhar’s preorder to prove that every closed subroot system in a Kac-Moody root system admits a $\pi$-system and this $\pi$-system need not be finite in general. Moreover, for any closed subroot system $\Psi$ of $\Delta,$ we prove that there exists a unique $\pi$-system $\Pi(\Psi),$ which is contained in the set of positive roots. Since the subroot systems of a root system are not very ‘well behaved’, this is quite surprising and it generalizes the previously well-known fact that they simple systems and positive systems determine each other at the level of the subroot system.
Using this unique $\pi$-system $\Pi(\Psi)$, we prove that for a real closed subroot system $\Psi,$ the real roots of a root generated subalgebra $\mathfrak g(\Psi)$ is equal to $\Psi$. This result was a much-awaited one in the literature because almost after $70$ years of Dynkin’s result, Roy and Venkatesh (Transform. Groups 2019) proved that the same is true for an affine root system. These two results provide a bridge between the algebraic and combinatorial side which shows that the root-generated subalgebras are in bijection with the real closed subroot systems which are in turn in one-to-one correspondence with the $\pi$-systems contained in the positive roots of a Kac-Moody algebra.
In the last part of our analysis of regular subalgebras generated by root vectors, we prove that for any closed subroot system $\Psi,$ the root generated subalgebra is isomorphic to a quotient of the derived subalgebra of the Kac-Moody algebra corresponding to the (infinite) Cartan matrix defined by the unique $\pi$-system of the closed subroot system $\Psi,$ by an ideal contained in the centre of the algebra. This result is a generalization of the existing results when the $\pi$-system is linearly independent and the ideal is zero when the $\pi$-system is linearly independent also follows from our result. In particular, as long as the roots are concerned, to understand the root generated subalgebras, it is enough to consider the derived algebras of Kac-Moody algebras $\mathfrak g’(A)$ corresponding to a(n infinite) GCM $A.$ Classification of regular subalgebras of an affine Kac-Moody Lie algebra is an interesting problem in its own right. Barnea et al. started such classification in $1998.$ Later Felikson et al. used combinatorics of root systems to classify the regular subalgebras in 2008, more precisely the root generated subalgebras of an affine Kac-Moody algebra. We took a completely different approach, namely, using the classification of the closed subroot system of a real affine root system given by Roy and Venkatesh, we classify the regular subalgebras of affine Kac-Moody Lie algebras with a symmetric set of roots and we get the classification of root generated subalgebras as a Corollary. Moreover, we also classify the maximal symmetric regular subalgebras we show a bijective correspondence between the maximal real closed subroot systems of the affine Lie algebra and the maximal symmetric regular subalgebras different from $[\mathfrak g,\mathfrak g].$ Which also shows that in the affine case, given a maximal closed subroot system $\Psi$ of $\Delta,$ the poset (with set inclusion as the partial order)
\begin{equation} A_\Psi:={\mathfrak s:\Delta(\mathfrak s)^{\mathrm{re}}=\Psi} \end{equation}
contains a unique maximal element.
Title: APRG Seminar: An encoding retrieving the edge universality of Wigner matrices
Speaker: Debapratim Banerjee (Ashoka University)
Date: Fri, 12 Jul 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
In this talk, we give a new combinatorial proof of classical edge universality of Wigner matrices without assuming the entries of the matrices are symmetrically distributed around 0. We complete this proof by giving a new encoding of the Wigner words and a counting strategy which works for traces of very high powers of the matrix. In this talk, we shall introduce the encoding, describe the class of words which capture the randomness and finally give some insight about the proof for general non vanishing odd moments.
Title: Eigenfunctions Seminar: An overview of Fuchs’ problem
Speaker: Sunil Chebolu (Illinois State University, Normal, USA)
Date: Fri, 05 Jul 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
More than 50 years ago, Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. After introducing the problem and known results in the first part of the talk, I will present an overview of my joint work with Keir Lockridge on this problem. We answered this question for various classes of groups including indecomposable abelian groups, dihedral groups, quaternion groups, and some $p$-groups. This work also gave us several new characterizations of Mersenne primes and Fermat primes.
Title: APRG Seminar: Isomorphisms between dense random graphs
Speaker: Lutz Warnke (University of California, San Diego, USA)
Date: Thu, 04 Jul 2024
Time: 11:15 am
Venue: LH-1, Mathematics Department
Applied benchmark tests for the famous ‘subgraph isomorphism problem’ empirically discovered interesting phase transitions in random graphs. This motivates our rigorous study of two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities $p_1,p_2$. In particular, (i) we prove a sharp threshold result for the appearance of $G_{n,p_1}$ as an induced subgraph of $G_{N,p_2}$, (ii) we show two-point concentration of the size of the maximum common induced subgraph of $G_{N,p_1}$ and $G_{N,p_2}$, and (iii) we show that the number of induced copies of $G_{n,p_1}$ in $G_{N,p_2}$ has an unusual ‘squashed lognormal’ limiting distribution.
These results resolve several open problems of Chatterjee and Diaconis, and confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble. The proofs are based on careful refinements of the first and second moment method, using extra twists to (a) take some non-standard behaviors into account, and (b) work around the large variance issues that prevent standard applications of the second moment method, using in particular pseudorandom properties and multi-round exposure arguments to tame the variance.
Based on joint work with my PhD students Erlang Surya and Emily Zhu; see arXiv:2305.04850.
Title: APRG Seminar: Algorithmic Discrepancy Theory: A Tale of Combinatorics, Probability, Convex Geometry and Algorithms
Speaker: Abhishek Shetty (University of California, Berkeley, USA)
Date: Thu, 04 Jul 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
Discrepancy theory is a well-studied area in mathematics, concerned with the question of partitioning geometry and combinatorial objects in balanced subsets. Starting from seminal works of Spencer, Banaszczyk, Gluskin amongst others, deep connections between the area and other areas in mathematics such as convex geometry and probability were established. But, for most of its history, the arguments establishing the existence of good partitions were non-constructive, even believed to be fundamentally non-algorithmic.
But, the past decade has seen a flurry of work in algorithmic discrepancy theory, leading to efficient algorithms for several of the most famous settings in discrepancy theory. Perhaps surprisingly, these algorithmic techniques have further strengthened the connection between discrepancy and other areas such as convex geometry and probability.
In this talk, we will survey recent results and techniques in algorithmic discrepancy with the aim to convey connections to various areas. Time permitting, we will end with natural conjectures that would lead to progress on long standing conjectures in discrepancy theory.
No prior exposure to algorithms, computer science or discrepancy will be assumed.
Title: Number Theory Seminar: Towards a refinement of the Bloch-Kato conjecture
Speaker: Sunil Chebolu (Illinois State University, Normal, USA)
Date: Wed, 03 Jul 2024
Time: 11 am
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
Let $F$ be a field that has a primitive $p$-th root of unity. According to the Bloch–Kato conjecture, now a theorem by Voevodsky and Rost, the norm-residue map \begin{equation} k_*(F)/pk_*(F) \rightarrow H^*(F, \mathbb{F}_p) \end{equation} from the reduced Milnor $K$-theory to the Galois cohomology of $F$ is an isomorphism of $\mathbb{F}_p$-algebras.
This isomorphism gives a presentation of the rather mysterious Galois cohomology ring through generators and relations. In joint work with Jan Minac, Cihan Okay, Andy Schultz, and Charlotte Ure, we have obtained a second cohomology refinement of the Bloch–Kato conjecture. Using this we can characterize the maximal $p$-extension of $F$, as the “decomposing field” for the cohomology of the absolute Galois group.
Title: Algebra & Combinatorics Seminar: Buchsbaumness and Castelnuovo-Mumford regularity of non-smooth monomial curves
Speaker: Ngo Viet Trung (Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam)
Date: Mon, 01 Jul 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such rings always have finite Macaulayfication. We show how to characterize the Buchsbaumness and the Castelnuovo–Mumford regularity of these rings by means of their finite Macaulayfication, and we use this method to study the Buchsbaumness and to estimate the Castelnuovo–Mumford regularity of large classes of non-smooth monomial curves in terms of the given monomials.
Title: APRG Seminar: Mated trees and a Peano curve in the directed landscape
Speaker: Manan Bhatia (MIT, Boston, USA)
Date: Fri, 28 Jun 2024
Time: 2:15 pm
Venue: LH-1, Mathematics Department
A recurrent theme encountered in many models of random geometry is that of two trees glued to one another with a space-filling curve snaking in between them. In this talk, we first recall a few examples of this, namely, Brownian geometry, Liouville quantum gravity, and the Brownian web. Subsequently, we discuss the construction of a pair of interlaced trees and the corresponding Peano curve in the directed landscape, the conjectural universal scaling limit of models in the Kardar-Parisi-Zhang universality class. Finally, we look at the question of determining the precise Holder and variation regularity of this space-filling curve and discuss some of the ideas involved in the proof. Based on the works arxiv:2304.03269 (joint with Riddhipratim Basu) and arxiv:2301.07704.
Title: Algebra & Combinatorics Seminar: Local vanishing for toric varieties
Speaker: Sridhar Venkatesh (University of Michigan, Ann Arbor, USA)
Date: Thu, 27 Jun 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
Let $f:Y \to X$ be a log resolution of singularities which is an isomorphism over the smooth locus of $X$, and the exceptional locus $E$ is a simple normal crossing divisor on $Y$. We prove vanishing (and non-vanishing) results for the higher direct images of differentials on $Y$ with log poles along $E$ in the case when $X$ is a toric variety. Our consideration of these sheaves is motivated by the notion of $k$-rational singularities introduced by Friedman-Laza. This is joint work with Anh Duc Vo and Wanchun Shen.
Title: PhD Thesis defence: Existence and implications of positively curved metrics on holomorphic vector bundles
Speaker: Arindam Mandal (IISc Mathematics)
Date: Wed, 26 Jun 2024
Time: 3 pm
Venue: LH-3, Mathematics Department
This thesis is divided into two parts. In the first part, we study interpolating and uniformly flat hypersurfaces in complex Euclidean space. The study of interpolation and sampling in the Bargmann-Fock spaces on the complex plane started with the work of K. Seip in 1992. In a series of papers, Seip and his collaborators have entirely characterized the interpolating and sampling sequences for the Bargmann-Fock spaces on the complex plane. This problem has also been studied for the Bargmann-Fock spaces on the higher dimensional complex Euclidean spaces. Very few results on interpolating and sampling hypersurfaces in higher dimensions are known. We have proven certain hypersurfaces are not interpolating in dimensions 2 and 3. Cerd'{a}, Schuster and Varolin have defined uniformly flat smooth hypersurfaces and proved that uniform flatness is one of the sufficient conditions for smooth hypersurfaces to be interpolating and sampling in higher dimensions. We have studied uniformly flat hypersurfaces in dimensions greater than or equal to two and proved a complete characterization of them. In dimension two, we provided sufficient conditions for a smooth algebraic hypersurface to be uniformly flat in terms of its projectivization.
The second part deals with the existence of a Griffiths positively curved metric on the Vortex bundle. Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we have the notions of Nakano positivity, Griffiths positivity, and ampleness. All these notions of positivity are equivalent for line bundles. In general, Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, J. P. Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. We have studied these systems on the Vortex bundle.
Title: PhD Thesis colloquium: The monopole-dimer model and eccentricities for the Cartesian product of graphs
Speaker: Anita Arora (IISc Mathematics)
Date: Mon, 24 Jun 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
This thesis comprises two main parts. The details of the two parts are as follows:
The first part of the thesis deals with the monopole-dimer model. The dimer (resp. monomer-dimer) model deals with weighted enumeration of perfect matchings (resp. matchings). The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. A more general model called the loop-vertex model has also been defined for an oriented graph and the partition function in this case can also be written as a determinant. However, this model depends on the orientation of the graph. The monopole-dimer model interprets the loop-vertex model independent of the orientation for planar graphs with Pfaffian orientation. The first part of the thesis focuses on the extension of the monopole-dimer model for planar graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof. We show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs so long as they are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs a la Kasteleyn and Temperley–Fischer, which turns out to be fourth power of a polynomial when all grid lengths are even. Further, we generalise this product formula to higher dimensions, again obtaining an explicit product formula. We also discuss about the asymptotic formulas for the free energy and monopole densities.
In 1999, Lu and Wu evaluated the partition function of the dimer model on two-dimensional grids embedded on a Möbius strip and a Klein bottle. We first prove a product formula for the partition function of the monopole-dimer model for the higher dimensional grid graphs with cylindrical and toroidal boundary conditions. We then consider the monopole-dimer model on high-dimensional Möbius and Klein grids, and evaluate the partition function for three-dimensional Möbius and Klein grids. Further, we show that the formula does not generalise for the higher dimensions in any natural way. Finally, we present a relation between the product formulas for three-dimensional grids with cylindrical and Möbius boundary conditions, generalising a result of Lu and Wu.
Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph of $G$, denoted $Ec(G)$, is a graph defined on the vertices of $G$ in which two vertices are adjacent if one is eccentric to the other. In the second part of the thesis, we find the structure and the girth of the eccentric graph of trees, and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of the Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a complete classification of when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and the Cartesian product of two cycles. Finally, we determine the conditions under which the eccentricity matrix of the Cartesian product of trees becomes invertible.
Title: MS Thesis colloquium: On the study of Toeplitz operators on Hardy Space over a multiply connected domain
Speaker: Sushobhan Das (IISc Mathematics)
Date: Fri, 21 Jun 2024
Time: 10 am
Venue: Hybrid - LH-3, Department of Mathematics and Microsoft Teams (online)
The classical theory of Toeplitz operators on Hardy space over the unit disk is a well-developed area in Operator Theory. If we substitute the domain disk $\Delta$ with a bounded multiply connected domain $D$, where $\partial D$ consists of finitely many smooth closed curves, what kinds of difficulties arise? This question motivates us to explore the theory for Toeplitz operators on Hardy space over a multiply connected domain $D$. In 1974, M.B. Abrahamse’s Ph.D. thesis made significant contributions in this topic, extending well-known results for the disk like characterizations of commutator ideals for the Banach Algebra generated by Toeplitz operators with continuous $\mathbb{C}(\partial D)$ or $H^\infty + C(\partial D)$ symbols, and the characterization of Fredholm operators with $H^\infty+C$ symbols to those for the multiply connected domain $D$. Also, he came up with the striking reduction theorem, which roughly says that modulo compact operators, the Toeplitz operator defined on the Hardy space over a multiply connected domain $D$, can be written as the direct sum of Toeplitz operators defined on the Hardy space over the unit disk.
In this talk, we will provide the definition of the Hardy Space $H^p$ over multiple connected domains $D$, where $1 \leq p \leq \infty$, and build some prerequisites to present the aforementioned characterization theorems obtained by Abrahamse for the case of multiple connected domains $D$. We will present the proofs of some of these theorems originally done by Abrahamse. Following that, we will examine the proof of the reduction theorem and explore some of its applications.
Title: APRG Seminar: On Hardy Spaces associated with the Twisted Laplacian and sharp estimates for the corresponding Wave Operator
Speaker: Jotsaroop Kaur (IISER Mohali)
Date: Fri, 21 Jun 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We define atomic Hardy space $H^p_{\mathcal{L}, at}(\mathbb{C}^n), 0<p\leq 1$ for the
twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy
space defined using the maximal function corresponding to the heat
semigroup $e^{-t\mathcal{L}},t>0$. We also prove sharp $L^p, 0<p\leq 1$
estimates for $\left(\mathcal{L}\right)^{-\beta/2}e^{i\sqrt{\mathcal{L}}}$. More
precisely we prove that it is a bounded operator on $H^p_{\mathcal{L},
at}(\mathbb{C}^n)$ when $\beta\geq (2n-1)\left(1/p-1/2\right)$.
Title: PROMYS Guest Lecture: What are Ramsey numbers?
Speaker: Rehana Patel (Wesleyan University, Middletown, USA)
Date: Fri, 14 Jun 2024
Time: 11:45 am
Venue: LH-5, Mathematics Department
Finite Ramsey theory is the study of structure that becomes unavoidable in large finite objects. In this talk, we will provide a brief taste of this rich and beautiful subject. We will start with the following question: In any group of six people, can we always find three who know one another or three who don’t? A far-reaching generalization of this question was first answered in a paper on logic by Frank Ramsey in 1928. Our approach to it will involve graph theory and combinatorics, with a dash of probability. No prerequisites will be needed to understand the talk.
Title: PhD Thesis defence: On existence and regularity of some complex Hessian equations on Kahler and transverse Kahler manifolds.
Speaker: P Sivaram (IISc Mathematics)
Date: Fri, 14 Jun 2024
Time: 11 am
Venue: Hybrid - LH-3, Department of Mathematics and Google meet (online)
The thesis consists of two parts. In the first part, we study the modified J-flow, introduced by Li-Shi. Analogous to the Lejmi-Szekelyhidi conjecture for the J-equation, Takahashi has conjectured that the solution to the modified J-equation exists if and only if some intersection numbers are positive and has verified the conjecture for toric manifolds. We study the behaviour of the modified J-flow on the blow-up of the projective spaces for rotationally symmetric metrics using the Calabi ansatz and obtain another proof of Takahashi’s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case i.e. when the positivity conditions fail. We prove that the flow develops singularities along a co-dimension one sub-variety. Moreover, away from this singular set, the flow converges to a solution of the modified J-equation, albeit with a different slope.
In the second part, we will describe a new proof of the regularity of conical Ricci flat metrics on Q-Goerenstein T-varieties. Such metrics arise naturally as singular models for Gromov-Hausdorff limits of Kahler-Einstein manifolds. The regularity result was first proved by Berman for toric manifolds and by Tran-Trung Nghiem in general. Nghiem adapted the pluri-potential theoretic approach of Kolodziej to the transverse Kahler setting. We instead adapt the purely PDE approach to $L^\infty$ estimates due to Guo-Phong et al. to the transverse Kahler setting, and thereby obtain a purely PDE proof of the regularity result.
Title: Geometry & Topology Seminar: Doing Morse theory without Smale condition
Speaker: Ipsita Datta (ETH Zurich)
Date: Wed, 12 Jun 2024
Time: 04:00 pm
Venue: LH-1 LH-3
We can apply Obstruction bundle gluing introduced by Hutchings and Taubes in Embedded contact homology to the setting of Morse theory. The goal is to understand this gluing technique in a simpler set up and analyse Morse theory with assumptions weaker than Smale condition on surfaces. This is work in progress with Yuan Yao.
Title: PhD Thesis colloquium: The Momentum Construction Method for Higher Extremal Kähler and Conical Higher cscK Metrics
Speaker: Rajas Sandeep Sompurkar (IISc Mathematics)
Date: Fri, 07 Jun 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
This Ph.D. thesis consists of two parts. In both the parts we study two new notions of canonical Kahler metrics introduced by Pingali viz. ‘higher extremal Kahler metric’ and ‘higher constant scalar curvature Kahler (higher cscK) metric’ both of whose definitions are analogous to the definitions of extremal Kahler metric and constant scalar curvature Kähler (cscK) metric respectively. On a compact Kahler manifold a higher extremal Kähler metric is a Kahler metric whose corresponding top Chern form equals its corresponding volume form multiplied by a smooth real-valued function whose gradient is a holomorphic vector field, while a higher cscK metric is a Kahler metric whose top Chern form is a real constant multiple of its volume form or equivalently whose top Chern form is harmonic. In both the parts we consider a special family of minimal ruled complex surfaces called as ‘pseudo-Hirzebruch surfaces’ which are the projective completions of holomorphic line bundles of non-zero degrees over compact Riemann surfaces of genera greater than or equal to two. These surfaces have got some nice symmetries in terms of their fibres and the zero and infinity divisors which enable the use of the momentum construction method of Hwang-Singer (a refinement of the Calabi ansatz procedure) for finding explicit examples of various kinds of canonical metrics on them.
In the first part of this Ph.D. thesis we will prove by using the momentum construction method that on a pseudo-Hirzebruch surface every Kahler class admits a higher extremal Kahler metric which is not a higher cscK metric. The construction of the required metric boils down to solving an ODE depending on a parameter on a closed and bounded interval with some boundary conditions, but the ODE is not directly integrable and requires a very delicate analysis for getting the existence of a solution satisfying all the boundary conditions. Then by doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude from this that higher cscK metrics (momentum-constructed or otherwise) do not exist in any Kahler class on this Kahler surface. We will briefly see the analogy of this problem with the related problem of constructing extremal Kähler metrics which are not cscK metrics on a pseudo-Hirzebruch surface which has been previously studied by Tønnesen-Friedman and Apostolov et al..
In the second part of this Ph.D. thesis we will see that if we relax the smoothness condition on our metrics a bit and allow our metrics to develop ‘conical singularities’ along at least one of the zero and infinity divisors of a pseudo-Hirzebruch surface then we do get ‘conical higher cscK metrics’ in each Kahler class of the Kahler surface by the momentum construction method. Even in this case the construction of the required metric boils down to solving a very similar ODE on the same interval but with different parameters and slightly different boundary conditions. We will show that our constructed metrics are conical Kahler metrics satisfying the strongest condition for conical metrics viz. the ‘polyhomogeneous condition’ of Jeffres-Mazzeo-Rubinstein, and we will interpret the conical higher cscK equation globally on the surface in terms of currents by using Bedford-Taylor theory. We will then employ the top ‘log Bando-Futaki invariant’ to obtain the linear relationship between the cone angles of the conical singularities of the metrics at the zero and infinity divisors of the surface.
Title: Algebra & Combinatorics Seminar: Ulrich subvarieties and non-existence of low rank Ulrich bundles on complete intersections
Speaker: Debaditya Raychaudhury (University of Arizona, Tucson, USA)
Date: Wed, 05 Jun 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
We characterize the existence of an Ulrich vector bundle on a variety $X\subset{\bf P}^N$ in terms of the existence of a subvariety satisfying certain conditions. Then we use this fact to prove that $(X,\mathcal{O}_X(a))$ where $X$ is a complete intersection of dimension $n\geq 4$, which if n = 4, is either ${\bf P}^4$ with $a\geq 2$, or very general with $a\geq 1$ and not of type (2), (2, 2), does not carry any Ulrich bundles of rank $r\leq 3$. Work in collaboration with A.F. Lopez.
Title: PROMYS Guest Lecture: Sums of n squares
Speaker: Raman Parimala (Emory University, Atlanta, USA)
Date: Mon, 03 Jun 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
We discuss representation of integers as a sum of $n$ squares. We explain the quaternionic composition law for sums of four squares and a proof of a theorem of Lagrange on which positive integers can be expressed as a sum of four squares. We outline general connections to the theory of quadratic forms.
Title: PROMYS Guest Lecture: Counting potato print patterns
Speaker: Tulsi Srinivasan (Azim Premji University, Bengaluru)
Date: Fri, 31 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
You might have carved a piece of potato to create a stamp to print with. How many patterns can we get from a single potato stamp? One approach to answering this question sheds light on the rich connections between objects and their symmetries, and leads us to a more general counting strategy.
Title: APRG Seminar: Rational dilation and constrained algebras
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Fri, 31 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
A set $\Omega$ is a spectral set for an operator $T$ if the spectrum of $T$ is contained in $\Omega$, and von Neumann’s inequality holds for $T$ with respect to the algebra $R(\Omega)$ of rational functions with poles off of the closure of $\Omega$. It is a complete spectral set if for all $n \in \mathbb{N}$, the same is true for $M_n(\mathbb C) \otimes R(\Omega)$. The rational dilation problem asks, if $\Omega$ is a spectral set for $T$, is it a complete spectral set for $T$? There are natural multivariable versions of this. There are a few cases where rational dilation is known to hold (e.g., over the disk and bidisk), and some where it is known to fail, for example over the Neil parabola, a distinguished variety in the bidisk. The Neil parabola is an example of a variety naturally associated to a constrained subalgebra of the disk algebra, namely $\mathbb{C} + z^2 A(\mathbb D)$. This talk discusses why rational dilation fails for a large class of such varieties associated to constrained algebras.
Title: APRG Seminar: Products of positive operators
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Thu, 30 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
The study of the class L+2 of Hilbert space operators which are the product of two bounded positive operators first arose in physics in the early ’60s. On finite dimensional Hilbert spaces, it is not hard to see that an operator is in this class if and only if it is similar to a positive operator. We extend the exploration of L+2 to separable infinite dimensional Hilbert spaces, where the structure is much richer, connecting (but not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The (generalized) spectral properties of elements of L+2 are also outlined, as well as membership in L+2 among various special classes of operators, including algebraic and compact operators.
Title: APRG Seminar: The operator Fejer-Riesz theorem in two variables, II
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Wed, 29 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
Here the proof that a positive (but not necessarily strictly positive) trigonometric polynomial with matrix coefficients can be written as a finite sum of hermitian squares of analytic polynomials is sketched. The difficulties in the case of such polynomials with coefficients which are operators on an infinite dimensional Hilbert space is also briefly discussed.
Title: APRG Seminar: The operator Fejer-Riesz theorem in two variables, I
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Tue, 28 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
The Fejer–Riesz thorem states that a positive (i.e., non-negative) trigonometric polynomial of degree $d$ on the unit circle is the hermitian square of an analytic polynomial of the same degree. Rosenblum extended this to polynomials with operator coefficients. The goal of these talks will be to outline a proof of a similar theorem in two variables. Since the techniques used in some proofs of the single variable case play an important role in the two variable proof, this particular talk concentrates primarily on these ideas. An application to strictly positive operator valued multivariable trigonometric polynomials is also considered.
Title: APRG Seminar: The scaling limit of the colored asymmetric simple exclusion process
Speaker: Milind Hegde (Columbia University, New York City, USA)
Date: Fri, 24 May 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
In the colored asymmetric simple exclusion process (ASEP), one places a particle of “color” $-k$ at each integer site $k \in \mathbb{Z}$. Particles attempt to swap places to the left with rate $q \in [0,1)$ and to the right with rate 1; the swap succeeds if the initiating particle has a higher color than the other particle (thus the particles tend to get more ordered over time). We will discuss the space-time scaling limit of this process (as well as a related discrete analog known as the stochastic six-vertex model), captured via a height function given by certain colored particle counts. The limit lies in the Kardar-Parisi-Zhang universality class, and is given by the Airy sheet and directed landscape, which were first constructed in 2018 by Dauvergne-Ortmann-Virág as limits in a very different setting – of fluctuations of a model of a random directed metric. The Yang-Baxter equation and line ensembles (collections of random non-intersecting curves) with certain Gibbs or spatial Markov properties will play fundamental roles in our discussion. This is based on joint work with Amol Aggarwal and Ivan Corwin.
Title: APRG Seminar: Asymptotic analysis of an optimal control problem for the Stokes system in a perforated domain
Speaker: Bidhan Chandra Sardar (IIT Ropar)
Date: Fri, 24 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
This talk focuses on the asymptotic analysis of an optimal control problem constrained by the stationary Stokes equations within a periodically perforated domain. The problem involves distributive controls applied to the interior region, where the Stokes operator includes oscillating coefficients for the state equations. We aim to demonstrate the convergence of the solutions of the considered optimal control problem to those of the limit OCP governed by the stationary Stokes equations and to establish the convergence of the associated cost functional.
Title: APRG Seminar: Optimal control and homogenization of semi-linear parabolic problems with highly oscillatory coefficients in a pillar-type domain
Speaker: Bidhan Chandra Sardar (IIT Ropar)
Date: Fri, 24 May 2024
Time: 10 am
Venue: LH-1, Mathematics Department
This talk considers an optimal control problem governed by a semi-linear heat equation within a two-dimensional pillar-type domain $\Omega_{\epsilon}$ .The problem features highly oscillatory periodic coefficients in both the state equation and the cost function $A_\epsilon$ and $B_\epsilon$. Our objective is to analyze the convergence of the optimal solutions (as $\epsilon \to 0$ ) and to identify the limit of the optimal control problem in a fixed domain that effectively captures the impact of the oscillatory coefficients.
Title: PhD Thesis colloquium: Holomorphic mappings and Kobayashi geometry of domains
Speaker: Annapurna Banik (IISc Mathematics)
Date: Fri, 24 May 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we shall study certain aspects of the geometry of the Kobayashi (pseudo)distance and the Kobayashi (pseudo)metric for domains in $\mathbb{C}^n$. We will focus on the following themes: on the interaction between Kobayashi geometry and the extension of holomorphic mappings, and on certain negative-curvature-type properties of Kobayashi hyperbolic domains equipped with their Kobayashi distances.
In the initial part of this talk, we shall present a couple of results on local continuous extension of proper holomorphic mappings $F:D \to \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $bD$ and $b\Omega$. These results are motivated by a well-known work by Forstneric–Rosay. However, our results allow us to have much lower regularity, for the patches of $bD, b\Omega$ that are relevant, than in earlier results in the literature. Moreover, our assumptions allow $b\Omega$ to contain boundary points of infinite type.
We will also discuss another type of extension phenomenon for holomorphic mappings, namely, Picard-type extension theorems. Well-known works by Kobayashi, Kiernan, and Joseph–Kwack have showed that Picard-type extension results hold true when the target spaces of the relevant holomorphic mappings belong to a class of Kobayashi hyperbolic complex manifolds – viewed as complex submanifolds embedded in some ambient complex manifold – with certain analytical properties. Beyond some classical examples, identifying such a target manifold by its geometric properties is, in general, hard. Restricting to $\mathbb{C}^n$ as the ambient space, we provide some geometric conditions on $b\Omega$, for any unbounded domain $\Omega \varsubsetneq \mathbb{C}^n$, for a Picard-type extension to hold true for holomorphic mappings into $\Omega$. These conditions are suggested, in part, by an explicit lower bound for the Kobayashi metric of a certain class of bounded domains. We establish the latter estimates using the regularity theory for the complex Monge–Ampere equation. The notion that allows us to connect these estimates with Picard-type extension theorems is called “visibility”.
In the concluding part of this talk, we will explore the notion of visibility for its own sake. For a Kobayashi hyperbolic domain $\Omega \varsubsetneq \mathbb{C}^n$, $\Omega$ being a visibility domain is a notion of negative curvature of $\Omega$ as a metric space equipped with the Kobayashi distance $K_{\Omega}$ and encodes a specific way in which $(\Omega, K_{\Omega})$ resembles the Poincare disc model of the hyperbolic plane. The earliest examples of visibility domains, given by Bharali–Zimmer, are pseudoconvex. In fact, all examples of visibility domains in the literature are, or are conjectured to be, pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.
Title: Algebra & Combinatorics Seminar: Equivariant cohomology in a tensor category
Speaker: Martin Andler (Université de Versailles, France)
Date: Wed, 22 May 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
In the 1950s, topologists introduced the notion of equivariant cohomology $H_G(E)$ for a topological space $E$ with an action by a compact group $G$. If the action is free, $H_G(E)$ should be $H(E/G)$, and be computed using de Rham cohomology. In 1950, even before the concept of equivariant cohomology had been formulated, Henri Cartan introduced a complex of equivariant differential forms for a compact Lie group acting on a differential manifold $E$, and proved a result amounting to stating that the cohomology of that complex computes $H_G(E)$. In 1999, Guillemin and Sternberg reformulated Cartan’s work in terms of a supersymmetric extension of the Lie algebra of $G$.
Our aim is to reconsider such considerations, by replacing vector spaces by a $k$-linear symmetric monoidal category, requiring that this category contain an odd unit to account for the supersymmetric dimension plus some further properties, and considering modules of a rigid Lie algebra object in that category. In that context, we obtain a version of Koszul’s homotopy isomorphism theorem, and recover as a consequence some known results as the acyclicity of the Koszul resolution. (Joint work with Siddhartha Sahi.)
Title: The Development of Mathematics in France since the late XIXth Century
Speaker: Martin Andler (Université de Versailles, France)
Date: Wed, 22 May 2024
Time: 11 am
Venue: LH-1, Mathematics Department
The strength of the French mathematical school goes back to the XVIIth century, with major figures like Descartes, Viète, Fermat or Pascal. But history shows that periods when French mathematics thrived alternated with less fruitful times. The factors are diverse, ranging from the role of singular geniuses to social and political causes: structure of higher education and research institutions, political upheavals, wars…
We will focus on the last 150 years: in the period before 1870, Germany had become the dominant scientific power, major advances were being made in England and Italy, to the great dismay of French scientists. In 1870, France was defeated by Prussia, the Emperor was overthrown, and a very favorable period started, allowing the emergence of a remarkable generation of French mathematicians. World War I had a disastrous effect on science (not only on science, of course), abruptly bringing the momentum to a halt. The reconstruction took some years: it is only in the 1950s that French mathematics flourished one again. By that time, however, mathematics had become much more specialized, and applied mathematics were left behind. It took again many years to reach a more balanced landscape where the pure and applied parts can thrive.
Title: PROMYS Guest Lecture: Rounding sequences and counting by congruence classes
Speaker: Caleb Shor (Western New England University, Springfield, USA)
Date: Tue, 21 May 2024
Time: 11:45 am
Venue: LH-1, Mathematics Department
For $n$ a natural number, consider the sequence of $n$ rational numbers $n/1, n/2, n/3, \dots, n/n$. Round each to the nearest integer to obtain sequence of $n$ integers. How many are odd?
In this talk we will see how knowledge of sums of squares and a result of Gauss will help to lead us to a somewhat surprising result. Time permitting, we will discuss similar results.
Title: PROMYS Guest Lecture: Who cut my cheese?
Speaker: Purvi Gupta (IISc Mathematics)
Date: Fri, 17 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Given a box packed with identical cubes of cheese, what is the maximum damage one can cause with a single straight cut through the box? This seemingly simple puzzle represents an old but recurrent mathematical theme that slices through numerous fields such as number theory, functional analysis, probability theory, and computational complexity theory. The cross-sections of convex bodies hold many mysteries, some of which continue to puzzle mathematicians today. We will focus on the deceptively simple case of the cube to demonstrate some of these ideas and open questions. No cheese will be harmed in the making of this talk.
Title: Geometry & Topology Seminar: Constructing smoothings of stable maps
Speaker: Mohan Swaminathan (Stanford University)
Date: Fri, 17 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
For positive integers $n$, $g$ and $d$, the moduli space $M(n,g,d)$ of degree d holomorphic maps to $\mathbb{CP}^n$ from non-singular projective curves of genus g is smooth and irreducible for $d > 2g-2.$ It is contained as an open subset within the compact moduli space $K(n,g,d)$ of “stable maps”, i.e., degree d holomorphic maps to $\mathbb{CP}^n$ from at-worst-nodal projective curves of arithmetic genus $g.$ An unfortunate feature of this very natural compactification is that $M(n,g,d)$ is far from being dense in $K(n,g,d)$. Concretely, this means that many stable maps are not “smoothable”, i.e., they don’t arise as limits of non-singular ones. In my talk, I will explain this phenomenon and a new sufficient condition for smoothability of stable maps, obtained in joint work with Fatemeh Rezaee.
Title: PhD Thesis colloquium: On Two complex Hessian equations and convergence of corresponding flows
Speaker: Ramesh Mete (IISc Mathematics)
Date: Fri, 10 May 2024
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
This dissertation consists of three parts, and two important types of complex hessian equations, namely – the J-equation and the deformed Hermitian Yang Mills (dHYM) equation.
In the first part, the main aim is to find out some appropriate “singular” solutions of the equations when they don’t admit smooth solutions (or equivalently, when the so-called “Nakai criteria” fails) - this is the so-called unstable case. An algebro-geometric characterization of the slopes for both the equations is formulated – which we call the “minimal J-slope” in the case of the J-equation and the “maximal dHYM-slope” for the dHYM equation. On compact Kahler surfaces we show that there exists a unique closed $(1,1)$- Kahler current that solves the “weak version” of the equations (i.e. the wedge product is replaced by the “non-pluripolar product”) with the modified slopes. In the higher dimensional case, we conjecture analogous existence and uniqueness results.
In the second part, the convergence behavior of the J-flow is studied on certain generalized projective bundles using Calabi symmetry. For the bundles an “invariant version” of the minimal J-slope is introduced. Furthermore, we show that the flow converges to some unique limit in the weak sense of currents, and the limiting current solves the J-equation with the invariant minimal J-slope. This result resolves our conjecture for J-equation on these examples with symmetry.
In the third part, the convergence behavior of a dHYM flow, called the “cotangent flow”, is studied in the unstable case on the blowup of $\mathbb{CP}^2$ or $\mathbb{CP}^3$. Analogous to our results in the second part, it is shown that this flow converges to some unique limit in the unstable case, and the limiting current solves the dHYM equation with the (invariant) maximal dHYM-slope.
Title: PROMYS Guest Lecture: Are we there yet?
Speaker: Jaclyn Lang (Temple University, Philadelphia, USA)
Date: Fri, 10 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
How far is it from Bangalore to Chennai? Is there a single correct answer to this question? In this talk we will explore different notions of distance as well as why you might choose one over another depending on the context. This will take us on a brief sight seeing tour through geometry, graph theory, and number theory.
Title: Geometry & Topology Seminar: Spectral Theory of the Sub-Riemannian Laplacian
Speaker: Nikhil Savale (University of Cologne)
Date: Mon, 22 Apr 2024
Time: 04:00 pm
Venue: LH-1
Sub-Riemannian (sR) geometry is the geometry of bracket-generating metric distributions on a manifold. Peculiar phenomena in sR geometry include the exotic Hausdorff dimension describing the growth rate of the volumes of geodesic balls. As well as abnormal geodesics that do not satisfy any variational equation. In this talk I will survey my results, to appear in a forthcoming book, which show how both these phenomena are reflected in the spectral theory of the hypoelliptic Laplacian in sR geometry.
Title: MS Thesis colloquium: Higher categories of n-bordisms
Speaker: Arghan Dutta (IISc Mathematics)
Date: Fri, 19 Apr 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
In this talk, we will discuss the notion of a complete Segal space – a model of an infinity category, and then study the infinity category of $n$-bordisms.
Higher category theory is the generalization of category theory to a context where there are not only morphisms between objects, but generally $k$-morphisms between $(k−1)$-morphisms, for all $k \in \N$. The theory of higher categories or $(\infty, 1)$-categories, as it is sometimes called, however, can be very intractable at times. That is why there are now several models which allow us to understand what a higher category should be. Among these models is the theory of quasi-categories, introduced by Bordman and Vogt, and much studied by Joyal and Lurie. There are also other very prominent models such as simplicial categories (Dwyer and Kan), relative categories (Dwyer and Kan), and Segal categories (Hirschowitz and Simpson). One of those models, complete Segal spaces, was introduced by Charles Rezk in his seminal paper “A model for the homotopy theory of homotopy theory”. Later they were shown to be a model for $(\infty, 1)$-categories.
One major application of higher category theory and one of the driving forces in developing it has been extended topological quantum field theory. This has recently led to what may become one of the central theorems of higher category theory, the proof of the cobordism hypothesis, conjectured by Baez and Dolan. Lurie suggested passing to $(\infty, n)$-categories for a proof of the Cobordism Hypothesis in arbitrary dimension $n$. However, finding an explicit model for such a higher category poses one of the difficulties in rigorously defining these $n$-dimensional TFTs, which are called “fully extended”. Our focus will be on the $(\infty, 1)$-category $\mathrm{Bord}_n^{(n -1)}$, a variant of the fully extended $\mathrm{Bord}_{n}$. Our goal is to sketch a detailed construction of the $(\infty, 1)$-category of $n$-bordisms as a complete Segal space.
Title: PhD Thesis colloquium: Invariants Associated with Complete Nevanlinna-Pick Spaces
Speaker: Abhay Jindal (IISc Mathematics)
Date: Thu, 18 Apr 2024
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we shall talk about two invariants associated with complete Nevanlinna-Pick (CNP) spaces. One of the invariants is an operator-valued multiplier of a given CNP space, and another invariant is a positive real number. These two invariants are called characteristic function and curvature invariant, respectively. The origin of these concepts can be traced back to the classical theory of contractions by Sz.-Nagy and Foias.
We extend the theory of Sz.-Nagy and Foias about the characteristic function of a contraction to a commuting tuple $(T_{1}, \dots, T_{d})$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. Surprisingly, there is a converse, which roughly says that if a kernel $k$ admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain. So, what can be said if $(T_{1}, \dots,T_{d})$ is $1/k$-contractive when $k$ is an irreducible unitarily invariant kernel, but does not have the complete Nevanlinna-Pick property? We shall see that if $k$ has a complete Nevanlinna-Pick factor $s$, then much can be retrieved.
We associate with a $1/k$-contraction its curvature invariant. The instrument that makes this possible is the characteristic function. We present an asymptotic formula for the curvature invariant. In the special case when the $1/k$-contraction is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of the $1/k$-contraction specifically when its characteristic function is a polynomial.
Title: PhD Thesis colloquium: Maps between Non-compact Surfaces
Speaker: Sumanta Das (IISc Mathematics)
Date: Wed, 17 Apr 2024
Time: 4 pm
Venue: LH-1, Department of Mathematics
In this talk, we discuss proper maps between two non-compact surfaces, with a particular emphasis on facts stemming from two fundamental questions in topology: whether every homotopy equivalence between two $n$-manifolds is homotopic to a homeomorphism, and whether every degree-one self-map of an oriented manifold is a homotopy equivalence.
Topological rigidity is the property that every homotopy equivalence between two closed $n$-manifolds is homotopic to a homeomorphism. This property refines the notion of homotopy equivalence, implying homeomorphism for a particular class of spaces. According to Nielsen’s results from the 1920s, compact surfaces exhibit topological rigidity. However, topological rigidity fails in dimensions three and above, as well as for compact bordered surfaces.
We prove that all non-compact, orientable surfaces are properly rigid. In fact, we prove a stronger result: if a homotopy equivalence between any two noncompact, orientable surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided that the surfaces are neither the plane nor the punctured plane. As an application, we also prove that any $\pi_1$-injective proper map between two non-compact surfaces is properly homotopic to a finite-sheeted covering map, given that the surfaces are neither the plane nor the punctured plane.
An oriented manifold $M$ is said to be Hopfian if every self-map $f\colon M\to M$ of degree one is a homotopy equivalence. This is the natural topological analog of Hopfian groups. H. Hopf posed the question of whether every closed, oriented manifold is Hopfian. We prove that every oriented infinite-type surface is non-Hopfian. Consequently, an oriented surface $S$ is of finite type if and only if every proper self-map of $S$ of degree one is homotopic to a homeomorphism.
Title: Number Theory Seminar: Zig-zag holds for Galois representations
Speaker: Eknath Ghate (TIFR Mumbai)
Date: Fri, 12 Apr 2024
Time: 2 pm
Venue: LH-1
It is an open problem to describe the shape of the reductions of local Galois representations attached to cusp forms at primes away from the level, or more generally, the shape of the reductions of two-dimensional crystalline representations. Partial results go back to Deligne, Fontaine and Edixhoven. One folklore conjecture (attributed to Breuil, Buzzard and Emerton) is that if the weight is even and the slope is fractional, then the reduction is always irreducible.
In this talk we shall state and prove our zig-zag conjecture which deals with large exceptional weights and half-integral slopes. These weights fall squarely outside the scope of the above conjecture. The conjecture states that the reduction in these cases is given by an alternating sequence of irreducible and reducible representations depending on the size of two auxiliary parameters. Special cases of zig-zag have been proved over the years by various authors using Langlands correspondences.
The present general proof uses the reverse of a recent limiting argument due to Chitrao-Ghate-Yasuda in the Colmez-Chenevier rigid analytic blow up space of trianguline representations to reduce the study of the reduction of crystalline representations to results on the reductions of semi-stable representations due to Breuil-Mezard, Guerberoff-Park and most recently Chitrao-Ghate.
Title: APRG Seminar: Pointwise localisation and weighted inequalities for Rubio de Francia square function
Speaker: Luz Roncal (BCAM - Basque Center for Applied Mathematics, Spain)
Date: Wed, 10 Apr 2024
Time: 4 pm
Venue: Microsoft Teams (online)
Let us denote by Rubio de Francia square function the square function formed by frequency projections on a collection of disjoint intervals of the real line. J. L. Rubio de Francia established in 1985 that this operator is bounded on $L^p$ for $p\ge 2$ and on $L^p(w)$, for $p>2$, with weights $w$ in the Muckenhoupt class $A_{p/2}$. What happens in the endpoint $L^1(w)$ for $w \in A_1$ was left open, and Rubio de Francia conjectured the validity of the estimate in this endpoint.
In this talk we will show a new pointwise sparse estimate for the Rubio de Francia square function. Such a bound implies quantitative weighted estimates which, in some cases, improve the available results. We will also confirm that the $L^2(w)$ conjecture is verified for radially decreasing even $A_1$ weights, and in full generality for the Walsh group analogue to the Rubio de Francia square function. In general, the $L^2$ weighted inequality is still an open problem.
Joint work with Francesco Di Plinio, Mikel Flórez-Amatriain, and Ioannis Parissis.
The video of this talk is available on the IISc Math Department channel.
Title: Algebra & Combinatorics Seminar: Epimorphism theorems and allied topics
Speaker: Amartya Kumar Dutta (ISI Kolkata)
Date: Tue, 09 Apr 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
In the area of Affine Algebraic Geometry, there are several problems on polynomial rings which are easy to state but difficult to investigate. Late Shreeram S. Abhyankar was the pioneer in investigating a class of such problems known as Epimorphism Problems or Embedding Problems. In this non-technical survey talk, we shall highlight some of the contributions of Abhyankar, Moh, Suzuki, Sathaye, Russell, Bhatwadekar and other mathematicians.
Title: In praise of Khahara
Speaker: Amartya Kumar Dutta (ISI Kolkata)
Date: Mon, 08 Apr 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
It was in ancient India that zero received its first acceptance as an integer in its own right. There was an awareness about its perils and yet ancient Indian mathematicians not only embraced zero as an integer but allowed it to participate in all four arithmetic operations, including as a divisor in a division.
But division by zero is strictly forbidden in the present edifice of mathematics. Verses from mathematical giants like Brahmagupta and Bhaskaracharya referring to numbers with “zero in the denominator” shock the modern reader. Certain examples in the Bijaganita of Bhaskaracharya appear as absurd nonsense.
But then there was a time when square roots of negative numbers were considered non-existent and forbidden; even the validity of subtracting a bigger number from a smaller number (i.e., the existence of negative numbers) took a long time to gain universal acceptance. Is it possible that we have simply bound ourselves to a certain safe convention and that there could be other approaches (“mathematical models” in fancy language) in which the ideas of Brahmagupta and Bhaskaracharya, and even the examples of Bhaskaracharya, will appear not only valid but even natural?
Enterprising modern mathematicians have created elaborate legal (or technical) machinery to overcome the limitations imposed by the prohibition against use of zero in the denominator. The most familiar are the methods of calculus with its concept of limit, results like l’Hopital’s rule, and a language which enables one to express intuitive ideas like $\frac{1}{0} = \infty$ through legally permitted euphemisms. Less well-known are the devices of commutative algebra and algebraic geometry like “localisation” which describes a legal structure for directly writing fractions with zero in the denominator without any subterfuge, and the more sophisticated ideas of “valuation theory” which admit multiple levels of infinities and thereby provide higher-dimensional analogues of l’Hopital’s rule.
In this talk we shall present an algebraic model proposed by Prof. Avinash Sathaye for understanding Bhaskaracharya’s treatment of khahara, (numbers with) zero in the denominator. A crucial ingredient of this model is the important concept of “idempotent” in modern abstract algebra (elements $e$ satisfying the relation $e^2=e$). To historians of mathematics who have tried to interpret Bhaskaracharya’s khahara in the light of calculus, the examples in Bijaganiita appear as absurdities. But all difficulties disappear in the light of the algebraic treatment based on idempotents. A verse from the commentary of Kr.s.n.adaivaj˜na indicates that idempotence was indeed envisaged as a natural property of numbers like zero and its reciprocal, the khahara.
Prof. Sathaye’s interpretation of Bhaskaracharya’s khahara also gives a new meaning to certain mysterious utterances of Ramanujan recorded by P.C. Mahalanobis. In the light of valuation theory, Bhaskaracharya’s khahara not only deserves our praise, perhaps they indicate unexplored possibilities!
Title: Eigenfunctions Seminar: Brownian objects
Speaker: Sanchayan Sen (IISc Mathematics)
Date: Fri, 05 Apr 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
The Brownian motion is the scaling limit of random walks where the step distribution has finite second moment. Various random objects constructed from the Brownian motion, e.g., the Brownian continuum random tree and the Brownian map, arise naturally in the study of random trees, graphs, and maps. In the first talk, we will give a gentle introduction to these objects. In the second talk, we will discuss some recent advances in establishing certain Brownian objects as the scaling limits of different models of random discrete structures.
Title: PhD Thesis defence: Dominating surface-group representations via Fock-Goncharov coordinates
Speaker: Pabitra Barman (IISc Mathematics)
Date: Thu, 04 Apr 2024
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
Let $S_{g,k}$ be a connected oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$ be two representations. $\rho_2$ is said to dominate $\rho_1$ if there exists $\lambda \le 1$ such that $\ell_{\rho_1}(\gamma) \le \lambda \cdot \ell_{\rho_2}(\gamma)$ for all $\gamma \in \pi_1(S_{g,k})$, where $\ell_{\rho}(\gamma)$ denotes the translation length of $\rho(\gamma)$ in $\mathbb{H}^3$. In 2016, Deroin–Tholozan showed that for a closed surface $S$ and a non-Fuchsian representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{R})$ that strictly dominates $\rho$. In 2023, Gupta–Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher-rank representations.
For a representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, the Hilbert length of a curve $\gamma\in \pi_1(S_{g,k})$ for $n >2$ is defined as \begin{equation} \ell_{\rho}(\gamma):=\ln \Bigg| \frac{\lambda_n}{\lambda_1} \Bigg|, \end{equation} where $\lambda_n$ and $\lambda_1$ are the largest and smallest eigenvalues of $\rho(\gamma)$ in modulus respectively. We show that for any generic representation $\rho : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock-Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$ representation. Weighted planar networks and the Collatz–Wielandt formula for totally positive matrices play a crucial role.
Let $X_n$ be the symmetric space of $PSL_n(\mathbb{C})$. The translation length of $A\in PSL_n(\mathbb{C})$ in $X_n$ is given as \begin{equation} \tau(A)= \sum_{i=1}^{n}\log |\lambda_i(A)|^2, \end{equation} where $\lambda_i(A)$ are the eigenvalues of $A$. We show that the same $j$ dominates $\rho$ with respect to the translation length at the origin as well. Lindström’s Lemma for planar networks and Weyl’s Majorant Theorem are some of the key ingredients of the proof.
In both cases, if $S_{g,k}$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.
Title: PhD Thesis defence: Geometry of Normed Linear Spaces in Light of Birkhoff-James Orthogonality
Speaker: Babhrubahan Bose (IISc Mathematics)
Date: Thu, 04 Apr 2024
Time: 3 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
Two points $x$ and $y$ in a normed linear space $\mathbb{X}$ are said to be Birkhoff-James orthogonal (denoted by $x\perp_By$) if $|x+\lambda y|\geq|x|~~\text{for every scalar}~\lambda.$ James proved that in a normed linear space of dimension more than two, Birkhoff-James orthogonality is symmetric if and only if the parallelogram law holds. Motivated by this result, Sain introduced the concept of pointwise symmetry of Birkhoff-James orthogonality in a normed linear space.
In this talk, we shall try to understand the geometry of normed spaces in the light of Birkhoff-James orthogonality. After introducing the basic notations and terminologies, we begin with a study of the geometry of the normed algebra of holomorphic maps in a neighborhood of a curve and establish a relationship among the extreme points of the closed unit ball, Birkhoff-James orthogonality, and zeros of holomorphic maps.
We next study Birkhoff-James orthogonality and its pointwise symmetry in Lebesgue spaces defined on arbitrary measure spaces and natural numbers. We further find the onto isometries of the sequence spaces using the pointwise symmetry of the orthogonality.
We shall then study the geometry of tensor product spaces and use the results to study the relationship between the symmetry of orthogonality and the geometry (for example, extreme points and smooth points) of certain spaces of operators. Our work in this section is motivated by the famous Grothendieck inequality.
Finally, we study the geometry of $\ell_p$ and $c_0$ direct sums of normed spaces ($1\leq p<\infty$). We shall characterize the smoothness and approximate smoothness of these spaces along with Birkhoff-James orthogonality and its pointwise symmetry. As a consequence of our study we answer a question pertaining to the approximate smoothness of a space, raised by Chmeilienski, Khurana, and Sain.
Title: Algebra & Combinatorics Seminar: Approximate packing of independent transversals in locally sparse graphs
Speaker: Debsoumya Chakraborti (Mathematics Institute, University of Warwick, UK)
Date: Wed, 03 Apr 2024
Time: 4 pm
Venue: LH-1, Mathematics Department (Joint with the APRG Seminar)
Consider a multipartite graph $G$ with maximum degree at most $n-o(n)$, parts $V_1,\ldots,V_k$ have size $|V_i|=n$, and every vertex has at most $o(n)$ neighbors in any part $V_i$. Loh and Sudakov proved that any such $G$ has an independent set, referred to as an ‘independent transversal’, which contains exactly one vertex from each part $V_i$. They further conjectured that the vertex set of $G$ can be decomposed into pairwise disjoint independent transversals. We resolve this conjecture approximately by showing that $G$ contains $n-o(n)$ pairwise disjoint independent transversals. As applications, we give approximate answers to questions on packing list colorings and multipartite Hajnal-Szemerédi theorem. We use probabilistic methods, including a ‘two-layer nibble’ argument. This talk is based on joint work with Tuan Tran.
Title: PhD Thesis defence: Some aspects of weighted kernel functions on planar domains
Speaker: Aakanksha Jain (IISc Mathematics)
Date: Mon, 01 Apr 2024
Time: 4:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we discuss various aspects of weighted kernel functions on planar domains. We focus on two key kernels, namely, the weighted Bergman kernel and the weighted Szegő kernel.
For a planar domain and an admissible weight function on it, we discuss some aspects of the corresponding weighted Bergman kernel. First, we see a precise relation between the weighted Bergman kernel and the classical Bergman kernel near a smooth boundary point of the domain. Second, the weighted kernel gives rise to weighted metrics in the same way as the classical Bergman kernel does. Motivated by work of Mok, Ng, Chan–Yuan and Chan–Xiao–Yuan among others, we talk about the nature of holomorphic isometries from the unit disc with respect to the weighted Bergman metrics arising from weights of the form $K(z,z)^{-d}$, where $K$ denotes the classical Bergman kernel and $d$ is a non-negative integer. Specific examples that we discuss in detail include those in which the isometry takes values in polydisk or a cartesian product of a disc and a unit ball, where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, we also present the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above.
In the next part of the talk, we discuss properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell’s work, the starting point is a weighted Kerzman–Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), we show that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, we show that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell’s list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.
Title: PhD Thesis colloquium: On regular subalgebras of Kac-Moody algebras and related combinatorics
Speaker: Md. Irfan Habib (IISc Mathematics)
Date: Thu, 28 Mar 2024
Time: 3 pm
Venue: Hybrid - LH-1, Department of Mathematics and Microsoft Teams (online)
In this thesis, we study two aspects of Kac-Moody algebras. One is to understand the subalgebras that can be embedded inside a Kac-Moody algebra as subalgebras generated by real root vectors. The other one is to explicitly classify the regular subalgebras and the maximal regular subalgebras of an untwisted affine Kac-Moody algebra.
Dynkin classified the semisimple regular subalgebras of a finite-dimensional semisimple Lie algebra back in $1949.$ One of the key tools he used for the classification is $\pi$-systems. For non-finite Kac-Moody algebras, $\pi$-system became an integral part of und erstanding the embedding of different types of algebras in a Kac-Moody algebra. Till now all the articles existing in the literature, which study $\pi$-systems, assume that the $\pi$-systems are either linearly independent or finite. It seems that our work is the first one to address infinite $\pi$ systems in the context of the embedding problem. This paves a way for us to understand the infinite (linearly independent) $\pi$-systems for Borcherds Kac-Moody algebras and understand the embedding problem in that setting. We used Deodhar’s preorder to prove that every closed subroot system in a Kac-Moody root system admits a $\pi$-system and this $\pi$-system need not be finite in general. Moreover, for any closed subroot system $\Psi$ of $\Delta,$ we prove that there exists a unique $\pi$-system $\Pi(\Psi),$ which is contained in the set of positive roots. Since the subroot systems of a root system are not very ‘well behaved’, this is quite surprising and it generalizes the previously well-known fact that they simple systems and positive systems determine each other at the level of the subroot system.
Using this unique $\pi$-system $\Pi(\Psi)$, we prove that for a real closed subroot system $\Psi,$ the real roots of a root generated subalgebra $\mathfrak g(\Psi)$ is equal to $\Psi$. This result was a much-awaited one in the literature because almost after $70$ years of Dynkin’s result, Roy and Venkatesh (Transform. Groups 2019) proved that the same is true for an affine root system. These two results provide a bridge between the algebraic and combinatorial side which shows that the root-generated subalgebras are in bijection with the real closed subroot systems which are in turn in one-to-one correspondence with the $\pi$-systems contained in the positive roots of a Kac-Moody algebra.
In the last part of our analysis of regular subalgebras generated by root vectors, we prove that for any closed subroot system $\Psi,$ the root generated subalgebra is isomorphic to a quotient of the derived subalgebra of the Kac-Moody algebra corresponding to the (infinite) Cartan matrix defined by the unique $\pi$-system of the closed subroot system $\Psi,$ by an ideal contained in the centre of the algebra. This result is a generalization of the existing results when the $\pi$-system is linearly independent and the ideal is zero when the $\pi$-system is linearly independent also follows from our result. In particular, as long as the roots are concerned, to understand the root generated subalgebras, it is enough to consider the derived algebras of Kac-Moody algebras $\mathfrak g’(A)$ corresponding to a(n infinite) GCM $A.$ Classification of regular subalgebras of an affine Kac-Moody Lie algebra is an interesting problem in its own right. Barnea et al. started such classification in $1998.$ Later Felikson et al. used combinatorics of root systems to classify the regular subalgebras in 2008, more precisely the root generated subalgebras of an affine Kac-Moody algebra. We took a completely different approach, namely, using the classification of the closed subroot system of a real affine root system given by Roy and Venkatesh, we classify the regular subalgebras of affine Kac-Moody Lie algebras with a symmetric set of roots and we get the classification of root generated subalgebras as a Corollary. Moreover, we also classify the maximal symmetric regular subalgebras we show a bijective correspondence between the maximal real closed subroot systems of the affine Lie algebra and the maximal symmetric regular subalgebras different from $[\mathfrak g,\mathfrak g].$ Which also shows that in the affine case, given a maximal closed subroot system $\Psi$ of $\Delta,$ the poset (with set inclusion as the partial order)
\begin{equation} A_\Psi:={\mathfrak s:\Delta(\mathfrak s)^{\mathrm{re}}=\Psi} \end{equation}
contains a unique maximal element.
Title: APRG Seminar: Restricted mean value property on Riemannian manifolds
Speaker: Utsav Dewan (ISI Kolkata)
Date: Wed, 27 Mar 2024
Time: 4 pm
Venue: Microsoft Teams (online)
A continuous function in an Euclidean domain is harmonic if and only if it satisfies the spherical mean value property for all spheres contained in that domain. But what happens if a continuous function satisfies instead the following ‘restricted mean value property’: for each point in the domain it satisfies the mean value property precisely on one such sphere (centered at the point). Then is the function still going to be harmonic? This is the classical ‘one-circle problem’ posed by Littlewood. We will see some results dealing with sufficient conditions in terms of the boundary behavior of the function for the above problem to have an affirmative answer in the setting of (1) domains in Riemannian manifolds and (2) Hadamard manifolds of pinched negative sectional curvature, extending classical results of Fenton for the Euclidean unit disc. This is based on a joint work with Prof. Kingshook Biswas.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Poles of Quotients of $L$-functions and primitivity of $L$-functions of $\mathrm{GL}_3$
Speaker: Ravi Raghunathan (IIT Bombay)
Date: Fri, 22 Mar 2024
Time: 2 pm
Venue: LH-1
Given two distinct cuspidal automorphic $L$-functions (of $\mathrm{GL}_n$ and $\mathrm{GL}_m$ over $\mathbb{Q}$) one expects that their quotients will have infinitely poles, but this is surprisingly hard to prove. In this talk, I will discuss my recent work on the case $m=n-2$ and the primitivity of the $L$-functions of cuspidal automorphic $L$-functions of $\mathrm{GL}_3$. These methods also work for Artin $L$-functions and, more generally, for the $L$-functions of Galois representations under further hypotheses.
Title: Number Theory Seminar: On values of the digamma function
Speaker: Siddhi Pathak (CMI)
Date: Fri, 15 Mar 2024
Time: 2 pm
Venue: LH-1
One of the central themes in number theory is the study of special values of $L$-functions, in particular, the investigation of their transcendental nature and algebraic relations among them. A special function governing linear relations among the values $L(1,\chi)$ as $\chi$ varies over Dirichlet characters modulo $q$, is the digamma function, which is the logarithmic derivative of the gamma function. In this talk, we discuss the arithmetic nature and related properties of values of the digamma function at rational arguments, and emphasize their connection with a seemingly unrelated conjecture of Erdos, which is still open.
Title: APRG Seminar: Hardy and Rellich identities and inequalities for Grushin operators via spherical vector fields and Bessel pairs
Speaker: Jotsaroop Kaur (IISER Mohali)
Date: Wed, 13 Mar 2024
Time: 4 pm
Venue: Microsoft Teams (online)
We establish Hardy, Hardy-Rellich and Rellich identities and inequalities with sharp constants for Grushin vector fields. We provide explicit remainder terms which substantially improve those known in the literature. This is based on a joint work with Debdip Ganguly and Prasun Roychowdhury.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: $p$-adic deformations of automorphic forms and Iwasawa Theory
Speaker: Mladen Dimitrov (University of Lille, France)
Date: Thu, 07 Mar 2024
Time: 11.30 am
Venue: LH-1
Families of $p$-adic cusp forms were first introduced by Hida, later leading to the construction of the eigencurve by Coleman and Mazur. Generalizations to reductive groups of higher rank, called eigenvarieties, are rigid analytic spaces providing the correct setup for the study of $p$-adic deformations of automorphic forms. In order to obtain arithmetic applications, such as constructing $p$-adic $L$-functions or proving explicit reciprocity laws for Euler systems, one needs to perform a meaningful limit process requiring to understand the geometry of the eigenvariety at the point corresponding to the $p$-stabilization of the automorphic form we are interested in.
While the geometry of an eigenvariety at points of cohomological weight is well understood thanks to classicality results, the study at classical points which are limit of discrete series (such as weight $1$ Hilbert modular forms or weight $(2,2)$ Siegel modular forms) is much more involved and the smoothness at such points is a crucial input in the proof of many cases of the Bloch–Kato Conjecture, the Iwasawa Main Conjecture and Perrin-Riou’s Conjecture.
Far more fascinating is the study of the geometry at singular points, especially at those arising as intersection between irreducible components of the eigenvariety, as those are related to trivial zeros of adjoint $p$-adic $L$-functions.
In this talk we will illustrate this philosophy based on ideas of Joël Bellaïche.
Title: APRG Seminar: Singular integral operators along codimension one subspaces depending on n-1 variables
Speaker: Mikel Florez Amatriain (BCAM – Basque Center for Applied Mathematics, Spain)
Date: Wed, 06 Mar 2024
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, I will report a work in progress in which we show $L^p$ bounds for singular integral operators formed by $(n-1)$-dimensional Hörmander-Mihlin multipliers. In our case, the multipliers act depending on $(n-1)$-dimensional variable subspaces, which depend only on the first $n-1$ variables.
We prove $L^p$ boundedness for these operators for $p>3/2$. Assuming that the frequency support of the function is contained in an annulus, we can show $L^p$ boundedness for $p>1$.
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Waring's problem
Speaker: Ramachandran Balasubramanian (IMSc, Chennai)
Date: Fri, 01 Mar 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Define $g(k) = \min \{ s :$ every positive integer can be written as a sum of $k$th powers of natural numbers with atmost $s$ summands$\}$. Lagrange proved that $g(2) = 4$. Waring conjectured that $g(3) = 9, g(4) = 19$ and so on.
In fact, in this question, there has been a lot of contribution from Indian mathematicians. The method of attacking this problem is called the circle method and it originates from a seminal paper of Hardy and Ramanujan. The final result owes a lot to the contributions of S.S. Pillai. The analogous question over number fields was settled by C.P. Ramanujam. We shall explain their contributions toward this problem.
Title: Number Theory Seminar: Linear congruence relations for exponents of Borcherds products
Speaker: Badri Vishal Pandey (University of Cologne, Germany)
Date: Fri, 01 Mar 2024
Time: 10 am
Venue: LH-1
For all positive powers of primes $p \geq 5$, we prove the existence of infinitely many linear congruences between the exponents of twisted Borcherds products arising from a suitable scalar-valued weight $1/2$ weakly holomorphic modular form or a suitable vector-valued harmonic Maassform. To this end, we work with the logarithmic derivatives of these twisted Borcherds products, and offer various numerical examples of non-trivial linear congruences between them modulo $p=11$. In the case of positive powers of primes $p = 2, 3$, we obtain similar results by multiplying the logarithmic derivative with a Hilbert class polynomial as well as a power of the modular discriminant function. Both results confirm a speculation by Ono. (joint work with Andreas Mono).
Title: APRG Seminar: Boundedness of pseudo-differential operators on radial sections of line bundles over the Poincaré upper half plane SL(2)/SO(2)
Speaker: Tapendu Rana (Ghent University, Belgium)
Date: Wed, 28 Feb 2024
Time: 4 pm
Venue: Microsoft Teams (online)
For a given function $a(x,\xi)$ on $\mathbb{R}^n \times \mathbb{R}^n$, consider the pseudo-differential operator $a(x,D)$ defined by
\begin{equation} a(x,D) (f)(x) =\int_{\mathbb{R}^n} a(x,\xi) \widehat f(\xi) e^{2\pi i x\cdot \xi} d\xi, \end{equation}
where $\widehat{f}$ denotes the Fourier transform of a function $f$. Let $S^0$ be the set of all smooth functions $a: \mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{C}$ satisfying
\begin{equation} \left| \frac{\partial^\beta_x}{\partial_{\xi} ^\alpha} a (x,\xi)\right| \leq {C_{\alpha,\beta} }\, {( 1+ |\xi| )^{-|\alpha|}} \end{equation}
for all $x,\xi \in \mathbb{R}^n $ and for all multi indices $\alpha$ and $\beta$. It is well known that for $a\in S^0$, the associated
pseudo-differential operator $a(x,D)$ extends to a bounded operator on $L^p(\mathbb{R}^n)$ to itself, for $1<p<\infty$.
In this talk, we will discuss an analogue of this result on radial sections of line bundles over the Poincaré upper half plane. More precisely, we will focus on the group $G=\mathrm{SL}(2,\mathbb{R})$, where we will explore the boundedness properties of the pseudo-differential operator defined on functions of fixed $K=\mathrm{SO}(2)$-type in $G$. Additionally, we will explore the case where the symbol exhibits restricted regularity in the spatial variable.
This talk is based on a joint work with Michael Ruzhansky.
The video of this talk is available on the IISc Math Department channel.
Title: Algebra & Combinatorics Seminar: Monomial expansions for q-Whittaker polynomials
Speaker: Ratheesh T V (IMSc Chennai)
Date: Fri, 23 Feb 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We consider the monomial expansion of the $q$-Whittaker polynomials given by the fermionic formula and via the inv and quinv statistics. We construct bijections between the parametrizing sets of these three models which preserve the $x$- and $q$-weights, and which are compatible with natural projection and branching maps. We apply this to the limit construction of local Weyl modules and obtain a new character formula for the basic representation of $\widehat{\mathfrak{sl}_n}$.
Title: APRG Seminar: Counting rational points near flat hypersurfaces
Speaker: Rajula Srivastava (Hausdorff Center / Universitaet Bonn / Max Planck Institute, Germany)
Date: Wed, 21 Feb 2024
Time: 4 pm
Venue: Microsoft Teams (online)
How many rational points with denominator of a given size lie within a given distance from a compact hypersurface? In this talk, we shall describe how the geometry of the surface plays a key role in determining this count, and present a heuristic for the same. In a recent breakthrough, J.J. Huang proved that this guess is indeed true for hypersurfaces with non-vanishing Gaussian curvature. What about hypersurfaces with curvature only vanishing up to a finite order, at a single point? We shall offer a new heuristic in this regime which also incorporates the contribution arising from “local flatness”. Further, we will describe how several ideas from Harmonic Analysis can be used to establish the indicated estimates for hypersurfaces of this type immersed by homogeneous functions. Based on joint work with N. Technau.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: A note on Linear Complementarity via zero sum two person games
Speaker: T.E.S. Raghavan (University of Illinois at Chicago, USA)
Date: Tue, 20 Feb 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
The matrix $M$ of a linear complementarity problem can be viewed as a payoff matrix of a two-person zero-sum game. Lemke’s algorithm can be successfully applied to reach a complementary solution or infeasibility when the game satisfies the following conditions: (i) The value of $M$ is equal to zero. (ii) For all principal minors of $M^T$ (transpose of $M$) the value is non-negative. (iii) For any optimal mixed strategy $y$ of the maximizer either $y_i>0$ or $(My)_i>0$ for each coordinate $i$.
Title: Eigenfunctions Seminar: Nonlinear Calderon-Zygmund theory and differential forms
Speaker: Swarnendu Sil (IISc Mathematics)
Date: Fri, 16 Feb 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
For any $1 < q <\infty,$ standard representation formulas and the Calderon–Zygmund estimates imply $u \in W^{2,q}_{\text{loc}}\left(\mathbb{R}^{n}\right)$ if $\Delta u \in L^{q}_{\text{loc}}(\mathbb{R}^{n}).$ Combined with the Sobolev–Morrey embeddings for $q>n,$ we deduce that $\nabla u$ is locally Hölder continuous. However, as soon as we pass from the linear case to the quasilinear operator, we no longer have any representation formula for the solution of the following problem \begin{equation} {-}{\rm div}\left(\left\lvert \nabla u \right\rvert^{p-2}\nabla u\right) = f \end{equation} if $p \neq 2$ and CZ estimates for second derivatives of the solution are not yet known. In fact, the solution can fail to be $C^{2}$ even when $f \equiv 0.$
However, one can still establish Hölder continuity of the gradient whenever
${\rm div}\left(\left\lvert \nabla u \right\rvert^{p-2}\nabla u\right) \in L^{q}_{\text{loc}}$ and $q>n.$ These type of results are often called
“Nonlinear Calderon–Zygmund theory”, as the regularity for the gradient is the same, i.e. “as if” Calderon–Zygmund estimates for second derivatives are
valid! This result relies heavily on a fundamental regularity result, commonly known as the DeGiorgi–Nash–Moser estimate, for $p$-harmonic functions.
However, such regularity results are specific to equations and are in general false for elliptic systems. In another groundbreaking work, Uhlenbeck extended
gradient Hölder continuity estimates for solutions to special type of systems, which includes the homogeneous $p$-Laplacian systems.
In this lecture, I would sketch the main ideas involved to establish nonlinear Calderon–Zygmund theory for scalar equations and elliptic systems with Uhlenbeck
structures. In the second half, I would discuss how to extend these estimates to the following $p$-Laplacian type system for vector-valued differential forms
\begin{equation}
d^{\ast}\left(\left\lvert d u \right\rvert^{p-2}d u\right) = f.
\end{equation}
This includes systems which are, strictly speaking, even non-elliptic.
Title: APRG Seminar: The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Data
Speaker: Ravi Prakash (Universidad de Concepcion, Concepcion, Chile)
Date: Thu, 15 Feb 2024
Time: 3:30 pm
Venue: LH-1, Mathematics Department
This Talk is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of developments, although breakthroughs are expected in the not-too-distant future.
Nonlinear constitutive relationships which, at a given point in the space, present a behaviour for large arguments that is described by monomials of order $p$ and $q$ is considered in this presentation.
The original contribution this work makes is that the nonlinear problem can be approximated by a weighted $p$-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p$-Laplacian in inverse problems with nonlinear materials. Moreover, when $p=2$, this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials.
The main result of this work is that for “large” Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted $p$-Laplace problem.
Title: APRG Seminar: Elliptic operators in rough domains
Speaker: José María Martell Berrocal (ICMAT, Madrid, Spain)
Date: Thu, 15 Feb 2024
Time: 2 pm
Venue: Microsoft Teams (online)
Solvability of the Dirichlet problem with data in $L^p$ for some finite $p$ for elliptic operators, such as the Laplacian, amounts to showing that the associated elliptic/harmonic measure satisfies a Reverse Hölder inequality. Under strong connectivity assumptions, it has been proved that such a solvability is equivalent to the fact that that all bounded null-solutions of the operator in question satisfy Carleson measure estimates. In this talk, we will give a historical overview of this theory and present some recent results in collaboration with M. Cao and P. Hidalgo where, without any connectivity, we characterize certain weak Carleson measure estimates for bounded null-solutions in terms of a Corona decomposition for the elliptic measure. This extends the previous theory to non-connected settings where, as a consequence of our method, we establish Fefferman-Kenig-Pipher perturbation results.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Whittaker functions on covers of p-adic groups and quantum groups at roots of unity
Speaker: Manish Patnaik (University of Alberta, Edmonton, Canada)
Date: Fri, 09 Feb 2024
Time: 2 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
Since the work of Kubota in the late 1960s, it has been known that certain Gauss sum twisted (multiple) Dirichlet series are closely connected to a theory of automorphic functions on metaplectic covering groups. The representation theory of such covering groups was then initiated by Kazhdan and Patterson in the 1980s, who emphasized the role of a certain non-uniqueness of Whitattaker functionals.
Motivated on the one hand by the recent theory of Weyl group multiple Dirichlet series, and on the other by the so-called “quantum” geometric Langlands correspondence, we explain how to connect the representation theory of metaplectic covers of $p$-adic groups to an object of rather disparate origin, namely a quantum group at a root of unity. This gives us a new point of view on the non-uniqueness of Whittaker functionals and leads, among other things, to a Casselman–Shalika type formula expressed in terms of (Gauss sum) twists of “$q$”-Littlewood–Richardson coefficients, objects of some combinatorial interest.
Joint work with Valentin Buciumas.
Title: PhD Thesis colloquium: On existence and regularity of some complex Hessian equations on Kahler and transverse Kahler manifolds
Speaker: P. Sivaram (IISc Mathematics)
Date: Thu, 08 Feb 2024
Time: 3:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3 LH-1, Mathematics Department
The thesis consists of two parts. In the first part, we study the modified J-flow, introduced by Li-Shi. Analogous to the Lejmi-Szekelyhidi conjecture for the J-equation, Takahashi has conjectured that the solution to the modified J-equation exists if and only if some intersection numbers are positive and has verified the conjecture for toric manifolds. We study the behaviour of the modified J-flow on the blow-up of the projective spaces for rotationally symmetric metrics using the Calabi ansatz and obtain another proof of Takahashi’s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case ie. when the positivity conditions fail. We prove that the flow develops singularities along a co-dimension one sub-variety. Moreover, away from this singular set, the flow converges to a solution of the modified J-equation, albeit with a different slope.
In the second part, we will describe a new proof of the regularity of conical Ricci flat metrics on Q-Gorenstein T-varieties. Such metrics arise naturally as singular models for Gromov-Hausdorff limits of Kahler-Einstein manifolds. The regularity result was first proved by Berman for toric manifolds and by Tran-Trung Nghiem in general. Nghiem adapted the pluri-potential theoretic approach of Kolodziej to the transverse Kahler setting. We instead adapt the purely PDE approach to $L^\infty$ estimates due to Guo-Phong et al. to the transverse Kahler setting, and thereby obtain a purely PDE proof of the regularity result.
Title: APRG Seminar: Multilinear maximal averages defined on non-degenerate hypersurfaces
Speaker: Kalachand Shuin (Seoul National University, South Korea)
Date: Wed, 07 Feb 2024
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, we will explore the $L^{p}$-boundedness of both bilinear and multilinear maximal averages defined on non-degenerate hypersurfaces. Additionally, we will delve into the $L^2(\mathbb{R}^d)\times L^2(\mathbb{R}^d)\times\cdots\times L^2(\mathbb{R}^d) \to L^{2/m}(\mathbb{R}^d)$ estimates for $m$-linear maximal averages, focusing on hypersurfaces with $1\leq \kappa < md-1$ non-zero principal curvatures.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Euclidean algorithms are Gaussian over imaginary quadratic fields
Speaker: Jungwon Lee (University of Warwick, UK)
Date: Mon, 05 Feb 2024
Time: 2 pm
Venue: LH-1
Baladi and Vallée shows the limit Gaussian distribution of the length of continued fractions as a random variable on the set of rational numbers with bounded denominators based on ergodic methods. We give an analogue of the result for complex continued fractions over imaginary quadratic number fields and discuss applications in value distribution of $L$-functions of $\mathrm{GL}_2$ (joint with Dohyeong Kim and Seonhee Lim).
Title: PhD Thesis colloquium: Existence and implications of positively curved metrics on holomorphic vector bundles
Speaker: Arindam Mandal (IISc Mathematics)
Date: Tue, 30 Jan 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
This thesis is divided into two parts. In the first part, we study interpolating and uniformly flat hypersurfaces in complex Euclidean space. The study of interpolation and sampling in the Bargmann–Fock spaces on the complex plane started with the work of K. Seip in 1992. In a series of papers, Seip and his collaborators have entirely characterized the interpolating and sampling sequences for the Bargmann–Fock spaces on the complex plane. This problem has also been studied for the Bargmann–Fock spaces on the higher dimensional complex Euclidean spaces. Very few results about the interpolating and sampling hypersurfaces in higher dimensions are known. We have proved certain hypersurfaces are not interpolating in dimensions 2 and 3. Cerda, Schuster and Varolin have defined uniformly flat smooth hypersurfaces and proved that uniform flatness is one of the sufficient conditions for smooth hypersurfaces to be interpolating and sampling in higher dimensions. We have studied the uniformly flat hypersurfaces in dimensions greater than or equal to two and proved a complete characterization of it. In dimension two, we provided sufficient conditions for a smooth hypersurface to be uniformly flat in terms of its projectivization.
The second part deals with the existence of a Griffiths positively curved metric on the Vortex bundle. Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we have the notions of Nakano positivity, Griffiths positivity, and ampleness. All these notions of positivity are equivalent for line bundles. In general, Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, J. P. Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. We have studied these systems on the Vortex bundle.
Title: APRG Seminar: Geometric spaces at finite resolution
Speaker: Walter van Suijlekom (Radboud Universiteit, Nijmegen, Netherlands and IASc Jubilee Chair)
Date: Mon, 29 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
After a gentle introduction to the spectral approach to geometry, we extend the framework in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to finite resolution. In our approach, the traditional role played by operator algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc.
We illustrate our methods in concrete examples obtained by spectral truncations of the circle and of metric spaces up to finite resolution. The former yield operator systems of finite-dimensional Toeplitz matrices, and the latter give suitable subspaces of the compact operators. We also analyze the cones of positive elements and the pure-state spaces for these operator systems, which turn out to possess a very rich structure.
Title: Algebra & Combinatorics Seminar: Some combinatorial structures realized by commutative rings
Speaker: Rameez Raja (NIT Srinagar)
Date: Wed, 24 Jan 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
There are many ways to associate a graph (combinatorial structure) to a commutative ring $R$ with unity. One of the ways is to associate a zero-divisor graph $\Gamma(R)$ to $R$. The vertices of $\Gamma(R)$ are all elements of $R$ and two vertices $x, y \in R$ are adjacent in $\Gamma(R)$ if and only if $xy = 0$. We shall discuss Laplacian of a combinatorial structure $\Gamma(R)$ and show that the representatives of some algebraic invariants are eigenvalues of the Laplacian of $\Gamma(R)$. Moreover, we discuss association of another combinatorial structure (Young diagram) with $R$. Let $m, n \in \mathbb{Z}_{>0}$ be two positive integers. The Young’s partition lattice $L(m,n)$ is defined to be the poset of integer partitions $\mu = (0 \leq \mu_1 \leq \mu_2 \leq \cdots \leq \mu_m \leq n)$. We can visualize the elements of this poset as Young diagrams ordered by inclusion. We conclude this talk with a discussion on Stanley’s conjecture regarding symmetric saturated chain decompositions (SSCD) of $L(m,n)$.
Title: APRG Seminar: Partial convexity conditions and the d/d(zbar) problem
Speaker: Debraj Chakrabarti (Central Michigan University, Mount Pleasant, USA)
Date: Wed, 24 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
On a so-called Stein manifold the $\overline{\partial}$-problem can be solved in each degree $(p,q)$ where $q\geq 1$, or in other words the Dolbeault cohomology vanishes in these degrees. Sufficient conditions on complex manifolds which ensure that the Dobeault cohomology in degree $(p,q)$ is finite dimensional or vanishes have been studied since Andreotti-Grauert, who introduced the notions of $q$-convex/$q$-complete manifolds, which generalize Steinness. For manifolds with boundary, Hormander and Folland-Kohn introduced the condition now called $Z(q)$ which ensures finite-dimensionality of the cohomology in degree $q$ as well as $\frac{1}{2}$ estimates for the $\overline{\partial}$-Neumann operator. These conditions ($q$-convexity/completeness and $Z(q)$) are biholomorphically invariant characteristics of the underlying complex manifold.
In the context of Hermitian manifolds, a different type of sufficient condition implies that the $L^2$-cohomology in degree $(p,q)$-vanishes. Here one assumes that the sum of any $q$-eigenvalues is positive, and this also leads to the vanishing of the $L^2$-cohomology via the Bochner-Kohn-Morrey formula. These conditions are not biholomorphically invariant (they depend on the choice of the metric).
In this report on ongoing joint work with Andy Raich and Phil Harrington, we discuss the relationship between the two types of the condition. We give new sufficient conditions for the vanishing of the $L^2$-cohomology in degree $(p,q)$ in a domain in a complex manifold and discuss to what extent the conditions are necessary.
Title: APRG Seminar: Quantitative FKG for crossings
Speaker: Ritvik Radhakrishnan (ETH, Zurich, Switzerland)
Date: Wed, 17 Jan 2024
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Consider critical Bernoulli bond percolation on $\mathbb{Z}^2$. We show that the two arm exponent is strictly larger than twice the one arm exponent. This answers a question of Schramm and Steif (2010), and shows that their proof of the existence of exceptional times on the triangular lattice also applies to the square lattice. We use an interpolation formula via noise to obtain asymptotic correlation of crossings and apply this at each scale to obtain the strict inequality of arm exponents. This talk is based on joint work with Vincent Tassion.
Title: PhD Thesis defence: Characters of classical groups twisted by roots of unity
Speaker: Nishu Kumari (IISc Mathematics)
Date: Mon, 15 Jan 2024
Time: 9:30 am
Venue: LH-1, Mathematics Department
This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.
Throughout the thesis, we fix an integer $t \geq 2$ and a primitive $t$’th root of unity $\omega$. We first consider the irreducible characters of representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. This characterization turns out to depend on the $t$-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores, and show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.
We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the last $m$ to $y, \omega y, \dots, \omega^{m-1} y$. For the other families, we take the same specializations but with $m=1$. Our motivation for studying these are the conjectures of Wagh–Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.
The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at $\omega^p x_i$ and $\omega^q y_j$, for $0 \leq p, q \leq t-1$, $1 \leq i \leq n$, and $1 \leq j \leq m$. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.
For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee–Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson–Pfannerer–Rubey–Uhlin (Forum Math. Sigma, 2021).
Title: Number Theory Seminar: Fake Projective Planes
Speaker: Gopal Prasad (University of Michigan, Ann Arbor, USA)
Date: Fri, 12 Jan 2024
Time: 12 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
Title: Geometry & Topology Seminar: On the dense existence of compact invariant sets
Speaker: Rohil Prasad (UC Berkeley)
Date: Wed, 10 Jan 2024
Time: 11:00 am
Venue: LH-1
This is joint work in progress with Dan Cristofaro-Gardiner. We explore the topological dynamics of Reeb flows beyond periodic orbits and find the following rather general phenomenon. For any Reeb flow for a torsion contact structure on a closed 3-manifold, any point is arbitrarily close to a proper compact invariant subset of the flow. Such a statement is false if the invariant subset is required to be a periodic orbit. Stronger results can also be proved that parallel theorems of Le Calvez-Yoccoz, Franks, and Salazar for homeomorphisms of the 2-sphere. In fact, we can also extend their results to Hamiltonian diffeomorphisms of closed surfaces of any genus.
Title: Algebra & Combinatorics Seminar: 'Almost' Representations and Group Stability
Speaker: Bharatram Rangarajan (Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel)
Date: Wed, 10 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
Consider the following natural robustness question: is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This classical question of stability goes all the way back to Turing and Ulam, and can be posed for different target groups, and different notions of distance. Group stability has been an active line of study in recent years, thanks to its connections to major open problems like the existence of non-sofic and non-hyperlinear groups, the group Connes embedding problem and the recent breakthrough result MIP*=RE, apart from property testing and error-correcting codes.
In this talk, I will survey some of the main results, techniques, and questions in this area.
Title: Geometry & Topology Seminar: A Complete Geodesic Metric on Finite Energy Spaces in Big Cohomology Class.
Speaker: Prakhar Gupta (University of Maryland)
Date: Mon, 08 Jan 2024
Time: 04:00 pm
Venue: LH-1
In this talk, I will describe a complete geodesic metric $d_p$ on the finite energy space $\mathcal{E}^p(X,\theta)$ for $p\geq 1$ where $\theta$ represents a big cohomology class. This work generalizes the complete geodesic metrics in the Kahler setting to the big setting. When p=1, the metric $d_1$ in the Kahler setting has found various applications in the understanding of Kahler-Einstein and Constant Scalar Curvature Kahler metrics. In this talk, I’ll describe how to construct the metric and explain some properties that could have useful applications in the future.
Title: Eigenfunctions Seminar: Macdonald polynomials, interpolation polynomials, and binomial coefficients
Speaker: Siddhartha Sahi (Rutgers University, USA)
Date: Fri, 05 Jan 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
The Macdonald polynomials are a homogeneous basis for the algebra of symmetric polynomials, which generalize many important families of special functions, such as Schur polynomials, Hall-Littlewood polynomials, and Jack polynomials.
The interpolation polynomials, introduced by F. Knop and the speaker, are an inhomogeneous extension of Macdonald polynomials, which are characterized by very simple vanishing properties.
The binomial coefficients are special values of interpolation polynomials, which play a central role in the higher rank $q$-binomial theorem of A. Okounkov.
We will give an elementary self-contained introduction to all three objects, and discuss some recent results, open problems, and applications.
Title: APRG Seminar: Commuting tuple of operators homogeneous under the unitary group
Speaker: Surjit Kumar (IIT, Madras)
Date: Thu, 04 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
Let $\mathbb B_d$ be the open unit ball in $\mathbb C^d$ and $\boldsymbol T$ be a commuting $d$-tuple of bounded linear operators
on a complex separable Hilbert space $\mathcal H$. Let $\mathcal U(d)$ be the linear group of unitary transformations acting on
$\mathbb C^d$ by the rule: $\boldsymbol z \mapsto u\cdot \boldsymbol z$, $\boldsymbol z \in \mathbb C^d$, where $u\cdot \boldsymbol z$
is the usual matrix product. We say that $\boldsymbol T$ is $\mathcal U(d)$-homogeneous if $u \cdot \boldsymbol T$ is unitarily
equivalent to $\boldsymbol T$ for all $u\in \mathcal U(d)$.
In this talk, we describe $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol M$ of multiplication by the coordinate functions acting
on a reproducing kernel Hilbert space $\mathcal H_K(\mathbb B_d, \mathbb C^n) \subseteq {\rm Hol}(\mathbb B_d, \mathbb C^n)$, where
$n$ is the dimension of the joint kernel of $\boldsymbol T^*$. The case $n=1$ is well understood, here, we focus on the case $n=d.$ We
describe this class of $\mathcal U(d)$-homogeneous operators, equivalently, non-negative definite kernels quasi invariant under the
action of the group $\mathcal U(d).$ As a result, we obtain criterion for boundedness, irreducibility and mutual unitary equivalence
among these operators.
This is a joint work with Soumitra Ghara, Gadadhar Misra and Paramita Pramanick.
Title: APRG Seminar: Lyapunov exponents for random matrix products
Speaker: Siddhartha Sahi (Rutgers University, USA)
Date: Wed, 03 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
We consider probability measures on $GL(n,\mathbb{R})$ that are invariant under the left action of the orthogonal group $O(n,\mathbb{R})$ and satisfy a mild integrability condition. For any such measure we consider the following two quantities: (a) the mean of the log of the absolute value of the eigenvalues of the matrices and (b) the Lyapunov exponents of random products of matrices independently drawn with respect to the measure. Our main result is a lower bound for (a) in terms of (b).
This lower bound was conjectured by Burns-Pugh-Shub-Wilkinson (2001), and special cases were proved by Dedieu-Shub (2002), Avila-Bochi (2003) and Rivin (2005). We give a proof in complete generality by using some results from the theory of spherical functions and Jack polynomials.
This is joint work with Diego Armentano, Gautam Chinta, and Michael Shub. (arXiv:2206.01091), (Ergodic theory and Dynamical systems, to appear).
Title: Algebra & Combinatorics Seminar: The Du Bois complex and some associated singularities
Speaker: Sridhar Venkatesh (University of Michigan, Ann Arbor, USA)
Date: Thu, 28 Dec 2023
Time: 11:30 am
Venue: LH-1, Mathematics Department
For a smooth variety $X$ over $\mathbb{C}$, the de Rham complex of $X$ is a powerful tool to study the geometry of $X$ because of results such as the degeneration of the Hodge-de Rham spectral sequence (when $X$ is proper). For singular varieties, it follows from the work of Deligne and Du Bois that there is a substitute called the Du Bois complex which satisfies many of the nice properties enjoyed by the de Rham complex in the smooth case. In this talk, we will discuss some classical singularities associated with this complex, namely Du Bois and rational singularities, and some recently introduced refinements, namely $k$-Du Bois and $k$-rational singularities. This is based on joint work with Wanchun Shen and Anh Duc Vo.
Title: PhD Thesis defence: Harmonic map heat flow and framed surface-group representations
Speaker: Gobinda Sau (IISc Mathematics)
Date: Thu, 28 Dec 2023
Time: 11 am
Venue: Microsoft Teams (online)
This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.
The focus of the first part is when the domain is the complex plane. In this case, given a finite twisted ideal polygon, there exists a harmonic map heat flow $u_t$ such that the image of $u_t$ is asymptotic to that polygon for all $t\in[0,\infty)$. Moreover, we prove that given any twisted ideal polygon in $\mathbb{H}^3$ with \textit{rotational symmetry}, there exists a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.
In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an irreducible representation $\rho$ of its fundamental group into $\text{PSL}_2(\mathbb{C})$, a seminal theorem of Donaldson asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$. In this thesis, we consider domain surfaces that are non-compact, namely \textit{marked and bordered surfaces} (introduced in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of marked points that are either punctures or marked points on boundary components. Our main result in this part is: given an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (\text{PSL}_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from $\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$. In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.
Title: Slides Kerala School of Mathematics
Speaker: A. Raghuram (Fordham University, New York City, USA)
Date: Thu, 28 Dec 2023
Time: 2 pm
Venue: LH-1, Mathematics Department
The basic ideas of Calculus started with Archimedes, and reached a highly developed form in the 17th Century with Newton and Leibniz often being credited as its inventors. What was not so well-known until only a few decades ago is that between the 14th and 17th Century there was an unbroken lineage of profound mathematicians working in Kerala who had independently discovered many of the great themes of Calculus. This talk is an introduction to the lives and works of some of the prominent members of the Kerala School of Mathematics. Most of the talk will be accessible to a general audience. Only towards the end of the talk some elementary mathematics will be assumed to explain a few of their contributions.
Title: Algebra & Combinatorics Seminar: Matrix factorisations of polynomials
Speaker: Ravindra Girivaru (University of Missouri, St. Louis, USA)
Date: Wed, 20 Dec 2023
Time: 4 pm
Venue: LH-3, Mathematics Department
A matrix factorisation of a polynomial $f$ is an equation $AB = f \cdot {\rm I}_n$ where $A,B$ are $n \times n$ matrices with polynomial entries and ${\rm I}_n$ is the identity matrix. This question has been of interest for more than a century and has been studied by mathematicians like L.E. Dickson. I will discuss its relation with questions arising in algebraic geometry about the structure of subvarieties in projective hypersurfaces.
Title: Algebra & Combinatorics Seminar: A simple proof for the characterization of chordal graphs using Horn hypergeometric series
Speaker: R. Venkatesh (IISc Mathematics)
Date: Wed, 13 Dec 2023
Time: 11:30 am
Venue: LH-3, Mathematics Department
Let $G$ be a finite simple graph (with no loops and no multiple edges), and let $I_G(x)$ be the multi-variate independence polynomial of $G$. In 2021, Radchenko and Villegas proved the following interesting characterization of chordal graphs, namely $G$ is chordal if and only if the power series $I_G(x)^{-1}$ is Horn hypergeometric. In this talk, I will give a simpler proof of this fact by computing $I_G(x)^{-1}$ explicitly using multi-coloring chromatic polynomials. This is a joint work with Dipnit Biswas and Irfan Habib.
Title: Algebra & Combinatorics Seminar: The structure and representation theory of quantum toroidal algebras
Speaker: Duncan Laurie (University of Oxford, UK)
Date: Fri, 08 Dec 2023
Time: 3 pm
Venue: LH-3, Mathematics Department
Quantum toroidal algebras are the next class of quantum affinizations after quantum affine algebras, and can be thought of as “double affine quantum groups”. However, surprisingly little is known thus far about their structure and representation theory in general.
In this talk we’ll start with a brief recap on quantum groups and the representation theory of quantum affine algebras. We shall then introduce and motivate quantum toroidal algebras, before presenting some of the known results. In particular, we shall sketch our proof of a braid group action, and generalise the so-called Miki automorphism to the simply laced case.
Time permitting, we shall discuss future directions and applications including constructing representations of quantum toroidal algebras combinatorially, written in terms of Young columns and Young walls.
Title: PhD Thesis colloquium: Some aspects of weighted kernel functions on planar domains
Speaker: Aakanksha Jain (IISc Mathematics)
Date: Fri, 08 Dec 2023
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we discuss various aspects of weighted kernel functions on planar domains. We focus on two key kernels, namely, the weighted Bergman kernel and the weighted Szegő kernel.
For a planar domain $D \subset \mathbb C$ and an admissible weight function $\mu$ on it, we discuss some aspects of the corresponding weighted Bergman kernel $K_{D, \mu}$. First, we see a precise relation between $K_{D, \mu}$ and the classical Bergman kernel $K_D$ near a smooth boundary point of $D$. Second, the weighted kernel $K_{D, \mu}$ gives rise to weighted metrics in the same way as the classical Bergman kernel does. Motivated by work of Mok, Ng, Chan–Yuan and Chan–Xiao–Yuan among others, we talk about the nature of holomorphic isometries from the disc $\mathbb D \subset \mathbb C$ with respect to the weighted Bergman metrics arising from weights of the form $\mu = K_{\mathbb D}^{-d}$ for some integer $d \geq 0$. Specific examples that we discuss in detail include those in which the isometry takes values in $\mathbb D^n$ and $\mathbb D \times \mathbb B^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, we also present the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above.
In the next part of the talk, we discuss properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell’s work, the starting point is a weighted Kerzman–Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), we show that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, we show that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell’s list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.
Title: Geometry & Topology Seminar: Cubulating hyperbolic mapping tori
Speaker: Suraj Krishna (Technion, Israel)
Date: Wed, 06 Dec 2023
Time: 04:00 pm
Venue: LH-3
A group is cubulated if it acts properly and cocompactly on a CAT(0) cube complex, which is a generalisation of a product of trees. Some well-known examples are free groups, surface groups and fundamental groups of closed hyperbolic 3-manifolds. I will show in the talk that semidirect products of hyperbolic groups with $\mathbb{Z}$ which are again hyperbolic are cubulated, and give some consequences.
Two prominent examples of our setup are
- mapping tori of fundamental groups of closed hyperbolic surfaces over pseudo-Anosov automorphisms, and
- mapping tori of free groups over atoroidal automorphisms.
Both these classes of groups are known to be cubulated by outstanding works. Our proof uses these two noteworthy results as building blocks and places them in a unified framework. Based on joint work with François Dahmani and Jean Pierre Mutanguha.
Title: Algebra & Combinatorics Seminar: A glimpse into the world of Rogers-Ramanujan identities
Speaker: Shashank Kanade (University of Denver, USA)
Date: Fri, 01 Dec 2023
Time: 11 am
Venue: LH-3, Mathematics Department
I will give a gentle introduction to the combinatorial Rogers–Ramanujan identities. While these identities are over a century old, and have many proofs, the first representation-theoretic proof was given by Lepowsky and Wilson about four decades ago. Now-a-days, these identities are ubiquitous in several areas of mathematics and physics. I will mention how these identities arise from affine Lie algebras and quantum invariants of knots.
Title: Number Theory Seminar: Stark-Heegner cycles over arbitrary number fields
Speaker: Guhan Venkat (Ashoka University)
Date: Thu, 30 Nov 2023
Time: 2 pm
Venue: LH-1
In his seminal paper in 2001, Henri Darmon proposed a systematic construction of $p$-adic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of local ($p$-adic) cohomology classes/cycles in the $p$-adic Galois representation attached to a cuspidal cohomological automorphic representation of $\mathrm{PGL}_2$ over any number field, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato conjecture. This work generalises previous constructions of Rotger-Seveso for elliptic cusp forms and earlier joint work with Williams for Bianchi cusp forms. Time permitting, I will also talk about the plectic analogues of these objects.
Title: Algebra & Combinatorics Seminar: Branching multiplicity of symplectic groups as $SL_2$ representations
Speaker: Sagar Shrivastava (TIFR, Mumbai)
Date: Tue, 28 Nov 2023
Time: 11:30 am
Venue: LH-1, Mathematics Department
Branching rules are a systematic way of understanding the multiplicity of irreducible representations in restrictions of representations of Lie groups. In the case of $GL_n$ and orthogonal groups, the branching rules are multiplicity free, but the same is not the case for symplectic groups. The explicit combinatorial description of the multiplicities was given by Lepowsky in his PhD thesis. In 2009, Wallach and Oded showed that this multiplicity corresponds to the dimension of the multiplicity space, which was a representation of $SL_2$ $(=Sp(2))$. In this talk, we give an alternate proof of the same without invoking any partition function machinery. The only assumption for this talk would be the Weyl character formula.
Title: Number Theory Seminar: Congruences and period relations: a new case
Speaker: Jacques Tilouine (LAGA, Universite Paris 13, Paris, France)
Date: Mon, 20 Nov 2023
Time: 2 pm
Venue: LH-3
In a joint work in progress with K. Prasanna, we
study period relations for the base change to $\mathrm{GL}_4$ of a
cohomological cuspidal representation on $\mathrm{GSp}_4$.
An unexpected period occurs in the period relations.
Title: Geometry & Topology Seminar: Criteria for simplicity
Speaker: Arpan Kabiraj (IIT Palakkad)
Date: Mon, 20 Nov 2023
Time: 04:00 pm
Venue: LH-1
In 80’s Goldman introduced a Lie bracket structure on the free homotopy classes of oriented closed curves on an oriented surface known as the Goldman Lie bracket. In this talk, I will give a brief overview of Goldman Lie algebra and discuss two criteria for a homotopy class of a curve to be simple in terms of the Goldman Lie bracket.
Title: Algebra & Combinatorics Seminar: Ideals in enveloping algebras of affine Kac-Moody algebras
Speaker: Rekha Biswal (NISER, Bhubaneswar)
Date: Fri, 17 Nov 2023
Time: 11:30 am
Venue: LH-3, Mathematics Department
In this talk, I will discuss about the structure of ideals in enveloping algebras of affine Kac–Moody algebras and explain a proof of the result which states that if $U(L)$ is the enveloping algebra of the affine Lie algebra $L$ and “$c$” is the central element of $L$, then any proper quotient of $U(L)/(c)$ by two sided ideals has finite Gelfand–Kirillov dimension. I will also talk about the applications of the result including the fact that $U(L)/(c-\lambda)$ for non zero $\lambda$ is simple. This talk is based on joint work with Susan J. Sierra.
Title: PhD Thesis colloquium: Dominating surface-group representations via Fock-Goncharov coordinates
Speaker: Pabitra Barman (IISc Mathematics)
Date: Fri, 17 Nov 2023
Time: 11:15 am
Venue: LH-1, Mathematics Department
Let $S$ be an oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S) \rightarrow PSL_2(\mathbb{C})$ be two representations. $\rho_2$ is said to dominate $\rho_1$ if there exists $\lambda \le 1$ such that $\ell_{\rho_1}(\gamma) \le \lambda \cdot \ell_{\rho_2}(\gamma)$ for all $\gamma \in \pi_1(S)$, where $\ell_{\rho}(\gamma)$ denotes the translation length of $\rho(\gamma)$ in $\mathbb{H}^3$. In 2016, Deroin–Tholozan showed that for a closed surface $S$ and a non-Fuchsian representation $\rho : \pi_1(S) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S) \rightarrow PSL_2(\mathbb{R})$ that strictly dominates $\rho$. In 2023, Gupta–Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher rank representations.
For a representation $\rho : \pi_1(S) \rightarrow PSL_n(\mathbb{C})$ where $n >2$, the Hilbert length of a curve $\gamma\in \pi_1(S)$ is defined as \begin{equation} \ell_{\rho}(\gamma):=\ln \Bigg| \frac{\lambda_n}{\lambda_1} \Bigg|, \end{equation} where $\lambda_n$ and $\lambda_1$ are the largest and smallest eigenvalues of $\rho(\gamma)$ in modulus respectively. We show that for any generic representation $\rho : \pi_1 (S) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock–Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$-representations. Weighted planar networks and the Collatz–Wielandt formula for totally positive matrices play a crucial role.
Let $ X_n$ be the symmetric space of $PSL_n(\mathbb{C})$. The translation length of $A\in PSL_n(\mathbb{C})$ in $X_n$ is given as \begin{equation} \ell_{X_n}(A)= \sum_{i=1}^{n}\log (\sigma_i(A))^2, \end{equation} where $\sigma_i(A)$ are the singular values of $A$. We show that the same $j$ dominates $\rho$ in the translation length spectrum as well. Lindström’s Lemma for planar networks is one of the key ingredients of the proof.
In both cases, if $S$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.
Title: Eigenfunctions Seminar: Slides Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
Speaker: Hariharan Narayanan (TIFR, Mumbai)
Date: Fri, 17 Nov 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Associated to two given sequences of eigenvalues is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as the number of eigenvalues tends to infinity.
Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdös–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. This is joint work with Scott Sheffield and Terence Tao.
Title: Algebra & Combinatorics Seminar: Aldous-type spectral gap results for the complete monomial group
Speaker: Subhajit Ghosh (Bar-Ilan University, Ramat-Gan, Israel)
Date: Wed, 15 Nov 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
Let us consider the continuous-time random walk on $G\wr S_n$, the complete monomial group of degree $n$ over a finite group $G$, as follows: An element in $G\wr S_n$ can be multiplied (left or right) by an element of the form
- $(u,v)_G:=(\mathbf{e},\dots,\mathbf{e};(u,v))$ with rate $x_{u,v}(\geq 0)$, or
- $(g)^{(w)}:=(\dots,\mathbf{e},g,\mathbf{e},\dots;\mathbf{id})$ with rate $y_w\alpha_g\; (y_w \gt 0,\;\alpha_g=\alpha_{g^{-1}}\geq 0)$,
such that $\{(u,v)_G,(g)^{(w)} : x_{u,v} \gt 0,\; y_w\alpha_g \gt 0,\;1\leq u \lt v \leq n,\;g\in G,\;1\leq w\leq n\}$ generates $G\wr S_n$. We also consider the continuous-time random walk on $G\times\{1,\dots,n\}$ generated by one natural action of the elements $(u,v)_G,1\leq u \lt v\leq n$ and $g^{(w)},g\in G,1\leq w\leq n$ on $G\times\{1,\dots,n\}$ with the aforementioned rates. We show that the spectral gaps of the two random walks are the same. This is an analogue of the Aldous’ spectral gap conjecture for the complete monomial group of degree $n$ over a finite group $G$.
Title: APRG Seminar: Pseudo-differential calculi and entropy estimates with Orlicz spaces and Orlicz modulation spaces
Speaker: Joachim Toft (Linnaeus University, Vaxjo, Sweden)
Date: Wed, 15 Nov 2023
Time: 11 am
Venue: LH-1, Mathematics Department
A convex function $\Phi$ from $[0,\infty]$ to $[0,\infty]$ with properties \begin{equation} \Phi (0)=0,\qquad \lim_{t\to \infty}\Phi (t)=\Phi (\infty )=\infty , \end{equation} is called a Young function. For any Young function $\Phi$, the Orlicz space $L^\Phi$ is a Banach space, and consists of all measurable functions $f$ such that $\Phi (t\cdot |f|)\in L^1$ for some $t>0$. By choosing $\Phi$ in suitable ways we gain the definition of any (Banach) Lebesgue space $L^p$, as well as sums of such spaces like $L^p+L^q$, $p,q\in [1,\infty ]$. In particular, the family of Orlicz spaces contain any Lebesgue space.
The Orlicz modulation space $M^{\Phi}$ is obtained by imposing $L^\Phi$ norm conditions of the short-time Fourier transforms of the involved functions and distributions. In the same way we may discuss Orlicz modulation spaces $M^{\Phi ,\Psi}$ of mixed normed types. Again, by choosing the Young functions $\Phi$ and $\Psi$ in suitable ways, $M^{\Phi ,\Psi}$ becomes the classical Feichtinger’s modulation space $M^{p,q}$.
In the talk we explain some basic properties and give some examples on interesting Orlicz spaces and Orlicz modulation spaces. We also explain some classical results on pseudo-differential operators acting on Lebesgue or modulation spaces, and give examples on how such results can be extended to the framework of Orlicz spaces and Orlicz modulation spaces.
As another example we discuss suitable Orlicz modulation spaces and the entropy functional $f\mapsto E_\phi (f)$ with $\phi$ as the coherent state, considered by E. H. Lieb when discussing kinetic energy in quantum systems. Here we find an Orlicz modulation space $M^\Phi$ which satisfies \begin{equation} M^{p_1}\subsetneq M^\Phi \subsetneq M^{p_2},\qquad p_1<\frac 12,\ p_2\ge \frac 12 \end{equation} for which $E_\phi$ is continuous on $M^{p_1}$ and $M^\Phi$, but discontinuous on $M^{p_2}$. We hope that this should shed some light on how to find suitable Banach spaces when dealing with non-linear functionals.
The talk is based on joint works with A. Gumber, E. Nabizadeh Morsalfard, N. Rana, S. Öztop and R. Üster.
Title: APRG Seminar: Fractional Fourier transform, harmonic oscillator propagators and Strichartz estimates
Speaker: Joachim Toft (Linnaeus University, Vaxjo, Sweden)
Date: Tue, 14 Nov 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Using the Bargmann transform, we give a proof of that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces, and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on modulation spaces. Especially we extend some results in our recent works and those of Bhimani, Cordero, Gröchenig, Manna, Thangavelu, and others. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable on so-called Pilipovic spaces and their distribution spaces. Especially we show that fractional Fourier transforms of any complex order can be defined, and that these transforms are continuous on any Pilipovic space and corresponding distribution space, which are not Gelfand–Shilov spaces. (The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.)
Title: PhD Thesis colloquium: Geometry of Normed Linear Spaces in Light of Birkhoff-James Orthogonality
Speaker: Babhrubahan Bose (IISc Mathematics)
Date: Fri, 10 Nov 2023
Time: 2:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
Two points $x$ and $y$ in a normed linear space $\mathbb{X}$ are said to be Birkhoff-James orthogonal (denoted by $x\perp_By$) if $|x+\lambda y|\geq|x|~~\text{for every scalar}~\lambda.$ James proved that in a normed linear space of dimension more than two, Birkhoff-James orthogonality is symmetric if and only if the parallelogram law holds. Motivated by this result, Sain introduced the concept of pointwise symmetry of Birkhoff-James orthogonality in a normed linear space.
In this talk, we shall try to understand the geometry of normed spaces in the light of Birkhoff-James orthogonality. After introducing the basic notations and terminologies, we begin with a study of the geometry of the normed algebra of holomorphic maps in a neighborhood of a curve and establish a relationship among the extreme points of the closed unit ball, Birkhoff-James orthogonality, and zeros of holomorphic maps.
We next study Birkhoff-James orthogonality and its pointwise symmetry in Lebesgue spaces defined on arbitrary measure spaces and natural numbers. We further find the onto isometries of the sequence spaces using the pointwise symmetry of the orthogonality.
We shall then study the geometry of tensor product spaces and use the results to study the relationship between the symmetry of orthogonality and the geometry (for example, extreme points and smooth points) of certain spaces of operators. Our work in this section is motivated by the famous Grothendieck inequality.
Finally, we study the geometry of $\ell_p$ and $c_0$ direct sums of normed spaces ($1\leq p<\infty$). We shall characterize the smoothness and approximate smoothness of these spaces along with Birkhoff-James orthogonality and its pointwise symmetry. As a consequence of our study we answer a question pertaining to the approximate smoothness of a space, raised by Chmeilienski, Khurana, and Sain.
Title: Geometry & Topology Seminar: The Einstein-Bogomolnyi metrics on Riemann sphere
Speaker: Chengjian Yao (Shanghai Tech)
Date: Mon, 06 Nov 2023
Time: 04:00 pm
Venue: MS Teams (online)
Einstein-Bogomolnyi metrics, which physically models the Cosmic Strings, solve the Einstein’s Fields Equation coupled with an Abelian gauge field and a Higgs field. In this talk, I will present a general existence and uniqueness theorem for Einstein-Bogomolnyi metrics on Riemann sphere. I will also discuss the behaviors of the metrics as the volume approaches the lower bound and infinity respectively, and the moduli space problem. Part of this talk is based on the joint work with Luis-Alvarez, Garcia-Fernandez, Garcia-Prada and Pingali.
Title: Eigenfunctions Seminar: Representations of $p$-adic groups over close local fields
Speaker: Radhika Ganapathy (IISc Mathematics)
Date: Fri, 03 Nov 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
In the first part of the talk, we will discuss the main statement of local class field theory that describes the abelian extensions of a non-archimedean local field $F$ in terms of the arithmetic of the field $F$. Then we will discuss the statement of the local Langlands conjectures, a vast generalization of local class field theory, that gives a (conjectural) parametrization of the irreducible complex representations of $G(F)$, where $G$ is a connected, reductive group over $F$, in terms of certain Galois representations. We will then discuss a philosophy of Deligne and Kazhdan that loosely says that to obtain such a parametrization for representations of $G(F’)$, with $F’$ of characteristic $p$, it suffices to obtain such a parametrization for representations of $G(F)$ for all local fields $F$ of characteristic $0$. In the second half of the talk, we will mention some instances where the Deligne-Kazhdan philosophy has been applied successfully to obtain a Langlands parametrization of irreducible representations of $G(F’)$ in characteristic $p$ and focus on some recent work on variants/generalizations of the work of Kazhdan.
Title: PhD Thesis defence: On commuting isometries and commuting isometric semigroups
Speaker: Shubham Rastogi (IISc Mathematics)
Date: Mon, 30 Oct 2023
Time: 12:15 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
The famous Wold decomposition gives a complete structure of an isometry on a Hilbert space. Berger, Coburn, and Lebow (BCL) obtained a structure for a tuple of commuting isometries acting on a Hilbert space. In this talk, we shall discuss a structure of a pair of commuting $C_0$-semigroups of isometries and obtain a BCL type result.
The right-shift-semigroup $\mathcal S^\mathcal E=(S^\mathcal E_t)_{t\ge 0}$ on $L^2(\mathbb R_+,\mathcal E)$ for any Hilbert space
$\mathcal E$ is defined as
\begin{equation}
(S_t^\mathcal E f)(x) = \begin{cases}
f(x-t) &\text{if } x\ge t,\\
0 & \text{otherwise,}
\end{cases}
\end{equation}
for $f\in L^2(\mathbb R_+,\mathcal E).$
Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for
a $C_0$-semigroup of isometries. The factorizations of the unilateral shift have been explored by BCL, we are interested in examining
the factorizations of the right-shift-semigroup.
Firstly, we shall discuss the contractive $C_0$-semigroups which commute with the right-shift-semigroup. Then, we give a complete
description of the pairs $(\mathcal V_1,\mathcal V_2)$ of commuting $C_0$-semigroups of contractions which satisfy $\mathcal S^\mathcal
E=\mathcal V_1\mathcal V_2$, (such a pair is called as a factorization of $\mathcal S^\mathcal E$), when $\mathcal E$ is a finite
dimensional Hilbert space.
Next, we discuss the Taylor joint spectrum for a pair of commuting isometries $(V_1,V_2)$ using the defect operator $C(V_1,V_2)$ defined as \begin{equation} C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+ V_1V_2V_2^*V_1^*. \end{equation} We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.
Title: Eigenfunctions Seminar: Slides Non-normal matrices: spectral instability, pseudospectrum, and random perturbation
Speaker: Anirban Basak (ICTS Bangalore)
Date: Fri, 20 Oct 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Non-normal matrices are ubiquitous in various branches of science, such as fluid dynamics, mathematical physics, partial differential equations, and many more. Non-normality causes notorious sensitivity of the eigenvalues, and the eigenvalue analysis often turns out to be misleading. These motivate the study of pseudospectrum, and the spectral properties of random perturbation of non-normal matrices. In the first part of the talk, we will introduce these issues and their resolutions through some fun experiments and simulations. In the latter half, we will move to describe spectral properties of random perturbations of non-normal Toeplitz matrices, where over the last few years a coherent theory has emerged.
Title: Geometry & Topology Seminar: Flat geometry and representations
Speaker: Gianluca Faraco (University of Milano Bicocca)
Date: Wed, 18 Oct 2023
Time: 11:00 am
Venue: LH-1
We discuss about flat structures on surfaces of finite type $S_{g,n}$, possibly with punctures. For a given representation $\chi\colon \pi_1(S_{g,n})\to \textnormal{Aff}(\mathbb C)$, we wonder if there exists a flat structure having the given representations as the holonomy representation. For closed surfaces $(n=0)$, holonomy representations has been determined by works of Haupt for representations in $\mathbb C$ and subsequently by Ghazouani for a generic representation in $\textnormal{Aff}(\mathbb C)$. It turns out that for surfaces of hyperbolic type, i.e. $2-2g-n<0$, the resulting structures must have special points, called branched points, around which the geometry fails to be modelled on $\mathbb C$. In the present seminar we discuss the case of punctured surfaces and provide conditions under which a representation $\chi$ is a holonomy representation of some flat structure. In this case, being surfaces no longer closed, it is even possible to find flat structures with no branched points. This is a joint work with Subhojoy Gupta and partially with Shabarish Chenakkod.
Title: Geometry & Topology Seminar: The Demailly systems with vortex ansatz
Speaker: Arindam Mandal (IISc)
Date: Mon, 16 Oct 2023
Time: 4:00 pm
Venue: LH-1
Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we can define the notions of Nakano positivity, Griffiths positivity and ampleness. All these notions of positivity are equivalent for line bundles. In general, Nakano positivity implies Griffiths positivity and Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. In this talk, I will briefly discuss about the solution of these systems on the vortex bundle using method of continuity.
Title: Geometry & Topology Seminar: Existence of higher extremal metrics on a minimal ruled surface
Speaker: Rajas Sompurkar (IISc)
Date: Mon, 09 Oct 2023
Time: 4:00 pm
Venue: LH-1
In this talk we will first see the definitions of ‘higher extremal Kahler metric’ and ‘higher constant scalar curvature Kahler (hcscK) metric’, both of which are analogous to the definitions of extremal Kahler metric and constant scalar curvature Kahler (cscK) metric respectively. Informally speaking, on a compact K ̈ahler manifold a higher extremal Kahler metric is a Kahler metric whose corresponding top Chern form and volume form differ by a smooth real-valued function whose gradient is a holomorphic vector field, and an hcscK metric is a Kahler metric whose top Chern form and volume form differ by a real constant or equivalently whose top Chern form is harmonic. We will then prove that on a special type of minimal ruled complex surface, which is an example of a ‘pseudo-Hirzebruch surface’, every Kahler class admits a higher extremal Kahler metric which is constructed by using the well-known momentum construction method involving the Calabi ansatz procedure. We will then check that this specific higher extremal Kahler metric yielded by the momentum construction method cannot be an hcscK metric. By doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude that hcscK metrics do not exist in any Kahler class on this Kahler surface. We will then see briefly what changes in the calculations in the momentum construction method when we take a general pseudo- Hirzebruch surface which is basically the projectivization of a certain kind of rank two holomorphic vector bundle over a compact Riemann surface of genus greater than or equal to two. It can be seen that the results about the existence of higher extremal Kahler metrics and the non-existence of hcscK metrics obtained in the special case of our minimal ruled surface can be generalized to all pseudo-Hirzebruch surfaces. If time permits we will see the motivation for studying this problem and its analogy with the related and previously well- studied problem of constructing extremal Kahler metrics on a pseudo-Hirzebruch surface.
Title: Eigenfunctions Seminar: Relative Trace Formula, Periods and non-vanishing of L-values
Speaker: Dinakar Ramakrishnan (California Institute of Technology, Pasadena/LA, USA)
Date: Fri, 06 Oct 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Of fundamental importance in number theory is the question of non-vanishing of central L-values of L-functions. One approach, explained in the talk, is to make use of the Relative trace formula (which will be introduced from scratch); a basic example of interest involves twists of L-functions of classical modular forms. If time permits, we will explain the recent work with Michel and Yang on $U(2)$-twists of $U(3)$ L-functions.
Title: APRG Seminar: Existence of minimizers for weighted $L^p$-Hardy inequalities
Speaker: Ujjal Das (Technion, Haifa, Israel)
Date: Wed, 04 Oct 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
We study the spectral gap phenomena for weighted $L^p$-Hardy inequalities on $C^{1,\gamma}$-domain with a compact boundary, where $\gamma\in (0,1]$. We show that the weighted Hardy constant is attained by some appropriate minimizer if and only if the spectral gap (the difference between the weighted Hardy constant and the weighted Hardy constant at infinity ) is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers. In this talk, we will try to understand how the ideas in criticality theory help us to extend the spectral gap phenomena from $C^2$-domains to $C^{1,\gamma}$-domains. This talk is based on the joint work with Yehuda Pinchover, Baptiste Devyver.
Title: Geometry & Topology Seminar: A Hodge theoretic projective structure on compact Riemann surfaces
Speaker: Indranil Biswas (Shiv Nadar University)
Date: Wed, 04 Oct 2023
Time: 11:00 am
Venue: LH-1
Any compact Riemann surface is shown to have a canonical projective structure (which is different from the canonical one given by the uniformization theorem). Some properties of this projective structure are established. (Joint work with Elisabetta Colombo, Paola Frediani and Gian Pietro Pirola.)
Title: APRG Seminar: Lower eigenvalue bounds of the Laplacian (and bi-Laplacian)
Speaker: Carsten Carstensen (Humboldt Universitat, Berlin, Germany)
Date: Tue, 03 Oct 2023
Time: 10:30 am
Venue: LH-1, Mathematics Department
Recent advances in the nonconforming FEM approximation of elliptic PDE eigenvalue problems include the guaranteed lower eigenvalue bounds (GLB) and its adaptive finite element computation. The first part of the talk explains the derivation of GLB for the simplest second-order (and fourth-order) eigenvalue problems with relevant applications, e.g., for the localization of the critical load in the buckling analysis of the Kirchhoff plates. The second part mentions an optimal adaptive mesh-refining algorithm for the effective eigenvalue computation for the Laplace (and bi-Laplace) operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of nonlinear approximation classes. Numerical experiments in the third part of the presentation shows benchmarks in which the naive adaptive mesh-refining and the post processed GLB do not lead to efficient GLB. The fourth part outlines a new extra-stabilised scheme based on extended Crouzeix-Raviart (resp. Morley) finite elements that directly computes approximations as GLB and that allows optimal convergence rates at the same time.
The presentation is on joint work with Dr. Sophie Puttkammer.
Title: Number Theory Seminar: About the unit $1^1.2^2.3^3.4^4....((N-1)/2)^{(N-1)/2}$ modulo a prime number $N$
Speaker: Loïc Merel (Institut de Mathematiques de Jussieu, Paris, France)
Date: Thu, 28 Sep 2023
Time: 11.30 am
Venue: LH-1
Let $N$ be a prime number $>3$. Mazur has defined, from the theory of modular forms, a unit $u$ in $\mathbb{Z}/N$. This unit turned out to be, up to a $6$-th root of unity,$\prod_{k=1}^{(N-1)/2}k^k$.
In this talk we will describe how the unit is connected to various objects in number theory. For instance:
–The unit $u$ can be understood as a derivative of the zeta function at $-1$, (despite living in a finite field).
– Lecouturier has shown that this unit is the discriminant of the Hasse polynomial: $\sum_{i=0}^{(N-1)/2}a_i X^i$ modulo $N$, where $a_i$ is the square of the $i$-th binomial coefficient in degree $N$.
– Calegari and Emerton have related $u$ to the class group of the quadratic field $\mathbb{Q}(\sqrt{-N})$.
For every prime number $p$ dividing $N-1$, It is important to determine when $u$ is a $p$-th power in $(\mathbb{Z}/N)^*$.
If time allows, I will describe the connections to modular forms and Galois representations, and the general theory that Lecouturier has developed from this unit. For instance,when $u$ is not a $p$-th power, a certain Hecke algebra acting on modular forms is of rank $1$ over the ring of $p$-adic integers $\mathbb{Z}_p$ (the original motivation of Mazur). The unit plays an important role in the developments around the conjecture of Harris and Venkatesh.
Title: Algebra & Combinatorics Seminar: Unique factorization for tensor products of parabolic Verma modules
Speaker: V. Sathish Kumar (IMSc, Chennai)
Date: Fri, 22 Sep 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
Let $\mathfrak g$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak h$. We prove that unique factorization holds for tensor products of parabolic Verma modules. We prove more generally a unique factorization result for products of characters of parabolic Verma modules when restricted to certain subalgebras of $\mathfrak h$. These include fixed point subalgebras of $\mathfrak h$ under subgroups of diagram automorphisms of $\mathfrak g$. This is joint work with K.N. Raghavan, R. Venkatesh and S. Viswanath.
Title: APRG Seminar: Applications of Hahn-Banach Theorem to sequence spaces and variational inequality
Speaker: Sudarsan Nanda (KIIT Bhubaneswar and IIT Kharagpur)
Date: Wed, 20 Sep 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
Application of the Hahn-Banach Theorem to the space of bounded sequences with a specific sub linear functional $p$ defined on it gives rise to linear functionals which are dominated by $p$ and are extensions of limits of convergent sequences. These are called Banach Limits and were studied by Banach (1932), and their uniqueness is called almost convergence and was characterised by Lonentz (1948).
In the present lecture we will discuss about the absolute analogue of almost convergence which generalizes lp spaces.
The two concepts of variational inequality and complementarily problems are essentially the same concepts which are studied by two different groups of mathematicians: applied mathematics on one hand and operations researchers on the other hand. The proof existence of variational inequality problem uses Hahn-Banach Theorem or Fixed Point Theorem.
In this lecture we will discuss about the existence of solutions of the complementarily problem, under the most general conditions on the operator and the cone.
Title: APRG Seminar: Almost sharp lower bound for the nodal volume of harmonic functions
Speaker: Lakshmi Priya M.E. (Tel Aviv University, Tel Aviv, Israel)
Date: Thu, 14 Sep 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
In this talk, I will discuss the relation between the growth of harmonic functions and their nodal volume. Let $u:\mathbb{R}^n \rightarrow \mathbb{R}$ be a harmonic function, where $n\geq 2$. One way to quantify the growth of $u$ in the ball $B(0,1) \subset \mathbb{R}^n$ is via the doubling index $N$, defined by \begin{equation} \sup_{B(0,1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|. \end{equation} I will present a result, obtained jointly with A. Logunov and A. Sartori, where we prove an almost sharp result, namely: \begin{equation} \mathcal{H}^{n-1}({u=0} \cap B(0,2)) \gtrsim_{n,\varepsilon} N^{1-\varepsilon}, \end{equation} where $\mathcal{H}^{n-1}$ denotes the $(n-1)$ dimensional Hausdorff measure.
Title: Formalization of $p$-adic $L$-functions in Lean 3
Speaker: Ashvni Narayanan (London school of Geometry and Number Theory; Sydney University)
Date: Wed, 13 Sep 2023
Time: 4:00 pm
Venue: LH-1, Mathematics Department
The Kubota-Leopoldt $p$-adic $L$-function is an important concept in number theory. It takes special values in terms of generalized Bernoulli numbers, and helps solve Kummer congruences. It is also used in Iwasawa theory. Formalization of $p$-adic $L$-functions has been done for the first time in a theorem prover called Lean 3. In this talk, we shall briefly introduce the concept of formalization of mathematics, the theory behind $p$-adic $L$-functions, and its formalization.
Title: Number Theory Seminar: Iwasawa theory for Rankin-Selberg Convolution at an Eisenstein prime
Speaker: Ravitheja Vangala (IISc)
Date: Wed, 06 Sep 2023
Time: 11.30 AM
Venue: LH-1
Let $p$ be an odd prime, $f$ be a $p$-ordinary newform of weight $k$ and $h$ be a normalized cuspidal $p$-ordinary Hecke eigenform of weight $\ell < k$. Let $p$ be an Eisenstein prime for $h$ i.e. the residual Galois representation of $h$ at $p$ is reducible. In this talk, we show that the $p$-adic $L$-function and the characteristic ideal of the $p^{\infty}$-Selmer group of the Rankin-Selberg convolution of $f$, $h$ generate the same ideal modulo $p$ in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$ holds modulo $p$. This is a joint work with Somnath Jha and Sudhanshu Shekhar.
Title: Geometry & Topology Seminar: Measured foliations at infinity of quasi-Fuchsian manifolds close to the Fuchsian locus
Speaker: Diptaishik Choudhury (University of Luxembourg)
Date: Mon, 04 Sep 2023
Time: 4:00 pm
Venue: MS teams (online)
Given a closed, oriented surface with genus greater that 2, we study quasi-Fuchsian hyperbolic 3-manifolds homeomorphic to this surface times the interval. Different properties of these manifolds have been carefully studied in previous important works on 3 manifold geometry and topology and some interesting questions about them still remain to be answered. In this talk, we will focus on a new geometric invariant associated to them which we call the measured foliations at infinity. These are horizontal measured foliations of a holomorphic quadratic differential ( the Schwarzian derivative ) associated canonically with each of the two connected component of the boundary at infinity of a quasi-Fuchsian manifold. We ask whether given any pair of measured foliations (F,G) on a surface, is there a quasi-Fuchsian manifold with F and G as it measured foliations at infinity. The answer is affirmative under certain assumptions; first, (F,G) satisfy the property of being an “arational filling pair” and second, the quasi-Fuchsian manifold should be very close to being “Fuchsian” . The goal of this talk would be introducing the concepts and outlining the proof idea.
Title: Algebra & Combinatorics Seminar: A Frobenius version of Tian's Alpha invariant
Speaker: Swaraj Pande (University of Michigan, Ann Arbor, USA)
Date: Wed, 30 Aug 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department (Joint with the Geometry & Topology Seminar)
The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing “log canonical threshold” with “F-pure threshold”, a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature.
Title: Geometry & Topology Seminar: SU(2) Representations of Three-Manifold groups
Speaker: Deeparaj Bhat (MIT)
Date: Mon, 28 Aug 2023
Time: 4:00 pm
Venue: LH-1 (or MS Teams (online) since the author has been keeping unwell)
By the resolution of the Poincare conjecture in 3D, we know that the only closed three-manifold with the trivial fundamental group is the three-sphere. In light of it, one can ask the following question: Suppose M is a closed three-manifold with the property that the only representation $\pi_1(M)\rightarrow SU(2)$ is the trivial one. Does this imply that $\pi_1(M)$ is trivial? The class of manifolds $M$ for which this question is interesting (and open) are integer homology spheres. We prove a result in this direction: the half-Dehn surgery on any non-trivial fibered knot $K$ in $S^3$ admits an irreducible representation. The proof uses instanton floer homology. I will give a brief introduction to instanton floer homology and sketch the strategy. This is based on work in progress, some jointly with Zhenkun Li and Fan Ye.
Title: Algebra & Combinatorics Seminar: The lattice of nil-Hecke algebras over reflection groups
Speaker: Sutanay Bhattacharya (University of California, San Diego, USA)
Date: Mon, 21 Aug 2023
Time: 11:30 am
Venue: LH-1, Mathematics Department
Associated to every reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, obtained by killing all “sufficiently long” braid words, as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal.
Motivated by the classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture, and continuing beyond the previous work of Khare, we attempt to obtain a classification of the finite-dimensional nil-Hecke algebras for all reflection groups $W$. These include the usual nil-Coxeter algebras for $W$ of finite type, their “fully commutative” analogues for $W$ of FC-finite type, three exceptional algebras (of types $F_4$,$H_3$,$H_4$), and three exceptional series (of types $B_n$ and $A_n$, two of them novel). We further uncover combinatorial bases of algebras, both known (fully commutative elements) and novel ($\overline{12}$-avoiding signed permutations), and classify the Frobenius nil-Hecke algebras in the aforementioned cases. (Joint with Apoorva Khare.)
Title: PhD Thesis colloquium: Harmonic map heat flow and framed surface-group representations
Speaker: Gobinda Sau (IISc Mathematics)
Date: Fri, 11 Aug 2023
Time: 11:15 am
Venue: LH-1, Mathematics Department
This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.
The focus of the first part is when the domain is the complex plane. In this case, given a finite cyclic configuration of points $P \subset \partial\mathbb{H}^3=\mathbb{CP}^1$, we construct a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ that is asymptotic to a twisted ideal polygon with ideal vertices contained in $P$. Moreover, we prove that given any ideal twisted polygon in $\mathbb{H}^3$ with rotational symmetry, there exists a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.
In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an irreducible representation $\rho$ of its fundamental group into $PSL_2(\mathbb{C})$, a seminal theorem of Donaldson asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$. In this thesis, we consider domain surfaces that are non-compact, namely marked and bordered surfaces (introduced in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of marked points that are either punctures or marked points on boundary components. Our main result in this part is: given an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (PSL_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from $\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$.
In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.
Title: APRG Seminar: Fan-Theobald-von Neumann systems
Speaker: M. Seetharama Gowda (University of Maryland in Baltimore County, USA)
Date: Fri, 04 Aug 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
Motivated by optimization considerations and the (matrix theory) inequalities of Ky Fan and von Neumann,
we introduce a Fan-Theobald-von Neumann system as a triple $(V,W,\lambda)$, where $V$ and $W$ are real
inner product spaces and $\lambda:V\rightarrow W$ is a (nonlinear) map satisfying the following condition:
For all $c,u\in V$,
$$\max \{\langle c,x\rangle: x\in [u] \}=\langle \lambda(c),\lambda(u)\rangle,$$
where $[u]:= \{x:\lambda(x)=\lambda(u)\}$.
This simple formulation happens to be equivalent to the Fenchel conjugate formula of the form $(\phi\circ \lambda)^*=\phi^*\circ \lambda$ and a subdifferential formula in some settings and becomes useful in addressing linear/distance optimization problems over “spectral sets” which are of the form $\lambda^{-1}(Q)$, where $Q$ is a subset of $W$. Three standard examples of FTvN systems are: $(\mathbb{R}^n,\mathbb{R}^n,\lambda)$ with $\lambda(x):=x^\downarrow$ (the decreasing rearrangement of the vector $x\in \mathbb{R}^n$); $({\cal H}^n,\mathbb{R}^n,\lambda)$, where ${\cal H}^n$ is the space of $n$ by $n$ complex Hermitian matrices with $\lambda$ denoting the eigenvalue map; and $(M_n,\mathbb{R}^n, \lambda)$, where $M_n$ is the space of $n$ by $n$ complex matrices with $\lambda$ denoting the singular value map. Other examples come from Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). In the general framework of Fan-Theobald-von Neumann systems, we introduce and elaborate on the concepts of commutativity, automorphisms, majorization, etc. We will also talk about “transfer principles” where properties (such as convexity) of $Q$ are carried over to $\lambda^{-1}(Q)$, leading to a generalization of a celebrated convexity theorem of Chandler Davis.
Title: APRG Seminar: Regularity results for n-Laplace systems with antisymmetric potential
Speaker: Armin Schikorra (University of Pittsburgh, USA)
Date: Wed, 26 Jul 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat.
I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione. Joint work with Dorian Martino.
Title: PhD Thesis colloquium: Characters of classical groups twisted by roots of unity
Speaker: Nishu Kumari (IISc Mathematics)
Date: Wed, 12 Jul 2023
Time: 11 am
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.
Throughout the thesis, we fix an integer $t \geq 2$ and a primitive $t$’th root of unity $\omega$. We first consider the irreducible characters of representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. This characterization turns out to depend on the $t$-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores, and show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.
We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the last $m$ to $y, \omega y, \dots, \omega^{m-1} y$. For the other families, we take the same specializations but with $m=1$. Our motivation for studying these are the conjectures of Wagh–Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.
The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at $\omega^p x_i$ and $\omega^q y_j$, for $0 \leq p, q \leq t-1$, $1 \leq i \leq n$, and $1 \leq j \leq m$. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.
For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee–Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson–Pfannerer–Rubey–Uhlin (Forum Math. Sigma, 2021).
Title: PhD Thesis defence: On some canonical metrics on holomorphic vector bundles over Kähler manifolds
Speaker: Kartick Ghosh (IISc Mathematics)
Date: Thu, 06 Jul 2023
Time: 3 pm
Venue: Hybrid - Google Meet (online) and LH-3, Mathematics Department
This thesis consists of two parts. In the first part, we introduce coupled K¨ahler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Amp`ere vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the K¨ahlerness of the symplectic form which arises in the moment map interpretation of Calabi-Yang-Mills equations.
Title: PhD Thesis defence: Interaction of distinguished varieties and the Nevanlinna-Pick interpolation problem in some domains
Speaker: Poornendu Kumar (IISc Mathematics)
Date: Tue, 04 Jul 2023
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.
In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.
In the second part of the talk, we shall discuss the uniqueness of the solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb G$. The uniqueness set is the largest set in $\mathbb G$ where all the solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in $\mathbb G$. The algebraic variety is called the uniqueness variety. We shall see if an $N$-point solvable Nevanlinna–Pick problem is such that it has no solutions of supremum norm less than one and that each of the $(N-1)$-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the $N$-point problem contains a distinguished variety containing all the initial nodes, this is called the Sandwich Theorem. Finally, we shall see the converse of the Sandwich Theorem.
Title: PhD Thesis colloquium: On commuting isometries and commuting isometric semigroups
Speaker: Shubham Rastogi (IISc Mathematics)
Date: Fri, 30 Jun 2023
Time: 4 pm
Venue: LH-3, Mathematics Department
The famous Wold decomposition gives a complete structure of an isometry on a Hilbert space. Berger, Coburn, and Lebow (BCL) obtained a structure for a tuple of commuting isometries acting on a Hilbert space. In this talk, we shall discuss the structures of the pairs of commuting $C_0$-semigroups of isometries in generality as well as under certain additional assumptions like double commutativity or dual double commutativity.
The right-shift-semigroup $\mathcal S^\mathcal E=(S^\mathcal E_t)_{t\ge 0}$ on $L^2(\mathbb R_+,\mathcal E)$ for any Hilbert space
$\mathcal E$ is defined as
\begin{equation}
(S_t^\mathcal E f)(x) = \begin{cases}
f(x-t) &\text{if } x\ge t,\\
0 & \text{otherwise,}
\end{cases}
\end{equation}
for $f\in L^2(\mathbb R_+,\mathcal E).$
Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for
a $C_0$-semigroup of isometries. The factorizations of the unilateral shift have been explored by BCL, we are interested in examining
the factorizations of the right-shift-semigroup.
Firstly, we shall discuss the contractive $C_0$-semigroups which commute with the right-shift-semigroup. Then, we give a complete
description of the pairs $(\mathcal V_1,\mathcal V_2)$ of commuting $C_0$-semigroups of contractions which satisfy $\mathcal S^\mathcal
E=\mathcal V_1\mathcal V_2$, (such a pair is called as a factorization of $\mathcal S^\mathcal E$), when $\mathcal E$ is a finite
dimensional Hilbert space.
Next, we discuss the Taylor joint spectrum for a pair of commuting isometries $(V_1,V_2)$ using the defect operator $C(V_1,V_2)$ defined as \begin{equation} C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+ V_1V_2V_2^*V_1^*. \end{equation} We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.
Title: PhD Thesis defence: Correlations in multispecies asymmetric exclusion process
Speaker: Nimisha Pahuja (IISc Mathematics)
Date: Fri, 16 Jun 2023
Time: 10 am
Venue: LH-1, Mathematics Department
This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:
We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.
Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.
We then talk about multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.
Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.
Title: Eigenfunctions Seminar: Understanding number fields through the distributions of their arithmetic invariants
Speaker: Ila Varma (University of Toronto, Toronto, Canada)
Date: Thu, 15 Jun 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
The most fundamental objects in number theory are number fields, field extensions of the rational numbers that are finite dimensional as vector spaces over $\mathbb{Q}$. Their arithmetic is governed heavily by certain invariants such as the discriminant, Artin conductors, and the class group; for example, the ring of integers inside a number field has unique prime factorization if and only if its class group is trivial. The behavior of these invariants is truly mysterious: it is not known how many number fields there are having a given discriminant or conductor, and it is an open conjecture dating back to Gauss as to how many quadratic fields have trivial class group.
Nonetheless, one may hope for statistical information regarding these invariants of number fields, the most basic such question being “How are such invariants distributed amongst number fields of degree $d$?” To obtain more refined asymptotics, one may fix the Galois structure of the number fields in question. There are many foundational conjectures that predict the statistical behavior of these invariants in such families; however, only a handful of unconditional results are known. In this talk, I will describe a combination of algebraic, analytic, and geometric methods to prove many new instances of these conjectures, including some joint results with Altug, Bhargava, Ho, Shankar, and Wilson.
Title: Algebra & Combinatorics Seminar: Inverse problem for electrical networks
Speaker: Terrence George (University of Michigan, Ann Arbor, USA)
Date: Wed, 14 Jun 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
I will discuss how the inverse problem of recovering conductances in an electrical network from its response matrix can be solved using an automorphism of the positive Grassmannian called the twist.
Title: APRG Seminar: Contraction and pattern formation in disordered actomyosin networks
Speaker: Dietmar Oelz (University of Queensland, Brisbane, Australia)
Date: Tue, 13 Jun 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
The origins of disordered actomyosin network contraction such as in the cellular cortex remain an active topic of research. We derive an agent-based mathematical model for the evolution of two-dimensional networks. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use simulations of disordered networks and find that both protein friction and actin filament bending are sufficient for contraction.
Asymptotic analysis of a special case of this model implies that bending induces a geometric asymmetry that enables motors to move faster close to filament plus-ends, inhibiting expansion.
We also explore a minimal model for pattern formation through biased turnover of actin filaments. The resulting discrete-time interacting particle system can be interpreted as voter model with continuous opinion space. We fully characterise the asymptotic shape of solutions which are characterised by transient clusters.
Title: APRG Seminar: Mathematical modelling of tumor growth and mechanical behaviour
Speaker: G P Raja Sekhar (IIT, Kharagpur)
Date: Fri, 09 Jun 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
In this talk, we first introduce the basic structure of tumors and consequently present some fundamental modelling aspects of tumor growth based on ODE / PDE models. We then introduce the biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. We introduce what is called in-vivo and in-vitro tumors considering an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are saturated by a physiological extracellular fluid. The mass and momentum equations for both the phases are coupled due to the interaction term. Well-posedness results will be discussed in brief. The criterion for necrosis will be shown in terms of the nutrient transport.
Title: PROMYS Guest Lecture: Prime numbers
Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)
Date: Wed, 07 Jun 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
Prime numbers have been studied by Humankind for centuries and have applications in Internet Cryptography. We will outline this connection and also talk about how prime numbers give rise to different number systems.
Title: Geometry & Topology Seminar: Counting Buckyballs
Speaker: Philip Engel (University of Georgia, Athens, USA)
Date: Wed, 07 Jun 2023
Time: 2 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
A “buckyball” or “fullerene” is a trivalent graph embedded in the sphere, all of whose rings have length 5 or 6. The term originates from the most famous buckyball, “Buckminsterfullerene,” a molecule composed of 60 carbon atoms. In this talk, I will explain why there are exactly 1203397779055806181762759 buckyballs with 10000 carbon atoms.
Title: APRG Seminar: Non-local conservation laws modeling traffic flow and crowd dynamics
Speaker: Aekta Aggarwal (Indian Institute of Management, Indore)
Date: Thu, 01 Jun 2023
Time: 4 pm
Venue: LH-1, Mathematics Department
Nonlocal conservation laws are gaining interest due to their wide range of applications in modeling real world phenomena such as crowd dynamics and traffic flow. In this talk, the well-posedness of the initial value problems for certain class of nonlocal conservation laws, scalar as well as system, will be discussed and monotone finite volume approximations for such PDEs will be proposed. Strong compactness of the proposed numerical schemes will be presented and their convergence to the entropy solution will be proven. Some numerical results illustrating the established theory will also be presented.
Title: PROMYS Guest Lecture: Divisibility tests, recurring decimals, and Artin's conjecture
Speaker: Apoorva Khare (IISc Mathematics)
Date: Wed, 31 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
If we had two extra thumbs, how would we check if “2024” is divisible by eleven? Or by “11”? We will see a simple test in any base $B$, i.e. usable by species having any number of fingers (whether shaped like hot-dogs or not); and for any divisor $d$. That is, the test works for everything ($d$), everywhere ($B$), all at once.
We will then move to recurring decimals. Note that 1/3 = 0.3333… and 1/3x3 = 0.1111… have the same number of digits - one - in their recurring parts. (Is 3 the only prime with this property in base 10?) More generally, we will see how many digits $1/d$ has in its recurring “decimal” expansion, for us or for any species as above.
Finally, for a species with a given number of fingers (= digits!), are there infinitely many primes $p$ for which the recurring part of $1/p$ has $p-1$ digits? (E.g. for us, 1/7 has the decimal recurring string (142857).) And what does this have to do with Gauss, Fermat, and one of the Bernoullis? Or with Artin and a decimal number starting with 0.3739558136… ? I will end by mentioning why this infinitude of primes holds for at least one species among humans (10), emus (6), ichthyostega (14), and computers (2) - but, we don’t know which one!
Title: Eigenfunctions Seminar: Tetrahedra with rational dihedral angles
Speaker: Kiran S. Kedlaya (University of California, San Diego, USA)
Date: Thu, 25 May 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
We classify similarity classes of tetrahedra whose dihedral angles are all rational multiples of $\pi$ (when measured in radians), answering a question of Conway-Jones from 1976. In the process, we also classify collections of vectors in $\mathbb{R}^3$ whose pairwise angles are rational. The proof uses a mixture of theoretical arguments, exact computations in computer algebra, and floating-point numerical computations. (Joint with Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein.)
Title: PROMYS Guest Lecture: Zeta functions and Euler products
Speaker: Alina Bucur (University of California, San Diego, USA)
Date: Wed, 24 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
We will define one of the most famous functions in all of mathematics, the Riemann zeta function, whose properties are the subject of one of the Millenium Problems. We will also look at some of its analogues for other objects.
Title: PROMYS Guest Lecture: Cyclotomic polynomials and roots of unity
Speaker: Kiran S. Kedlaya (University of California, San Diego, USA)
Date: Mon, 22 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
We introduce an important family of polynomials, the cyclotomic polynomials, whose roots are the roots of unity of a fixed order. We explore the structure of these polynomials and the number fields that they generate, including a brief look at Gauss sums.
Title: PROMYS Guest Lecture: Could Euclid trisect every angle? How about us?
Speaker: Parthanil Roy (ISI, Bangalore)
Date: Wed, 17 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
This talk will be a lucid introduction to the formal mathematics behind Euclidean Constructions, which we all learn in our middle school curriculum. The rules, regulations and restrictions of this type of construction will be discussed in detail. An alternative will also be suggested. We shall also find out how a completely geometric question can be answered using purely algebraic techniques giving rise to an elegant theory introduced in the nineteenth century by a famous French mathematician named Évariste Galois.
Title: Geometry & Topology Seminar: On manifolds homeomorphic to the n-sphere
Speaker: Somnath Basu (IISER-Kolkata)
Date: Thu, 11 May 2023
Time: 3:00 pm
Venue: LH-1
We shall discuss Reeb’s Theorem and basic differential topology of Morse functions. This was used by Milnor to prove the existence of exotic spheres in 7 dimensions. We shall propose a generalization of Reeb’s Theorem and discuss a proof of it. This is joint work with Sachchidanand Prasad.
Title: APRG Seminar: Approximation algorithms for the volumes of spectrahedra
Speaker: Mohan Ravichandran (Bogazici University, Istanbul, Turkey)
Date: Thu, 11 May 2023
Time: 2 pm
Venue: LH-1, Mathematics Department
The problem of algorithmically computing the volumes of convex bodies is a well studied problem in combinatorics and theoretical computer science. The best known results are perhaps those concerning the use of Markov Chain Monte Carlo techniques for approximately computing the volumes of general convex bodies. There are also results of a different kind: Deterministic (approximate) computation of the volumes of (certain)polytopes. In this direction, Alexander Barvinok and John Hartigan gave an algorithm based upon the Maximum Entropy heuristic from Statistical Physics that provides good approximations for certain classes of polytopes, that includes the transportation polytopes.
The Maximum Entropy heuristic, originally introduced by Jaynes in 1957 says the following: Suppose one is faced with an unknown probability distribution over a product space. Further suppose we know the expectations of a certain number of random variables with respect to this measure. Then the Maximum Entropy heuristic says that it ‘is natural’ to work with that probability distribution that has max entropy subject to the given linear constraints. Barvinok and Hartigan’s work uses this idea and combines it with some fundamental results about the computability of entropies of these max entropy distributions.
In this talk, I will show how to adapt this approach to Spectrahedra, which are a naturally occurring class of convex sets, defined as slices of the cone of Positive Semidefinite matrices. The case of spectrahedra shows up several surprises. As a byproduct of this work it will follow that central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. I will then give several examples to illustrate the utility of this method.
This is joint work with Jonathan Leake (U Waterloo) and Mahmut Levent Dogan (T U Berlin).
Title: APRG Seminar: Homogenization questions inspired by machine learning and the semi-supervised learning problem
Speaker: Raghavendra Venkatraman (Courant Institute / New York University, USA)
Date: Wed, 03 May 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
This talk comprises two parts. In the first part, we revisit the problem of pointwise semi-supervised learning (SSL). Working on random geometric graphs (a.k.a point clouds) with few “labeled points”, our task is to propagate these labels to the rest of the point cloud. Algorithms that are based on the graph Laplacian often perform poorly in such pointwise learning tasks since minimizers develop localized spikes near labeled data. We introduce a class of graph-based higher order fractional Sobolev spaces (H^s) and establish their consistency in the large data limit, along with applications to the SSL problem. A crucial tool is recent convergence results for the spectrum of the graph Laplacian to that of a weighted Laplace-Beltrami operator in the continuum.
Obtaining optimal convergence rates for such spectra has so-far been an open question in stochastic homogenization. In the rest of the talk, we answer this question by obtaining optimal, state-of-the-art results for the case of a Poisson point cloud on a bounded domain in Euclidean space with Dirichlet or Neumann boundary conditions.
The first half is joint work with Dejan Slepcev (CMU), and the second half is joint work with Scott Armstrong (Courant).
Title: PhD Thesis defence: Homogenization of PDEs on oscillating boundary domains with L1 data and Optimal control problems
Speaker: Renjith T. (IISc Mathematics)
Date: Thu, 27 Apr 2023
Time: 11 am
Venue: LH-3, Mathematics Department
This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems. The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and L1 data in a circular oscillating domain. As the source term we are considering is in L1, we will examine the renormalized solutions. The second problem we will investigate is an optimal control problem governed by a second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the oscillations occur in $m$ directions, with $1< m < n$. We will explore the asymptotic behavior of this problem by homogenizing the corresponding optimality systems.
Title: Geometry & Topology Seminar: Maximal submanifolds in pseudo-hyperbolic space and applications
Speaker: Andrea Seppi (University of Grenoble)
Date: Mon, 24 Apr 2023
Time: 4:00 pm
Venue: MS Teams (online)
The Asymptotic Plateau Problem is the problem of existence of submanifolds of vanishing mean curvature with prescribed boundary “at infinity”. It has been studied in the hyperbolic space, in the Anti-de Sitter space, and in several other contexts. In this talk, I will present the solution of the APP for complete spacelike maximal p-dimensional submanifolds in the pseudo-hyperbolic space of signature (p,q). In the second part of the talk, I will discuss applications of this result in Teichmüller theory and for the study of Anosov representations. This is joint work with Graham Smith and Jérémy Toulisse.
Title: Number Theory Seminar: Class field theory for varieties over local fields
Speaker: Jitendra Rathore (TIFR Mumbai)
Date: Thu, 20 Apr 2023
Time: 11 AM
Venue: LH-1
The (tame) class field theory for a smooth variety $X$ is the
study of describing the abelianized (tame) {'e}tale fundamental group of
$X$in terms of some groups which are defined using algebraic cycles of $X$.
In this talk, we study the tame class field theory for smooth varieties
over local fields. We will begin with defining few notions and recalling
various results from the past to overview the historical background of the
subject. We will then study abelianized tame fundamental group denoted as
$\pi^{ab,t}_{1}(X)$, with the help of reciprocity map $\rho^{t}_{X} :
C^{t}(X) \rightarrow \pi^{ab,t}_{1}(X)$ and will describe the kernel and
topological cokernel of this map. This talk is based on a joint work with
Prof. Amalendu Krishna and Dr. Rahul Gupta.
Title: Geometry & Topology Seminar: New Higher Genus Minimal Lagrangians in CP^{2}
Speaker: Charles Ouyang (University of Massachusetts)
Date: Mon, 17 Apr 2023
Time: 9:00 pm
Venue: MS Teams (online)
Minimal Lagrangian tori in CP^{2} are the expected local model for particular point singularities of Calabi-Yau 3-folds and numerous examples have been constructed. In stark contrast, very little is known about higher genus examples, with the only ones to date due to Haskins-Kapouleas and only in odd genus. Using loop group methods we construct new examples of minimal Lagrangian surfaces of genus 1/2(k-1)(k-2) for large k. In particular, we construct the first examples of such surfaces with even genus. This is joint work with Sebastian Heller and Franz Pedit.
Title: Geometry & Topology Seminar: Metric geometry of Einstein 4-manifolds with special holonomy.
Speaker: Ruobing Zhang (Princeton University)
Date: Mon, 10 Apr 2023
Time: 8:30 pm
Venue: MS Teams (online)
This talk focuses on the recent resolutions of several well-known conjectures in studying the Einstein 4-manifolds with special holonomy. The main results include the following.
(1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.
(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have an ALX model geometry with optimal asymptotic rate.
(3) Any gravitational instanton is biholomorphic to a dense open subset of some compact algebraic surface.
Title: APRG Seminar: Mathematics for Nanomedicine: From Accelerated acquisitions, Advance Image Processing, to Patient Specific Models
Speaker: Rahul Kumar (CIC biomaGUNE, San Sebastian, Spain)
Date: Mon, 10 Apr 2023
Time: 3 pm
Venue: Microsoft Teams (Online)
Nanomedicine is an offshoot of nanotechnology that involves many disciplines, including the manipulation and manufacturing of materials, imaging, diagnosis, monitoring, and treatment. An efficient iterative reconstruction algorithm,together with Total Variation (TV), and a good mathematical model, can be used to enhance the spatial resolution and predictive capabilities. In this webinar, I will start with our current results using integrated approach for predicting efficient biomarkers for Acute respiratory distress syndrome (ARDS) and then move to PDE based (Total variation flow) approach for Image denoising which can have promising applications in denoising medical images from different modalities. In principle, I will be discussing the below-mentioned topics and their important concepts in dealing with the main markers of cardiovascular diseases, specifically Pulmonary Hypertension.
1. 4D FlowMRI Data Assimilation: Integrated approach reveals new biomarkers for Experimental ARDS conditions. The purpose of this study is to characterize flow patterns and several other hemodynamic parameters (WSS, OSI, Helicity) using computational fluid dynamics model by combining imaging data from 4D-Flow MRI with hemodynamic pressure and flow waveforms from control and hypertensive subjects (related to acute respiratory distress syndrome). This work mainly concerns how to facilitate bench-bedside approach using integrated approach by combining CFD and AI.
2. An adaptive $C^0$ interior penalty discontinuous galerkin approximation of second order total variation problems.
Singular nonlinear fourth order boundary value problems have significant applications in image processing and material science.
We consider an adaptive $C^0$ Interior Penalty Discontinuous Galerkin (C0IPDG) method for the numerical solution of singular
nonlinear fourth order boundary value problems arising from the minimization of functionals involving the second order total
variation. The mesh adaptivity will be based on an aposteriori error estimator that can be derived by duality arguments. The
fourth order elliptic equation reads as follows:
\begin{align}
u + \lambda \nabla \cdot \nabla \cdot \frac{D^2 u}{|D^2 w|} = & \ 0 \quad \mbox{in} \ Q := \Omega, \\
u = & \ 0 \quad \mbox{on} \ \Gamma,\\
n_{\Gamma} \cdot\frac{D^2 u} {n_{\Gamma}} = & \ 0 \quad \mbox{on} \ {\Gamma}.
\end{align}
Title: Computer Proofs and Artificial Intelligence in Mathematics
Speaker: Kevin Buzzard (Imperial College, London, UK), Viraj Kumar (IISc), and others
Date: Thu, 06 Apr 2023
Time: 3:30 pm - 5:00 pm
Venue: LH-1, Mathematics Department
ChatGPT and other advances in Artificial Intelligence have become popular sensations. In parallel with this has been an enormous advance in the digitization of mathematics through Interactive Theorem Provers and their libraries. Artificial Intelligence has started entering mathematics through these and other routes.
This session will have some presentations/demos about present use of Computer Proofs, Artificial Intelligence together and separately in Mathematics and related fields (including software), both in research and in teaching. After that everyone is welcome to discuss their work, ideas, wish-lists etc related to these themes.
Title: Geometry & Topology Seminar: Tight contact structures on Seifert fibered 3-manifolds.
Speaker: Tanushree Shah (Glasgow University)
Date: Mon, 03 Apr 2023
Time: 2:00 pm
Venue: MS Teams (online)
I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.
Title: Eigenfunctions Seminar: Mathematics and the computer
Speaker: Kevin Buzzard (Imperial College, London, UK)
Date: Mon, 03 Apr 2023
Time: 4 pm
Venue: Faculty Hall, Main Building
For decades, mathematicians have been using computers to calculate. More recently there has been some interest in trying to get them to reason. What is the difference? An example of a calculation: compute the first one million prime numbers. An example of reasoning: prove that there are infinitely many prime numbers. Tools like ChatGPT can prove things like this, because they have seen many proofs of it on the internet. But can computers help researchers to come up with new mathematics? Hoping that a computer will automatically prove the Riemann Hypothesis is still science fiction. But new tools and methods are becoming available. I will give an overview of the state of the art.
(This is a Plenary talk in the EECS Research Students’ Symposium)
Title: PhD Thesis colloquium: Interaction of distinguished varieties and the Nevanlinna-Pick interpolation problem in some domains
Speaker: Poornendu Kumar (IISc Mathematics)
Date: Fri, 31 Mar 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.
In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.
In the second part of the talk, we shall discuss the uniqueness of the solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb G$. The uniqueness set is the largest set in $\mathbb G$ where all the solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in $\mathbb G$. The algebraic variety is called the uniqueness variety. We shall see if an $N$-point solvable Nevanlinna–Pick problem is such that it has no solutions of supremum norm less than one and that each of the $(N-1)$-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the $N$-point problem contains a distinguished variety containing all the initial nodes, this is called the Sandwich Theorem. Finally, we shall see the converse of the Sandwich Theorem.
Title: Number Theory Seminar: $p \neq q$ Iwasawa Theory
Speaker: Debanjana Kundu (Fields Institute, Toronto, Canada)
Date: Fri, 31 Mar 2023
Time: 2 PM
Venue: LH-1
Let $K$ be an imaginary quadratic field of class number $1$ such that both $p$ and $q$ split in $K$. We show that under appropriate hypotheses, the $p$-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $\mathbb{Z}_q$-extension of $K$. This is joint work with Antonio Lei.
Title: Algebra & Combinatorics Seminar: Harish-Chandra modules over full toroidal Lie algebras and higher-dimensional Virasoro algebras
Speaker: Souvik Pal (ISI, Bangalore)
Date: Thu, 30 Mar 2023
Time: 4 pm
Venue: LH-1, Mathematics Department
The Virasoro algebra, which can be realized as a central extension of (complex) polynomial vector fields on the unit circle, plays a key role in the representation theory of affine Lie algebras, as it acts on almost every highest weight module for the affine Lie algebra. This remarkable phenomenon eventually led to constructing the affine-Virasoro algebra, which is a semi-direct product of the affine Lie algebra and the Virasoro algebra with a common extension. The representation theory of the affine-Virasoro algebra has been studied extensively and is an extremely well-developed classical object.
In this talk, we shall consider a natural higher-dimensional analogue of the affine-Virasoro algebra, popularly known as the full toroidal Lie algebra in the literature and henceforth classify the irreducible Harish-Chandra modules over this Lie algebra. As a by-product, we also obtain the classification of all possible irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, thereby proving Eswara Rao’s conjecture (conjectured in 2004). These directly generalize the well-known result of O. Mathieu for the classical Virasoro algebra and also the recent work of Billig–Futorny for the higher rank Witt algebra.
Title: Geometry & Topology Seminar: Relatively hyperbolic groups and convex projective structures
Speaker: Mitul Islam (Heidelberg University)
Date: Wed, 29 Mar 2023
Time: 2:00 pm
Venue: LH-1
Studying discrete subgroups of linear groups using a preserved geometric structure has a long tradition, for instance, using real hyperbolic geometry to study discrete subgroups of SO(n,1). Convex projective structures, a generalization of real hyperbolic structures, has recently received much attention in the context of studying discrete subgroups of PGL(n). In this talk, I will discuss convex projective structures and discuss results (joint with A. Zimmer) on relatively hyperbolic groups that preserve convex projective structures. In particular, I will discuss a complete characterization of relative hyperbolicity in terms of the geometry of the projective structure.
Title: Algebra & Combinatorics Seminar: On quasi Steinberg characters of complex reflection groups
Speaker: Ashish Mishra (Federal University of Para, Belem, Brazil)
Date: Wed, 29 Mar 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
Consider a finite group $G$ and a prime number $p$ dividing the order of $G$. A $p$-regular element of $G$ is an element whose order is coprime to $p$. An irreducible character $\chi$ of $G$ is called a quasi $p$-Steinberg character if $\chi(g)$ is nonzero for every $p$-regular element $g$ in $G$. The quasi $p$-Steinberg character is a generalization of the well-known $p$-Steinberg character. A group, which does not have a non-linear quasi $p$-Steinberg character, can not be a finite group of Lie type of characteristic $p$. Therefore, it is natural to ask for the classification of all non-linear quasi $p$-Steinberg characters of any finite group $G$. In this joint work with Digjoy Paul and Pooja Singla, we classify quasi $p$-Steinberg characters of all finite complex reflection groups.
Title: Number Theory Seminar: On the $p$-adic $L$-function of automorphic overconvergent $F$-isocrystals
Speaker: Fabien Trihan (Sophia University, Japan)
Date: Tue, 28 Mar 2023
Time: 10.30 AM
Venue: LH-2
We report on new ideas of Ki-Seng Tan and myself towards the construction of a $p$-adic $L$-function associated to an automorphic overconvergent $F$-isocrystal over a curve over a finite field. This function should be of interest in the Iwasawa theory for such coefficients.
Title: Geometry & Topology Seminar: Asymptotics of SL(3,R)-Hitchin representations along rays of cubic differentials
Speaker: Andrea Tamburelli (Rice University, USA)
Date: Mon, 27 Mar 2023
Time: 4:00 pm
Venue: MS Teams (online)
Hitchin’s theory of Higgs bundles associated holomorphic differentials on a Riemann surface to representations of the fundamental group of the surface into a Lie group. We study the geometry common to representations whose associated holomorphic differentials lie on a ray. In the setting of SL(3,R), we provide a formula for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Mike Wolf.
Title: Number Theory Seminar: $p$-adic adjoint $L$-functions for Hilbert modular forms
Speaker: Baskar Balasubramanyam (IISER Pune)
Date: Thu, 23 Mar 2023
Time: 10.30 AM
Venue: LH-2
Let $F$ be a totally real field. Let $\pi$ be a cuspidal cohomological automorphic representation for $\mathrm{GL}_2/F$. Let $L(s, \mathrm{Ad}^0, \pi)$ denote the adjoint $L$-function associated to $\pi$. The special values of this $L$-function and its relation to congruence primes have been studied by Hida, Ghate and Dimitrov. Let $p$ be an integer prime. In this talk, I will discuss the construction of a $p$-adic adjoint $L$-function in neighbourhoods of very decent points of the Hilbert eigenvariety. As a consequence, we relate the ramification locus of this eigenvariety to the zero set of the $p$-adic $L$-functions. This was first established by Kim when $F=\mathbb{Q}$. We follow Bellaiche’s description of Kim’s method, generalizing it to arbitrary totally real number fields. This is joint work with John Bergdall and Matteo Longo.
Title: Eigenfunctions Seminar: Slides From random permutations and random matrices to random growth: an invitation to the fascinating mathematics of the KPZ universality class
Speaker: Riddhipratim Basu (ICTS Bangalore)
Date: Fri, 17 Mar 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
From the longest increasing subsequence in a random permutation to the shortest distance in a randomly weighted two dimensional Euclidean lattice, a large class of planar random growth models are believed to exhibit shared large scale features of the so-called Kardar-Parisi-Zhang (KPZ) universality class. Over the last 25 years, intense mathematical activity has led to a lot of progress in the understanding of these models, and connections to several other topics such as algebraic combinatorics, random matrices and partial differential equations have been unearthed. I shall try to give an elementary introduction to this area, describe some of what is known as well as many questions that remain open.
Title: PhD Thesis colloquium: Correlations in multispecies asymmetric exclusion processes
Speaker: Nimisha Pahuja (IISc Mathematics)
Date: Fri, 17 Mar 2023
Time: 11 am
Venue: LH-1, Mathematics Department
This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:
We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.
Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.
We then talk about the multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.
Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.
Title: Number Theory Seminar: The Siegel-Veech transform in dynamics and number theory
Speaker: Anish Ghosh (TIFR Mumbai)
Date: Wed, 15 Mar 2023
Time: 2 PM
Venue: LH-1
The Siegel-Veech transform is a basic tool in homogeneous as well as Teichmuller dynamics. I will introduce the transform and explain how it can be used in counting problems.
Title: Geometry & Topology Seminar: Minimal slopes and singular solutions for some complex Hessian equations
Speaker: Ved Datar (IISc, Bangalore)
Date: Mon, 13 Mar 2023
Time: 4:00 pm
Venue: LH-1
It is well known that solvability of the complex Monge- Ampere equation on compact Kaehler manifolds is related to the positivity of certain intersection numbers. In fact, this follows from combining Yau’s celebrated resolution of the Calabi conjecture, with Demailly and Paun’s generalization of the classical Nakai-Mozhesoin criteria. This correspondence was recently extended to a broad class of complex non-linear PDEs including the J-equation and the deformed Hermitian-Yang-Mills (dHYM) equations by the work of Gao Chen and others (including some at IISc). A natural question to ask is whether solutions (necessarily singular) exist in any reasonable sense if the Nakai criteria fails. Results of this nature are ubiquitous in Kaehler geometry - existence of weak Kaehler-Einstein metrics on normal varieties and Hermitian-Einstein metrics on reflexive sheaves to name a couple. Much closer to the present theme, is the work of Boucksom-Eyssidieux-Guedj-Zeriahi on solving the complex Monge-Ampere equation in big classes. In the talk, I will first speak about some joint and ongoing work with Ramesh Mete and Jian Song, that offers a reasonably complete resolution in complex dimension two, at least for the J-equation and the dHYM equations. Next, I will discuss some conjectures on what one can expect in higher dimensions.
Title: PhD Thesis defence: Local Projection Stabilization Methods for the Oseen Problem
Speaker: Rahul Biswas (IISc Mathematics)
Date: Fri, 10 Mar 2023
Time: 4:30 pm
Venue: LH-1, Mathematics Department
Convection dominated fluid flow problems show spurious oscillations when solved using the usual Galerkin finite element method (FEM). To suppress these un-physical solutions we use various stabilization methods. In this thesis, we discuss the Local Projection Stabilization (LPS) methods for the Oseen problem.
This thesis mainly focuses on three different finite element methods each serving a purpose of its own. First, we discuss the a priori analysis of the Oseen problem using the Crouzeix-Raviart (CR1) FEM. The CR1/P0 pair is a well-known choice for solving mixed problems like the Oseen equations since it satisfies the discrete inf-sup condition. Moreover, the CR1 elements are easy to implement and offer a smaller stencil compared with conforming linear elements (in the LPS setting). We also discuss the CR1/CR1 pair for the Oseen problem to achieve a higher order of convergence.
Second, we discuss a posteriori analysis for the Oseen problem using the CR1/P0 pair using a dual norm approach. We define an error estimator and prove that it is reliable and discuss an efficiency estimate that depends on the diffusion coefficient.
Next, we focus on formulating an LPS scheme that can provide globally divergence free velocity. To achieve this, we use the $H(div;\Omega)$ conforming Raviart-Thomas (${\rm RT}^k$) space of order $k \geq 1$. We show a strong stability result under the SUPG norm by enriching the ${\rm RT}^k$ space using tangential bubbles. We also discuss the a priori error analysis for this method.
Finally, we develop a hybrid high order (HHO) method for the Oseen problem under a generalized local projection setting. These methods are known to allow general polygonal meshes. We show that the method is stable under a “SUPG-like” norm and prove a priori error estimates for the same.
Title: PhD Thesis colloquium: On some canonical metrics on holomorphic vector bundles over Kähler manifolds
Speaker: Kartick Ghosh (IISc Mathematics)
Date: Fri, 10 Mar 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
This thesis consists of two parts. In the first part, we introduce coupled Kähler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using the Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of the Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Ampère vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the Kählerness of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations.
Title: APRG Seminar: Basins of attraction and their uniformization
Speaker: Kaushal Verma (IISc Mathematics)
Date: Wed, 01 Mar 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
This will be an introductory talk on some matters relating to Fatou-Bieberbach domains and uniformizing stable manifolds.
Title: Number Theory Seminar: Harmonic Analysis and Number Theory
Speaker: Devadatta Hegde (University of Minnesota, Twin Cities, USA)
Date: Wed, 01 Mar 2023
Time: 2 PM
Venue: LH-1
In the late 1950s, an important problem in number theory was to extend the notion of $L$-functions attached to cuspforms on the upper-half
plane to automorphic forms on reductive groups. Langlands’s work on non-abelian Harmonic analysis, namely the problem of the spectral decomposition of automorphic forms, led him to a general notion of $L$-functions
attached to cuspforms. We give an introduction to the spectral decomposition of automorphic forms and discuss some contemporary problems.
Title: Number Theory Seminar: Crystalline representations and Wach modules in the relative case
Speaker: Abhinandan (University of Tokyo, Japan)
Date: Tue, 28 Feb 2023
Time: 10.30 AM
Venue: LH-2
Over an unramified extension $F/\mathbb{Q}_p$, by the works of Fontaine, Wach, Colmez and Berger, it is well-known that a crystalline representation of the absolute Galois group of $F$ is of finite height. Moreover, in this case, to a crystalline representation one can functorially attach a lattice inside the associated etale $(\varphi, \Gamma)$-module called the Wach module. Berger showed that the aforementioned functor induces an equivalence between the category of crystalline representations and Wach modules. Furthermore, this categorical equivalence admits an integral refinement. In this talk, our goal is to generalize the notion of Wach modules to relative $p$-adic Hodge theory. For a “small” unramified base (in the sense of Faltings) and its etale fundamental group, we will generalize the result of Berger to an equivalence between crystalline representations and relative Wach modules as well as establish its integral refinement.
Title: APRG Seminar: Schur polynomials: from smooth functions and determinants, to symmetric functions and all characters
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 24 Feb 2023
Time: 3 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
Cauchy’s determinantal identity (1840s) expands via Schur polynomials the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the rank-one matrix $(u_i v_j)$. This theme has resurfaced in the 2010s in analysis (following a 1960s computation by Loewner), in the quest to find polynomials $p(t)$ with a negative coefficient that entrywise preserve positivity. A key novelty here has been the application of Schur polynomials, which essentially arise from the expansion of $\det(p[{\bf u}{\bf v}^T])$, to positivity.
In the first half of the talk, I will explain the above journey from matrix positivity to determinantal identities and Schur polynomials; then go beyond, to the expansion of $\det(f[{\bf u}{\bf v}^T])$ for all power series $f$. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.) In the second half, joint with Siddhartha Sahi, I will explain how to extend the above determinantal identities to (a) any subgroup $G$ of signed permutations; (b) any character of $G$, or even complex class function; (c) any commutative ground ring $R$; and (d) any power series over $R$.
Title: Number Theory Seminar: Galois representations at the boundary of the eigencurve
Speaker: Aditya Karnataki (CMI)
Date: Wed, 22 Feb 2023
Time: 2 PM
Venue: LH-1
Andreatta, Iovita, and Pilloni have proven the existence of an adic eigencurve, which includes characteristic $p$ points at the boundary. In joint work with Ruochuan Liu, using the theory of Crystalline periods, we show that the Galois representations associated to these points satisfy an appropriate trianguline property.
Title: Algebra & Combinatorics Seminar: 2-Categorification of Category O and 3d Mirror Symmetry
Speaker: Justin Hilburn (Perimeter Institute, Waterloo, Canada)
Date: Fri, 17 Feb 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
In 1976 Bernstein, Gelfand, and Gelfand introduced Category $\mathcal{O}$ for a semi-simple Lie algebra $\mathfrak{g}$. This is roughly the smallest sub-category of $\mathfrak{g}$-mod containing the Verma modules and such that the simple modules have projective covers. After work of Beilinson–Bernstein and Beilinson–Ginzburg–Soergel it became clear that the the good homological properties of this category were due to the fact that it can be identified with a category of perverse sheaves on the flag variety $G/B$.
In this talk I will show how this story fits into the physics of 3d mirror symmetry. This leads to conjectural 2-categorifications of category $\mathcal{O}$ that can be computed explicitly for $\mathfrak{g} = \mathfrak{sl}_2$.
Title: Algebra & Combinatorics Seminar: Weak faces of highest weight modules and root systems
Speaker: G. Krishna Teja (HRI, Prayagraj)
Date: Fri, 10 Feb 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
The geometry, and the (exposed) faces, of $X$ a “Root polytope” or “Weyl polytope” over a complex simple Lie algebra $\mathfrak{g}$, have been studied for many decades for various applications, including by Satake, Borel–Tits, Casselman, and Vinberg among others. This talk focuses on two recent combinatorial analogues to these classical faces, in the discrete setting of weight-sets $X$.
Chari et al [Adv. Math. 2009, J. Pure Appl. Algebra 2012] introduced and studied two combinatorial subsets of $X$ a root system or the weight-set wt $V$ of an integrable simple highest weight $\mathfrak{g}$-module $V$, for studying Kirillov–Reshetikhin modules over the specialization at $q=1$ of quantum affine algebras $U_q(\hat{\mathfrak{g}})$ and for constructing Koszul algebras. Later, Khare [J. Algebra 2016] studied these subsets under the names “weak-$\mathbb{A}$-faces” (for subgroups $\mathbb{A}\subseteq (\mathbb{R},+)$) and “$212$-closed subsets”. For two subsets $Y\subseteq X$ in a vector space, $Y$ is said to be $212$-closed in $X$, if $y_1+y_2=x_2+x_2$ for $y_i\in Y$ and $x_i\in X$ implies $x_1,x_2\in Y$.
In finite type, Chari et al classified these discrete faces for $X$ root systems and wt $V$ for all integrable $V$, and Khare for all (non-integrable) simple $V$. In the talk, we extend and completely solve this problem for all highest weight modules $V$ over any Kac–Moody Lie algebra $\mathfrak{g}$. We classify, and show the equality of, the weak faces and $212$-closed subsets in the three prominent settings of $X$: (a) wt $V$ $\forall V$, (b) the hull of wt $V$ $\forall V$, (c) wt $\mathfrak{g}$ (consisting of roots and 0). Moreover, in the case of (a) (resp. of (b)), such subsets are precisely the weights falling on the exposed faces (resp. the exposed faces) of the hulls of wt $V$.
Title: Eigenfunctions Seminar: Slides Cones, and spectral properties of positive operators in real Banach spaces via Game theory
Speaker: T.E.S. Raghavan (University of Illinois at Chicago, USA)
Date: Fri, 27 Jan 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
While statistical decision theory led me to game theory, certain war duel models, and the close connection between the Perron–Frobenius theorem and game theory led me to the works of M.G. Krein on special classes of cones, and spectral properties of positive operators. The influence of Professors V.S. Varadarajan, K.R Parthasarathy and S.R.S Varadhan in early 60’s at ISI is too profound to many of us as young graduate students in 1962-66 period. The talk will highlight besides the theorems, the teacher-student interactions of those days.
Title: APRG Seminar: Free energy of the diluted Shcherbina–Tirozzi model with quadratic Hamiltonian
Speaker: Ratul Biswas (University of Minnesota, Minneapolis, USA)
Date: Wed, 18 Jan 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
The study of diluted spin glasses may help solve some problems in computer science and physics. In this talk, I shall introduce the diluted Shcherbina–Tirozzi (ST) model with a quadratic Hamiltonian, for which we computed the free energy at all temperatures and external field strengths. In particular, we showed that the free energy can be expressed in terms of the weak limits of the quenched spin variances and identified these weak limits as the unique fixed points of a recursive distributional operator. The talk is based on a joint work with Wei-Kuo Chen and Arnab Sen.
Title: PhD Thesis colloquium: Homogenization of PDEs on oscillating boundary domains with L1 data and Optimal control problems
Speaker: Renjith T. (IISc Mathematics)
Date: Tue, 17 Jan 2023
Time: 11 am
Venue: LH-1, Mathematics Department
This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal
control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems.
The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and $L^1$
data in a circular oscillating domain. As the source term we are considering is in $L^1$, we will examine the
renormalized solutions. The second problem we will investigate is an optimal control problem governed by a
second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the
oscillations occur in $m$ directions, with $1<m<n$. We will explore the asymptotic behavior of this problem by
homogenizing the corresponding optimality systems.
Title: Eigenfunctions Seminar: Stark’s conjectures and refinements
Speaker: Mahesh Kakde (IISc Mathematics)
Date: Fri, 13 Jan 2023
Time: 3 – 5 pm (with a 15 minute break at 4:00)
Venue: LH-1, Mathematics Department
In the first half of the talk I will recall two classical theorems - Dirichlet’s class number formula and Stickelberger’s theorem. Stark and Brumer gave conjectural generalisations of these statements. We will see formulations of some of these conjectures. In the second half of the talk we will restrict to a special case of the Brumer-Stark conjecture. Here p-adic techniques can be used to resolve the conjecture. We will see a sketch of this proof. This is joint work with Samit Dasgupta.
Title: Algebra & Combinatorics Seminar: Free root spaces of Borcherds-Kac-Moody Lie superalgebras
Speaker: Shushma Rani (IISER, Mohali)
Date: Wed, 11 Jan 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
Let $\mathfrak g$ be a Borcherds–Kac–Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak g$ is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph $G$. By Chevalley relations we get a triangular decomposition $\mathfrak g = \mathfrak n_+ \oplus \mathfrak h \oplus \mathfrak n_{-}$, and each root space $\mathfrak g_{\alpha}$ is either contained in $\mathfrak n_+$ or $\mathfrak n_{-}$. In particular, each $\mathfrak g_{\alpha}$ involves only the relations (2) and (3). In this talk, we will discuss the root spaces of $\mathfrak g$ which are independent of the Serre relations. We call these roots free roots of $\mathfrak g$. Since these root spaces involve only commutation relations coming from the graph $G$ we can study them combinatorially using heaps of pieces and construct two different bases for these root spaces of $\mathfrak g$.
Title: Geometry & Topology Seminar: Thom conjecture in higher dimensions
Speaker: Marko Slapar (University of Ljubljana, Slovenia)
Date: Wed, 11 Jan 2023
Time: 2:00 pm
Venue: LH-1, Mathematics Department (Joint with the APRG Seminar)
The Thom conjecture, proven by Kronheimer and Mrowka in 1994, states that complex curves in $\mathbb{C}{\rm P}^2$ are genus minimizers in their homology class. We will show that an analogous statement does not hold for complex hypersurfaces in $\mathbb{C}{\rm P}^3$. This is joint work with Ruberman and Strle.
Title: Geometry & Topology Seminar: The virtual intersection theory of isotropic Quot schemes
Speaker: Shubham Sinha (University of California, San Diego, USA)
Date: Tue, 10 Jan 2023
Time: 4:00 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
The intersection theory of the Grassmannian, known as Schubert calculus, is an important development in geometry, representation theory and combinatorics. The Quot scheme is a natural generalization of the Grassmannian. In particular, it provides a compactification of the space of morphisms from a smooth projective curve C to the Grassmannian. The intersection theory of the Quot scheme can be used to recover Vafa-Intriligator formulas, which calculate explicit expressions for the count of maps to the Grassmannian subject to incidence conditions with Schubert subvarieties.
The symplectic (or orthogonal) Grassmannian parameterizes isotropic subspaces of a vector space endowed with symplectic (or symmetric) bilinear form. I will present explicit formulas for certain intersection numbers of the symplectic and the orthogonal analogue of Quot schemes. Furthermore, I will compare these intersection numbers with the Gromov–Ruan–Witten invariants of the corresponding Grassmannians.
Title: Number Theory Seminar: Uniform irreducibility of Galois action on the $\ell$-primary part of Abelian $3$-folds of Picard type
Speaker: Mladen Dimitrov (University of Lille, France)
Date: Tue, 10 Jan 2023
Time: 10.30 AM
Venue: LH-1
Half a century ago Manin proved a uniform version of Serre’s celebrated result on the openness of the Galois image in the automorphisms of the $\ell$-adic Tate module of any non-CM elliptic curve over a given number field. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension. Namely, we establish a uniform irreducibility of Galois acting on the $\ell$-primary part of principally polarized Abelian $3$-folds of Picard type without CM factors, under some technical condition which is void in the semi-stable case. A key part of the argument is representation theoretic and relies on known cases of the Gan-Gross-Prasad Conjectures.
Title: Number Theory Seminar: A two dimensional version of the delta method and applications to quadratic forms
Speaker: Pankaj Vishe (Durham University, UK)
Date: Thu, 22 Dec 2022
Time: 11.30 AM
Venue: LH-1
We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over $\mathbb{Q}$ defined over at least $10$ variables. This is a joint work with Simon Myerson (warwick) and Junxian Li (Bonn).
Title: Algebra & Combinatorics Seminar: Legendre Pairs: old and new results/conjectures and the road ahead
Speaker: Ilias S. Kotsireas (Wilfrid Laurier University, Waterloo, Canada)
Date: Fri, 16 Dec 2022
Time: 2:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
We shall discuss Legendre Pairs, an interesting combinatorial object related to the Hadamard conjecture. We shall demonstrate the exceptional versatility of Legendre Pairs, as they admit several different formulations via concepts from disparate areas of Mathematics and Computer Science. We shall mention old and new results and conjectures within the past 20+ years, as well as potential future avenues for investigation.
The video of this talk is available on the IISc Math Department channel.
Title: Algebra & Combinatorics Seminar: Complex dynamics: degenerations, and irreducibility problems
Speaker: Rohini Ramadas (Warwick Mathematics Institute, UK)
Date: Wed, 14 Dec 2022
Time: 2 pm
Venue: LH-1, Mathematics Department
$\mathrm{Per}_n $ is an affine algebraic curve, defined over $\mathbb Q$, parametrizing (up to change of coordinates) degree-2 self-morphisms of $\mathbb P^1$ with an $n$-periodic ramification point. The $n$-th Gleason polynomial $G_n$ is a polynomial in one variable with $\mathbb Z$-coefficients, whose vanishing locus parametrizes (up to change of coordinates) degree-2 self-morphisms of $\mathbb C$ with an $n$-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is $\mathrm{Per}_n$ connected? (2) Is $G_n$ irreducible over $\mathbb Q$?
We show that if $G_n$ is irreducible over $\mathbb Q$, then $\mathrm{Per}_n$ is irreducible over $\mathbb C$, and is therefore connected. In order to do this, we find a $\mathbb Q$-rational smooth point of a projective completion of $\mathrm{Per}_n$. This $\mathbb Q$-rational smooth point represents a special degeneration of degree-2 morphisms, and as such admits an interpretation in terms of tropical geometry.
(This talk will be pitched at a broad audience.)
Title: Algebra & Combinatorics Seminar: Cross-ratios and perfect matchings
Speaker: Rob Silversmith (Warwick Mathematics Institute, UK)
Date: Wed, 14 Dec 2022
Time: 10 am
Venue: LH-1, Mathematics Department
Given a bipartite graph $G$ (subject to a constraint), the “cross-ratio degree” of G is a non-negative integer invariant of $G$, defined via a simple counting problem in algebraic geometry. I will discuss some natural contexts in which cross-ratio degrees arise. I will then present a perhaps-surprising upper bound on cross-ratio degrees in terms of counting perfect matchings. Finally, time permitting, I may discuss the tropical side of the story.
Title: Colloquium: Schur Multipliers and Classification of finite dimensional nilpotent Lie superalgebras
Speaker: Saudamini Nayak (Kalinga Institute of Industrial Technology, Bhubaneswar)
Date: Mon, 12 Dec 2022
Time: 4 pm
Venue: Microsoft Teams (Online)
The theory of Lie superalgebras have many applications in various areas of Mathematics and Physics. Kac gives a comprehensive description of mathematical theory of Lie superalgebras, and establishes the classification of all finite dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero. In the last few years the theory of Lie superalgebras has evolved remarkably, obtaining many results in representation theory and classification. Most of the results are extension of well known facts of Lie algebras. But the classification of all finite dimensional nilpotent Lie superalgebras is still an open problem like that of finite dimensional nilpotent Lie algebras. Till today nilpotent Lie superalgebras $L$ of $\dim L \leq 5$ over real and complex fields are known.
Batten introduced and studied Schur multiplier and cover of Lie algebras and later on studied by several authors. We have extended these notation to Lie superalgebra case. Given a free presentation $ 0 \longrightarrow R \longrightarrow F \longrightarrow L \longrightarrow 0 $ of Lie superalgebra $L$ we define the multiplier of $L$ as $\mathcal{M}(L) = \frac{[F,F]\cap R}{[F, R]}$. In this talk we prove that for nilpotent Lie superalgebra $L = L_{\bar{0}} \oplus L_{\bar{1}}$ of dimension $(m\mid n)$ and $\dim L^2= (r\mid s)$ with $r+s \geq 1$, \begin{equation} \dim \mathcal{M}(L)\leq \frac{1}{2}\left[(m + n + r + s - 2)(m + n - r -s -1) \right] + n + 1. \end{equation} Moreover, if $r+s = 1$, then the equality holds if and only if $ L \cong H(1, 0) \oplus A(m-3 \mid n)$ where $A(m-3 \mid n)$ is an abelian Lie superalgebra of dimension $(m-3 \mid n)$, and $H(1, 0)$ is special Heisenberg Lie superalgebra of dimension $(3 \mid 0)$. Then we define the function $s(L)$ as \begin{equation} s(L)= \frac{1}{2}(m+n-2)(m+n-1)+n+1-\dim \mathcal{M}(L). \end{equation} Clearly $s(L) \geq 0$ and structure of $L$ with $s(L)=0$ is known. We obtain classification all finite dimensional nilpotent Lie superalgebras with $s(L) \leq 2$.
We hope, this leads to a complete classification of the finite dimensional nilpotent Lie superalgebras of dimension $6,7$.
Title: Colloquium: Powers of ideals in combinatorics
Speaker: S. Selvaraja (Chennai Mathematical Institute)
Date: Wed, 07 Dec 2022
Time: 4 pm
Venue: Microsoft Teams (Online)
In this talk, I will discuss the relation of square-free monomial ideals to combinatorics. In particular, I will explain some combinatorial invariants of hypergraphs that can be used to describe the Castelnuovo–Mumford regularity and componentwise linearity of different kinds of powers of squarefree monomial ideals.
Title: A higher dimensional analog of Margulis’ construction of expanders
Speaker: Arghya Mondal (Chennai Mathematical Institute)
Date: Fri, 02 Dec 2022
Time: 11:00 am
Venue: LH-1
Expanders are a family of finite graphs that are sparse but highly connected. The first explicit examples of expanders were quotients of a Cayley graph of a discrete group with Property (T) by finite index subgroups. This was due to Margulis. In recent years, higher dimensional generalizations of expander graphs (family of simplicial complexes of a fixed dimension) have received much attention. I will talk about a generalization of Margulis’ group theoretic construction that replaces expanders by one of its higher analogs.
Title: Number Theory Seminar: Relating the conjectures of Artin-Tate and BSD
Speaker: Niranjan Ramachandran (University of Maryland, College Park, USA)
Date: Thu, 01 Dec 2022
Time: 11 AM
Venue: LH-1
I will report on recent work with Lichtenbaum and Suzuki on a new proof of the relation between the arithmetic of an elliptic curve over function fields and surfaces over finite fields.
Title: APRG Seminar: Bi-spatial random attractors and ergodicity for stochastic Navier–Stokes equations on the whole space
Speaker: Manil T. Mohan (IIT, Roorkee)
Date: Thu, 01 Dec 2022
Time: 3:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
We discuss the random dynamics and asymptotic analysis of 2D Navier–Stokes equations. We consider two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a linear multiplicative white noise of Ito type on the whole space. We prove that non-autonomous 2D SNSE generates a bi-spatial continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $L^2(\mathbb{R}^2)$, then there exists a unique bi-spatial $(L^2(\mathbb{R}^2), \mathbb{H}^1(\mathbb{R}^2))$-pullback random attractor for non-autonomous SNSE which is compact and attracting not only in $L^2$-norm but also in $\mathbb{H}^1$-norm. Next, we discuss the existence of an invariant measure for the random cocycle associated with autonomous SNSE which is a consequence of the existence of random attractors. We prove the uniqueness of invariant measures by using the linear multiplicative structure of the noise coefficient and exponential stability of solutions.
Title: Geometry & Topology Seminar: Bergman-Szegő kernel asymptotics in weakly pseudoconvex finite type cases
Speaker: Nikhil Savale (Universität zu Köln, Germany)
Date: Wed, 30 Nov 2022
Time: 4:00 pm
Venue: Microsoft Teams (online) (Joint with the APRG Seminar)
We construct a pointwise Boutet de Monvel-Sjostrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman’s boundary asymptotics of the Bergman kernel to weakly pseudo-convex domains in dimension two. Next we present an application where we prove that a weakly pseudoconvex two dimensional domain of finite type with a Kähler-Einstein Bergman metric is biholomorphic to the unit ball. This extends earlier work of Fu-Wong and Nemirovski-Shafikov. Based on joint works with C.Y. Hsiao and M. Xiao.
Title: Colloquium: Higher order accurate numerical schemes for hyperbolic conservation laws
Speaker: Rakesh Kumar (IISER, Thiruvananthapuram)
Date: Mon, 28 Nov 2022
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
The system of hyperbolic conservation laws is the first order partial differential equations of the form \begin{equation} \frac{\partial \mathbf{u}}{\partial t}+\sum_{\alpha=1}^d \frac{\partial \mathbf{f}_{\alpha}(\mathbf{u})}{\partial x_{\alpha}} =0,~~~~~~ (\mathbf{x},t)\in \Omega \times (0,T], \qquad \qquad \qquad (1) \end{equation} subject to initial data \begin{equation} \mathbf{u}(\mathbf{x},0)=\mathbf{u}_0(\mathbf{x}), \end{equation} where $\mathbf{u}=(u_1,u_2,\ldots, u_m)\in \mathbb{R}^m$ are the conserved variables and $\mathbf{f}_{\alpha}:\mathbb{R}^m \rightarrow \mathbb{R}^m$, $\alpha=1,2,\ldots,d$ are the Cartesian components of flux. It is well-known that the classical solution of (1) may cease to exist in finite time, even when the initial data is sufficiently smooth. The appearance of shocks, contact discontinuities and rarefaction waves in the solution profile make difficult to devise higher-order accurate numerical schemes because numerical schemes may develop spurious oscillations or sometimes blow up of the solution may occur.
In this talk, we will discuss recently developed Weighted Essentially Non-oscillatory (WENO) and hybrid schemes for hyperbolic conservation laws. These schemes compute the solution accurately while maintaining the high resolution near the discontinuities in a non-oscillatory manner.
Title: MS Thesis defence: Attaching Galois Representations to Modular Forms of weight 2
Speaker: Mansimar Singh (IISc Mathematics)
Date: Mon, 28 Nov 2022
Time: 9 am
Venue: Microsoft Teams (online)
The aim of this talk is to understand $\ell$-adic Galois representations and associate them to normalized Hecke eigenforms of weight $2$. We will also associate these representations to elliptic curves over $\mathbb{Q}$. This will enable us to state the Modularity Theorem. We will also mention its special case which was proved by Andrew Wiles and led to the proof of Fermat’s Last Theorem.
We will develop most of the central objects involved - modular forms, modular curves, elliptic curves, and Hecke operators, in the talk. We will directly use results from algebraic number theory and algebraic geometry.
Title: Number Theory Seminar: Serre weights of certain mod $p$ Hilbert modular forms
Speaker: Abhik Ganguli (IISER Mohali)
Date: Thu, 24 Nov 2022
Time: 11.30 AM
Venue: LH-5
Let $F$ be a totally real field and $p$ be an odd prime unramified in $F$. We will give an overview of the problem of determining the explicit mod $p$ structure of a modular $p$-adic Galois representation and determining the associated local Serre weights. The Galois representations are attached to Hilbert modular forms over $F$, more precisely to eigenforms on a Shimura curve over $F$. The weight part of the Serre’s modularity conjecture for Hilbert modular forms relates the local Serre weights at a place $v|p$ to the structure of the mod $p$ Galois representation at the inertia group over $v$. Thus, local Serre weights give good information on the structure of the modular mod $p$ Galois representation. The eigenforms considered are of small slope at a fixed place $\mathbf{p}|p$, and with certain constraints on the weight over $\mathbf{p}$. This is based on a joint work with Shalini Bhattacharya.
Title: Geometry & Topology Seminar: The Optimal Symplectic Connection equation and stability of fibrations
Speaker: Lars Martin Sektnan (Chalmers)
Date: Wed, 23 Nov 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
Abstract: A fundamental problem in complex geometry is to construct canonical metrics, such as Hermite-Einstein (HE) metrics on vector bundles and constant scalar curvature Kähler (cscK) metrics on Kahler manifolds. On a given vector bundle/manifold, such a metric may or may not exist, in general. The existence question for such metrics has been found to have deep connections to algebraic geometry. In the case of vector bundles, the Hitchin-Kobayashi correspondence proved by Uhlenbeck–Yau and Donaldson show that the existence of a HE metric is captured by the notion of slope stability for the vector bundle. In the case of manifolds, the still open Yau-Tian-Donaldson conjecture relates the existence of cscK metrics to K-stability of the underlying polarised variety.
Together with Ruadhaí Dervan, I started a research programme where we study canonical metrics, called Optimal Symplectic Connections, and a notion of stability, on fibrations. We proposed a Hitchin-Kobayashi/Yau-Tian-Donaldson type conjecture in this setting as well. In the case when the fibration is the projectivisation of a vector bundle, we recover the Hermite-Einstein and slope stability notions, respectively, and as such the theory can be seen as a generalisation of the classical bundle theory to more general fibrations. There has recently been great progress on this topic both on the differential and algebraic side, through works of Hallam, McCarthy, Ortu, Hattori, Spotti and Engberg, in addition to the joint works with Dervan. The aim of this talk is to give an introduction to and overview of the status of this programme.
Title: Colloquium: Complex adjacency spectra of (multi)digraphs
Speaker: Gopinath Sahoo (Bennett University, Noida)
Date: Fri, 18 Nov 2022
Time: 4 pm
Venue: Microsoft Teams (Online)
Given any graph, we can uniquely associate a square matrix which stores informations about its vertices and how they are interconnected. The goal of spectral graph theory is to see how the eigenvalues and eigenvectors of such a matrix representation of a graph are related to the graph structure. We consider here (multi)digraphs and define a new matrix representation for a multidigraph and named it as the complex adjacency matrix.
The relationship between the adjacency matrix and the complex adjacency matrix of a multidigraph are established. Furthermore, some of the advantages of the complex adjacency matrix over the adjacency matrix of a multidigraph are observed. Besides, some of the interesting spectral properties (with respect to the complex adjacency spectra) of a multidigraph are established. It is shown that not only the eigenvalues, but also the eigenvectors corresponding to the complex adjacency matrix of a multidigraph carry a lot of information about the structure of the multidigraph.
Title: Geometry & Topology Seminar: Reciprocal geodesics and dihedral subgroups of lattices in PSL(2, R)
Speaker: Viveka Erlandsson (U Bristol)
Date: Wed, 16 Nov 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
Abstract: I will discuss the growth of the number of infinite dihedral subgroups of lattices G in PSL(2, R). Such subgroups exist whenever the lattice has 2-torsion and they are related to so-called reciprocal geodesics on the corresponding quotient orbifold. These are closed geodesics passing through an even order orbifold point, or equivalently, homotopy classes of closed curves having a representative in the fundamental group that’s conjugate to its own inverse. We obtain the asymptotic growth of the number of reciprocal geodesics (or infinite dihedral subgroups) in any orbifold, generalizing earlier work of Sarnak and Bourgain-Kontorivich on the growth of the number of reciprocal geodesics on the modular surface. Time allowing, I will explain how our methods also show that reciprocal geodesics are equidistributed in the unit tangent bundle. This is joint work with Juan Souto.
Title: Colloquium: Nonlocal Calderón Problem
Speaker: Tuhin Ghosh (Universität Bielefeld, Germany)
Date: Wed, 16 Nov 2022
Time: 4 pm
Venue: Microsoft Teams (Online)
In this talk, we will discuss the Calderón type inverse problem of determining the coefficients of the nonlocal operators. In the mathematical literature, the method of Electrical Impedance Tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium is known as Calderón’s problem. We will introduce the nonlocal analog of it and further study the connection with the local analog as well.
Title: APRG Seminar: Higher-Order Graphon Theory: Fluctuations and Degeneracies
Speaker: Bhaswar Bhattacharya (University of Pennsylvania / Wharton School, Philadelphia, USA)
Date: Wed, 16 Nov 2022
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network and are the central objects in graph limit (graphon) theory. In this talk we will derive the higher-order fluctuations (asymptotic distributions) of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can have both Gaussian or non-Gaussian components, depending on a notion of regularity of subgraphs, where the non-Gaussian component is an infinite weighted sum of centered chi-squared random variables with the weights determined by the spectral properties of the graphon. We will also discuss various structure theorems and open questions about degeneracies of the limiting distribution and connections to quasirandom graphs.
(Joint work with Anirban Chatterjee and Svante Janson.)
Title: Number Theory Seminar: Integrality of smoothed $p$-adic Artin $L$-functions
Speaker: Bence Forras (University of Duisburg-Essen, Germany)
Date: Wed, 16 Nov 2022
Time: 11.30 AM
Venue: LH-1
We introduce a smoothed version of the equivariant $S$-truncated
$p$-adic Artin $L$-function for one-dimensional admissible $p$-adic Lie
extensions of number fields. Integrality of this smoothed $p$-adic
$L$-function, conjectured by Greenberg, has been verified for pro-$p$
extensions (assuming the Equivariant Iwasawa Main Conjecture) as well as
$p$-abelian extensions (unconditionally). Integrality in the general case
is also expected to hold, and is the subject of ongoing research.
Title: Colloquium: GTS Poset and Laplacian Immanants of Trees
Speaker: Mukesh Kumar Nagar (Punjab Engineering College, Chandigarh)
Date: Mon, 14 Nov 2022
Time: 4 pm
Venue: Microsoft Teams (Online)
A poset denoted $\mathsf{GTS}_n$ on the set of unlabeled trees with $n$ vertices was defined by Csikvàri. He showed that several tree parameters are monotonic as one goes up this $\mathsf{GTS}_n$ poset. Let $T$ be a tree on $n$ vertices and let $\mathcal{L}_q^T$ be the $q$-analogue of its Laplacian. For all $q\in \mathbb{R}$, I will discuss monotonicity of the largest and the smallest eigenvalues of $\mathcal{L}_q^T$ along the $\mathsf{GTS}_n$ poset.
For a partition $\lambda \vdash n$, let the normalized immanant of $\mathcal{L}_q^T$ indexed by $\lambda$ be denoted as $\overline{\mathrm{Imm}}_{\lambda}(\mathcal{L}_q^T)$. Monotonicity of $\overline{\mathrm{Imm}}_{\lambda}(\mathcal{L}_q^T)$ will be discussed when we go up along $\mathsf{GTS}_n$ or when we change the size of the first row in the hook partition $(\lambda=k,1^{n-k})$ and the two row partition $\lambda=(n-k,k)$. We will also discuss monotocity of each coefficients in the $q$-Laplacian immanantal polynomials $\overline{\mathrm{Imm}}_{\lambda}(xI-\mathcal{L}_q^T)$ when we go up along $\mathsf{GTS}_n$. At the end of this talk, I will discuss our ongoing research projects and future plans.
This is a joint work with Prof. A. K. Lal (IITK) and Prof. S. Sivaramakrishnan (IITB).
Title: APRG Seminar: Sequence of Toeplitz algebras converge to odd spheres for the quantum Gromov-Hausdorff distance
Speaker: Sushil Singla (IISc Mathematics)
Date: Fri, 11 Nov 2022
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
Marc Rieffel had introduced the notion of quantum Gromov-Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions of two sphere in this distance, that one finds in many scattered places in the theoretical physics literature. The compact quantum metric spaces and convergence in the quantum Gromov-Hausdorff distance has been explored by a lot of mathematicians in the last two decades. We will define compact quantum metric space structure on the sequence of Toeplitz algebras on generalized Bergman space and prove that it converges to the space of continuous function on odd spheres in the quantum Gromov-Hausdorff distance. This is a joint work with Prof. Tirthankar Bhattacharyya.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Bers’ simultaneous uniformization theorem and the intersection of Poincaré holonomy varieties
Speaker: Shinpei Baba (Osaka University)
Date: Wed, 09 Nov 2022
Time: 4:00 pm
Venue: LH-1 (In person)
The Poincaré holonomy variety (or $sl(2, C)$-oper) is the set of holonomy representations of all complex projective structures on a Riemann surface. It is a complex analytic subvariety of the $PSL(2, C)$ character variety of the underlying topological surface. In this talk, we consider the intersection of such subvarieties for different Riemann surface structures, and we prove the discreteness of such an intersection. As a corollary, we reprove Bers’ simultaneous uniformization theorem, without any quasiconformal deformation theory.
Title: Number Theory Seminar: Counting matrices with Diophantine restrictions on the entries
Speaker: Satadal Ganguly (ISI Kolkata)
Date: Wed, 09 Nov 2022
Time: 11.30 AM
Venue: LH-1
It is a natural question to count matrices $A$ with integer entries in an expanding box of side length $x$ with $\det(A) = r$, a fixed integer; or with the characteristic polynomial of $A = f$, a fixed integer polynomial; and there are several results in the literature on these problems. Most of the existing results, which use either Ergodic methods or Harmonic Analysis, give asymptotics for the number of such matrices as $x$ goes to infinity and in the only result we have been able to find that gives a bound on the error term, the bound is not very satisfactory. The aim of this talk will be to present an ongoing joint work with Rachita Guria in which, for the easiest case of $2 \times 2$ matrices, we have been able to obtain reasonable bounds for the error terms for the above problems by employing elementary Fourier Analysis and results from the theory of Automorphic Forms.
Title: Colloquium: On inhomogeneous analogue of Thue-Roth's type inequality
Speaker: Veekesh Kumar (NISER, Bhubaneswar)
Date: Fri, 04 Nov 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
For a real number $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. The study of the sequence $\|\alpha^n\|$ for $\alpha > 1$ naturally arises in various contexts in number theory. For example, it is not known that the sequence $\|e^n\|$ tends to zero as $n$ tends to infinity. Also, the growth of the sequence $\|(3/2)^n\|$ is linked to the famous Waring’s problem. This was the motivation for Mahler in 1957 to prove that for any non-integral rational number $\alpha > 1$ and any real number $c$ with $0 < c < 1$, the inequality $\|\alpha^n\| < c^n$ has only finitely many solutions in $n\in\mathbb{N}$. Mahler also asked the characterization of all algebraic numbers satisfying the same property as the non-integral rational numbers. In 2004, Corvaja and Zannier proved a Thue-Roth-type inequality with moving targets and as consequence, they completely answered the above question of Mahler. In this talk, we will explore this theme and will present recent result, building on the earlier works of Corvaja and Zannier, establishing an inhomogeneous Thue-Roth’s type theorem with moving targets.
Title: Algebra & Combinatorics Seminar: A $q$-analog of the adjacency matrix of the $n$-cube
Speaker: Subhajit Ghosh (Bar-Ilan University, Ramat-Gan, Israel)
Date: Fri, 04 Nov 2022
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Let $q$ be a prime power and define $(n)_q:=1+q+q^2+\cdots+q^{n-1}$, for a non-negative integer $n$. Let $B_q(n)$ denote the set of all subspaces of $\mathbb{F}_q^n$, the $n$-dimensional $\mathbb{F}_q$-vector space of all column vectors with $n$ components.
Define a $B_q(n)\times B_q(n)$ complex matrix $M_{q,n}$ with entries given by
\begin{equation}
M_{q,n}(X,Y):=
\begin{cases}
1&\text{ if }Y\subseteq X, \dim(Y)=\dim(X)-1,\\
q^{\dim(X)}&\text{ if }X\subseteq Y, \dim(Y)=\dim(X)+1,\\
0&\text{ otherwise.}
\end{cases}
\end{equation}
We think of $M_{q,n}$ as a $q$-analog of the adjacency matrix of the
$n$-cube. We show that the eigenvalues of $M_{q,n}$ are
\begin{equation}
(n-k)_q - (k)_q\text{ with multiplicity }\binom{n}{k}_q,\quad k=0,1,\dots,n,
\end{equation}
and we write down an explicit canonical eigenbasis of $M_{q,n}$. We give
a weighted count of the number of rooted spanning trees in the $q$-analog
of the $n$-cube.
This talk is based on a joint work with M. K. Srinivasan.
Title: APRG Seminar: Stability of the Pohozaev obstruction and Non-existence
Speaker: Saikat Mazumdar (IIT, Bombay)
Date: Wed, 02 Nov 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we will consider the issues of non-existence of solutions to a Yamabe type equation on bounded Euclidean domains (dim>2). The leading order terms of this equation are invariant under conformal transformations which leads to the classical Pohozaev identity. This in turn gives non-existence of solutions to the PDE when the domain is star-shaped with respect to the origin.
We show that this non-existence is surprisingly stable under perturbations, which includes situations not covered by the Pohozaev obstruction, if the boundary of the domain has a positive curvature. In particular, we show that there are no positive variational solutions to our PDE under $C^1$-perturbations of the potential when the domain is star-shaped with respect to the origin and the mean curvature of the boundary at the origin is positive. The proof of our result relies on sharp blow-up analysis. This is a joint work with Nassif Ghoussoub (UBC, Vancouver) and Frédéric Robert (Institut Élie Cartan, Nancy).
Title: Number Theory Seminar: Short second moment and subconvexity for $\mathrm{GL}(3)$ $L$-functions
Speaker: Keshav Aggarwal (Alfred Renyi Institute of Mathematics, Budapest, Hungary)
Date: Wed, 02 Nov 2022
Time: 5 pm
Venue: Microsoft Teams (Online)
We bound a short second moment average of $\mathrm{GL}(3)$ and $\mathrm{GL}(3) \times \mathrm{GL}(1)$ $L$-functions. These yield $t$-aspect and depth aspect subconvexity bounds respectively, and improve upon the earlier subconvexity exponents. This moment estimate provides an analogue for cusp forms of Ivic’s bound for the sixth moment of the zeta function, and is the first time a short second moment has been used to obtain a subconvex bound in higher rank. This is a joint work with Ritabrata Munshi and Wing Hong Leung.
Title: Geometry & Topology Seminar: Long curves on hyperbolic surfaces and the geometry of their complements
Speaker: Aaron Calderon (U Chicago)
Date: Wed, 02 Nov 2022
Time: 6:30 pm
Venue: Microsoft Teams (online)
In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections as well as a qualitative strengthening of her theorem that describes what these curves (and their complements) actually look like. This is joint work with Francisco Arana-Herrera.
Title: MS Thesis defence: Local Langlands correspondence for GL(1) and GL(2)
Speaker: Thummala Vamsi Krishna (IISc Mathematics)
Date: Fri, 28 Oct 2022
Time: 10 am
Venue: Microsoft Teams (online)
In the first part of the talk we will discuss the main statement of local class field theory and discuss the statement of Local Langlands correspondence for $GL_2(K)$, where $K$ is a non-archimedean local field. In the process, we will also introduce all the objects in the statement of correspondence. We will then discuss a brief sketch of the proof of the main statement of local class field theory.
Title: Geometry & Topology Seminar: Geometric inequalities in spaces of nonpositive curvature
Speaker: Mohammad Ghomi (Georgia Tech)
Date: Wed, 26 Oct 2022
Time: 6:30 pm
Venue: Microsoft Teams (online)
We will discuss total mean curvatures, i.e., integrals of symmetric functions of the principle curvatures, of hypersurfaces in Riemannian manifolds. These quantities are fundamental in geometric variational problems as they appear in Steiner’s formula, Brunn-Minkowski theory, and Alexandrov-Fenchel inequalities. We will describe a number of new inequalities for these integrals in non positively curved spaces, which are obtained via Reilly’s identities, Chern’s formulas, and harmonic mean curvature flow. As applications we obtain several new isoperimetric inequalities, and Riemannian rigidity theorems. This is joint work with Joel Spruck.
Title: PhD Thesis colloquium: Local Projection Stabilization Methods for the Oseen Problem
Speaker: Rahul Biswas (IISc Mathematics)
Date: Fri, 21 Oct 2022
Time: 12 pm
Venue: LH-1, Mathematics Department
Convection dominated fluid flow problems show spurious oscillations when solved using the usual Galerkin finite element method (FEM). To suppress these un-physical solutions we use various stabilization methods. In this thesis, we discuss the Local Projection Stabilization (LPS) methods for the Oseen problem.
This thesis mainly focuses on three different finite element methods each serving a purpose of its own. First, we discuss the a priori analysis of the Oseen problem using the Crouzeix-Raviart (CR1) FEM. The CR1/P0 pair is a well-known choice for solving mixed problems like the Oseen equations since it satisfies the discrete inf-sup condition. Moreover, the CR1 elements are easy to implement and offer a smaller stencil compared with conforming linear elements (in the LPS setting). We also discuss the CR1/CR1 pair for the Oseen problem to achieve a higher order of convergence.
Second, we discuss the a posteriori analysis for the Oseen problem using the CR1/P0 pair using a dual norm approach. We define an error estimator and prove that it is reliable and discuss an efficiency estimate that depends on the diffusion coefficient.
Next, we focus on formulating an LPS scheme that can provide globally divergence free velocity. To achieve this, we use the $H(div;\Omega)$ conforming Raviart-Thomas (${\rm RT}^k$) space of order $k \geq 1$. We show a strong stability result under the SUPG norm by enriching the ${\rm RT}^k$ space using tangential bubbles. We also discuss the a priori error analysis for this method.
Finally, we develop a hybrid high order (HHO) method for the Oseen problem under a generalized local projection setting. These methods are known to allow general polygonal meshes. We show that the method is stable under a “SUPG-like” norm and prove a priori error estimates for the same.
Title: Geometry & Topology Seminar: J-equation on holomorphic vector bundles
Speaker: Ryosuke Takahashi (Kyushu University)
Date: Wed, 19 Oct 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
In general, the equivalence of the stability and the solvability of an equation is an important problem in geometry. In this talk, we introduce the J-equation on holomorphic vector bundles over compact Kahler manifolds, as an extension of the line bundle case and the Hermitian-Einstein equation over Riemann surfaces. We investigate some fundamental properties as well as examples. In particular, we give algebraic obstructions called the (asymptotic) J-stability in terms of subbundles on compact Kahler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime.
Title: Number Theory Seminar: Dimensions of admissible representations of reductive $p$-adic groups
Speaker: Marie-France Vigneras (Institut de Mathematiques de Jussieu, Paris, France)
Date: Wed, 19 Oct 2022
Time: 5 pm
Venue: Zoom (Online)
I will answer some questions (admissibility, dimensions of invariants by Moy-Prasad groups)
on representations of reductive $p$-adic groups and on Hecke algebras modules raised in my paper for the 2022-I.C.M. Noether lecture.
Title: APRG Seminar: Mean value property in limit for eigenfunctions of the Laplace–Beltrami operator on symmetric spaces: II
Speaker: Muna Naik (Harish-Chandra Research Institute, Prayagraj)
Date: Wed, 12 Oct 2022
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
This talk will be a continuation of my previous talk. In this talk, I will present the proof of a result stated in my earlier talk, which characterizes eigenfunctions of the Laplace–Beltrami operator through sphere averages as the radius of the sphere tends to infinity in a rank one symmetric space of noncompact type.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: The distinction problem for symmetric spaces
Speaker: Dipendra Prasad (IIT Bombay)
Date: Wed, 12 Oct 2022
Time: 2 pm
Venue: LH 1
If $\theta$ is an involution on a group $G$ with fixed points $H$,
it is a question of considerable interest to classify irreducible representations of $G$
which carry an $H$-invariant linear form. We will discuss some cases of this
question paying attention to finite dimensional representation of compact groups
where it is called the Cartan-Helgason theorem.
Title: MS Thesis Defence: Asymmetric Super-Heston-rough volatility model with Zumbach effect as a scaling limit of quadratic Hawkes processes
Speaker: Priyanka Chudasama (IISc Mathematics)
Date: Thu, 06 Oct 2022
Time: 11 am
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
Modelling price variation has always been of interest, from options pricing to risk management. It has been observed that the high-frequency financial market is highly volatile, and the volatility is rough. Moreover, we have the Zumbach effect, which means that past trends in the price process convey important information on future volatility. Microscopic price models based on the univariate quadratic Hawkes process can capture the Zumbach effect and the rough volatility behaviour at the macroscopic scale. But they fail to capture the asymmetry in the upward and downward movement of the price process. Thus, to incorporate asymmetry in price movement at micro-scale and rough volatility and the Zumbach effect at macro-scale, we introduce the bivariate Modified-quadratic Hawkes process for upward and downward price movement. After suitable scaling and shifting, we show that the limit of the price process in the Skorokhod topology behaves as so-called Super-Heston-rough model with the Zumbach effect.
Title: Geometry & Topology Seminar: Classification of two-dimensional shrinking gradient Kähler-Ricci solitons
Speaker: Charles Cifarelli (Université de Nantes)
Date: Wed, 28 Sep 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
We will present some recent work on the classification of shrinking gradient Kähler-Ricci solitons on complex surfaces. In particular, we classify all non-compact examples, which together with previous work of Tian, Wang, Zhu, and others in the compact case gives the complete classification. This is joint work with R. Bamler, R. Conlon, and A. Deruelle.
Title: Number Theory Seminar: Limitations to equidistribution in arithmetic progressions
Speaker: Akshaa Vatwani (IIT Gandhinagar)
Date: Wed, 28 Sep 2022
Time: 11.30 AM
Venue: LH 1
It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri-Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions “on average” for moduli $q$ in the range $q \le x^{1/2 -\epsilon }$ for any $\epsilon>0$. In 1989, building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli $q$ is extended to $q \le x/ (\log x)^B$ for any $B>1$. We discuss variants of this result and give some applications. This is joint work with Aditi Savalia.
Title: PhD Thesis defence: Weights of highest weight modules over Kac–Moody algebras
Speaker: G V Krishna Teja (IISc Mathematics)
Date: Thu, 22 Sep 2022
Time: 2 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
This thesis explores highest weight modules $V$ over complex semisimple and Kac-Moody algebras. The first part of the talk addresses (non-integrable) simple highest weight modules $V = L(\lambda)$. We provide a “minimum” description of the set of weights of $L(\lambda)$, as well as a “weak Minkowski decomposition” of the set of weights of general $V$. Both of these follow from a “parabolic” generalization of the partial sum property in root systems: every positive root is an ordered sum of simple roots, such that each partial sum is also a root.
Second, we provide a positive, cancellation-free formula for the weights of arbitrary highest weight modules $V$. This relies on the notion of “higher order holes” and “higher order Verma modules”, which will be introduced and discussed in the talk.
Third, we provide BGG resolutions and Weyl-type character formulas for the higher order Verma modules in certain cases - these involve a parabolic Weyl semigroup. Time permitting, we will discuss about weak faces of the set of weights, and their complete classification for arbitrary $V$.
Title: Geometry & Topology Seminar: Existence of twisted Kahler-Einstein metrics in big classes
Speaker: Tamas Darvas (University of Maryland)
Date: Wed, 21 Sep 2022
Time: 6:30 pm
Venue: Microsoft Teams (online)
We prove existence of twisted Kähler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when -K_X is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory. This is joint work with Kewei Zhang.
Title: Number Theory Seminar: Sturm-type bound for square-free Fourier coefficients of Hilbert modular forms
Speaker: Rishabh Agnihotri (IISc)
Date: Wed, 21 Sep 2022
Time: 11.30 AM
Venue: LH-1
Hilbert modular forms are generalization of classical modular forms over totally real number fields. The Fourier coefficients of a modular form are of great importance owing to their rich arithmetic and algebraic properties. In the theory of modular forms one of the classical problems is to determine a modular form by a subset of all Fourier coefficient. In this talk, we discuss about to determination of a Hilbert modular form by the Fourier coefficients indexed by square-free integral ideals. In particular, we talk about the following result.
Given any $\epsilon>0$, a non zero Hilbert cusp form $\mathbf{f}$ of weight $k=(k_1,k_2,\ldots, k_n)\in (\mathbb{Z}^{+})^n$ and square-free level $\mathfrak{n}$ with Fourier coefficients
$C(\mathbf{f},\mathfrak{m})$, then there exists a square-free integral ideal $\mathfrak{m}$ with $N(\mathfrak{m})\ll k_0^{3n+\epsilon} N(\mathfrak{m})^{\frac{6n^2 +1}{2}+\epsilon}$ such that $C(\mathbf{f},\mathfrak{m})\neq 0$. The implied constant depend on $\epsilon , F.$
Title: Number Theory Seminar: Lifting global mod $p$ Galois representations
Speaker: Najmuddin Fakhruddin (TIFR, Mumbai)
Date: Wed, 14 Sep 2022
Time: 12 PM
Venue: LH-1
Let $F$ be a global field and $\Gamma_F$ its absolute Galois group. Given
a continuous representation $\bar{\rho}: \Gamma_F \to G(k)$, where $G$ is a split
reductive group and $k$ is a finite field, it is of interest to know when $\bar{\rho}$ lifts
to a representation $\rho: \Gamma_F \to G(O)$, where $O$ is a complete discrete
valuation ring of characteristic zero with residue field $k$. One would also like to control
the local behaviour of $\rho$ at places of $F$, especially at primes dividing $p = \mathrm{char}(k)$
(if $F$ is a number field). In this talk I will give an overview of a method developed in joint work with
Chandrashekhar Khare and Stefan Patrikis which allows one to construct such lifts in many cases.
Title: APRG Seminar: Mean value property in limit for eigenfunctions of the Laplace–Beltrami operator on symmetric spaces
Speaker: Muna Naik (Harish-Chandra Research Institute, Prayagraj)
Date: Wed, 14 Sep 2022
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
In rank one symmetric space of noncompact type, we shall talk about the characterization of all eigenfunctions of the Laplace–Beltrami operator through sphere and ball averages as the radius of the sphere or ball tends to infinity.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Moduli space of abelian surfaces with fixed $3$-torsion representation
Speaker: Shiva Chidambaram (MIT, USA)
Date: Wed, 07 Sep 2022
Time: 11.30 AM
Venue: LH-3
The modularity lifting theorem of Boxer-Calegari-Gee-Pilloni established for the first time the existence of infinitely many modular abelian surfaces $A / \mathbb{Q}$ upto twist with $\text{End}_{\mathbb{C}}(A) = \mathbb{Z}$. We render this explicit by first finding some abelian surfaces whose associated mod-$p$ representation is residually modular and for which the modularity lifting theorem is applicable, and then transferring modularity in a family of abelian surfaces with fixed $3$-torsion representation. Let $\rho: G_{\mathbb{Q}} \rightarrow GSp(4,\mathbb{F}_3)$ be a Galois representation with cyclotomic similitude character. Then, the transfer of modularity happens in the moduli space of genus $2$ curves $C$ such that $C$ has a rational Weierstrass point and $\mathrm{Jac}(C)[3] \simeq \rho$. Using invariant theory, we find explicit parametrization of the universal curve over this space. The talk will feature demos of relevant code in Magma.
Title: Geometry & Topology Seminar: Counterexamples to the Labourie conjecture
Speaker: Peter Smillie (Universität Heidelberg)
Date: Wed, 07 Sep 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
In 2006, Labourie defined a map from a bundle over Teichmuller space to the Hitchin component of the representation variety $Rep(\pi_1(S),PSL(n,R))$, and conjectured that it is a homeomorphism for every $n$ (it was known for $n =2,3$). I will describe some of the background to the Labourie conjecture, and then show that it does not hold for any $n >3$. Having shown that Labourie’s map is more interesting than a mere homeomorphism, I will describe some new questions and conjectures about how it might look.
Title: Algebra & Combinatorics Seminar: Zeta functions on graphs
Speaker: Junaid Hasan (University of Washington, Seattle, USA)
Date: Fri, 02 Sep 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
This talk is based on the work of Stark and Terras (Zeta functions of Finite graphs and Coverings I, II, III). In this talk we start with an introduction to zeta functions in various branches of mathematics. Our focus is mainly on zeta functions on finite undirected connected graphs. We obtain an analogue of the prime number theorem, but for graphs, using the Ihara Zeta Function. We also introduce edge and path zeta functions and show interesting results.
Title: Geometry & Topology Seminar: Positive sectional curvature and Ricci flow
Speaker: Anusha Krishnan (University of Münster, Germany)
Date: Tue, 30 Aug 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
The preservation of positive curvature conditions under the Ricci flow has been an important ingredient in applications of the flow to solving problems in geometry and topology. Works by Hamilton and others established that certain positive curvature conditions are preserved under the flow, culminating in Wilking’s unified, Lie algebraic approach to proving invariance of positive curvature conditions. Yet, some questions remain. In this talk, we describe positive sectional curvature metrics on $\mathbb{S}^4$ and $\mathbb{C}P^2$, which evolve under the Ricci flow to metrics with sectional curvature of mixed sign. This is joint work with Renato Bettiol.
Title: Number Theory Seminar: Finite-group actions on reductive groups and buildings II: the unauthorized sequel
Speaker: Jeff Adler (American University, USA)
Date: Mon, 29 Aug 2022
Time: 12 PM
Venue: LH-1
Let $k$ be a nonarchimedian local field, $\widetilde{G}$ a connected reductive $k$-group, $\Gamma$ a finite group of automorphisms of $\widetilde{G}$, and $G:= (\widetilde{G}^\Gamma)^\circ$ the connected part
of the group of $\Gamma$-fixed points of $\widetilde{G}$.
The first half of my talk will concern motivation: a desire for a more explicit understanding of base change and other liftings of representations. Toward this end, we adapt some results of Kaletha-Prasad-Yu. Namely, if one assumes that the residual characteristic of $k$ does not divide the order of $\Gamma$, then they show, roughly speaking, that $G$ is reductive, the building $\mathcal{B}(G)$ of $G$ embeds in the set of $\Gamma$-fixed points of $\mathcal{B}(\widetilde{G})$, and similarly for reductive quotients of parahoric subgroups.
We prove similar statements, but under a different hypothesis on $\Gamma$. Our hypothesis does not imply that of K-P-Y, nor vice versa. I will include some comments on how to resolve such a totally unacceptable situation.
(This is joint work with Joshua Lansky and Loren Spice.)
Title: Algebra & Combinatorics Seminar: Closures, Coverings, and Complexity
Speaker: S. Venkitesh (IIT, Bombay)
Date: Fri, 26 Aug 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
The polynomial method is an ever-expanding set of algebraic techniques, which broadly entails capturing combinatorial objects by algebraic means, specifically using polynomials, and then employing algebraic tools to infer their combinatorial features. While several instances of the polynomial method have been part of the combinatorialist’s toolkit for decades, development of this method has received more traction in recent times, owing to several breakthroughs like (i) Dvir’s solution (2009) to the finite-field Kakeya problem, followed by an improvement by Dvir, Kopparty, Saraf, and Sudan (2013), (ii) Guth and Katz (2015) proving a conjecture by Erdös on the distinct distances problem, (iii) solutions to the capset problem by Croot, Lev, and Pach (2017), and Ellenberg and Gijswijt (2017), to name a few.
One of the ways to employ the polynomial method is via the classical algebraic objects – (affine) Zariski closure, (affine) Hilbert function, and Gröbner basis. Owing to their applicability in several areas like computational complexity, combinatorial geometry, and coding theory, an important line of enquiry is to understand these objects for ‘structured’ sets of points in the affine space. In this talk, we will be mainly concerned with Zariski closures of symmetric sets of points in the Boolean cube.
Firstly, we will look at a combinatorial characterization of Zariski closures of all symmetric sets, and its application to some hyperplane and polynomial covering problems for the Boolean cube, over any field of characteristic zero. We will also briefly look at Zariski closures over fields of positive characteristic, although much less is known in this setting. Secondly, we will see a simple illustration of a ‘closure statement’ being used as a technique for proving bounds on the complexity of approximating Boolean functions by polynomials. We will conclude with some open questions on Zariski closures motivated by problems on these two fronts.
Some parts of this talk will be based on the works: https://arxiv.org/abs/2107.10385, https://arxiv.org/abs/2111.05445, https://arxiv.org/abs/1910.02465.
Title: Algebra & Combinatorics Seminar: The rank spectral sequence on Quillen's Q construction
Speaker: Bruno Kahn (Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France)
Date: Thu, 25 Aug 2022
Time: 12 pm
Venue: LH-1, Mathematics Department
I will explain a generalisation of the constructions Quillen used to prove that the $K$-groups of rings of integers are finitely generated. It takes the form of a ‘rank’ spectral sequence, converging to the homology of Quillen’s $Q$-construction on the category of coherent sheaves over a Noetherian integral scheme, and whose $E^1$ terms are given by homology of Steinberg modules. Computing its $d^1$ differentials is a challenge, which can be approached through the universal modular symbols of Ash-Rudolph.
Title: Geometry & Topology Seminar: The Thomas-Yau conjecture
Speaker: Yang Li (MIT)
Date: Wed, 24 Aug 2022
Time: 6:30 pm
Venue: Microsoft Teams (online)
The Thomas-Yau conjecture is an open-ended program to relate special Lagrangians to stability conditions in Floer theory, but the precise notion of stability is subject to many interpretations. I will focus on the exact case (Stein Calabi-Yau manifolds), and deal only with almost calibrated Lagrangians. We will discuss how the existence of destabilising exact triangles obstructs special Lagrangians, under some additional assumptions, using the technique of integration over moduli spaces.
Title: APRG Seminar: The Koranyi Spherical Maximal Function on Heisenberg groups
Speaker: Rajula Srivastava (Univ. of Wisconsin at Madison, USA / Univ. Bonn + MPIM, Germany)
Date: Wed, 24 Aug 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we discuss the problem of obtaining sharp $L^p\to L^q$ estimates for the local maximal operator associated with averaging over dilates of the Koranyi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra “twist” due to the Heisenberg group law), the geometry of the Koranyi sphere (in particular, the flatness at the poles) and an “imbalanced” scaling argument encapsulated by a new type of Knapp example, which we shall describe in detail.
Title: Number Theory Seminar: p-adic variation of the Asai--Flach Euler system
Speaker: Arshay Sheth (University of Warwick, UK)
Date: Wed, 17 Aug 2022
Time: 11.30 AM
Venue: LH-1
Euler systems are cohomological tools that play a crucial role in the study of special values of $L$-functions; for instance, they have been used to prove cases of the Birch–Swinnerton-Dyer conjecture and have recently been used to prove cases of the more general Bloch–Kato conjecture. A fundamental technique in these recent advances is to show that Euler systems vary in $p$-adic families. In this talk, we will first give a general introduction to the theme of $p$-adic variation in number theory and introduce the necessary background from the theory of Euler systems; we will then explain the idea and importance of $p$-adically varying Euler systems, and finally discuss current work in progress on $p$-adically varying the Asai–Flach Euler system, which is an Euler system arising from quadratic Hilbert modular eigenforms.
Title: Number Theory Seminar: Abelian Varieties not isogenous to Jacobians
Speaker: Ananth Shankar (University of Wisconsin-Madison, USA)
Date: Wed, 10 Aug 2022
Time: 11.30 AM
Venue: LH-1
I will talk about recent work pertaining to the existence of abelian varieties not isogenous to Jacobians over fields of both characteristic zero and p. This is joint work with Jacob Tsimerman.
Title: APRG Seminar: Properties of cyclic functions
Speaker: Jeet Sampat (Washington University in St. Louis, USA)
Date: Mon, 08 Aug 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
For a natural number $n$ and $1 \leq p < \infty$, consider the Hardy space $H^p(D^n)$ on the unit polydisk. Beurling’s theorem characterizes all shift cyclic functions in $H^p(D^n)$ when $n = 1$. Such a theorem is not known to exist in most other analytic function spaces, even in the one variable case. Therefore, it becomes natural to ask what properties these functions satisfy to understand them better. The goal of this talk is to showcase some important properties of cyclic functions in two different settings.
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Fix $1 \leq p,q < \infty$ and natural numbers $m, n$. Let $T : H^p(D^n) \to H^q(D^m)$ be a bounded linear operator. Then $T$ preserves cyclic functions i.e., $Tf$ is cyclic whenever $f$ is, if and only if $T$ is a weighted composition operator.
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Let $H$ be a normalized complete Nevanlinna-Pick (NCNP) space, and let $f, g$ be functions in $H$ such that $fg$ also lies in $H$. Then, $f$ and $g$ are multiplier cyclic if and only if $fg$ is multiplier cyclic.
We also extend (1) to a large class of analytic function spaces. Both properties generalize all previously known results of this type.
Title: APRG Seminar: Homogenization of optimal control problem governed by Stokes system in a pillar-type domain
Speaker: Bidhan Chandra Sardar (IIT Ropar)
Date: Wed, 13 Jul 2022
Time: 11:30 am
Venue: LH-1, Mathematics Department
In this talk, we consider the optimal control problem (OCP) governed by the steady Stokes system in a two-dimensional domain $\Omega_{\epsilon}$ with a rapidly oscillating boundary prescribed with Neumann boundary condition and Dirichlet boundary conditions on the rest of the boundary. We aim to study the convergences analysis of the optimal solution (as $\epsilon\to 0$) and identify the limit OCP problem in a fixed domain.
Title: PhD Thesis colloquium: The Bergman Kernel of Siegel Modular Forms: Bounds on the Sup-norm
Speaker: K Hariram (IISc Mathematics)
Date: Mon, 11 Jul 2022
Time: 3 pm
Venue: LH-1, Mathematics Department (Hybrid mode)
The primary goal of this dissertation is to establish bounds for the sup-norm of the Bergman kernel of Siegel modular forms. Upper and lower bounds for them are studied in the weight as well as level aspect. We get the optimal bound in the weight aspect for degree 2 Siegel modular forms of weight $k$ and show that the maximum size of the sup-norm $k^{9/2}$. For higher degrees, a somewhat weaker result is provided. Under the Resnikoff-Saldana conjecture (refined with dependence on the weight), which provides the best possible bounds on Fourier coefficients of Siegel cusp forms, our bounds become optimal. Further, the amplification technique is employed to improve the generic sup-norm bound for an individual Hecke eigen-forms however, with the sup-norm being taken over a compact set of the Siegel’s fundamental domain instead. In the level aspect, the variation in sup-norm of the Bergman kernel for congruent subgroups $\Gamma_0^2(p)$ are studied and bounds for them are provided. We further consider this problem for the case of Saito-Kurokawa lifts and obtain suitable results.
Title: Geometry & Topology Seminar: Extending Taubes' Gromov invariant to Calabi--Yau 3-folds
Speaker: Mohan Swaminathan (Princeton/Stanford)
Date: Thu, 30 Jun 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
I will describe the construction of an integer-valued symplectic invariant counting embedded pseudo-holomorphic curves in a Calabi–Yau 3-fold in certain cases. This may be seen as an analogue of the Gromov invariant defined by Taubes for symplectic 4-manifolds. The construction depends on a detailed bifurcation analysis of the moduli space of embedded curves along generic paths of almost complex structures. This is based on joint work with Shaoyun Bai.
Title: APRG Seminar: Geometry of random geodesics - 3
Speaker: Arjun Krishnan (University of Rochester, USA)
Date: Wed, 22 Jun 2022
Time: 2:00-3:40 pm (with a 10 minute break 2:45-2:55)
Venue: LH-1, Mathematics Department (and Microsoft Teams)
First-passage percolation is a canonical example of a random metric on the lattice $\mathbb{Z}^d$. It is also conjecturally in the KPZ universality class for growth models. This is a three-part talk, in which we will cover the following topics:
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Overview of geodesics in first-passage percolation; their asymptotic geometry and KPZ behavior; bigeodesics and their connections to the random Ising model.
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Busemann functions, their construction and their properties; encoding geodesic behavior using Busemann functions.
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Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.
Title: MS Thesis colloquium: Attaching Galois Representations to Modular Forms of weight 2
Speaker: Mansimar Singh (IISc Mathematics)
Date: Mon, 20 Jun 2022
Time: 10 am
Venue: Microsoft Teams (online)
The aim of this talk is to understand $\ell$-adic Galois representations and associate them to normalized Hecke eigenforms of weight $2$. We will also associate these representations to elliptic curves over $\mathbb{Q}$. This will enable us to state the Modularity Theorem. We will also mention its special case which was proved by Andrew Wiles and led to the proof of Fermat’s Last Theorem.
We will develop most of the central objects involved - modular forms, modular curves, elliptic curves, and Hecke operators, in the talk. We will directly use results from algebraic number theory and algebraic geometry.
Title: APRG Seminar: Geometry of random geodesics - 2
Speaker: Arjun Krishnan (University of Rochester, USA)
Date: Mon, 20 Jun 2022
Time: 2:00-3:40 pm (with a 10 minute break 2:45-2:55)
Venue: LH-1, Mathematics Department (and Microsoft Teams)
First-passage percolation is a canonical example of a random metric on the lattice $\mathbb{Z}^d$. It is also conjecturally in the KPZ universality class for growth models. This is a three-part talk, in which we will cover the following topics:
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Overview of geodesics in first-passage percolation; their asymptotic geometry and KPZ behavior; bigeodesics and their connections to the random Ising model.
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Busemann functions, their construction and their properties; encoding geodesic behavior using Busemann functions.
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Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.
Title: APRG Seminar: Geometry of random geodesics - 1
Speaker: Arjun Krishnan (University of Rochester, USA)
Date: Fri, 17 Jun 2022
Time: 2:00-3:40 pm (with a 10 minute break 2:45-2:55)
Venue: LH-1, Mathematics Department (and Microsoft Teams)
First-passage percolation is a canonical example of a random metric on the lattice $\mathbb{Z}^d$. It is also conjecturally in the KPZ universality class for growth models. This is a three-part talk, in which we will cover the following topics:
-
Overview of geodesics in first-passage percolation; their asymptotic geometry and KPZ behavior; bigeodesics and their connections to the random Ising model.
-
Busemann functions, their construction and their properties; encoding geodesic behavior using Busemann functions.
-
Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.
Title: PhD Thesis colloquium: Weights of highest weight modules over Kac–Moody algebras
Speaker: G V Krishna Teja (IISc Mathematics)
Date: Thu, 16 Jun 2022
Time: 1:30 pm
Venue: Microsoft Teams (online)
This thesis explores highest weight modules $V$ over complex semisimple and Kac-Moody algebras. The first part of the talk addresses (non-integrable) simple highest weight modules $V = L(\lambda)$. We provide a “minimum” description of the set of weights of $L(\lambda)$, as well as a “weak Minkowski decomposition” of the set of weights of general $V$. Both of these follow from a “parabolic” generalization of the partial sum property in root systems: every positive root is an ordered sum of simple roots, such that each partial sum is also a root.
Second, we provide a positive, cancellation-free formula for the weights of arbitrary highest weight modules $V$. This relies on the notion of “higher order holes” and “higher order Verma modules”, which will be introduced and discussed in the talk.
Third, we provide BGG resolutions and Weyl-type character formulas for the higher order Verma modules in certain cases - these involve a parabolic Weyl semigroup. Time permitting, we will discuss about weak faces of the set of weights, and their complete classification for arbitrary $V$.
Title: MS Thesis colloquium: Local Langlands correspondence for GL(1) and GL(2)
Speaker: Thummala Vamsi Krishna (IISc Mathematics)
Date: Tue, 14 Jun 2022
Time: 10:30 am
Venue: LH-1, Mathematics Department
In the first part of the talk we will discuss the main statement of local class field theory and sketch a proof of it. We will then discuss the statement of local Langlands correspondence for $GL_2(K)$, where $K$ is a non archimedian local field. In the process we will also introduce all the objects that go in the statement of the correspondence.
Title: APRG Seminar: Braided quantum symmetries of graph C*-algebras
Speaker: Sutanu Roy (NISER, Bhubaneswar)
Date: Wed, 08 Jun 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we will explain the existence of a universal braided compact quantum group acting on a graph $C^*$-algebra in the twisted monoidal category of $C^*$-algebras equipped with an action of the circle group. To achieve this we construct a braided version of the free unitary quantum group. Finally, we will compute this universal braided compact quantum group for the Cuntz algebra. This is a joint work in progress with Suvrajit Bhattacharjee and Soumalya Joardar.
Title: Number Theory Seminar: On the space of irreducible polynomials in many variables
Speaker: Asvin G (University of Wisconsin-Madison, USA)
Date: Thu, 19 May 2022
Time: 11 am
Venue: LH-1, Department of Mathematics
(Joint work with Andy O’Desky) There is a very classical formula counting the number of irreducible polynomials in one variable over a finite field. We study the analogous question in many variables and generalize Gauss’ formula. Our techniques can be used to answer many other questions about the space of irreducible polynomials in many variables such as it’s euler characteristic or euler hodge-deligne polynomial. To prove these results, we define a generalization of the classical ring of symmetric functions and use natural basis in it to help us compute the answer to the above questions.
Title: Algebra & Combinatorics Seminar: Combinatorial Brill-Noether theory via lattice points and polyhedra
Speaker: Madhusudan Manjunath (IIT, Bombay)
Date: Tue, 17 May 2022
Time: 2 pm
Venue: LH-1, Mathematics Department
We start by considering analogies between graphs and Riemann surfaces. Taking cue from this, we formulate an analogue of Brill–Noether theory on a finite, undirected, connected graph. We then investigate related conjectures from the perspective of polyhedral geometry.
Title: APRG Seminar: Amenable and hyperbolic groups, stable random fields, and von Neumann algebras
Speaker: Parthanil Roy (ISI, Bangalore)
Date: Wed, 11 May 2022
Time: 4 pm
Venue: Microsoft Teams (online) (Joint with the Geometry & Topology Seminar)
Random fields indexed by amenable groups arise naturally in machine learning algorithms for structured and dependent data. On the other hand, mixing properties of such fields are extremely important tools for investigating asymptotic properties of any method/algorithm in the context of space-time statistical inference. In this work, we find a necessary and sufficient condition for weak mixing of a left-stationary symmetric stable random field indexed by an amenable group in terms of its Rosinski representation. The main challenge is ergodic theoretic - more precisely, the unavailability of an ergodic theorem for nonsingular (but not necessarily measure preserving) actions of amenable groups even along a tempered Følner sequence. We remove this obstacle with the help of a truncation argument along with the seminal work of Lindenstrauss (2001) and Tempelman (2015), and finally applying the Maharam skew-product. This work extends the domain of application of the speaker’s previous paper connecting stable random fields with von Neumann algebras via the group measure space construction of Murray and von Neumann (1936). In particular, weak mixing has now become $W^*$-rigid properties for stable random fields indexed by any amenable group, not just $\mathbb{Z}^d$. We have also shown that many stable random fields generated by natural geometric actions of hyperbolic groups on various negatively curved spaces are actually mixing and hence weakly mixing.
This talk is based on an ongoing joint work with Mahan Mj (TIFR Mumbai) and Sourav Sarkar (University of Cambridge).
The video of this talk is available on the IISc Math Department channel. Here are the slides.
Title: Number Theory Seminar: Distribution and spacing statistics of Sato-Tate sequences in short intervals
Speaker: Kaneenika Sinha (IISER Pune)
Date: Fri, 06 May 2022
Time: 11 am
Venue: Google Meet (Online)
A conjecture of Katz and Sarnak predicts that the distribution of spacings between ``straightened” Hecke angles (corresponding to Fourier coefficients of Hecke newforms) matches that of a uniformly distributed, random sequence in the unit interval. This comparison is made with the help of local spacing statistics, such as the level spacing distribution and various types of correlations of the Hecke angles. In previous joint work with Baskar Balasubramanyam and ongoing joint work with my PhD student Jewel Mahajan, we have provided evidence in favour of this conjecture, by showing that the pair correlation function of the Hecke angles, averaged over families of Hecke newforms, is expected to be Poissonnian, with variance converging to zero as we take larger and larger families. In this talk, we will explore various types of questions arising in the study of the local behaviour of sequences of Hecke angles, and explain the above-mentioned results.
Title: Algebra & Combinatorics Seminar: Splitting subspaces and the Touchard-Riordan formula
Speaker: Amritanshu Prasad (IMSc, Chennai)
Date: Fri, 06 May 2022
Time: 3 pm
Venue: LH-1, Mathematics Department
Let $T$ be a linear endomorphism of a $2m$-dimensional vector space. An $m$-dimensional subspace $W$ is said to be $T$-splitting if $W$ intersects $TW$ trivially.
When the underlying field is finite of order $q$ and $T$ is diagonal with distinct eigenvalues, the number of splitting subspaces is essentially the the generating function of chord diagrams weighted by their number of crossings with variable $q$. This generating function was studied by Touchard in the context of the stamp folding problem. Touchard obtained a compact form for this generating function, which was explained more clearly by Riordan.
We provide a formula for the number of splitting subspaces for a general operator $T$ in terms of the number of $T$-invariant subspaces of various dimensions. Specializing to diagonal matrices with distinct eigenvalues gives an unexpected and new proof of the Touchard–Riordan formula.
This is based on joint work with Samrith Ram.
Title: Geometry & Topology Seminar: Fundamental groups of open manifolds with nonnegative Ricci curvature
Speaker: Jiaying Pan (Fields Institute)
Date: Wed, 04 May 2022
Time: 9:00 pm
Venue: Microsoft Teams (online)
We survey the recent progress on the fundamental group of open manifolds with nonnegative Ricci curvature. This includes finite generation and virtual abelianness/nilpotency of the fundamental groups.
Title: APRG Seminar: Minimal dilations for commuting contractions and operator model
Speaker: Sourav Pal (IIT, Bombay)
Date: Thu, 28 Apr 2022
Time: 3 pm
Venue: LH-1, Mathematics Department
For commuting contractions $T_1,\ldots ,T_n$ acting on a Hilbert space $\mathcal{H}$ with $T=\prod_{i=1}^{n}T_i$, we find a necessary and sufficient condition under which $(T_1,\ldots ,T_n)$ dilates to commuting isometries $(V_1,\ldots ,V_n)$ or commuting unitaries $(U_1,\ldots ,U_n)$ acting on the minimal isometric dilation space or the minimal unitary dilation space of $T$ respectively, where $V=\prod_{i=1}^{n}V_i$ and $U=\prod_{i=1}^{n}U_i$ are the minimal isometric and the minimal unitary dilations of $T$ respectively. We construct both Schäffer and Sz. Nagy-Foias type isometric and unitary dilations for $(T_1,\ldots ,T_n)$. Also, a special minimal isometric dilation is constructed where the product $T$ is a $C_0$ contraction, that is $T^{*n}\to 0$ strongly as $n\to \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\ldots ,T_n)$. When the product $T$ is a $C_0$ contraction, the dilation of $(T_1,\ldots ,T_n)$ leads to a natural factorization of $T$ in terms of compression of Toeplitz operators with linear analytic symbols.
Title: Number Theory Seminar: Generalized Lambert series
Speaker: Atul Dixit (IIT Gandhinagar)
Date: Thu, 28 Apr 2022
Time: 11 am
Venue: Microsoft Teams (Online)
Lambert series lie at the heart of modular forms and the theory of the Riemann zeta function. The early pioneers in the subject were Ramanujan and Wigert. We discuss Ramanujan’s formula for odd zeta values and its generalizations and analogues obtained by the speaker with his co-authors culminating into a recent transformation for $\sum_{n=1}^{\infty}\sigma_a(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. We will discuss several applications of this result. A formula of Wigert and its recent analogue found by Soumyarup Banerjee, Shivajee Gupta and the author will be discussed and its application in the zeta-function theory will be given. This talk is an amalagamation of results of the author on this topic from various papers co-authored with Bibekananda Maji, Rahul Kumar, Rajat Gupta, Soumyarup Banerjee and Shivajee Gupta.
Title: Number Theory Seminar: The correlation coefficient in representation theory
Speaker: U. K. Anandavardhanan (IIT Bombay)
Date: Fri, 22 Apr 2022
Time: 3 pm
Venue: LH 1
Given a group $G$ and two Gelfand subgroups $H$ and $K$ of $G$, associated to an irreducible representation $\pi$ of $G$, there is a notion of $H$ and $K$ being correlated with respect to $\pi$ in $G$. This notion is defined by Benedict Gross in 1991. We discuss this theme and give some details in a specific example (which is joint work with Arindam Jana).
Title: APRG Seminar: Hirschman–Widder densities and their preservers
Speaker: Alexander Belton (Lancaster University, Lancaster, UK)
Date: Wed, 20 Apr 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Hirschman–Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, that is, integrable functions which give rise to totally positive Toeplitz kernels. This talk will introduce this class of densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.
This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: Quasi-analytic functions, spherical means, and uncertainty principles on Heisenberg groups and symmetric spaces
Speaker: Pritam Ganguly (IISc Mathematics)
Date: Mon, 18 Apr 2022
Time: 2 pm
Venue: LH-1, Mathematics Department (Hybrid mode)
This thesis has two parts. The first part revolves around certain theorems related to an uncertainty principle and quasi-analyticity. In contrast, the second part reflects a different mathematical theme, focusing on the classical problem of $L^p$ boundedness of spherical maximal function on the Heisenberg group.
The highlights of the first part are as follows: An uncertainty principle due to Ingham (proved initially on $\mathbb{R}$) investigates the best possible decay admissible for the Fourier transform of a function that vanishes on a nonempty open set. One way to establish such a result is to use a theorem of Chernoff (proved originally on $\mathbb{R}^n$), which provides a sufficient condition for a smooth function to be quasi-analytic in terms of a Carleman condition involving powers of the Laplacian. In this part of this thesis, we aim to prove various analogues of theorems of Ingham and Chernoff in different contexts such as the Heisenberg group, Hermite and special Hermite expansions, rank one Riemannian symmetric spaces, and Euclidean space with Dunkl setting. More precisely, we prove various analogues of Chernoff’s theorem for the full Laplacian on the Heisenberg group, Hermite and special Hermite operators, Laplace-Beltrami operators on rank one symmetric spaces of both compact and non-compact type, and Dunkl Laplacian. The main idea is to reduce the situation to the radial case by employing appropriate spherical means or spherical harmonics and then to apply Chernoff type theorems to the radial parts of the operators indicated above. Using those Chernoff type theorems, we then show several analogues of Ingham’s theorem for the spectral projections associated with those aforementioned operators. Furthermore, we provide examples of compactly supported functions with Ingham type decay in their spectral projections, demonstrating the sharpness of Ingham’s theorem in all of the relevant contexts mentioned above.
In the second part of this thesis, we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\mathbb{H}^n}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.
Title: APRG Seminar: Local and multilinear non-commutative de Leeuw theorems
Speaker: Amudhan Krishnaswamy Usha (Technische Universiteit Delft, Netherlands)
Date: Wed, 13 Apr 2022
Time: 4 pm
Venue: Microsoft Teams (online)
If $m$ is a function on a commutative group $G$, one may define an associated Fourier multiplier $T_m$, which acts on functions on the dual group. If this $T_m$ is a bounded linear map on the $L_p$ space of the dual group, is the restriction of $m$ to a subgroup $H$ also the symbol of a bounded multiplier on the $L_p$ space of the dual group of $H$? De Leeuw showed that this is indeed the case when $G=\mathbb{R}^n$, and others later extended this to all locally compact commutative groups. Moreover, the norm of the multiplier corresponding to the restricted symbol is bounded above by the norm of the original multiplier. For non-commutative groups, one may ask the same question by replacing “$L_p$ spaces of the dual group” with the non-commutative $L_p$ space of the group von Neumann algebra. Caspers, Parcet, Perrin and Ricard showed that the answer is still yes in the non-commutative case, provided $G$ has something called the “small-almost invariant neighbourhood property with respect to the subgroup $H$”.
In recent joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi, we prove a local version of this result, which removes this restriction (for a price). We show that the norm of the $L_p$ Fourier multiplier for the subgroup is bounded by some constant depending only on the support of the symbol $m$. This constant measures the failure of the small invariant neighbourhood property, and can be explicitly estimated for real reductive Lie groups. We also prove non-commutative multilinear versions of the De Leeuw theorems, and use these to construct examples of multilinear multipliers on the Heisenberg group. I will outline these results in my talk, and if time permits, describe some possible extensions.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Complex variable theorems for finding zeroes and poles of transcendental functions
Speaker: D V Giri (University of New Mexico and Pro-Tech, USA)
Date: Mon, 11 Apr 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
The problem of locating the poles and zeros of complex functions in a finite domain of the complex plane, occurs in many scientific disciplines e.g., dispersion relations in plasma physics, the singularity expansion method in electro-magnetic scattering or antenna problems.
The principle of the argument or the winding number is useful in finding the number of zeros of an analytic function in a given contour. A simple extension of this theorem yields relationships involving the locations of these zeros! The resulting equations can be solved very accurately for the zero locations, thus avoiding initial, guess values, which are required by many other techniques. Examples such as a 20th order polynomial, natural frequencies of a thin wire will be discussed.
This method has been extended to the problem of locating the zeros and poles of a complex meromorphic function $M(s)$ in a specified rectangular or square region of the complex plane. It is assumed that $M(s)$ has to be numerically computed. It is interesting to note that the word “meromorphic” is derived from the Greek meros $(\mu \varepsilon \rho \omicron \zeta)$ = fraction and morph $(\mu \omicron \rho \varphi \eta)$ = form, and means “like a fraction.” In keeping with the origin of the word “meromorphic,” the complex function $M(s)$ considered in this paper will be a ratio of two entire functions of the complex variables. The procedure developed here eliminates the usual 2-dimensional search and replaces it with a direct constructive method for determining the poles of $M(s)$ based on an application of Cauchy’s residue theorem. Two examples, i.e., 1) ratios of polynomials and 2) input impedance of a biconical antenna, are numerically illustrated.
Title: Number Theory Seminar: Fine Selmer group of elliptic curves
Speaker: Somnath Jha (IIT Kanpur)
Date: Fri, 08 Apr 2022
Time: 4 pm
Venue: Microsoft Teams (Online)
The ($p^{\infty}$) fine Selmer group (also called the $0$-Selmer group) of an elliptic curve is a subgroup of the usual $p^{\infty}$ Selmer group of an elliptic curve and is related to the first and the second Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of a number field $K$ is intricately related to Iwasawa’s $\mu$-invariant vanishing conjecture on the growth of $p$-part of the ideal class group of $K$ in the cyclotomic tower. In this talk, we will discuss the structure and properties of the fine Selmer group over certain $p$-adic Lie extensions of global fields. This talk is based on joint work with Sohan Ghosh and Sudhanshu Shekhar.
Title: APRG Seminar: Dimensionality and patterns with curvature
Speaker: Malabika Pramanik (University of British Columbia, Vancouver, Canada)
Date: Wed, 06 Apr 2022
Time: 8 pm
Venue: Microsoft Teams (online)
Hausdorff dimension is a notion of size ubiquitous in geometric measure theory. A set of large Hausdorff dimension contains many points, so it is natural to expect that it should contain specific configurations of interest. Yet many existing results in the literature point to the contrary. In particular, there exist full-dimensional sets $K$ in the plane with the property that if a point $(x_1, x_2)$ is in $K$, then no point of the form $(x_1, x_2 + t)$ lies in $K$, for any $t \neq 0$.
A recent result of Kuca, Orponen and Sahlsten shows that every planar set of Hausdorff dimension sufficiently close to 2 contains a two-point configuration of the form $(x1, x2)+\{(0, 0), (t, t^2)\}$ for some $t \neq 0$. This suggests that sets of sufficiently large Hausdorff dimension may contain patterns with “curvature”, suitably interpreted. In joint work with Benjamin Bruce, we obtain a characterization of smooth functions $\Phi : \mathbb{R} \to \mathbb{R}^d$ such that every set of sufficiently high Hausdorff dimension in $d$-dimensional Euclidean space contains a two point configuration of the form $\{x, x + \Phi(t)\}$, for some $t$ with $\Phi(t) \neq 0$.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: A qualitative description of the horoboundary of the Teichmüller metric
Speaker: Aitor Azemar (University of Glasgow)
Date: Wed, 06 Apr 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
The horofunction compactification of a metric space keeps track of the possible limits of balls whose centers go off to infinity. This construction was introduced by Gromov, and although it is usually hard to visualize, it has proved to be a useful tool for studying negatively curved spaces. In this talk I will explain how, under some metric assumptions, the horofunction compactification is a refinement of the significantly simpler visual compactification. I will then go over how this relation allows us to use the simplicity of the visual compactification to get geometric and topological properties of the horofunction compactification. Most of these applications will be in the context of Teichmüller spaces with respect to the Teichmüller metric, where the relation allows us to prove, among other things, that Busemann points are not dense within the horoboundary and that the horoboundary is path connected.
Title: Number Theory Seminar: Recent investigations of $L$-invariants of modular forms
Speaker: John Bergdall (Bryn Mawr College, USA)
Date: Fri, 01 Apr 2022
Time: 6 pm
Venue: Microsoft Teams (Online)
In this talk I will explain new research on $L$-invariants of modular forms, including ongoing joint work with Robert Pollack. $L$-invariants, which are $p$-adic invariants of modular forms, were discovered in the 1980’s, by Mazur, Tate, and Teitelbaum. They were formulating a $p$-adic analogue of Birch and Swinnerton-Dyer’s conjecture on elliptic curves. In the decades since, $L$-invariants have shown up in a ton of places: $p$-adic $L$-series for higher weight modular forms or higher rank automorphic forms, the Banach space representation theory of $\mathrm{GL}(2,\mathbb{Q}_p)$, $p$-adic families of modular forms, Coleman integration on the $p$-adic upper half-plane, and Fontaine’s $p$-adic Hodge theory for Galois representations. In this talk I will focus on recent numerical and statistical investigations of these $L$-invariants, which touch on many of the theories just mentioned. I will try to put everything into the context of practical questions in the theory of automorphic forms and Galois representations and explain what the future holds.
Title: MS Thesis colloquium: Asymmetric Super-Heston-rough volatility model with Zumbach effect as a scaling limit of quadratic Hawkes processes
Speaker: Priyanka Chudasama (IISc Mathematics)
Date: Wed, 30 Mar 2022
Time: 11 am
Venue: LH-1, Mathematics Department
Modelling price variation has always been of interest, from options pricing to risk management. It has been observed that the high-frequency financial market is highly volatile, and the volatility is rough. Moreover, we have the Zumbach effect, which means that past trends in the price process convey important information on future volatility. Microscopic price models based on the univariate quadratic Hawkes (hereafter QHawkes) process can capture the Zumbach effect and the rough volatility behaviour at the macroscopic scale. But they fail to capture the asymmetry in the upward and downward movement of the price process. Thus, to incorporate asymmetry in price movement at micro-scale and rough volatility and the Zumbach effect at macroscale, we introduce the bivariate Modified-QHawkes process for upward and downward price movement. After suitable scaling and shifting, we show that the limit of the price process in the Skorokhod topology behaves as so-called Super-Heston-rough model with the Zumbach effect.
Title: Geometry & Topology Seminar: Degenerating conic Kähler-Einstein metrics
Speaker: Henri Guenancia (CNRS, Toulouse)
Date: Wed, 30 Mar 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
I will discuss a recent joint work with Olivier Biquard about conic Kähler-Einstein metrics with cone angle going to zero. We study two situations, one in negative curvature (toroidal compactifications of ball quotients) and one in positive curvature (Fano manifolds endowed with a smooth anticanonical divisor) leading up to the resolution of a folklore conjecture involving the Tian-Yau metric.
Title: APRG Seminar: Cousins and ancestors of the arithmetic-geometric inequality
Speaker: Colin McSwiggen (Courant Institute, NYU, USA)
Date: Wed, 30 Mar 2022
Time: 7 pm
Venue: Microsoft Teams (online)
A fundamental and widely used mathematical fact states that the arithmetic mean of a collection of non-negative real numbers is at least as large as its geometric mean. This is the most basic example of a large family of inequalities between symmetric functions that have attracted the interest of combinatorialists in recent years. This talk will present recent joint work with Jon Novak at UC San Diego, which unifies many such inequalities as corollaries of a fundamental monotonicity property of spherical functions on symmetric spaces. We will also discuss conjectural extensions of these results to even more general objects such as Heckman-Opdam hypergeometric functions and Macdonald polynomials.
The talk will be accessible to a broad mathematical audience and will not assume any knowledge of symmetric spaces or symmetric functions. However, the second half of the talk will assume familiarity with basic constructions of Lie theory, such as root systems and the Iwasawa decomposition. Details of the relevant work can be found in this pre-print.
The video of this talk is available on the IISc Math Department channel.
Title: Algebra & Combinatorics Seminar: A combinatorial model for totally nonnegative partial flag varieties
Speaker: Maitreyee Kulkarni (Universität Bonn, Germany)
Date: Fri, 25 Mar 2022
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Postnikov defined the totally nonnegative Grassmannian as the part of the Grassmannian where all Plücker coordinates are nonnegative. This space can be described by the combinatorics of planar bipartite graphs in a disk, by affine Bruhat order, and by a host of other combinatorial objects. In this talk, I will recall some of this story, then talk about in progress joint work, together with Chris Fraser and Jacob Matherne, which hopes to extend this combinatorial description to more general partial flag varieties.
Title: Algebra & Combinatorics Seminar: Singular Hodge theory for combinatorial geometries
Speaker: Jacob P. Matherne (Universität Bonn, Germany)
Date: Fri, 25 Mar 2022
Time: 3 pm
Venue: LH-1, Mathematics Department
Here are two problems about hyperplane arrangements.
Problem 1: If you take a collection of planes in $\mathbb{R}^3$, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the “Top-Heavy Conjecture”, that Dowling and Wilson conjectured in 1974.
Problem 2: Given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it. These polynomials should have nonnegative coefficients.
Both of these problems were formulated for all matroids, and in the case of hyperplane arrangements they are controlled by the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement. For arbitrary matroids, no such variety exists; nonetheless, I will discuss a solution to both problems for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory. This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.
Title: Number Theory Seminar: On the Tamagawa Number Conjecture in char. $p>0$
Speaker: Fabien Trihan (Sophia University, Japan)
Date: Fri, 25 Mar 2022
Time: 12 pm
Venue: LH 1
We will talk on the an analogue of the Tamagawa Number conjecture, with coefficients over varieties over finite fields. This a joint work with O. Brinon (Bordeaux) and a work in progress.
Title: APRG Seminar: The Laplace transform in the Dunkl setting
Speaker: Margit Rösler (Universität Paderborn, Germany)
Date: Wed, 23 Mar 2022
Time: 4 pm
Venue: Microsoft Teams (online)
In the analysis on symmetric cones and the classical theory of hypergeometric functions of matrix argument, the Laplace transform plays an essential role. In an unpublished manuscript dating back to the 1980ies, I.G. Macdonald proposed a generalization of this theory, where the spherical polynomials of the underlying symmetric cone - such as the cone of positive definite matrices - are replaced by Jack polynomials with arbitrary index. He also introduced a Laplace transform in this context, but many of the statements in his manuscript remained conjectural. In the late 1990ies, Baker and Forrester took up these matters in their study of Calogero-Moser models, still at a rather formal level, and they noticed that they were closely related to Dunkl theory.
In this talk, we explain how Macdonald’s Laplace transform can be rigorously established within Dunkl theory, and we discuss several of its applications, including Riesz distributions and Laplace transform identities for the Cherednik kernel and for Macdonald’s hypergeometric series in terms of Jack polynomials.
Part of the talk is based on joint work with Dominik Brennecken.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Lagrangian Cobordisms and Enriched Knot Diagrams
Speaker: Ipsita Datta (IAS Princeton, USA)
Date: Wed, 23 Mar 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
We present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$. The obstructions arise from studying moduli spaces of holomorphic disks with corners and boundaries on immersed objects called Lagrangian tangles. The obstructions boil down to area relations and sign conditions on disks bound by knot diagrams of the boundaries of the Lagrangian. We present examples of pairs of knots that cannot be Lagrangian cobordant and knots that cannot bound Lagrangian disks.
Title: Geometry & Topology Seminar: Convergence of complex geometric measures to non-Archimedean measures
Speaker: Sanal Shivprasad (University of Michigan)
Date: Wed, 16 Mar 2022
Time: 9:00 pm
Venue: Microsoft Teams (online)
We consider certain degenerating families of complex manifolds, each carrying a canonical measure (for example, the Bergman measure on a compact Riemann surface of genus at least one). We show that the measure converges, in a suitable sense, to a measure on a non-Archimedean space, in the sense of Berkovich. No knowledge of non-Archimedean geometry will be assumed.
Title: PhD Thesis defence: Near-optimal Non-malleable Codes and Leakage Resilient Secret Sharing Schemes
Speaker: Sruthi Sekar (IISc Mathematics)
Date: Fri, 11 Mar 2022
Time: 9:30 am
Venue: Microsoft Teams (online)
Non-malleable codes (NMCs) are coding schemes that help in protecting crypto-systems under tampering attacks, where the adversary tampers the device storing the secret and observes additional input-output behavior on the crypto-system. NMCs give a guarantee that such adversarial tampering of the encoding of the secret will lead to a tampered secret, which is either same as the original or completely independent of it, thus giving no additional information to the adversary. The specific tampering model that we consider in this work, called the “split-state tampering model”, allows the adversary to tamper two parts of the codeword arbitrarily, but independent of each other. Leakage resilient secret sharing schemes help a party, called a dealer, to share his secret message amongst n parties in such a way that any $t$ of these parties can combine their shares to recover the secret, but the secret remains hidden from an adversary corrupting $< t$ parties to get their complete shares and additionally getting some bounded bits of leakage from the shares of the remaining parties.
For both these primitives, whether you store the non-malleable encoding of a message on some tamper-prone system or the parties store shares of the secret on a leakage-prone system, it is important to build schemes that output codewords/shares that are of optimal length and do not introduce too much redundancy into the codewords/shares. This is, in particular, captured by the rate of the schemes, which is the ratio of the message length to the codeword length/largest share length. This thesis explores the question of building these primitives with optimal rates.
The focus of this talk will be on taking you through the journey of non-malleable codes culminating in our near-optimal NMCs with a rate of 1/3.
Title: Number Theory Seminar: On Eulerian multiple zeta values and the block shuffle relations
Speaker: Nobuo Sato (National Taiwan University)
Date: Fri, 11 Mar 2022
Time: 4 pm
Venue: Microsoft Teams (Online)
Euler solved the famous Basel problem and discovered that Riemann zeta functions at positive even integers are rational multiples of powers of $\pi$. Multiple zeta values (MSVs) are a multi-dimensional generalization of the Riemann zeta values, and MZVs which are rational multiples of powers of $\pi$ is called Eulerian MZVs. In 1996, Borwein-Bradley-Broadhurst discovered a series of conjecturally Eulerian MZVs which together with the known Eulerian family seems to exhaust all Eulerian MZVs at least numerically. A few years later, Borwein-Bradley-Broadhurst-Lisonek discovered two families of interesting conjectural relations among MZVs generalizing the previous conjecture of Eulerian MZVs, which were later extended further by Charlton in light of alternating block structure. In this talk, I would like to present my recent joint work with Minoru Hirose concerning block shuffle relations that simultaneously resolve and generalize the conjectures of Charlton.
Title: Geometry & Topology Seminar: Minimal surfaces in products of trees
Speaker: Nathaniel Sagman (Caltech)
Date: Wed, 09 Mar 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
Recently, Markovic proved that there exists a maximal representation into (PSL(2,R))^3 such that the associated energy functional on Teichmuller space admits multiple critical points. In geometric terms, there is more than one minimal surface in the relevant homotopy class in the corresponding product of closed Riemann surfaces. This is related to an important question in Higher Teichmuller theory. In this talk, we explain that this non-uniqueness arises from non-uniqueness of minimal surfaces in products of trees. We plan to discuss energy minimizing properties for minimal maps into trees, as well as the geometry of the surfaces found in Markovic’s work. This is work in progress, joint with Vladimir Markovic.
Title: APRG Seminar: Blaschke Products, Level Sets, and Crouzeix's Conjecture
Speaker: Kelly Bickel (Bucknell University, USA)
Date: Wed, 09 Mar 2022
Time: 6 pm
Venue: Microsoft Teams (online)
This talk is motivated by interest in Crouzeix’s conjecture for compressions of the shift with finite Blaschke products as symbols. Specifically, in this setting, Crouzeix’s conjecture suggests a related, weaker conjecture about the behavior of level sets of finite Blaschke products. I’ll discuss this level set conjecture in several cases, though the main case of interest will involve uncritical finite Blaschke products. Here, the geometry of the numerical ranges of their associated compressions of the shift has allowed us to establish the conjecture in low degree situations (n=3, n=4, n =5 with a caveat). Time permitting, I’ll explain how these geometric results also give insights into Crouzeix’s conjecture for the associated compressed shifts. This talk is based on joint work with Pam Gorkin.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Another view of Ferrero--Washington Theorem
Speaker: Jungwon Lee (University of Warwick, UK)
Date: Fri, 04 Mar 2022
Time: 6 pm
Venue: Microsoft Teams (Online)
We reprove the main equidistribution instance in the Ferrero–Washington proof of the vanishing of cyclotomic Iwasawa $\mu$-invariant, based on the ergodicity of a certain $p$-adic skew extension dynamical system that can be identified with Bernoulli shift (joint with Bharathwaj Palvannan).
Title: Number Theory Seminar: A modular construction of unramified $p$-extensions of $\mathbb{Q}(N^{1/p})$
Speaker: Jaclyn Lang (Temple University, USA)
Date: Fri, 25 Feb 2022
Time: 6 pm
Venue: Microsoft Teams (Online)
In his 1976 proof of the converse to Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-$p$ extensions of the $p$-th cyclotomic field when $p$ is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-$p$ extensions of $\mathbb{Q}(N^{1/p})$ when $N$ is a prime that is congruent to $-1$ mod $p$. This answers a question posted on Frank Calegari’s blog.
Title: Geometry & Topology Seminar: Analytic stability conditions for polarised varieties
Speaker: Ruadhaí Dervan (University of Cambridge)
Date: Wed, 23 Feb 2022
Time: 4:00 pm
Venue: Microsoft Teams (online)
A conjectural correspondence due to Yau, Tian and Donaldson relates the existence of certain canonical Kähler metrics (“constant scalar curvature Kähler metrics”) to an algebro-geometric notion of stability (“K-stability”). I will describe a general framework linking geometric PDEs (“Z-critical Kähler metrics”) to algebro-geometric stability conditions (“Z-stability”), in such a way that the Yau-Tian-Donaldson conjecture is the classical limit of these new broader conjectures. The main result will prove that a special case of the main conjecture: the existence of Z-critical Kähler metrics is equivalent to Z-stability.
Title: APRG Seminar: The Cauchy-Szego projection and its commutator for domains in $\mathbb{C}^n$ with minimal smoothness: Optimal bounds
Speaker: Loredana Lanzani (Syracuse University, USA)
Date: Wed, 16 Feb 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Let $D\subset\mathbb{C}^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani & E. M. Stein states that the Cauchy–Szegö projection $\mathcal{S}_\omega$ maps $L^p(bD, \omega)$ to $L^p(bD, \omega)$ continuously for any $1<p<\infty$ whenever the reference measure $\omega$ is a bounded, positive continuous multiple of induced Lebesgue measure. Here we show that $\mathcal{S}_\omega$ (defined with respect to any measure $\omega$ as above) satisfies explicit, optimal bounds in $L^p(bD, \Omega_p)$, for any $1<p<\infty$ and for any $\Omega_p$ in the maximal class of $A_p$-measures, that is $\Omega_p = \psi_p\sigma$ where $\psi_p$ is a Muckenhoupt $A_p$-weight and $\sigma$ is the induced Lebesgue measure. As an application, we characterize boundedness in $L^p(bD, \Omega_p)$ with explicit bounds, and compactness, of the commutator $[b, \mathcal{S}_\omega]$ for any $A_p$-measure $\Omega_p$, $1<p<\infty$. We next introduce the notion of holomorphic Hardy spaces for $A_p$-measures, and we characterize boundedness and compactness in $L^2(bD, \Omega_2)$ of the commutator $[b,\mathcal{S}_{\Omega_2}]$ where $\mathcal{S}_{\Omega_2}$ is the Cauchy–Szegö projection defined with respect to any given $A_2$-measure $\Omega_2$. Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy–Szegö kernel, but these are unavailable in our setting of minimal regularity of $bD$; at the same time, recent techniques that allow to handle domains with minimal regularity, are not applicable to $A_p$-measures. It turns out that the method of quantitative extrapolation is an appropriate replacement for the missing tools.
This is joint work with Xuan Thinh Duong (Macquarie University), Ji Li (Macquarie University) and Brett Wick (Washington University in St. Louis).
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Non commutative cluster coordinates for Higher Teichmüller Spaces
Speaker: Daniele Alessandrini (Columbia University)
Date: Wed, 16 Feb 2022
Time: 6:30 pm (**Non-standard time!**)
Venue: Microsoft Teams (online)
In higher Teichmuller theory we study subsets of the character varieties of surface groups that are higher rank analogs of Teichmuller spaces, e.g. the Hitchin components, the spaces of maximal representations and the other spaces of positive representations. Fock-Goncharov generalized Thurston’s shear coordinates and Penner’s Lambda-lengths to the Hitchin components, showing that they have a beautiful structure of cluster variety. We applied a similar strategy to Maximal Representations and we found new coordinates on these spaces that give them a structure of non-commutative cluster varieties, in the sense defined by Berenstein-Rethak. This was joint work with Guichard, Rogozinnikov and Wienhard. In a project in progress we are generalizing these coordinates to the other sets of positive representations.
Title: Number Theory Seminar: Galois Families of Modular Forms
Speaker: Gabor Wiese (University of Luxembourg)
Date: Fri, 11 Feb 2022
Time: 3 pm
Venue: Microsoft Teams (Online)
Following a joint work with Sara Arias-de-Reyna and François Legrand, we present a new kind of families of modular forms. They come from representations of the absolute Galois group of rational function fields over $\mathbb{Q}$. As a motivation and illustration, we discuss in some details one example: an infinite Galois family of Katz modular forms of weight one in characteristic $7$, all members of which are non-liftable. This may be surprising because non-liftability is a feature that one might expect to occur only occasionally.
Title: Geometry & Topology Seminar: Quadratic differentials and applications to spherical geometry (joint work with Quentin Gendron)
Speaker: Guillaume Tahar (Weizmann Institute)
Date: Wed, 09 Feb 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Up to biholomorphic change of variable, local invariants of a quadratic differential at some point of a Riemann surface are the order and the residue if the point is a pole of even order. Using the geometric interpretation in terms of flat surfaces, we solve the Riemann-Hilbert type problem of characterizing the sets of local invariants that can be realized by a pair (X,q) where X is a compact Riemann surface and q is a meromorphic quadratic differential. As an application to geometry of surfaces with positive curvature, we give a complete characterization of the distributions of conical angles that can be realized by a cone spherical metric with dihedral monodromy.
Title: APRG Seminar: On the $L^p$-Fourier transform norm for compact extensions of locally compact groups
Speaker: Ali Baklouti (University of Sfax, Tunisia)
Date: Wed, 09 Feb 2022
Time: 4 pm
Venue: Microsoft Teams (online)
This is a continuation of a talk I gave at the University of Delhi in $2015.$ Let $G$ be a separable locally compact unimodular group of type I, and $\widehat G$ the unitary dual of $G$ endowed with the Mackey Borel structure. We regard the Fourier transform $\mathcal F$ as a mapping of $L^1(G)$ to a space of $\mu$-measurable field of bounded operators on $\widehat G$ defined for $\pi\in\widehat G$ by $ L^1(G)\ni f\mapsto \mathcal Ff : \mathcal Ff(\pi)=\pi(f), $ where $\mu$ denotes the Plancherel measure of $G$. The mapping $f \mapsto \mathcal F f$ extends to a continuous operator $\mathcal F^p : L^p(G) \to L^q(\widehat G)$, where $p\geq 1$ is real number and $q$ its conjugate. We are concerned in this talk with the norm of the linear map $\mathcal F^p$. We first record some results on the estimate of this norm for some classes of solvable Lie groups and their compact extensions and discuss the sharpness problem. We look then at the case where $G$ is a separable unimodular locally compact group of type I. Let $N$ be a unimodular closed normal subgroup of $G$ of type I, such that $G/N$ is compact. We show that $\Vert \mathscr F^p(G)\Vert \leq \Vert \mathscr F^p(N )\Vert$. In the particular case where $G=K\ltimes N$ is defined by a semi-direct product of a separable unimodular locally compact group $N$ of type I and a compact subgroup $K$ of the automorphism group of $N$, we show that equality holds if $N$ has a $K$-invariant sequence $(\varphi_j)_j$ of functions in $L^1(N)\cap L^p(N)$ such that ${\Vert \mathscr F\varphi_j \Vert_q}/{\Vert \varphi_j \Vert_p}$ tends to $\Vert \mathscr F^p(N )\Vert$ when $j$ goes to infinity.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Asymptotics and codimensions of modules over Iwasawa algebras
Speaker: Sujatha Ramdorai (University of British Columbia, Canada)
Date: Fri, 04 Feb 2022
Time: 3 pm
Venue: Microsoft Teams (Online)
Let $R$ be the Iwasawa algebra over a compact, $p$-adic, pro-$p$ group
$G$, where $G$ arises as a Galois group of number fields from Galois representations.
Suppose $M$ is a finitely generated $R$-module. In the late 1970’s , Harris studied the
asymptotic growth of the ranks of certain coinvariants of $M$ arising from the action
of open subgroups of $G$ and related them to the codimension of $M$. In this talk, we
explain how Harris’ proofs can be simplified and improved upon, with possible
applications to studying some natural subquotients of the Galois groups of number fields.
Title: APRG Seminar: Grushin pseudo-multipliers
Speaker: Rahul Garg (IISER, Bhopal)
Date: Wed, 02 Feb 2022
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, I shall talk about analogues of pseudo-differential operators (pseudo-multipliers) associated with the joint functional calculus for the Grushin operator. In particular, we shall discuss some sufficient conditions on a symbol function which imply $L^2$-boundedness of the associated Grushin pseudo-multiplier. This talk is based on a joint work (arXiv:2111.10098) with Sayan Bagchi.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: The congruence ideal associated to $p$-adic families of Yoshida lifts
Speaker: Bharathwaj Palvannan (IISc)
Date: Fri, 28 Jan 2022
Time: 3 pm
Venue: Microsoft Teams (Online)
This talk will be a report of work in progress with Ming-Lun Hsieh. Just as in classical Iwasawa theory where one studies congruences involving Hecke eigenvalues associated to Eisenstein series, we study congruences involving $p$-adic families of Hecke eigensystems associated to the space of Yoshida lifts of two Hida families. Our goal is to show that under suitable assumptions, the characteristic ideal of a dual Selmer group is contained inside the congruence ideal.
Title: Colloquium: Multiple zeta values and multiple Apéry-like sums
Speaker: P. Akhilesh (Kerala School of Mathematics)
Date: Fri, 21 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Multiple zeta values are the real numbers \begin{equation} \zeta({\bf a})= \sum_{n_1>\cdots>n_r>0}n_1^{-a_1}\cdots n_r^{-a_r}, \end{equation} where ${\bf a}=(a_1, \ldots ,a_r) $ is an admissible composition, i.e. a finite sequence of positive integers, with $a_1 \geqslant 2$ when $r\neq 0$.
The multiple Apéry-like sums defined by \begin{equation} \sigma({\bf a})=\sum_{n_1>\cdots>n_r>0}\left({2 n_1 \atop n_1}\right)^{-1}n_1^{-a_1}\cdots n_r^{-a_r} \end{equation} when ${\bf a}\neq\varnothing$ and by $\sigma(\varnothing)=1$. We show that for any admissible composition ${\bf a}$, there exists a finite formal $\bf Z$-linear combination $\sum \lambda_{\bf b} {\bf b}$ of admissible compositions such that \begin{equation} \zeta({\bf a})=\sum \lambda_{\bf b}\, \sigma({\bf b}). \end{equation} The simplest instance of this fact is the identity \begin{equation} \sum_{n=1}^{\infty}\frac{1}{n^2}=3\sum_{n=1}^{\infty}\frac{1}{\left({2n \atop n}\right)n^2} \end{equation} discovered by Euler, which expresses that $\zeta(2)=3\,\sigma(2)$. Note that multiple Apéry-like sums have the advantage on multiple zeta values to be exponentially quickly convergent.
This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give new integral formulas for multiple zeta values and Apéry-like sums. They enable us to give a short direct proof of Zagier’s formulas for $\zeta(2,\ldots,2,3,2,\ldots,2)$ (D. Zagier, Evaluation of the multiple zeta values $\zeta(2,\ldots,2,3,2,\ldots,2)$, Annals of Math. 175 (2012), 977–1000) as well as of similar ones in the context of Apéry-like sums.
Title: APRG Seminar: Restricted weak type $(1,1)$ estimates for averaging operators
Speaker: María J. Carro (Universidad Complutense de Madrid, Madrid, Spain)
Date: Wed, 19 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
There many operators in Harmonic Analysis which can be described as an average of a family of operators $\{T_j\}_j$ for which some boundedness properties are known. In particular, if $T_j$ are uniformly bounded on $L^p$, then the Minkowski integral inequality tells us that $T$ also satisfies this property. But things change completely if the information that we have is that $T_j$ are of weak type (1,1).
However, under certain condition on the operators $T_j$, the weak type boundedness of $T$ can be reached.
This is a joint work with my student Sergi Baena.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: On mapping class group of 4-manifolds
Speaker: Anubhav Mukherjee (Georgia Tech)
Date: Wed, 19 Jan 2022
Time: 9 pm
Venue: Microsoft Teams (online)
One interesting question in low-dimensional topology is to understand the structure of mapping class group of a given manifold. In dimension 2, this topic is very well studied. The structure of this group is known for various 3-manifolds as well (ref- Hatcher’s famous work on Smale conjecture). But virtually nothing is known in dimension 4. In this talk I will try to motivate why this problem in dimension 4 is interesting and how it is different from dimension 2 and 3. I will demonstrate some “exotic” phenomena and if time permits, I will talk a few words on my work with Jianfeng Lin where we used an idea motivated from this to disprove a long standing open problem about stabilizations of 4-manifolds.
Title: PhD Thesis colloquium: Quasi-analytic functions, spherical means, and uncertainty principles on Heisenberg groups and symmetric spaces
Speaker: Pritam Ganguly (IISc Mathematics)
Date: Tue, 18 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
This talk has two parts. The first part revolves around certain theorems related to an uncertainty principle and quasi-analyticity. On the other hand, the second part reflects a different mathematical theme, focusing on the classical problem of $L^p$ boundedness of spherical maximal function on the Heisenberg group.
The highlights of the first part are as follows: An uncertainty principle due to Ingham (proved initially on $\mathbb{R}$) investigates the best possible decay admissible for the Fourier transform of a function that vanishes on a nonempty open set. One way to establish such a result is to use a theorem of Chernoff (proved originally on $\mathbb{R}^n$), which provides a sufficient condition for a smooth function to be quasi-analytic in terms of a Carleman condition involving powers of the Laplacian. In this part of this talk, we plan to discuss various analogues of Chernoff’s theorem for the full Laplacian on the Heisenberg group, Hermite, and special Hermite operators, Laplace-Beltrami operators on rank one symmetric spaces of both compact and non-compact type, and Dunkl Laplacian. Using those Chernoff type theorems, we then show several analogues of Ingham’s theorem for the spectral projections associated with those aforementioned operators. Furthermore, we provide examples of compactly supported functions with Ingham type decay in their spectral projections, demonstrating the sharpness of Ingham’s theorem in all of the relevant contexts mentioned above.
In this second part of this talk, we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\mathbb{H}^n}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.
Title: Colloquium: Dickson-Siegel-Eichler-Roy elementary orthogonal transformations
Speaker: Ambily Ambattu Asokan (Cochin University of Science and Technology)
Date: Mon, 17 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Classical groups and their generalizations are central objects in Algebraic $K$-theory. Orthogonal groups are one type of classical groups. We shall discuss a generalized version of elementary orthogonal groups.
Let $R$ be a commutative ring in which $2$ is invertible. Let $Q$ be a non-degenerate quadratic space over $R$ of rank $n$ and let $\mathbb{H}(R)^m$ denote the hyperbolic space of rank $m$. We consider the elementary orthogonal transformations of the quadratic space $Q \perp \mathbb{H}(R)^m$. These transformations were introduced by Amit Roy in $1968$. Earlier forms of these transformations over fields were considered by Dickson, Siegel, Eichler and Dieudonné. We call the elementary orthogonal transformations as Dickson–Siegel–Eichler–Roy elementary orthogonal transformations or Roy’s elementary orthogonal transformations. The group generated by these transformations is called DSER elementary orthogonal group. We shall discuss the structure of this group.
As part of the solution to the famous Serre’s problem on projective modules, D. Quillen had proved the remarkable Local-Global criterion for a module $M$ to be extended. This result is known as Quillen’s Patching Theorem or Quillen’s Local-Global Principle. The Bass–Quillen conjecture is a natural generalization of Serre’s problem. In this talk, we shall see the solution of the quadratic version of the Bass–Quillen conjecture over an equicharacteristic regular local ring.
The DSER elementary orthogonal group is a normal subgroup of the orthogonal group. We shall also discuss some generalizations of classical groups over form rings and their comparison with the DSER elementary orthogonal group.
Title: APRG Seminar: On Wigner's theorem
Speaker: Peter Semrl (University of Ljubljana, Slovenia)
Date: Thu, 13 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Some recent improvements of Wigner’s unitary-antiunitary theorem will be presented. A connection with Gleason’s theorem will be explained.
The video of this talk is available on the IISc Math Department channel.
Title: Colloquium: On fractional counting processes
Speaker: Kuldeep Kumar Kataria (IIT Bhilai)
Date: Wed, 12 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Advances in various fields of modern studies have shown the limitations of traditional probabilistic models. The one such example is that of the Poisson process which fails to model the data traffic of bursty nature, especially on multiple time scales. The empirical studies have shown that the power law decay of inter-arrival times in the network connection session offers a better model than exponential decay. The quest to improve Poisson model led to the formulations of new processes such as non-homogeneous Poisson process, Cox point process, higher dimensional Poisson process, etc. The fractional generalizations of the Poisson process has drawn the attention of many researchers since the last decade. Recent works on fractional extensions of the Poisson process, commonly known as the fractional Poisson processes, lead to some interesting connections between the areas of fractional calculus, stochastic subordination and renewal theory. The state probabilities of such processes are governed by the systems of fractional differential equations which display a slowly decreasing memory. It seems a characteristic feature of all real systems. Here, we discuss some recently introduced generalized counting processes and their fractional variants. The system of differential equations that governs their state probabilities are discussed.
Title: Colloquium: Recent advances in mixed finite element approximation for poroelasticity
Speaker: Arbaz Khan (IIT Roorkee)
Date: Mon, 10 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear response of a porous elastic medium saturated with fluid and a diffusive fluid flow within it, assuming small deformations. This is the starting point for modeling human organs in computational medicine and for modeling the mechanics of permeable rock in geophysics. Finite element methods for Biot’s consolidation model have been widely studied over the past four decades.
In the first part of the talk, we discuss a posteriori error estimators for locking-free mixed finite element approximation of Biot’s consolidation model. The simplest of these is a conventional residual-based estimator. We establish bounds relating the estimated and true errors, and show that these are independent of the physical parameters. The other two estimators require the solution of local problems. These local problem estimators are also shown to be reliable, efficient and robust. Numerical results are presented that validate the theoretical estimates, and illustrate the effectiveness of the estimators in guiding adaptive solution algorithms.
In the second part of the talk, we discuss a novel locking-free stochastic Galerkin mixed finite element method for the Biot consolidation model with uncertain Young’s modulus and hydraulic conductivity field. After introducing a five-field mixed variational formulation of the standard Biot consolidation model, we discuss stochastic Galerkin mixed finite element approximation, focusing on the issue of well-posedness and efficient linear algebra for the discretized system. We introduce a new preconditioner for use with MINRES and establish eigenvalue bounds. Finally, we present specific numerical examples to illustrate the efficiency of our numerical solution approach.
Finally, we discuss some remarks related to non-conforming approximation of Biot’s consolidation model.
Title: Colloquium: Chromatic symmetric functions and the Borcherds–Kac–Moody Lie algebras
Speaker: G. Arunkumar (IIT Dharwad)
Date: Fri, 07 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Let $\mathfrak g$ be a Borcherds algebra with the associated graph $G$. We prove that the chromatic symmetric function of $G$ can be recovered from the Weyl denominators of $\mathfrak g$, and this gives a Lie theoretic proof of Stanley’s expression for chromatic symmetric function in terms of power sum symmetric functions. Also, this gives an expression for the chromatic symmetric function of $G$ in terms of root multiplicities of $\mathfrak g$. We prove a modified Weyl denominator identity for Borcherds algebras which is an extension of the celebrated classical Weyl denominator identity, and this plays an important role in the proof our results. The absolute value of the linear coefficient of the chromatic polynomial of $G$ is known as the chromatic discriminant of $G$. As an application of our main theorem, we prove that certain coefficients appearing in the above said expression of chromatic symmetric function is equal to the chromatic discriminant of $G$. Also, we find a connection between the Weyl denominators and the $G$-elementary symmetric functions. Using this connection, we give a Lie-theoretic proof of non-negativity of coefficients of $G$-power sum symmetric functions. I will also talk about the plethysms of chromatic symmetric functions.
Title: Colloquium: Intersection cohomology and torus actions of complexity one
Speaker: Narasimha Chary Bonala (Ruhr-Universität Bochum, Germany)
Date: Wed, 05 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Intersection cohomology is a cohomology theory for describing the topology of singular algebraic varieties. We are interested in studying intersection cohomology of complete complex algebraic varieties endowed with an action of an algebraic torus. An important invariant in the classification of torus actions is the complexity. It is defined as the codimension of a general torus orbit. Classification of torus actions is intimately related to questions of convex geometry.
In this talk, we focus on the calculation of the (rational) intersection cohomology Betti numbers of complex complete normal algebraic varieties with a torus action of complexity one. Intersection cohomology for the surface and toric cases was studied by Stanley, Fieseler–Kaup, Braden–MacPherson and many others. We suggest a natural generalisation using the geometric and combinatorial approach of Altmann, Hausen, and Süß for normal varieties with a torus action in terms of the language of divisorial fans.
Title: PhD Thesis defence: Homogenization of certain PDEs and associated optimal control problems on various rough domains
Speaker: Abu Sufian (IISc Mathematics)
Date: Tue, 04 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Several critical physical properties of a material are controlled by its geometric construction. Therefore, analyzing the effect of a material’s geometric structure can help to improve some of its beneficial physical properties and reduce unwanted behavior. This leads to the study of boundary value problems in complex domains such as perforated domain, thin domain, junctions of the thin domain of different configuration, domain with rapidly oscillating boundary, networks domain, etc.
This talk will discuss various homogenization problems posed on high oscillating domains. We discuss in detail one of the articles (see, Journal of Differential Equations 291 (2021): 57-89.), where the oscillatory part is made of two materials with high contrasting conductivities. Thus the low contrast material acts as near insulation in-between the conducting materials. Mathematically this leads to the study of degenerate elliptic PDE at the limiting scale. We also briefly explain another interesting article (see, ESAIM: Control, Optimisation, and Calculus of Variations 27 (2021): S4.), where the oscillations are on the curved interface with general cost functional.
In the first part of my talk, I will briefly discuss the periodic unfolding method and its construction as it is the main tool in our analysis.
The second part of this talk will briefly discuss the boundary optimal control problems subject to Laplacian and Stokes systems.
In the third part of the talk, we will discuss the homogenization of optimal control problems subject to a elliptic variational form with high contrast diffusivity coefficients. The interesting result is the difference in the limit behavior of the optimal control problem, which depends on the control’s action, whether it is on the conductive part or insulating part. In both cases, we derive the \two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete-scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We obtain the homogenized equation for the state, but the two-scale separation of the cost functional remains as an open question.
Title: Colloquium: Shift invariant subspaces of composition operators
Speaker: P. Muthukumar (ISI, Bangalore)
Date: Mon, 03 Jan 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Let $H^2$ denote the Hardy space on the open unit disk $\mathbb{D}$. For a given holomorphic self map $\varphi$ of $\mathbb{D}$, the composition operator $C_\varphi$ on $H^2$ is defined by $C_\varphi(f) = f \circ \varphi$. In this talk, we discuss about Beurling type invariant subspace of composition operators, that is common invariant subspaces of shift (multiplication) and composition operators. We will also consider the model spaces that are invariant under composition operators.
Title: Geometry & Topology Seminar: Fully nonlinear PDEs on complex manifolds
Speaker: George Mathew (Ohio State University, USA)
Date: Thu, 16 Dec 2021
Time: 11:00 am
Venue: LH-1, Mathematics Department
Since the Calabi conjecture was proved in 1978 by S.T. Yau, there has been extensive studies into nonlinear PDEs on complex manifolds. In this talk, we consider a class of fully nonlinear elliptic PDEs involving symmetric functions of partial Laplacians on Hermitian manifolds. This is closely related to the equation considered by Székelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Under fairly general assumptions, we derive apriori estimates and show the existence of solutions. In addition, we also consider the parabolic counterpart of this equation and prove the long-time existence and convergence of solutions.
Title: PhD Thesis defence: Finite Element Analysis of Dirichlet Boundary Control Problems Governed by Certain PDEs
Speaker: Ramesh Chandra Sau (IISc Mathematics)
Date: Wed, 15 Dec 2021
Time: 10 am
Venue: Microsoft Teams (online)
The study of the optimal control problems governed by partial differential equations(PDEs) have been a significant research area in applied mathematics and its allied areas. The optimal control problem consists of finding a control variable that minimizes a cost functional subject to a PDE. In this talk, I will present finite element analysis of Dirichlet boundary optimal control problems governed by certain PDEs. This talk will be divided into four parts.
In the first part, we discuss the Dirichlet boundary control problem, its physical interpretation, mathematical formulation, and some approaches (numerical) to solve it.
In the second part, we study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Poisson equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and co-state variables. We propose a finite element-based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. We present the analysis for $L^2$ cost functional, but this analysis can also be extended to the gradient cost functional problem. A priori error estimates of optimal order in the energy norm are derived up to the regularity of the solution.
In the third part, we discuss the Dirichlet boundary optimal control problem governed by the Stokes equation. We develop a finite element discretization by using $\mathbf{P}_1$ elements (in the fine mesh) for the velocity and control variable and $P_0$ elements (in the coarse mesh) for the pressure variable. We present a posteriori error estimators for the error in the state, co-state, and control variables. As a continuation of the second part, we extend our ideas to the linear parabolic equation in the last part of the presentation. The space discretization of the state and co-state variables is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. We use $H^1$-conforming 3D finite elements for the control variable. We present the error estimates of state, adjoint state, and control.
Title: PhD Thesis defence: Positivity preservers forbidden to operate on diagonal blocks
Speaker: Prateek Kumar Vishwakarma (IISc Mathematics)
Date: Fri, 03 Dec 2021
Time: 9 am
Venue: Microsoft Teams (online)
The question of which functions acting entrywise preserve positive
semidefiniteness has a long history, beginning with the Schur product
theorem [Crelle 1911], which implies that absolutely monotonic
functions (i.e., power series with nonnegative coefficients) preserve
positivity on matrices of all dimensions. A famous result of Schoenberg
and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are
no other such functions.
Motivated by modern applications, Guillot and Rajaratnam
[Trans. Amer. Math. Soc. 2015] classified the entrywise positivity
preservers in all dimensions, which act only on the off-diagonal entries.
These two results are at “opposite ends”, and in both cases the preservers
have to be absolutely monotonic.
The goal of this thesis is to complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
The talk will begin by discussing connections between metric geometry and positivity, also via positive definite functions. Following this, we present Schoenberg’s motivations in studying entrywise positivity preservers, followed by classical variants for matrices with entries in other real and complex domains. Then we shall see the result due to Guillot and Rajaratnam on preservers acting only on the off-diagonal entries, touching upon the modern motivation behind it. This is followed by its generalization in the thesis. Finally, we present the (remaining) main results in the thesis, and conclude with some of the proofs.
Title: Geometry & Topology Seminar: On theta-positive surface subgroups in PO(p,q)
Speaker: Maria Beatrice Pozzetti (Heidelberg University)
Date: Wed, 01 Dec 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
Surprisingly there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups only consisting of injective representations with discrete image. Guichard and Wienhard introduced the notion of $\Theta$ positive representations as a conjectural framework to explain this phenomena. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of $\Theta$ positive representations in PO(p,q). As an application we deduce that they indeed form connected components of character varieties.
Title: Number Theory Seminar: A Multiplicity one theorem and Gelfand criterion
Speaker: Shiv Prakash Patel (IIT Delhi)
Date: Fri, 26 Nov 2021
Time: 2 pm
Venue: Microsoft Teams (Online)
Let $H$ be a subgroup of a group $G$. For an irreducible representation $\sigma$ of $H$, the triple $(G,H, \sigma)$ is called a Gelfand triple if $\sigma$ appears at most once in any irreducible representation of $G$. Given a triple, it is usually difficult to determine whether a given triple is a Gelfand triple. One has a sufficient condition which is geometric in nature to determine if a given triple is a Gelfand triple, called Gelfand criterion. We will discuss some examples of the Gelfand triple which give us multiplicity one theorem for non-degenerate Whittaker models of ${\mathrm GL}_n$ over finite chain rings, such as $\mathbb{Z}/p^n\mathbb{Z}$.
This is a joint work with Pooja Singla.
Title: Geometry & Topology Seminar: Hermitian-Einstein metrics for reflexive sheaves on normal varieties
Speaker: Richard Wentworth (University of Maryland)
Date: Wed, 24 Nov 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
I will discuss some aspects of a singular version of the Donaldson-Uhlenbeck-Yau theorem for bundles and sheaves over normal complex varieties satisfying some conditions. Several applications follow, such as a characterization of the case of equality in the Bogomolov-Gieseker theorem. Such singular metrics also arise naturally under certain types of degenerations, and I will make some comments on the relationship between this result and the Mehta-Ramanathan restriction theorem.
Title: APRG Seminar: Geometric dilation and the quantum annulus
Speaker: James E. Pascoe (University of Florida, Gainesville, USA)
Date: Wed, 24 Nov 2021
Time: 5:30 pm
Venue: Microsoft Teams (online)
The von Neumann inequality says the value of a polynomial at a contractive operator is bounded by the norm of the polynomial on the disk. The von Neumann inequality is often proven using the Sz.-Nagy dilation theorem, which essentially says that one can model a contraction by a unitary operator. We adapt a technique of Nelson for proving the von Neumann inequality: one considers the singular value decomposition and then replaces the singular values with automorphisms of the disk to obtain a matrix valued analytic function which must attain its maximum on the boundary. Moreover, the matrix valued function involved in fact gives a minimal unitary dilation. With McCullough, we adapt Nelson’s trick to various other classes of operators to obtain their dilation theory, including the quantum annulus, row contractions and doubly commuting contractions. We conjecture a geometric relationship between Ando’s inequality and Gerstenhaber’s theorem.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Dilatation for random point-pushing pseudo-Anosovs
Speaker: Tarik Aougab (Haverford)
Date: Wed, 17 Nov 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
Choose a homotopically non-trivial curve “at random” on a compact orientable surface- what properties is it likely to have? We address this when, by “choose at random”, one means running a random walk on the Cayley graph for the fundamental group with respect to the standard generating set. In particular, we focus on self-intersection number and the location of the metric in Teichmuller space minimizing the geodesic length of the curve. As an application, we show how to improve bounds (due to Dowdall) on dilatation of a point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of the defining curve, for a “random” point-pushing map. This represents joint work with Jonah Gaster.
Title: APRG Seminar: Homogeneous quantum symmetries of finite spaces over the circle group
Speaker: Sutanu Roy (NISER, Bhubaneswar)
Date: Wed, 17 Nov 2021
Time: 4 pm
Venue: Microsoft Teams (online)
In 1998 Shuzhou Wang, in his pioneering work, introduced quantum symmetry groups of finite spaces motivated by a general question posed by Alain Connes: what is the quantum automorphism group of a space? By finite spaces, here we mean finite-dimensional C*-algebras. Wang’s results have initiated several fundamental developments in operator algebras, quantum groups and noncommutative geometry. Let us consider a generalised situation where we shall equip the finite spaces with a continuous action of the circle group. This talk aims to understand the object that captures the quantum symmetries of these systems and their applications.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: Hermitian metrics and singular Riemann surface foliations
Speaker: Sahil Gehlawat (IISc Mathematics)
Date: Wed, 17 Nov 2021
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, we will see an interplay between hermitian metrics and singular Riemann surface foliations. It will be divided into three parts. The first part of the talk is about the study of curvature properties of complete Kahler metrics on non-pseudoconvex domains. Examples of such metrics were constructed by Grauert in 1956, who showed that it is possible to construct complete Kahler metrics on the complement of complex analytic sets in a domain of holomorphy. In particular, he gave an explicit example of a complete Kahler metric (the Grauert metric) on $\mathbb{C}^n \setminus {0}$. We will confine ourselves to the study of such complete Kahler metrics. We will make some observations about the holomorphic sectional curvature of such metrics in two prototype cases, namely (i) $\mathbb{C}^n \setminus {0}$, $n>1$, and (ii) $(B^N)\setminus A$, where $A$ is an affine subspace. We will also study complete Kahler metrics using Grauert’s construction on the complement of a principal divisor in a domain of holomorphy and show that there is an intrinsic continuity in the construction of this metric, i.e., we can choose this metric in a continuous fashion if the corresponding principal divisors vary continuously in an appropriate topology.
The second part of the talk deals with Verjovsky’s modulus of uniformization that arises in the study of the leaf-wise Poincare metric on a hyperbolic singular Riemann surface lamination. This is a function defined away from the singular locus. One viewpoint is to think of this as a domain functional. Adopting this view, we will show that it varies continuously when the domains vary continuously in the Hausdorff sense. We will also give an analogue of the classical Domain Bloch constant by D. Minda for hyperbolic singular Riemann surface laminations.
In the last part of the talk, we will discuss a parametrized version of the Mattei-Moussu theorem, namely a holomorphic family of holomorphic foliations in $\mathbb{C}^2$ with an isolated singular point at the origin in the Siegel domain are holomorphically equivalent if and only if the holonomy maps of the horizontal separatrix of the corresponding foliations are holomorphically conjugate.
Title: Colloquium: Hyperbolic Polynomials, Helton-Vinnikov curves, and Symmetroids
Speaker: Papri Dey (University of Missouri, USA)
Date: Tue, 16 Nov 2021
Time: 5:30 pm
Venue: Microsoft Teams (online)
In this talk, I shall give a panoramic view of my research work. I shall introduce the notion of hyperbolic polynomials and discuss an algebraic method to test hyperbolicity of a multivariate polynomial w.r.t. some fixed point via sum-of-squares relaxation, proposed in my research work. An important class of hyperbolic polynomials are definite determinantal polynomials. Helton–Vinnikov curves in the projective plane are cut-out by hyperbolic polynomials in three variables. This leads to the computational problem of explicitly producing a symmetric positive definite linear determinantal representation for a given curve. I shall focus on two approaches to this problem proposed in my research work: an algebraic approach via solving polynomial equations, and a geometric-combinatorial approach via scalar product representations of the coefficients and its connection with polytope membership problem. The algorithms to solve determinantal representation problems are implemented in Macaulay2 as a software package DeterminantalRepresentations.m2. Then I shall briefly address the methodologies to find the degree and the defining equations of certain varieties which are obtained as the image of some given varieties of $\mathbb{P}_n$ under coordinate-wise power map, for example the $4 \times 4$ orthostochastic variety. Finally, I shall demonstrate a connection of symmetroids with the real degeneracy loci of matrices.
Title: Number Theory Seminar: Differential Characters of Anderson Modules and $z$-Isocrystals
Speaker: Arnab Saha (IIT Gandhinagar)
Date: Fri, 12 Nov 2021
Time: 2 pm
Venue: Microsoft Teams (Online)
The theory of $\delta$-geometry was developed by A. Buium based on the analogy with differential algebra where the analogue of a differential operator is played by a $\pi$-derivation $\delta$. A $\pi$-derivation $\delta$ arises from the $\pi$-typical Witt vectors and naturally associates with a lift of Frobenius $\phi$. In this talk, we will discuss the theory of $\delta$-geometry for Anderson modules. Anderson modules are higher dimensional generalizations of Drinfeld modules.
As an application of the above, we will construct a canonical $z$-isocrystal $\mathbb{H}(E)$ with a Hodge- Pink structure associated to an Anderson module $E$ defined over a $\pi$-adically complete ring $R$ with a fixed $\pi$-derivation $\delta$ on it. Depending on a $\delta$-modular parameter, we show that the $z$-isocrystal $\mathbb{H}(E)$ is weakly admissible in the case of Drinfeld modules of rank $2$. Hence, by the analogue of Fontaine’s mysterious functor in the positive characteristic case (as constructed by Hartl), one associates a Galois representation to such an $\mathbb{H}(E)$. The relation of our construction with the usual Galois representation arising from the Tate module of $E$ is currently not clear. This is a joint work with Sudip Pandit.
Title: Colloquium: Non-positive curvature in groups
Speaker: Radhika Gupta (Temple University, USA)
Date: Thu, 11 Nov 2021
Time: 6:30 pm
Venue: Microsoft Teams (online)
A finitely generated group can be viewed as the group of symmetries of a metric space, for example its Cayley graph. When the metric space has non-positive curvature, then the group satisfies some exceptional properties. In this talk, I will introduce two notions of non-positive curvature – CAT(0) and delta hyperbolic. I will present some results comparing groups acting on such spaces. I will also talk about the group of outer automorphisms of a free group, which itself is neither CAT(0) nor delta-hyperbolic, but still benefits a lot from the presence of non-positive curvature.
Title: APRG Seminar: Deep Learning for PDEs
Speaker: Siddhartha Mishra (ETH Zurich, Switzerland)
Date: Wed, 10 Nov 2021
Time: 5 pm
Venue: Microsoft Teams (online)
Machine Learning, particularly Deep Learning, algorithms are being increasingly used to approximate solutions of partial differential equations (PDEs). We survey recent results on different aspects of deep learning in the context of PDEs namely, 1) Supervised learning for high-dimensional parametrized PDEs 2) Operator learning for approximating infinite-dimensional operators which arise in PDEs and 3) Physics informed Neural Networks for approximating both forward and inverse problems for PDEs. We will highlight open questions in the analysis of deep learning algorithms for PDEs.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Singular metric with positive scalar curvature
Speaker: Man-Chun Lee (The Chinese University of Hong Kong)
Date: Wed, 10 Nov 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
A celebrated theorem of Gromov-Lawson and Schoen-Yau states that a n-torus cannot admit metrics with positive scalar curvature. Thus, the torus is of vanishing Yamabe type. In this talk, we will discuss its extension to metrics with some singularity. This is a joint work with L.-F. Tam.
Title: PhD Thesis defence: On certain invariant measures for correspondences, their analysis, and an application to recurrence
Speaker: Mayuresh Londhe (IISc Mathematics)
Date: Tue, 09 Nov 2021
Time: 10:30 am
Venue: Microsoft Teams (online)
In this thesis, we analyse certain dynamically interesting measures arising in holomorphic dynamics beyond the classical framework of maps. We will consider measures associated with semigroups and, more generally, with meromorphic correspondences, that are invariant in a specific sense. Our results are of two different flavours. The first type of results deal with potential-theoretic properties of the measures associated with certain polynomial semigroups, while the second type of results are about recurrence phenomena in the dynamics of meromorphic correspondences. The unifying features of these results are the use of the formalism of correspondences in their proofs, and the fact that the measures that we consider are those that describe the asymptotic distribution of the iterated inverse images of a generic point.
The first class of results involve giving a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh–Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which we can describe explicitly given a choice of a set of generators. In particular, we generalize the classical result of Brolin to certain finitely generated polynomial semigroups. To do so, we establish the continuity of the logarithmic potential for the Dinh–Sibony measure, which might also be of independent interest. Thereafter, we use the $F$-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.
The second class of results involves meromorphic correspondences. These are, loosely speaking, multi-valued analogues of meromorphic maps. We shall present an analogue of the Poincare recurrence theorem for meromorphic correspondences with respect to the measures alluded to above. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result on the invariance properties of the supports of the measures mentioned, and, as a corollary, give a geometric description of the support of such a measure.
Title: Number Theory Seminar: Frobenius trace distributions for Gaussian hypergeometric functions
Speaker: Neelam Saikia (University of Virginia, USA)
Date: Fri, 05 Nov 2021
Time: 8.30 pm
Venue: Microsoft Teams (Online)
In the 1980’s, Greene defined hypergeometric functions over finite fields using
Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric
series studied by Gauss, Kummer and others. These functions have played important roles in the study of Apery-style supercongruences, the Eichler-Selberg trace
formula, Galois representations, and zeta-functions of arithmetic varieties. In this
talk we discuss the distributions (over large finite fields) of natural families of these
functions. For the $_2F_1$ functions, the limiting distribution is semicircular, whereas
the distribution for the $_3F_2$ functions is Batman distribution.
Title: PhD Thesis defence: Hardy's inequalities for Grushin operator and Hermite multipliers on modulation spaces
Speaker: Rakesh Kumar (IISc Mathematics)
Date: Wed, 03 Nov 2021
Time: 4 pm
Venue: LH-1, Mathematics Department
We prove Hardy’s inequalities for the fractional power of Grushin operator $\mathcal{G}$ which is chased via two different approaches. In the first approach, we first prove Hardy’s inequality for the generalized sublaplacian. We first find Cowling–Haagerup type of formula for the fractional sublaplacian and then using the modified heat kernel, we find integral representations of the fractional generalized sublaplacian. Then we derive Hardy’s inequality for generalized sublaplacian. Finally using the spherical harmonics, applying Hardy’s inequality for individual components, we derive Hardy’s inequality for Grushin operator. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\mathbb{R}^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\mathbb{R}^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\mathcal{G}_s f$ in $L^p(\mathbb{R}^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy–Littlewood–Sobolev inequality for the Grushin operator.
Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\mathbb{R}^n)$. We find a relation between the boundedness of sublaplacian multipliers $m(\tilde{\mathcal{L}})$ on polarised Heisenberg group $\mathbb{H}^n_{pol}$ and the boundedness of Hermite multipliers $m(\mathcal{H})$ on modulation spaces $M^{p,q}(\mathbb{R}^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe those conditions on multipliers are more than required restrictive. We improve the results for the special case $p=q$ of the modulation spaces $M^{p,q}(\mathbb{R}^n)$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}(\mathbb{R}^n)$ and the boundedness of Fourier multipliers on torus $\mathbb{T}^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr"odinger equation related to Hermite on modulation spaces.
Title: Number Theory Seminar: Recent applications of delta symbol
Speaker: Ritabrata Munshi (ISI Kolkata)
Date: Fri, 29 Oct 2021
Time: 2 pm
Venue: Microsoft Teams (Online)
The delta symbol is the key in solving many different problems in the analytic theory of numbers. In recent years this has been used to solve various sub-convexity problems for higher rank $L$-functions. This talk will be a brief report on some new progresses. In particular, I will mention the results obtained in recent joint works with Roman Holowinsky & Zhi Qi and Sumit Kumar & Saurabh Singh.
Title: APRG Seminar: Local theory of stable polynomials and bounded rational functions of several variables
Speaker: Greg Knese (Washington University in St. Louis, USA)
Date: Wed, 27 Oct 2021
Time: 4:30 pm
Venue: Microsoft Teams (online)
We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess non-tangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial $p$, we analyze the ideal of numerators $q$ such that $q/p$ is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of $p$. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk. Joint work with Bickel, Pascoe, Sola.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Teichmüller spaces, quadratic differentials, and cluster coordinates
Speaker: Dylan Allegretti (University of British Columbia)
Date: Wed, 27 Oct 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
In the late 1980s, Nigel Hitchin and Michael Wolf independently discovered a parametrization of the Teichmüller space of a compact surface by holomorphic quadratic differentials. In this talk, I will describe work in progress on a generalization of their result. I will review the definition of the “enhanced Teichmüller space” which has been widely studied in the mathematical physics and cluster algebra literature. I will then describe a version of the result of Hitchin and Wolf which relates meromorphic quadratic differentials to the enhanced Teichmüller space. This builds on earlier work by a number of authors, including Wolf, Lohkamp, Gupta, and Biswas-Gastesi-Govindarajan.
Title: Number Theory Seminar: On equidistribution of Gauss sums of cuspidal representations of $GL_d(\mathbb{F}_q)$
Speaker: C. S. Rajan (TIFR Mumbai)
Date: Fri, 22 Oct 2021
Time: 2 pm
Venue: Microsoft Teams (Online)
We investigate the distribution of the angles of Gauss sums attached to the cuspidal representations of general linear groups over finite fields. In particular we show that they happen to be equidistributed with respect to the Haar measure. However, for representations of $PGL_2(\mathbb{F}_q)$, they are clustered around $1$ and $-1$ for odd $p$ and around $1$ for $p=2$. This is joint work with Sameer Kulkarni.
Title: PhD Thesis defence: On the geometry and operator theory of the bidisc and the symmetrized bidisc
Speaker: Anindya Biswas (IISc Mathematics)
Date: Thu, 21 Oct 2021
Time: 11:30 am
Venue: Microsoft Teams (online)
This work is concerned with the geometric and operator theoretic aspects of the bidisc and the symmetrized bidisc. First, we have focused on the geometry of these two domains. The symmetrized bidisc, a non-homogeneous domain, is partitioned into a collection of orbits under the action of its automorphism group. We investigate the properties of these orbits and pick out some necessary properties so that the symmetrized bidisc can be characterized up to biholomorphic equivalence. As a consequence, among other things, we have given a new defining condition of the symmetrized bidisc and we have found that a biholomorphic copy of the symmetrized bidisc defined by E. Cartan. This work on the symmetrized bidisc also helps us to develop a characterization of the bidisc. Being a homogeneous domain, the bidisc’s automorphism group does not reveal much about its geometry. Using the ideas from the work on the symmetrized bidisc, we have identified a subgroup of the automorphism group of the bidisc and observed the corresponding orbits under the action of this subgroup. We have identified some properties of these orbits which are sufficient to characterize the bidisc up to biholomorphic equivalence.
Turning to operator theoretic work on the domains, we have focused mainly on the Schur Agler class class on the bidisc and the symmetrized bidisc. Each element of the Schur Agler class on these domains has a nice representation in terms of a unitary operator, called the realization formula. We have generalized the ideas developed in the context of the bidisc and the symmetrized bidisc and applied it to the Nevanlinna problem and the interpolating sequences. It turns out, our generalization works for a number of domains, such as annulus and multiply connected domains, not only the bidisc and the symmetrized bidisc.
Title: APRG Seminar: Lebesgue space estimates for spherical maximal functions on Heisenberg groups
Speaker: Rajula Srivastava (University of Wisconsin at Madison, USA)
Date: Wed, 20 Oct 2021
Time: 6:30 pm
Venue: Microsoft Teams (online)
We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson–Sjölin–Hörmander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: On uniform estimates for complex Monge-Ampere equations
Speaker: Bin Guo (Rutgers University)
Date: Wed, 20 Oct 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
We will discuss the $L^\infty$ estimates for a class of fully nonlinear partial differential equations on a compact Kahler manifold, which includes the complex Monge-Ampere and Hessian equations. Our approach is purely based on PDE methods, and is free of pluripotential theory. We will also talk about some generalizations to the stability of MA and Hessian equations. This is based on joint works with D.H. Phong and F. Tong.
Title: Geometry & Topology Seminar: Connected components of strata of k-differentials
Speaker: Quentin Gendron (CIMAT)
Date: Wed, 13 Oct 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
The k-differentials are sections of the tensorial product of the canonical bundle of a complex algebraic curves. Fixing a partition (m_1,…,m_n) of k(2g-2), we can define the strata of k-differentials of type (m_1,…,m_n) to be the space of k-differentials on genus g curves with zeroes of orders m_i. After checking that the strata or not empty, the first interesting topological question about these strata is the classification of their connected component. In the case k=1, this was settled in an important paper of Kontsevich and Zorich. This result was extend to k=2 by Lanneau, with corrections of Chen-Möller. The classification is unknown for k greater or equal to 3 as soon as g is greater or equal to 2. In this talk, I will present partial results on this classification problem obtained together with Dawei Chen (arXiv:2101.01650) and in progress with Andrei Bogatyrev. In particular, I will highlight the way Pell-Abel equation appears in this problem.
Title: APRG Seminar: Injectivity of spherical means on $H$-type groups
Speaker: E.K. Narayanan (IISc Mathematics)
Date: Wed, 13 Oct 2021
Time: 4 pm
Venue: Microsoft Teams (online)
We consider three different spherical means on a Heisenberg type group. First, the standard spherical means, which is the average of a function over the spheres in the complement of the center of the group, second is the average over product of spheres in the center and its complement and the third one over spheres defined by a homogeneous norm on the group. We establish injectivity results for these means on $L^p$ spaces for the range $1 \leq p \leq 2m/(m-1)$ where $m$ is the dimension of the center. Our results extend and generalize S. Thangavelu’s results for spherical means on the Heisenberg group. (Joint work with P. K. Sanjay and K. T. Yasser)
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Families of $(\varphi, \tau)$-modules and Galois representations
Speaker: Aditya Karnataki (Beijing International Center for Mathematical Research)
Date: Fri, 08 Oct 2021
Time: 2 pm
Venue: Microsoft Teams (Online)
Let $K$ be a finite extension of $\mathbb{Q}_p$. The theory of $(\varphi, \Gamma)$-modules constructed by Fontaine provides a good category to study $p$-adic representations of the absolute Galois group $Gal(\bar{K}/K)$. This theory arises from a ‘‘devissage’’ of the extension $\bar{K}/K$ through an intermediate extension $K_{\infty}/K$ which is the cyclotomic extension of $K$. The notion of $(\varphi, \tau)$-modules generalizes Fontaine’s constructions by using Kummer extensions other than the cyclotomic one. It encapsulates the important notion of Breuil-Kisin modules among others. It is thus desirable to establish properties of $(\varphi, \tau)$-modules parallel to the cyclotomic case. In this talk, we explain the construction of a functor that associates to a family of $p$-adic Galois representations a family of $(\varphi, \tau)$-modules. The analogous functor in the $(\varphi, \Gamma)$-modules case was constructed by Berger and Colmez . This is joint work with Leo Poyeton.
Title: APRG Seminar: An analogue of Beurling's theorem for the Heisenberg group
Speaker: S. Thangavelu (IISc Mathematics)
Date: Wed, 06 Oct 2021
Time: 4 pm
Venue: Microsoft Teams (online)
A theorem attributed to Beurling for the Fourier transform pairs asserts that for any nontrivial function $f$ on $\mathbb{R}$ the bivariate function $ f(x) \hat{f}(y) e^{|xy|} $ is never integrable over $ \mathbb{R}^2.$ Well known uncertainty principles such as theorems of Hardy, Cowling–Price etc. follow from this interesting result. In this talk we explore the possibility of formulating (and proving!) an analogue of Beurling’s theorem for the operator valued Fourier transform on the Heisenberg group.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Uniqueness of certain cylindrical tangent cones
Speaker: Gabor Szekelyhidi (Notre Dame)
Date: Wed, 06 Oct 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over S^3 x S^3. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.
Title: Number Theory Seminar: Generic representations of $p$--adic unitary groups
Speaker: Santosh Nadimpalli (IIT Kanpur)
Date: Fri, 01 Oct 2021
Time: 2 pm
Venue: Microsoft Teams (Online)
In this talk, we will discuss genericity of cuspidal representations of $p$-adic unitary groups. Generic representations play a central role in the local Langlands correspondences and explicit knowledge of such representations will be useful in understanding the local Langlands correspondence in a more explicit way. After a brief review of $p$-adic unitary groups, their unipotent subgroups, Whittaker functionals and genericity of cuspidal representations in this context, we will discuss the arithmetic nature of the problem.
Title: PhD Thesis defence: Nodal sets of random functions
Speaker: Lakshmi Priya M. E. (IISc Mathematics)
Date: Fri, 01 Oct 2021
Time: 4:15 pm
Venue: Microsoft Teams (online)
This thesis is devoted to the study of nodal sets of random functions. The random functions and the specific aspect of their nodal set that we study fall into two broad categories: nodal component count of Gaussian Laplace eigenfunctions and volume of the nodal set of centered stationary Gaussian processes (SGPs) on $\mathbb{R}^d$, $d \geq 1$.
Gaussian Laplace eigenfunctions: Nazarov–Sodin pioneered the study of nodal component count for Gaussian Laplace eigenfunctions; they investigated this for random spherical harmonics (RSH) on the two-dimensional sphere $S^2$ and established exponential concentration for their nodal component count. An analogous result for arithmetic random waves (ARW) on the $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$, was established soon after by Rozenshein.
We establish concentration results for the nodal component count in the following three instances: monochromatic random waves (MRW) on growing Euclidean balls in $\R^2$; RSH and ARW, on geodesic balls whose radius is slightly larger than the Planck scale, in $S^2$ and $\mathbb{T}^2$ respectively. While the works of Nazarov–Sodin heavily inspire our results and their proofs, some effort and a subtler treatment are required to adapt and execute their ideas in our situation.
Stationary Gaussian processes: The study of the volume of nodal sets of centered SGPs on $\mathbb{R}^d$ is classical; starting with Kac and Rice’s works, several studies were devoted to understanding the nodal volume of Gaussian processes. When $d = 1$, under somewhat strong regularity assumptions on the spectral measure, the following results were established for the zero count on growing intervals: variance asymptotics, central limit theorem and exponential concentration.
For smooth centered SGPs on $\mathbb{R}^d$, we study the unlikely event of overcrowding of the nodal set in a region; this is the event that the volume of the nodal set in a region is much larger than its expected value. Under some mild assumptions on the spectral measure, we obtain estimates for probability of the overcrowding event. We first obtain overcrowding estimates for the zero count of SGPs on $\mathbb{R}$, we then deal with the overcrowding question in higher dimensions in the following way. Crofton’s formula gives the nodal set’s volume in terms of the number of intersections of the nodal set with all lines in $\mathbb{R}^d$. We discretize this formula to get a more workable version of it and, in a sense, reduce this higher dimensional overcrowding problem to the one-dimensional case.
Title: Geometry & Topology Seminar: Triangulated surfaces in moduli space
Speaker: Sahana Vasudevan (MIT)
Date: Wed, 29 Sep 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
Triangulated surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the study of random triangulated surfaces in the large genus setting, and proved results about the systole, diameter and Cheeger constant of random triangulated surfaces. Subsequently Mirzakhani proved analogous results for random hyperbolic surfaces. These results, along with many others, suggest that the geometry of random triangulated surfaces mirrors the geometry of random hyperbolic surfaces in the case of large genus asymptotics. In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space.
Title: APRG Seminar: Interpolation of Data by Smooth Functions
Speaker: Charles Fefferman (Princeton University, USA)
Date: Wed, 29 Sep 2021
Time: 6:30 pm
Venue: Microsoft Teams (online)
Let $X$ be your favorite Banach space of continuous functions on $\mathbb{R}^n$. Given a real-valued function $f$ defined on some (possibly awful) set $E$ in $\mathbb{R}^n$, how can we decide whether $f$ extends to a function $F$ in $X$? If such an $F$ exists, then how small can we take its norm? Can we make $F$ depend linearly on $f$? What can we say about the derivatives of $F$ at or near points of $E$ (assuming $X$ consists of differentiable functions)?
Suppose $E$ is finite. Can we compute a nearly optimal $F$? How many computer operations does it take? What if we demand merely that $F$ agree approximately with $f$? Suppose we allow ourselves to discard a few data points as “outliers”. Which points should we discard?
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: The Hermitian geometry of Strominger connection
Speaker: Fangyang Zheng (Chongqing Normal University)
Date: Fri, 24 Sep 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
In this talk we will discuss the geometry of Strominger connection of Hermitian manifolds, based on recent joint works with Quanting Zhao. We will focus on two special types of Hermitian manifolds: Strominger Kaehler-like (SKL) manifolds, and Strominger parallel torsion (SPT) manifolds. The first class means Hermitian manifolds whose Strominger connection (also known as Bismut connection) has curvature tensor obeying all Kaehler symmetries, and the second class means Hermitian manifolds whose Strominger conneciton has parallel torsion. We showed that any SKL manifold is SPT, which is known as (an equivalent form of) the AOUV Conjecture (namely, SKL implies pluriclosedness). We obtained a characterization theorem for SPT condition in terms of Strominger curvature, which generalizes the previous theorem. We will also discuss examples and some structural results for SKL and SPT manifolds.
Title: APRG Seminar: Double $N$-picture for the Poisson transform on $G/K$
Speaker: Bernhard Krötz (Universität Paderborn, Germany)
Date: Wed, 22 Sep 2021
Time: 4 pm
Venue: Microsoft Teams (online)
Title: Geometry & Topology Seminar: Uniform Poincare inequalities on measured metric spaces
Speaker: Soma Maity (IISER Mohali)
Date: Wed, 15 Sep 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $\mu.$ We establish a family of uniform Poincare inequalities on $(X,d,\mu)$ if it satisfies a local Poincare inequality ($P_{loc}$), and a condition on the growth of volume. Consequently, if $\mu$ is doubling and supports $(P_{loc})$ then it satisfies a uniform $(\sigma,\beta,\sigma)$-Poincare inequality. If $(X,d,\mu)$ is a Gromov-hyperbolic space, then using the volume comparison theorem introduced by Besson, Courtoise, Gallot, and Sambusetti, we obtain a uniform Poincare inequality with the exponential growth of the Poincare constant. Next, we relate the growth of Poincare constants to the growth of discrete subgroups of isometries of $X$, which act on it properly. This is Joint work with Gautam Nilakantan.
Title: Geometry & Topology Seminar: Stable cohomology of discrimination complements
Speaker: Ishan Banerjee (U Chicago)
Date: Wed, 08 Sep 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
Homological stability is an interesting phenomenon exhibited by many natural sequences of classifying spaces and moduli spaces like the moduli spaces of curves M_g and the moduli spaces of principally polarized abelian varieties A_g. In this talk I will explain some efforts to find similar phenomena in the cohomology of discrimination complements.
Title: Number Theory Seminar: mod $p$ local Langlands correspondence for $GL_2$
Speaker: Mihir Sheth (IISc Mathematics)
Date: Fri, 03 Sep 2021
Time: 2 pm
Venue: Microsoft Teams (Online)
Let $F$ be a non-archimedean local field of residue characteristic $p$. The classical local Langlands correspondence is a 1-1 correspondence between 2-dimensional irreducible complex representations of the Weil group of $F$ and certain smooth irreducible complex representations of $GL_2(F)$. The number-theoretic applications made it necessary to seek such correspondence of representations on vector spaces over a field of characteristic $p$. In this talk, however, I will show that for $F$ of residue degree $> 1$, unfortunately, there is no such 1-1 mod $p$ correspondence. This result is an elaboration of the arguments of Breuil and Paskunas to an arbitrary local field of residue degree $> 1$.
Title: Geometry & Topology Seminar: Steady gradient Ricci solitons with positive curvature operators
Speaker: Yi Lai (UC Berkeley)
Date: Wed, 01 Sep 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operators. We show that these solitons are non-collapsed.
Title: APRG Seminar: $L^p$ bounds for the helical maximal function
Speaker: David Beltran (University of Wisconsin at Madison, USA)
Date: Wed, 01 Sep 2021
Time: 6:30 pm
Venue: Microsoft Teams (online)
A natural 3-dimensional analogue of Bourgain’s circular maximal function theorem in the plane is the study of the sharp $L^p$ bounds in $\mathbb{R}^3$ for the maximal function associated with averages over dilates of the helix (or, more generally, of any curve with non-vanishing curvature and torsion). In this talk, we present a sharp result, which establishes that $L^p$ bounds hold if and only if $p>3$. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.
Title: APRG Seminar: A unified view of space-time covariance functions through Gelfand Pairs
Speaker: Christian Berg (University of Copenhagen, Denmark)
Date: Wed, 01 Sep 2021
Time: 4 pm
Venue: Microsoft Teams (online)
In Geostatistics one examines measurements depending on the location on the earth and on time. This leads to Random Fields of stochastic variables $Z(\xi,u)$ indexed by $(\xi,u)$ belonging to $\mathbb{S}^2\times \mathbb{R}$, where $\mathbb{S}^2$–the 2-dimensional unit sphere–is a model for the earth, and $\mathbb{R}$ is a model for time.
If the variables are real-valued, one considers a basic probability space $(\Omega,\mathcal F,P)$, where all the random variables $Z(\xi,u)$ are defined as measurable mappings from $\Omega$ to $\mathbb{R}$.
One is interested in isotropic and stationary random fields $Z(\xi,u),\;(\xi,u)\in\mathbb{S}^2 \times\mathbb{R}$, i.e., the situation where there exists a continuous function $f:[-1,1] \times \mathbb{R} \to \mathbb{R}$ such that the covariance kernel is given as
\begin{equation} \mbox{cov}(Z(\xi,u),Z(\eta,v))=f(\xi\cdot\eta,v-u),\quad \xi,\eta\in\mathbb{S}^2,\;u,v\in\mathbb{R}. \end{equation}
Here $\xi\cdot\eta=\cos(\theta(\xi,\eta))$ is the scalar product equal to cosine of the length of the geodesic arc (=angle) between $\xi$ and $\eta$.
We require with other words that the covariance kernel only depends on the geodesic distance between the points on the sphere and on the time difference.
Porcu and Berg (2017) gave a characterization of such kernels by having uniformly convergent expansions
\begin{equation} f(x,u)=\sum_{n=0}^\infty b_n(u)P_n(x), \quad \sum_{n=0}^\infty b_n(0)<\infty, \end{equation}
where $(b_n)$ is a sequence of real-valued characteristic (=continuous positive definite) functions on $\mathbb{R}$ and $P_n$ are the Legendre polynomials on $[-1,1]$ normalized as $P_n(1)=1$. The result can be generalized to spheres $\mathbb{S}^d$ of any dimension $d$ and $\mathbb{R}$ can be replaced by an arbitrary locally compact group.
In work of Peron, Porcu and Berg (2018) it was pointed out that the spheres can be replaced by compact homogeneous spaces $G/K$, where $(G,K)$ is a Gelfand pair.
We shall explain the theory of Gelfand pairs and also show how recent work of several people can be extended to this framework.
The presentation is largely based on the recent paper of the speaker with the same title as the talk published in Journal Fourier Analysis and Applications 26 (2020).
Title: PhD Thesis colloquium: Homogenization of certain PDEs and associated optimal control problems on various rough domains
Speaker: Abu Sufian (IISc Mathematics)
Date: Thu, 26 Aug 2021
Time: 11 am
Venue: Microsoft Teams (online)
Several critical physical properties of a material are controlled by its geometric construction. Therefore, analyzing the effect of a material’s geometric structure can help to improve some of its beneficial physical properties and reduce unwanted behavior. This leads to the study of boundary value problems in complex domains such as perforated domain, thin domain, junctions of the thin domain of different configuration, domain with rapidly oscillating boundary, networks domain, etc.
In this thesis colloquium, we will discuss various homogenization problems posed on high oscillating domains. We discuss in detail one of the articles (see, Journal of Differential Equations 291 (2021): 57-89.), where the oscillatory part is made of two materials with high contrasting conductivities. Thus the low contrast material acts as near insulation in-between the conducting materials. Mathematically this leads to the study of degenerate elliptic PDE at the limiting scale. We also briefly explain another interesting article (see, ESAIM: Control, Optimisation, and Calculus of Variations 27 (2021): S4.), where the oscillations are on the curved interface with general cost functional. Due to time constraints, we may not discuss other chapters of the thesis.
In the first part of my talk, I will briefly discuss the periodic unfolding method and its construction as it is the main tool in our analysis.
The second part of the talk will be homogenizing optimal control problems subject to the considered PDEs. The interesting result is the difference in the limit behavior of the optimal control problem, which depends on the control’s action, whether it is on the conductive part or insulating part. In both cases, we derive the two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete-scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We do obtain the homogenized equation for the state, but the two-scale separation of the cost functional remains as an open question.
Title: PhD Thesis colloquium: Near-Optimal Non-malleable Codes and Leakage Resilient Secret Sharing Schemes
Speaker: Sruthi Sekar (IISc Mathematics)
Date: Mon, 09 Aug 2021
Time: 2 pm
Venue: Microsoft Teams (online)
Non-malleable codes (NMCs) are coding schemes that help in protecting crypto-systems under tampering attacks, where the adversary tampers the device storing the secret and observes additional input-output behavior on the crypto-system. NMCs give a guarantee that such adversarial tampering of the encoding of the secret will lead to a tampered secret, which is either same as the original or completely independent of it, thus giving no additional information to the adversary. Leakage resilient secret sharing schemes help a party, called a dealer, to share his secret message amongst $n$ parties in such a way that any $t$ of these parties can combine their shares to recover the secret, but the secret remains hidden from an adversary corrupting $< t$ parties to get their complete shares and additionally getting some bounded bits of leakage from the shares of the remaining parties.
For both these primitives, whether you store the non-malleable encoding of a message on some tamper-prone system or the parties store shares of the secret on a leakage-prone system, it is important to build schemes that output codewords/shares that are of optimal length and do not introduce too much redundancy into the codewords/shares. This is, in particular, captured by the rate of the schemes, which is the ratio of the message length to the codeword length/largest share length. The research goal of the thesis is to improve the state of art on rates of these schemes and get near-optimal/optimal rates.
In this talk, I will specifically focus on leakage resilient secret sharing schemes, describe the leakage model, and take you through the state of the art on their rates. Finally, I will present a recent construction of an optimal (constant) rate, leakage resilient secret sharing scheme in the so-called “joint and adaptive leakage model” where leakage queries can be made adaptively and jointly on multiple shares.
Title: PhD Thesis colloquium: Positivity preservers forbidden to operate on diagonal blocks
Speaker: Prateek Kumar Vishwakarma (IISc Mathematics)
Date: Mon, 26 Jul 2021
Time: 4 pm
Venue: Microsoft Teams (online)
The question of which functions acting entrywise preserve positive
semidefiniteness has a long history, beginning with the Schur product
theorem [Crelle 1911], which implies that absolutely monotonic
functions (i.e., power series with nonnegative coefficients) preserve
positivity on matrices of all dimensions. A famous result of Schoenberg
and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are
no other such functions.
Motivated by modern applications, Guillot and Rajaratnam
[Trans. Amer. Math. Soc. 2015] classified the entrywise positivity
preservers in all dimensions, which act only on the off-diagonal entries.
These two results are at “opposite ends”, and in both cases the preservers
have to be absolutely monotonic.
The goal of this thesis is to complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
The talk will begin by discussing connections between metric geometry and positivity, also via positive definite functions. Following this, we present Schoenberg’s motivations in studying entrywise positivity preservers, followed by classical variants for matrices with entries in other real and complex domains. Then we shall see the result due to Guillot and Rajaratnam on preservers acting only on the off-diagonal entries, touching upon the modern motivation behind it. This is followed by its generalization in the thesis. Finally, we present the (remaining) main results in the thesis, and conclude with some of the proofs.
Title: PhD Thesis colloquium: Finite Element Analysis of Dirichlet Boundary Control Problems Governed by Certain PDEs
Speaker: Ramesh Chandra Sau (IISc Mathematics)
Date: Thu, 22 Jul 2021
Time: 4 pm
Venue: Microsoft Teams (online)
The study of the optimal control problems governed by partial differential equations (PDEs) have been a significant research area in the applied mathematics and its allied areas. The optimal control problem consists of finding a control variable that minimizes a cost functional subject to a PDE. In this talk, I will present finite element analysis of Dirichlet boundary optimal control problems governed by certain PDEs. This talk will be divided into three parts.
In the first part, we study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Poisson equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and co-state variables. We propose a finite element-based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. We present the analysis for $L^2$ cost functional, but this analysis can also be extended to the gradient cost functional problem. A priori error estimates of optimal order in the energy norm are derived up to the regularity of the solution.
In the second part, we discuss the Dirichlet boundary optimal control problem governed by the Stokes equation. We develop a finite element discretization by using $\mathbf{P}_1$ elements (in the fine mesh) for the velocity and control variable and $P_0$ elements (in the coarse mesh) for the pressure variable. We present a new a posteriori error estimator for the control error. This estimator generalizes the standard residual type estimator of the unconstrained Dirichlet boundary control problems by adding terms at the contact boundary that address the non-linearity. We sketch out the proof of the estimator’s reliability and efficiency.
As a continuation of the first part, we extend our ideas to the linear parabolic equation in the third part of this presentation. The space discretization of the state and co-state variables is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. We use $H^1$-conforming 3D finite elements for the control variable. We present a sketch to demonstrate the existence and uniqueness of the solution; and the error estimates of state, adjoint state, and control.
Title: APRG Seminar: Stable Random Fields, Patterson-Sullivan measures and Extremal Cocycle Growth
Speaker: Parthanil Roy (ISI, Bangalore)
Date: Wed, 21 Jul 2021
Time: 4 pm
Venue: Microsoft Teams (online)
We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on a suitable space $X$. The connection between extreme values and the indexing group $G$ is mediated by the action of $G$ on the limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth, which quantifies the distortion of measures on the boundary in comparison to the movement of points in the space $X$. We show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle $U(X/G)$ provided $X/G$ has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric $\alpha$-stable ($0 < \alpha < 2$) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups. (Joint work with Jayadev Athreya and Mahan Mj, under review in Probability Theory and Related Fields.)
Title: PhD Thesis colloquium: Hermitian metrics and singular Riemann surface foliations
Speaker: Sahil Gehlawat (IISc Mathematics)
Date: Fri, 16 Jul 2021
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, we will see an interplay between hermitian metrics and singular Riemann surface foliations. It will be divided into three parts. The first part of the talk is about the study of curvature properties of complete Kahler metrics on non-pseudoconvex domains. Examples of such metrics were constructed by Grauert in 1956 who showed that it is possible to construct complete Kahler metrics on the complement of complex analytic sets in a domain of holomorphy. In particular, he gave an explicit example of a complete Kahler metric (the Grauert metric) on $\mathbb{C}^n \setminus {0}$. We will confine ourselves to the study of such complete Kahler metrics. We will make some observations about the holomorphic sectional curvature of such metrics in two prototype cases, namely (i) $\mathbb{C}^n \setminus {0}$, $n>1$, and (ii) $(B^N)\setminus A$, where $A$ is an affine subspace. We will also study complete Kahler metrics using Grauert’s construction on the complement of a principal divisor in a domain of holomorphy and show that there is an intrinsic continuity in the construction of this metric, i.e., we can choose this metric in a continuous fashion if the corresponding principal divisors vary continuously in an appropriate topology.
The second part of the talk deals with Verjovsky’s modulus of uniformization that arises in the study of the leaf-wise Poincare metric on a hyperbolic singular Riemann surface lamination. This is a function defined away from the singular locus. One viewpoint is to think of this as a domain functional. Adopting this view, we will show that it varies continuously when the domains vary continuously in the Hausdorff sense. We will also give an analogue of the classical Domain Bloch constant by D. Minda for hyperbolic singular Riemann surface laminations.
In the last part of the talk, we will discuss a parametrized version of the Mattei-Moussu theorem namely, a holomorphic family of holomorphic foliations in $\mathbb{C}^2$ with an isolated singular point at the origin in the Siegel domain are holomorphically equivalent if and only if the holonomy maps of the horizontal separatrix of the corresponding foliations are holomorphically conjugate.
Title: PhD Thesis defence: Algorithmic and Hardness Results for Fundamental Fair-Division Problems
Speaker: Nidhi Rathi (IISc Mathematics)
Date: Wed, 14 Jul 2021
Time: 4 pm
Venue: Microsoft Teams (online)
The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges that arise in various settings of fair-division problems and complements the existential (and non-constructive) guarantees and various hardness results by way of developing efficient (approximation) algorithms and identifying computationally tractable instances.
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Our work in fair cake division develops several algorithmic results for allocating a divisible resource (i.e., the cake) among a set of agents in a fair/economically efficient manner. While strong existence results and various hardness results exist in this setup, we develop a polynomial-time algorithm for dividing the cake in an approximately fair and efficient manner. Furthermore, we identify an encompassing property of agents’ value densities (over the cake)—the monotone likelihood ratio property (MLRP)—that enables us to prove strong algorithmic results for various notions of fairness and (economic) efficiency.
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Our work in fair rent division develops a fully polynomial-time approximation scheme (FPTAS) for dividing a set of discrete resources (the rooms) and splitting the money (rents) among agents having general utility functions (continuous, monotone decreasing, and piecewise-linear), in a fair manner. Prior to our work, efficient algorithms for finding such solutions were know n only for a specific set of utility functions. We complement the algorithmic results by proving that the fair rent division problem (under genral utilities) lies in the intersection of the complexity classes, PPAD (Polynomial Parity Arguments on Directed graphs) and PLS (Polynomial Local Search).
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Our work respectively addresses fair division of rent, cake (divisible), and discrete (indivisible) goods in a partial information setting. We show that, for all these settings and under appropriate valuations, a fair (or an approximately fair) division among $n$ agents can be efficiently computed using only the valuations of $n-1$ agents. The $n$th (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.
Title: PhD Thesis colloquium: On certain invariant measures for correspondences, their analysis, and an application to recurrence
Speaker: Mayuresh Londhe (IISc Mathematics)
Date: Tue, 29 Jun 2021
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, we shall focus on certain dynamically interesting measures arising in holomorphic dynamics beyond the classical framework of maps. We will consider measures associated with semigroups and, more generally, with meromorphic correspondences, that are invariant in a specific sense. Our results are of two different flavours. The first type of results deal with potential-theoretic properties of the measures associated with certain polynomial semigroups, while the second type of results are about recurrence phenomena in the dynamics of meromorphic correspondences. The unifying features of these results are the use of the formalism of correspondences in their proofs, and the fact that the measures that we consider are those that describe the asymptotic distribution of the iterated inverse images of a generic point.
The first class of results involve giving a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh–Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which we can describe explicitly given a choice of a set of generators. In particular, we generalize the classical result of Brolin to certain finitely generated polynomial semigroups. To do so, we establish the continuity of the logarithmic potential for the Dinh–Sibony measure, whose proof will be sketched. If time permits, we will discuss bounds on the capacity and diameter of the Julia sets of such semigroups, for which we use the $F$-functional of Mhaskar and Saff.
The second class of results involves meromorphic correspondences. These are, loosely speaking, multi-valued analogues of meromorphic maps. We shall present an analogue of the Poincare recurrence theorem for meromorphic correspondences with respect to the measures alluded to above. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. If time permits, we shall also discuss a result on the invariance properties of the supports of the measures mentioned.
Title: PhD Thesis colloquium: On the geometry and operator theory of the bidisc and the symmetrized bidisc
Speaker: Anindya Biswas (IISc Mathematics)
Date: Wed, 23 Jun 2021
Time: 4 pm
Venue: Microsoft Teams (online)
This work is concerned with the geometric and operator theoretic aspects of the bidisc and the symmetrized bidisc. First, we have focused on the geometry of these two domains. The symmetrized bidisc, a non-homogeneous domain, is partitioned into a collection of orbits under the action of its automorphism group. We investigate the properties of these orbits and pick out some necessary properties so that the symmetrized bidisc can be characterized up to biholomorphic equivalence. As a consequence, among other things, we have given a new defining condition of the symmetrized bidisc and we have found that a biholomorphic copy of the symmetrized bidisc defined by E. Cartan. This work on the symmetrized bidisc also helps us to develop a characterization of the bidisc. Being a homogeneous domain, the bidisc’s automorphism group does not reveal much about its geometry. Using the ideas from the work on the symmetrized bidisc, we have identified a subgroup of the automorphism group of the bidisc and observed the corresponding orbits under the action of this subgroup. We have identified some properties of these orbits which are sufficient to characterize the bidisc up to biholomorphic equivalence.
Turning to operator theoretic work on the domains, we have focused mainly on the Schur Agler class class on the bidisc and the symmetrized bidisc. Each element of the Schur Agler class on these domains has a nice representation in terms of a unitary operator, called the realization formula. We have generalized the ideas developed in the context of the bidisc and the symmetrized bidisc and applied it to the Nevanlinna problem and the interpolating sequences. It turns out, our generalization works for a number of domains, such as annulus and multiply connected domains, not only the bidisc and the symmetrized bidisc.
Title: MS Thesis defence: Random graphs with given degree sequence
Speaker: Neeladri Maitra (IISc Mathematics)
Date: Mon, 14 Jun 2021
Time: 2 pm
Venue: Microsoft Teams (online)
In this talk, we focus on random graphs with a given degree sequence. In the first part, we look at uniformly chosen trees from the set of trees with a given child sequence. A non-negative sequence of integers $(c_1,c_2,\dots,c_l)$ with sum $l-1$ is a child sequence for a rooted tree $t$ on $l$ nodes, if for some ordering $v_1,v_2,\dots,v_l$ of the nodes of $t$, $v_i$ has exactly $c_i$ many children in $t$. We consider for each $n$, a child sequence $\mathbf{c}^{(n)}$, with sum $n-1$, and let $\mathbf{t}_n$ be the random tree having the uniform distribution on the set of all plane trees with $n$ vertices, which has $\mathbf{c}^{(n)}$ as their child sequence. Under the assumption that a finite number of vertices of $\mathbf{t}_n$ has large degrees, we show that the scaling limit of $\mathbf{t}_n$ is the Inhomogeneous Continuum Random Tree (ICRT), in the Gromov-Hausdorff topology. This generalizes a result of Broutin and Marckert from 2012, where they show the scaling limit to be the Brownian Continuum Random Tree (BCRT), under the assumption that no vertex in $\mathbf{t}_n$ has large degree.
In the second part, we look at vacant sets left by random walks on random graphs via simulations. Cerný, Teixeira and Windisch (2011) proved that for random $d$-regular graphs, there is a number $u_*$, such that if a random walk is run up to time $un$ with $u<u_*$, $n$ being the total number of nodes in the graph, a giant component of linear size, in the subgraph spanned by the nodes yet unvisited by the random walk, emerges. Whereas, if the random walk is tun up to time un with $u>u_*$, the size of the largest component, of the subgraph spanned by nodes yet unvisited by the walk, is $\text{o}(n)$. With the help of simulations, we try to look for such a phase transition for supercritical configuration models, with heavy-tailed degrees.
Title: APRG Seminar: On the minimum embedding dimension for polynomially convex embeddings
Speaker: Purvi Gupta (IISc Mathematics)
Date: Wed, 02 Jun 2021
Time: 4 pm
Venue: Microsoft Teams (online)
A compact subset of $\mathbb{C}^n$ is polynomially convex if it is defined by a family of polynomial inequalities. The minimum complex dimension into which all compact real manifolds of a fixed dimension admit smooth polynomially convex embeddings is not known (although there are some obvious bounds).
In this talk, we will discuss some recent improvements on the previously known bounds, especially focusing on the odd-dimensional case, where the embeddings cannot be produced by classical (local) perturbation techniques. This is joint work with R. Shafikov.
Title: APRG Seminar: The Laplacian with mixed Dirichlet-Neumann boundary conditions on Weyl chambers
Speaker: Krzysztof Stempak (Wrocław University of Science and Technology, Poland)
Date: Wed, 26 May 2021
Time: 5 pm
Venue: Zoom (online)
Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $\Omega_+=\Omega\cap C_+$ be the positive part of $\Omega$. We define a family ${-\Delta_{\eta}^+}$ of self-adjoint extensions of the Laplacian $-\Delta_{\Omega_+}$, labeled by homomorphisms $\eta\colon W\to {1,-1}$. In the construction of these $\eta$-Laplacians $\eta$-symmetrization of functions on $\Omega$ is involved. The Neumann Laplacian $-\Delta_{N,\Omega_+}$ is included and corresponds to $\eta\equiv 1$. If $H^{1}(\Omega)=H^{1}_0(\Omega)$, then the Dirichlet Laplacian $-\Delta_{D,\Omega_+}$ is either included and corresponds to $\eta={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $\Psi(-\Delta_{N,\Omega})$ and $\Psi(-\Delta_{\eta}^+)$, or $\Psi(-\Delta_{D,\Omega})$ and $\Psi(-\Delta_{D,\Omega_+})$, where $\Psi$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$.
In the talk, for simplicity, I will focus on the case $\Omega = \R^d$ (so $\Omega_+ = C_+$) and $\Psi = \Psi_t, t > 0$, where $\Psi_t(\lambda) = \exp(−t\lambda)$ for $\lambda > 0$. Then the integral kernels of $\Psi_t(-\Delta^{+}_{\eta})$, called the $\eta$-heat kernels, will be investigated in more detail.
Title: Geometry & Topology Seminar: Connections between Geometry and Analysis on Manifolds and Path Spaces
Speaker: Aaron Naber (Northwestern University)
Date: Wed, 26 May 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
In the last decades there have been many connections made between the analysis of a manifold M and the geometry of M. Said correctly, there are now many ways to make precise that well-behaved analysis on M is ’equivalent’ to the existence of lower bounds on Ricci curvature. Such ideas are the starting point for regularity theories and more abstract settings for analysis, including analysis on metric-measure spaces. We will begin this talk with an elementary review of these ideas. More recently it has become apparent analysis on the path space PM of a manifold is closely connected to two sided bounds on Ricci curvature. Again, said correctly one can make an equivalence that the analysis on PM is well behaved iff M has a two sided Ricci curvature bound. As a general phenomena, one see’s that analytic estimates on M lift to estimates on PM in the presence of two sided Ricci bounds. Our talk will mainly focus on explaining all the words in this abstract and giving some rough understanding of the broad ideas involved. Time allowing, we will briefly explain newer results with Haslhofer/Kopfer on differential harnack inequalities on path space.
Title: APRG Seminar: Recent developments in the mathematics of neural nets
Speaker: Anirbit Mukherjee (Wharton, University of Pennsylvania, USA)
Date: Fri, 21 May 2021
Time: 6 pm
Venue: Microsoft Teams (online)
A profound mathematical mystery of our times is to be able to explain the phenomenon of training neural nets i.e “deep-learning”. The dramatic progress of this approach in the last decade has gotten us the closest we have ever been to achieving “artificial intelligence”. But trying to reason about these successes - for even the simplest of nets - immediately lands us into a plethora of extremely challenging mathematical questions, typically about discrete stochastic processes. In this talk we will describe the various themes of our work in provable deep-learning.
We will start with a brief introduction to neural nets and then see glimpses of our initial work on understanding neural functions, loss functions for autoencoders and algorithms for exact neural training. Next, we will explain our recent result about how under mild distributional conditions we can construct an iterative algorithm which can be guaranteed to train a ReLU gate in the realizable setting in linear time while also keeping track of mini-batching - and its provable graceful degradation of performance under a data-poisoning attack. We will show via experiments the intriguing property that our algorithm very closely mimics the behaviour of Stochastic Gradient Descent (S.G.D.), for which similar convergence guarantees are still unknown.
Lastly, we will review this very new concept of “local elasticity” of a learning process and demonstrate how it appears to reveal certain universal phase transitions during neural training. Then we will introduce a mathematical model which reproduces some of these key properties in a semi-analytic way. We will end by delineating various exciting future research programs in this theme of macroscopic phenomenology with neural nets.
Title: Geometry & Topology Seminar: The Hitchin-Simpson equations over Kaehler Manifolds
Speaker: Siqi He (Simon Centre, Stonybrook)
Date: Wed, 19 May 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
The Hitchin-Simpson equations defined over a Kaehler manifold are first order, non-linear equations for a pair of a connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchin–Simpson equations with norms of these 1-forms unbounded. We will talk about two applications of this compactness theorem, one is the realization problem of the Taubes’ Z2 harmonic 1-form and another is the Hitchin’s WKB problem in higher dimensional. We will also discuss some open questions related to this question.
Title: APRG Seminar: The Schwartz correspondence for the complex motion group of $C^2$
Speaker: Bianca Di Blasio (Universita di Milano, Italy)
Date: Wed, 19 May 2021
Time: 4 pm
Venue: Microsoft Teams (online)
Let $(K\ltimes G,K)$ be a Gelfand pair, where $K\ltimes G$ is the semidirect product of a Lie group $G$ with polynomial growth and $K$ a compact group of automorphisms of $G$. Then the Gelfand spectrum $\Sigma$ of the commutative convolution algebra of $K$-invariant integrable functions on $G$ admits natural embeddings into $\mathbb{R}^n$ spaces as a closed subset. Let $\mathcal{S}(G)^K$ be the space of $K$-invariant Schwartz functions on $G$. Defining $\mathcal{S}(\Sigma)$ as the space of restrictions to $\Sigma$ of Schwartz functions on $\mathbb R^n$, we call Schwartz correspondence for $(K\ltimes G,K)$ the property that the spherical transform is an isomorphism of $\mathcal{S}(G)^K$ onto $\mathcal{S}(\Sigma)$. In all the cases studied so far, the Schwartz correspondence has been proved to hold true. These include all pairs $(K\ltimes G,K)$ with $K$ abelian and a large number of pairs with $G$ nilpotent. In this talk we show that the Schwartz correspondence holds for the pair $(K\ltimes G,K)$, where $G=U_2\ltimes \mathbb{C}^2$ is the complex motion group and $K={\rm Int}(U_2) $ is the group of inner automorphisms of $G$ induced by elements of $U_2$. This is one of the simplest pairs with $G$ non-nilpotent and $K$ non-abelian. This work arises from a collaboration with Francesca Astengo and Fulvio Ricci.
Title: APRG Seminar: On the Schwartz correspondence for Gelfand pairs of polynomial growth
Speaker: Francesca Astengo (Universita di Genova, Italy)
Date: Wed, 12 May 2021
Time: 4 pm
Venue: Microsoft Teams (online)
A result due to Hulanicki (and refined by Veneruso) states that if $m$ is a Schwartz function on $\mathbb{R}^2$ and $L, T$ are the Heisenberg sublaplacian and the central derivative, then the operator $m(L,i^{-1}T)$ has a Schwartz radial convolution kernel $k$. It is therefore natural to ask whether all Schwartz convolution kernels arise in this way. In collaboration with Bianca Di Blasio and Fulvio Ricci we are considering this kind of problem in the context of Gelfand Pairs of polynomial growth. In this talk I will discuss some old and new results.
Title: APRG Seminar: Riesz sequence of the system of left translates on the Heisenberg group
Speaker: R. Radha (IIT, Madras)
Date: Wed, 05 May 2021
Time: 4 pm
Venue: Microsoft Teams (online)
It is well known that the system of translates $\{T_k\phi:k\in\mathbb{Z}\}$ is a Riesz sequence in $L^2(\mathbb{R})$ if and only if there exist $A,B>0$ such that \begin{equation} A\leq\sum_{k\in\mathbb{Z}}|\widehat{\phi}(\xi+k)|^2\leq B\hspace{.5 cm}a.e.\ \xi\in[0,1], \end{equation} where $\widehat{\phi}$ denotes the Fourier transform of $\phi$. This result is very important in time-frequency analysis especially in constructing wavelet basis for $L^2(\mathbb{R})$ using multiresolution analysis technique and also in studying sampling problems in a shift-invariant space.
In this talk, we ask a similar question for the system of left translates $\{ L_\gamma\phi:\gamma\in\Gamma\}$ on the Heisenberg group $\mathbb{H}^n$, where $\phi\in L^2(\mathbb{H}^n)$ and $\Gamma$ is a lattice in $\mathbb{H}^n$. We take $\Gamma= \{(2k,l,m):k,l\in\mathbb{Z}^n,m\in\mathbb{Z}\}$ as the standard lattice in order to avoid computational complexity. Recently it has been proved that if $\phi\in L^2(\mathbb{H}^n)$ is such that \begin{equation} \sum_{r\in\mathbb{Z}}\left\langle \widehat{\phi}(\lambda+r),\widehat{L_{(2k,l,0)}\phi}(\lambda+r) \right\rangle_{\mathcal{B}_2}|\lambda+r|^n=0\ a.e.\ \lambda\in(0,1], \end{equation} for all $(k,l)\in\mathbb{Z}^{2n}\setminus\{(0,0)\}$, then $\{L_{(2k,l,m)}\phi:k,l\in\mathbb{Z}^n, m\in\mathbb{Z}\}$ is a Riesz sequence if and only if there exist $A,B>0$ such that \begin{equation} A\leq \sum_{r\in\mathbb{Z}}\left|\widehat{\phi}(\lambda+r)\right|_{\mathcal{B}_2}^2|\lambda+r|^n\leq B\ \ a.e.\ \lambda\in(0,1]. \end{equation} Here $\widehat{\phi}$ denotes the group Fourier transform of $\phi$ and $\mathcal{B}_2$ denotes the Hilbert space of Hilbert-Schmidt operators on $L^2(\mathbb{R}^n)$. In the absence of the above condition, the requirement of Riesz sequence is given in terms of the Gramian of the system $\{\tau\left(L_{(2k,l,0)}\phi\right)(\lambda):k,l\in\mathbb{Z}^n\}$ for $\lambda\in(0,1]$, where $\tau$ is the fiber map. We shall discuss these results in the talk along with the computational issues.
Title: Geometry & Topology Seminar: Moduli space of Kahler-Einstein metrics of negative scalar curvature
Speaker: Jian Song (Rutgers University)
Date: Wed, 05 May 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
Let K(n, V) be the space of n-dimensional compact Kahler-Einstein manifolds with negative scalar curvature and volume bounded above by V. We prove that any sequence in K(n, V) converges in pointed Gromov-Hausdorff topology to a finite union of complete Kahler-Einstein metric spaces without loss of volume, which is biholomorphic to an algebraic semi-log canonical model with its non-log terminal locus removed. We further show that the Weil-Petersson metric extends uniquely to a Kahler current with continuous local potentials on the KSB compactification of the moduli space of canonically polarized manifolds. In particular, the Weil-Petersson volume of the KSB moduli space is finite.
Title: APRG Seminar: Billiard flow and eigenfunction concentration on polyhedra
Speaker: Mayukh Mukherjee (IIT, Bombay)
Date: Wed, 28 Apr 2021
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, we discuss a basic (and somewhat classical) problem of Laplace eigenfunction mass concentration on convex polyhedra. We show quantitative mass concentration in a neighbourhood of the non-smooth part of the boundary, or the “pockets” of the billiard. On the way, we discuss several new dynamical properties of the billiard flow which are required for the proof.
Title: Geometry & Topology Seminar: Comparison geometry of holomorphic bisectional curvature, and limit spaces
Speaker: John Lott (UC Berkeley)
Date: Wed, 28 Apr 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
I will discuss the geometry of Kaehler manifolds with a lower bound on the holomorphic bisectional curvature, along with their pointed Gromov-Hausdorff limits. Some of the proofs use Ricci flow.
Title: Geometry & Topology Seminar: Tame and relatively elliptic CP1-structures on surfaces
Speaker: Lorenzo Ruffoni (Florida State Univ)
Date: Wed, 21 Apr 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
Projective geometry provides a common framework for the study of classical Euclidean, spherical, and hyperbolic geometry. A major difference with the classical case is that a projective structure is not completely determined by its holonomy representation. In general, a complete description of the space of structures with the same holonomy is still missing. We will consider certain structures on punctured surfaces, and we will discuss how to describe all of those with a given holonomy in the case of the thrice-punctured sphere. This is done in terms of a certain geometric surgery known as grafting. Our approach involves a study of the Möbius completion, and of certain meromorphic differentials on Riemann surfaces. This is joint work with Sam Ballas, Phil Bowers, and Alex Casella.
Title: APRG Seminar: Multi-parameter structures on $R^n$ and nilpotent groups
Speaker: Fulvio Ricci (SNS Pisa, Italy)
Date: Wed, 21 Apr 2021
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk we give a survey on a certain number of multi-parameter structures, on $\mathbb{R}^n$ and on nilpotent groups, that have first appeared in joint work of mine with A. Nagel, E. Stein and S. Wainger. They include flag and multi-norm structures.
These structures are intermediate between the one-parameter dilation structures of standard Calderón-Zygmund theory and the full n-parameter product structure. Each structure has its own type of maximal functions, singular integral operators, square functions, Hardy spaces.
Title: APRG Seminar: Wiener Tauberian Theorems on SL(2,R)
Speaker: Tapendu Rana (IIT, Bombay)
Date: Wed, 14 Apr 2021
Time: 4 pm
Venue: Microsoft Teams (online)
The celebrated Wiener Tauberian theorem asserts that for $ f \in L^1(\mathbb{R})$, the closed ideal generated by the function $f$ is equal to the whole of $ L^1(\mathbb{R})$ if and only if its Fourier transform $\hat f $ is nowhere vanishing on $\mathbb{R}$. The analogous result holds for locally compact abelian groups.
However in 1955, L. Ehrenpreis and F. I. Mautner observed that the corresponding result is not true for the commutative Banach algebra $L^1(G//K)$ of $K$-biinvariant functions on $G$ and proved Wiener Tauberian theorem with additional conditions, for $G= \mathrm{SL(2,\mathbb{R})}$ and $ K=\mathrm{SO}(2) $. Their result is ameliorated by Y. Ben Natan et al. In their paper, the authors studied the analog of the Wiener Tauberian theorem for the Banach algebra $ L^1( \mathrm{SL(2,\mathbb{R})} //\mathrm{SO}(2))$.
In this talk, we will discuss an analog of the Wiener Tauberian theorem for the Lorentz spaces $L^{p,1}(\mathrm {SL}(2, \mathbb{R}))$, $1\leq p<2$.
Title: Geometry & Topology Seminar: Uniform exponential growth for groups acting on CAT(0) cube complexes
Speaker: Radhika Gupta (Temple University)
Date: Wed, 14 Apr 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
‘Growth’ is a geometrically defined property of a group that can reveal algebraic aspects of the group. For instance, Gromov showed that a group has polynomial growth if and only if it is virtually nilpotent. In this talk, we will focus on growth of groups that act on a CAT(0) cube complex. Such spaces are combinatorial versions of the more general CAT(0) (negatively curved) spaces. For instance, the fundamental group of a closed hyperbolic 3-manifold acts non-trivially on a CAT(0) cube complex. Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then it either has ‘uniform exponential growth’ or it is virtually abelian. I will present some generalizations of their theorem. This is joint work with Kasia Jankiewicz and Thomas Ng.
Title: Geometry & Topology Seminar: Combining Rational maps and Kleinian groups via orbit equivalence - II
Speaker: Mahan MJ (TIFR)
Date: Fri, 09 Apr 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers’ boundary groups that are mateable in our framework. Broadly speaking, the first talk (by Sabyasachi Mukherjee) will emphasize the aspects to do with complex dynamics more, and the second (by Mahan Mj) will lay emphasis on the 1-dimensional dynamics aspects of the problem.
Title: Geometry & Topology Seminar: Combining Rational maps and Kleinian groups via orbit equivalence - I
Speaker: Sabyasachi Mukherjee (TIFR)
Date: Wed, 07 Apr 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers’ boundary groups that are mateable in our framework. Broadly speaking, the first talk (by Sabyasachi Mukherjee) will emphasize the aspects to do with complex dynamics more, and the second (by Mahan Mj) will lay emphasis on the 1-dimensional dynamics aspects of the problem.
Title: APRG Seminar: Heisenberg uniqueness pairs for the hyperbola and the Perron-Frobenius operators
Speaker: Deb Kumar Giri (IISc Mathematics)
Date: Wed, 07 Apr 2021
Time: 4 pm
Venue: Microsoft Teams (online)
The notion of Heisenberg uniqueness pair has been introduced by Hedenmalm and Montes-Rodriguez (Ann. of Math. 2011) as a version of the uncertainty principle, that is, a nonzero function and its Fourier transform both cannot be too small simultaneously. Let $\Gamma$ be a smooth curve or finite disjoint union of smooth curves in the plane and $\Lambda$ be any subset of the plane. Let $\mathcal X(\Gamma)$ be the space of all finite complex-valued Borel measures in the plane which are supported on $\Gamma$ and are absolutely continuous with respect to the arc length measure on $\Gamma.$ Let $\mathcal{AC}(\Gamma,\Lambda)=\{\mu\in \mathcal{X}(\Gamma) : \widehat\mu|_{\Lambda}=0\},$ then we say that $\Lambda$ is a Fourier uniqueness set for $\Gamma$ or $(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair, if $\mathcal{AC}(\Gamma,\Lambda)={0}.$
In this talk, we will discuss the following: Let $\Gamma$ be the hyperbola $\{(x,y)\in\mathbb R^2 : xy=1\}$ and $\Lambda_\beta^\theta$ be the lattice-cross defined by \begin{equation} \Lambda_\beta^\theta=\left((\mathbb Z+\{\theta\})\times\{0\}\right) \cup \left(\{0\}\times\beta\mathbb Z\right), \end{equation} where $\beta$ is a positive real and $\theta=1/{p}$, for some $p\in\mathbb N,$ then $\left(\Gamma,\Lambda_\beta^\theta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq{p}.$ Moreover, the space $\mathcal{AC}\left(\Gamma,\Lambda_\beta^\theta\right)$ is infinite-dimensional provided $\beta>p.$
Title: Geometry & Topology Seminar: Khovanov homotopy type
Speaker: Sucharit Sarkar (UCLA)
Date: Wed, 31 Mar 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
The Khovanov chain complex is a categorification of the Jones polynomial and is built using a functor from Kauffman’s cube of resolutions to Abelian groups. By lifting this functor to the Burnside category, one can construct a CW complex whose reduced cellular chain complex agrees with the Khovanov complex. This produces a Khovanov homotopy type whose reduced homology is Khovanov homology. I will present a general outline of this construction, starting with Khovanov’s functor. This work is joint with Tyler Lawson and Robert Lipshitz.
Title: APRG Seminar: Harmonic analysis and Hormander's multiplier theorem in the rational Dunkl setting
Speaker: Agnieszka Hejna (University of Wrocław, Poland)
Date: Wed, 31 Mar 2021
Time: 4 pm
Venue: Microsoft Teams (online)
Dunkl theory is a generalization of Fourier analysis and special function theory related to root systems and reflections groups. The Dunkl operators $T_{j}$, which were introduced by C. F. Dunkl in 1989, are deformations of directional derivatives by difference operators related to the reflection group. The goal of this talk will be to study harmonic analysis in the rational Dunkl setting. The first part will be devoted to some of results obtained in recent joint works with Jacek Dziubanski (2019, 2020).
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improved estimates of the heat kernel $h_t(\mathbf{x},\mathbf{y})$ of the Dunkl heat semigroup generated by Dunkl–Laplace operator $\Delta_k=\sum_{j=1}^{N}T_j^2$ expressed in terms of analysis on the spaces of homogeneous type;
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theorem regarding the support of Dunkl translations $\tau_{\mathbf{x}}\phi$ of $L^2$ compactly supported function $\phi$ (not necessarily radial).
The results listed above turn out to be useful tools in studying harmonic analysis in Dunkl setting. We will discuss this kind of applications in the second part of the talk. We will focus on a version of the classical Hormander’s multiplier theorem proved in joint work with Dziubanski (2019). If time permits, we will discussed how our tools can be used to for studying singular integrals of convolution type or Littlewood–Paley square functions in the Dunkl setting.
Title: PhD Thesis colloquium: Algorithmic and Hardness Results for Fundamental Fair-Division Problems
Speaker: Nidhi Rathi (IISc Mathematics)
Date: Fri, 26 Mar 2021
Time: 3 pm
Venue: Microsoft Teams (online)
The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges that arise in various settings of fair-division problems and complements the existential (and non-constructive) guarantees and various hardness results by way of developing efficient (approximation) algorithms and identifying computationally tractable instances.
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Our work in fair cake division develops several algorithmic results for allocating a divisible resource (i.e., the cake) among a set of agents in a fair/economically efficient manner. While strong existence results and various hardness results exist in this setup, we develop a polynomial-time algorithm for dividing the cake in an approximately fair and efficient manner. Furthermore, we identify an encompassing property of agents’ value densities (over the cake)—the monotone likelihood ratio property (MLRP)—that enables us to prove strong algorithmic results for various notions of fairness and (economic) efficiency.
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Our work in fair rent division develops a fully polynomial-time approximation scheme (FPTAS) for dividing a set of discrete resources (the rooms) and splitting the money (rents) among agents having general utility functions (continuous, monotone decreasing, and piecewise-linear), in a fair manner. Prior to our work, efficient algorithms for finding such solutions were know n only for a specific set of utility functions. We complement the algorithmic results by proving that the fair rent division problem (under genral utilities) lies in the intersection of the complexity classes, PPAD (Polynomial Parity Arguments on Directed graphs) and PLS (Polynomial Local Search).
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Our work respectively addresses fair division of rent, cake (divisible), and discrete (indivisible) goods in a partial information setting. We show that, for all these settings and under appropriate valuations, a fair (or an approximately fair) division among $n$ agents can be efficiently computed using only the valuations of $n-1$ agents. The $n$th (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.
Title: APRG Seminar: A note of Titchmarsh theorem and approximation theory on the sphere
Speaker: Radouan Daher (University of Hassan II, Morocco)
Date: Wed, 24 Mar 2021
Time: 4 pm
Venue: Microsoft Teams (online)
Our aim in this talk is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier-Laplace transform of a set of functions satisfying a generalized Lipschitz condition in the space $L_p, 1 < p \leq 2$ on the sphere. We also prove analogues of Jackson’s direct theorem for the moduli of smoothness of all orders constructed on the basis of spherical shift. Finally, we prove equivalence between moduli of smoothness and $K$-functional for the couple $(L^2 (\sigma^{m-1} ), W^r_2 (\sigma^{m-1} ))$.
This is joint work with S. El Ouadih, O. Tyr and F. Saadi.
Title: Geometry & Topology Seminar: Anosov representations of geometrically finite Fuchsian groups
Speaker: Andrew Zimmer (U Wisconsin-Madison)
Date: Wed, 24 Mar 2021
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
Anosov representations are representations of word hyperbolic groups into semisimple Lie groups with many good geometric properties. In this talk I will develop a theory of Anosov representation of geometrically finite Fuchsian groups (a special class of relatively hyperbolic groups). I will only discuss the case of representations into the special linear group and avoid general Lie groups. This is joint work with Canary and Zhang.
Title: APRG Seminar: A Laplacian on the unilateral shift space
Speaker: Shrihari Sridharan (IISER, Thiruvananthapuram)
Date: Wed, 17 Mar 2021
Time: 4 pm
Venue: Microsoft Teams (online)
Rough analysis, as undertaken and popularised by Robert Strichartz and Jun Kigami, deals with the construction of a Laplacian and the study of associated problems on certain fractal sets, embedded in some Euclidean space, thus naturally exploiting the Euclidean topology. In this talk, we generalise this study to abstract, totally disconnected, metric measure unilateral shift spaces. In particular, we discuss the construction of a Laplacian as a renormalised limit of difference operators defined on finite sets that approximate the entire space. We further propose a weak definition of this Laplacian, analogous to the one in calculus, by choosing test functions as those which have finite energy and vanish on various (appropriately defined) boundary sets. We then define the Neumann derivative of functions on these boundary sets and establish a relationship between the three important concepts in our analysis so far, namely, the Laplacian, the bilinear energy form and the Neumann derivative of a function.
This is a joint work with my doctoral student, Sharvari Neetin Tikekar.
Title: Geometry & Topology Seminar: The optimal exponent of certain Moser-Trudinger type inequality on polarized manifolds.
Speaker: Kewei Zhang (Beijing Normal University)
Date: Wed, 17 Mar 2021
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
In this talk I will present a quantization approach which directly relates Fujita-Odaka’s delta-invariant to the optimal exponent of certain Moser-Trudinger type inequality on polarized manifolds. As a consequence we obtain new criterions for the existence of twisted Kahler-Einstein metrics or constant scalar curvature Kahler metrics on possibly non-Fano manifolds.
Title: APRG Seminar: Strong ill-posedness for nonlinear Schrödinger equations
Speaker: Divyang Bhimani (IISER, Pune)
Date: Wed, 10 Mar 2021
Time: 4 pm
Venue: Microsoft Teams (online)
We consider nonlinear Schrödinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation (strong ill-posedness) results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. We shall also discuss similar results for fractional Hartree equation. The talk is based on a joint work with Remi Carles and Saikatul Haque.
Title: PhD Thesis colloquium: Nodal sets of random functions
Speaker: Lakshmi Priya M. E. (IISc Mathematics)
Date: Wed, 03 Mar 2021
Time: 4 pm
Venue: Microsoft Teams (online)
This thesis is devoted to the study of nodal sets of random functions. The random functions and the specific aspect of their nodal set that we study fall into two broad categories: nodal component count of Gaussian Laplace eigenfunctions and volume of the nodal set of centered stationary Gaussian processes (SGPs) on $\mathbb{R}^d$, $d \geq 1$.
Gaussian Laplace eigenfunctions: Nazarov–Sodin pioneered the study of nodal component count for Gaussian Laplace eigenfunctions; they investigated this for random spherical harmonics (RSH) on the two-dimensional sphere $S^2$ and established exponential concentration for their nodal component count. An analogous result for arithmetic random waves (ARW) on the $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$, was established soon after by Rozenshein.
We establish concentration results for the nodal component count in the following three instances: monochromatic random waves (MRW) on growing Euclidean balls in $\R^2$; RSH and ARW, on geodesic balls whose radius is slightly larger than the Planck scale, in $S^2$ and $\mathbb{T}^2$ respectively. While the works of Nazarov–Sodin heavily inspire our results and their proofs, some effort and a subtler treatment are required to adapt and execute their ideas in our situation.
Stationary Gaussian processes: The study of the volume of nodal sets of centered SGPs on $\mathbb{R}^d$ is classical; starting with Kac and Rice’s works, several studies were devoted to understanding the nodal volume of Gaussian processes. When $d = 1$, under somewhat strong regularity assumptions on the spectral measure, the following results were established for the zero count on growing intervals: variance asymptotics, central limit theorem and exponential concentration.
For smooth centered SGPs on $\mathbb{R}^d$, we study the unlikely event of overcrowding of the nodal set in a region; this is the event that the volume of the nodal set in a region is much larger than its expected value. Under some mild assumptions on the spectral measure, we obtain estimates for probability of the overcrowding event. We first obtain overcrowding estimates for the zero count of SGPs on $\mathbb{R}$, we then deal with the overcrowding question in higher dimensions in the following way. Crofton’s formula gives the nodal set’s volume in terms of the number of intersections of the nodal set with all lines in $\mathbb{R}^d$. We discretize this formula to get a more workable version of it and, in a sense, reduce this higher dimensional overcrowding problem to the one-dimensional case.
Title: APRG Seminar: An inverse problems approach to sharp estimates in harmonic analysis
Speaker: Chandan Biswas (IISc Mathematics)
Date: Wed, 03 Mar 2021
Time: 4 pm
Venue: Microsoft Teams (online)
One way to prove a theorem in analysis is to initially recognize what can go wrong. In this talk we will discuss a few recent results that follows a detailed version of this approach to establish existence of maximizers of some functionals.
Title: APRG Seminar: Optimal exponential integrability of maps with finite non convex energy
Speaker: Arka Mallick (Technion, Haifa, Israel)
Date: Wed, 24 Feb 2021
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, I would like to present some recent results regarding the behaviour of functions which are uniformly bounded under the action of a certain class of non-convex non-local functionals related to the degree of a map. In the literature, this class of functionals happens to be a very good substitute of the $L^p$ norm of the gradient of a Sobolev function. As a consequence various improvements of the classical Poincaré’s inequality, Sobolev’s inequality and Rellich-Kondrachov’s compactness criterion were established. This talk will be focused on addressing the gap between a certain exponential integrability and the boundedness for functions which are finite under the action of these class of non-convex functionals.
Title: APRG Seminar: Newton Polyhedra associated with global domains
Speaker: Joonil Kim (Yonsei University, Seoul, South Korea)
Date: Wed, 17 Feb 2021
Time: 4 pm
Venue: Zoom (online)
We consider a class of oscillatory integrals with polynomial phase functions $P$ over global domains $D$ in $\mathbb{R}^2$. As an analogue of Varchenko’s theorem in a global domain, we investigate the two main issues (i) whether the integral converges or not and (ii) how fast it decays. They are described in terms of a generalized notion of Newton polyhedra associated with $(P,D)$. Finally, we discuss its applications to the Strichartz Estimates associated with the general class of dispersive equations.
Title: APRG Seminar: Sharp $L^p-L^q$ estimate for the spectral projection associated with the twisted Laplacian
Speaker: Eunhee Jeong (Korea Institute for Advanced Study, South Korea)
Date: Wed, 10 Feb 2021
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, we are concerned with sharp estimate for the spectral projection $P_\mu$ associated with the twisted Laplacian in the Lebesgue spaces. We provide a complete characterization of the sharp $L^p-L^q$ bound for $P_\mu$, which is similar to that for the spectral projection associated with the Laplacian. As an application, we discuss the resolvent estimate for the twisted Laplacian. This talk is based on a joint work with Sanghyuk Lee and Jaehyeon Ryu.
Title: APRG Seminar: Sharp $L^p-L^q$ estimates for the Hermite spectral projection
Speaker: Sanghyuk Lee (Seoul National University, Republic of Korea)
Date: Wed, 03 Feb 2021
Time: 4 pm
Venue: Zoom (online)
This talk primarily concerns the sharp bound on the spectral projection of the Hermite operator in the $L^p$ spaces. In comparison with the spectral projection of the Laplacian, the sharp bound has not been not so well understood. We consider the estimate for the spectral Hermite projection in general $L^p-L^q$ framework and obtain various new sharp estimates in an extended range. Especially, we provide a complete characterization of the local estimate and prove the endpoint $L^2$–$L^{2(d+3)/(d+1)}$ estimate which has been left open since the work of Koch and Tataru. We also discuss application of the projection estimate to related problems, such as the resolvent estimate for the Hermite operator and Carleman estimate for the heat operator.
Title: PhD Thesis defence: Trace Estimate For The Determinant Operator and K-Homogeneous Operators
Speaker: Paramita Pramanick (IISc Mathematics)
Date: Mon, 01 Feb 2021
Time: 4 pm
Venue: Microsoft Teams (online)
For a commuting $d$-tuple of operators $\boldsymbol T=(T_1, \ldots , T_d)$ defined on a complex separable Hilbert space $\mathcal{H}$, let $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$ be the $d \times d$ block operator $\big (\big ( \big [ T_j^*,T_i] \big )\big )$ of commutators: $[T_j^*,T_i] := T_j^* T_i - T_i T_j^*$. We define an operator on the Hilbert space $\mathcal{H}$, to be designated the determinant operator, corresponding to the block operator $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$. We show that if the $d$-tuple is cyclic, the determinant operator is positive and the compression of a fixed set of words in $T_j^*$ and $T_i$ – to a nested sequence of finite dimensional subspaces increasing to $\mathcal{H}$ – does not grow very rapidly, then the trace of the determinant of the operator $\big (\big ( \big [ T_j^*,T_i] \big )\big )$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a certain small class of commuting $d$-tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
Let $\Omega$ be an irreducible classical bounded symmetric domain of rank $r$ in $\mathbb{C}^d$. Let $\mathbb{K}$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group $\mathbb{K}$ consisting of linear transformations acts naturally on any $d$-tuple $\mathbf{T}$ of commuting bounded linear operators by the rule: \begin{equation} k \cdot \mathbf{T} = \big( k_1(T_1, \dots, T_d), \dots, k_d(T_1, \dots, T_d) \big), \ k \in \mathbb{K}, \end{equation} where $k_1(\mathbf{z}), \dots, k_d(\mathbf{z})$ are linear polynomials. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\mathbf{T}$ is $\mathbb{K}$-homogeneous. We realize a certain class of $\mathbb{K}$-homogeneous $d$-tuples $\mathbf{T}$ as a $d$-tuple of multiplication by the coordinate functions $z_1, \dots, z_d$ on a reproducing kernel Hilbert space $\mathcal{H}_K$. (The Hilbert space $\mathcal{H}_K$ consisting of holomorphic functions defined on $\Omega$, with $K$ as reproducing kernel.) Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(\Omega)$. For an irreducible bounded symmetric domain $\Omega$ of rank 2, an explicit description of the operator $\sum_{i=1}^d T_i^* T_i$ is given. Based on this formula, a conjecture giving the form of this operator in any rank $r \geq 1$ was made. This conjecture was recently verified by H. Upmeier.
Title: APRG Seminar: On translation invariance of the mixed frame operator
Speaker: Anupam Gumber (IISc Mathematics)
Date: Wed, 27 Jan 2021
Time: 4 pm
Venue: Microsoft Teams (online)
A countable family $\{\psi_n: n \in \mathbb{N}\}$ of elements in a Hilbert space $\mathcal{H}$ constitutes a frame if there are constants $0< A\leq B < \infty$ s.t. $\forall f \in \mathcal{H}$ we have: \begin{equation} A\|f\|^2 \leq \sum\limits_n|\langle f,\psi_n \rangle|^2\leq B \|f\|^2, \end{equation} where $\langle\cdot, \cdot\rangle$ denotes an inner-product in $\mathcal{H}$. Frames were introduced by Duffin and Schaeffer in 1952 to deal with problems in nonharmonic Fourier series, and have been used more recently to obtain signal reconstruction for signals embedded in certain noises.
For a given pair of frames $\{\psi_n\}$ and $\{\varphi_n\}$ in $\mathcal{H}$, the associated mixed frame operator $S: \mathcal{H} \to \mathcal{H}; f \mapsto \sum_n \langle f, \psi_n \rangle\varphi_n$ is a bounded linear operator. The translation invariance of this operator plays a significant role in investigating reproducing formulas for frame pairs $\{\psi_n\}$ and $\{\varphi_n\}$ in $\mathcal{H}$.
In the present talk, we examine necessary and sufficient conditions for $S$ to be invariant under translations on $\mathbb{R}^n$ when $\{\psi_n\}$ and $\{\varphi_n\}$ belong to a special class of structured frame systems in $L^2(\mathbb{R}^n)$.
Title: APRG Seminar: On Riesz sets
Speaker: Nishant Chandgotia (TIFR-CAM, Bengaluru)
Date: Wed, 20 Jan 2021
Time: 4 pm
Venue: Microsoft Teams (online)
Given a finite (complex-valued) measure $\mu$ on the circle $\mathbb{R}/\mathbb{Z}$ we write supp$(\widehat{\mu})$ to denote the support of its Fourier coefficients. A subset $P$ of the integers is called a Riesz set if for all measures $\mu$ on $\mathbb{R}/\mathbb{Z}$ for which supp$(\widehat{\mu}) \subset P$, $\mu$ is absolutely continuous with respect to the Lebesgue measure. It is well-known that $P=-\mathbb{N}$, the negative integers, form a Riesz set given a famous result by F. and M. Riesz. Rudin proved that if one appends a lacunary set to the set of negative natural numbers, then it is a Riesz set. This idea was picked up by Meyer who wrote a beautiful paper on the subject, christened such sets (as Riesz sets) and proved, for instance, that appending the squares to the negative integers is still a Riesz set. For this proof a surprising (only initially though) use of the Bohr topology plays an important role.
Can we do this for the cubes? I believe that this is still open and the only progress made uses Fermat’s last theorem. I am not a specialist on the subject and have some rudimentary understanding of harmonic analysis. Most of what I have learnt is by talking to some specialists and reading a few papers/text books. Still, I will attempt to give you a primer on what I know and what I believe nobody does.
Title: APRG Seminar: Non-local ODE via conformal geometry
Speaker: Maria del Mar Gonzalez (Universidad Autónoma de Madrid, Spain)
Date: Wed, 13 Jan 2021
Time: 4 pm
Venue: Microsoft Teams (online)
We will consider some results for non-local ODE with good conformal properties. These include equations with a fractional Laplacian and a Hardy-type critical potential. It turns out that conformal geometry provides a powerful tool to handle such non-local ODE. We will classify the asymptotic behavior of solutions and give a generalized notion of Wronskian, for instance, together with some applications.
Title: Algebra & Combinatorics Seminar: Projective representations of discrete nilpotent groups
Speaker: Pooja Singla (IIT Kanpur)
Date: Fri, 08 Jan 2021
Time: 3 pm
Venue: Zoom (online)
The study of projective representations of a group has a long history starting from the work of Schur. Two essential ingredients to study the group’s projective representations are describing its Schur multiplier and representation group. In this talk, we describe these for the discrete Heisenberg groups. We also include a few general results regarding projective representations of finitely generated discrete nilpotent groups. This talk is based on the joint work with Sumana Hatui and E.K. Narayanan.
Title: APRG Seminar: Time-frequency analysis and suitable function spaces
Speaker: Hans Georg Feichtinger (Institute of Mathematics, University of Vienna, Austria)
Date: Wed, 06 Jan 2021
Time: 4 pm
Venue: Zoom (online)
The key-point of this talk will be some exploration of function spaces concepts arising from time-frequency analysis respectively Gabor Analysis. Modulation spaces and Wiener amalgams have proved to be indispensable tools in time-frequency analysis, but also for the treatment of pseudo-differential operators or Fourier integral operators.
More precisely, we will recall a short summary of the concepts of Wiener amalgam spaces and modulation spaces, as well as the concept of Banach Gelfand Triples, with the associated kernel theorem (in the spirit of the L. Schwartz kernel theorem). We will indicate in which sense these spaces allow to capture more precisely the mapping properties of operators which may be unbounded in the Hilbert space setting. The subfamily of translation and modulation invariant spaces plays a specific role, with naturally associated regularization operators involving smoothing by convolution and localization by pointwise multiplication.
Title: PhD Thesis defence: Total variation cutoff for random walks on some finite groups
Speaker: Subhajit Ghosh (IISc Mathematics)
Date: Mon, 21 Dec 2020
Time: 2 pm
Venue: Microsoft Teams (online)
This thesis studies the mixing times for three random walk models. Specifically these are the random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below:
The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by 3-cycles of the form $(i, n − 1, n)$ and $(i, n, n − 1)$. We call this the transpose top-2 with random shuffle. We find the spectrum of the transition matrix of this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $(n − 3/2)\log (n)$ for this shuffle.
The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group Bn generated by the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 \leq i \leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n \log (n)$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 \leq i < n$ also has a cutoff at $(n − 1/2)\log (n)$.
The random walk on the complete monomial group: Let $G_1 \subseteq \cdots \subseteq G_n \subseteq \cdots$ be a sequence of finite groups with $|G_1| > 2$. We study the properties of a random walk on the complete monomial group $G_n \wr S_n$ (wreath product of $G_n$ with $S_n$) generated by the elements of the form $(e,\dots, e, g; id)$ and $(e,\dots, e, g^{−1}, e,\dots, e, g; (i, n))$ for $g \in G_n$, $1 \leq i < n$. We call this the warp-transpose top with random shuffle on $G_n \wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n \log (n) + (1/2) n \log(|G_n| − 1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n \log (n)$ if $|G_n| = o( n^\delta )$ for all $\delta > 0$.
Title: Geometry & Topology Seminar: Bishop-Gromov inequality generalised
Speaker: Gerard Besson (Institut Fourier, Université de Grenoble)
Date: Mon, 21 Dec 2020
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
We describe a general context, related to metric spaces, in which a weak version of the celebrated Bishop-Gromov inequality is valid and suggest that this could serve of a synthetic version of a lower bound on the Ricci curvature.
Title: Algebra & Combinatorics Seminar: Multiplicities in Selmer groups and root numbers for Artin twists
Speaker: Sudhanshu Shekhar (IIT Kanpur)
Date: Fri, 18 Dec 2020
Time: 3 pm
Venue: Microsoft Teams (online)
Given a Galois extension of number fields $K/F$ and two elliptic curves $A$ and $B$ with equivalent residual Galois representation mod $p$, for an odd prime $p$, we will discuss the relation between the $p$-parity conjecture of $A$ twisted by $\sigma$ and that of $B$ twisted by $\sigma$ for an irreducible, self dual, Artin representation $\sigma$ of the Galois group of $K/F$.
This is a joint work with Somnath Jha and Tathagata Mandal.
Title: APRG Seminar: Variational bounds for spherical means
Speaker: Luz Roncal (BCAM – Basque Center for Applied Mathematics, Spain)
Date: Wed, 16 Dec 2020
Time: 4 pm
Venue: Microoft Teams (online)
We show $L^p\to L^q$ estimates for local and global $r$-variation operators associated to the family of spherical means. These can be understood as a strengthening of $L^p$-improving estimates for the spherical maximal function. Our bounds turn out to be sharp up to the endpoints (except for dimension 3) although we also provide positive results in certain endpoints. The results imply associated sparse domination and consequent weighted inequalities.
This is joint work with David Beltran, Richard Oberlin, Andreas Seeger, and Betsy Stovall.
Title: Geometry & Topology Seminar: Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces
Speaker: Qiongling Li (Chern institute, Nankai)
Date: Mon, 14 Dec 2020
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
On a Riemann surface with a holomorphic $r$-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a harmonic metric of the Higgs bundle for which the grading is orthogonal. In this talk, we focus on a general non-compact Riemann surface with an $r$-differential which is not necessarily meromorphic at infinity. We introduce the notion of complete solution of the Toda equation, and we prove the existence and uniqueness of a complete solution by using techniques for both Toda equations and harmonic bundles. Moreover, we show some quantitative estimates of the complete solution. This is joint work with Takuro Mochizuki (RIMS).
Title: Algebra & Combinatorics Seminar: On the simultaneous approximation of algebraic numbers
Speaker: Ravindranath Thangadurai (Harish-Chandra Research Institute, Prayagraj)
Date: Mon, 07 Dec 2020
Time: 3 pm
Venue: Microsoft Teams (online)
In 2004, Corvaja and Zannier proved an extension of Roth’s theorem on rational approximation of algebraic numbers. With a collaboration of Dr. Veekesh Kumar, we proved a simultaneous version of Corvaja and Zannier’s result. These results are applications of a strong form of the Subspace Theorem. In this talk, we shall discuss the motivation of Corvaja and Zannier’s result and our generalization.
Title: Geometry & Topology Seminar: Smooth asymptotics for collapsing Calabi-Yau metrics
Speaker: Hans Joachim Hein (Mathematisches Institute, Munster)
Date: Mon, 07 Dec 2020
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
Yau’s solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion (locally uniformly away from the singular fibers) of Calabi-Yau metrics collapsing along a holomorphic fibration of a fixed compact Calabi-Yau manifold. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
Title: Algebra & Combinatorics Seminar: Toppleable permutations, excedances and acyclic orientations
Speaker: Arvind Ayyer (IISc Mathematics)
Date: Fri, 04 Dec 2020
Time: 3 pm
Venue: Microsoft Teams (online)
Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (arXiv:1612.06816) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. For the most part of the talk, we will explain the main ideas in showing that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Time permitting, we will give some ideas showing that this is also the number of acyclic orientations with unique sink of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$.
This is joint work with P. Tetali (GATech) and D. Hathcock (CMU), and is available at arXiv:2010.11236.
Title: APRG Seminar: A heat equation approach to intertwining
Speaker: Nicola Garofalo (Universita di Padova, Italy)
Date: Thu, 03 Dec 2020
Time: 4 pm
Venue: Microsoft Teams (online)
(Note: unusual day.)
In this talk I present a heat semigroup approach to some intertwining formulas which arise in conformal CR geometry. This is recent joint work with G. Tralli.
Title: Geometry & Topology Seminar: Circle patterns on surfaces with complex projective structures
Speaker: Wai Yeung Lam (Université du Luxembourg)
Date: Mon, 30 Nov 2020
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. A natural question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, one can consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is diffeomorphic to the Teichmueller space via a natural projection. In this talk, we will report progress on the torus case.
Title: APRG Seminar: Homogeneous multipliers on Sobolev spaces
Speaker: Riju Basak (IISER, Bhopal)
Date: Wed, 25 Nov 2020
Time: 4 pm
Venue: Microsoft Teams (online)
In 1987, A. Bonami and S. Poornima proved that a non-constant function which is homogeneous of degree zero cannot be a Fourier multiplier on homogeneous Sobolev spaces. In this talk, we will discuss the Fourier multipliers on Heisenberg group $\mathbb{H}^n$ and Weyl multipliers on $\mathbb{C}^n$ acting on Sobolev spaces. We define a notion of homogeneity of degree zero for bounded operators on $L^{2}(\mathbb{R}^n)$ and establish analogous results for Fourier multipliers on Heisenberg group and Weyl multipliers on $\mathbb{C}^n$ acting on Sobolev spaces. This talk is based on the recent work with Rahul Garg and Sundaram Thangavelu.
Title: Geometry & Topology Seminar: The J-equation and the deformed Hermitian-Yang-Mills equation
Speaker: Gao Chen (University of Wisconsin-Madison)
Date: Mon, 23 Nov 2020
Time: 2:30 pm
Venue: MS teams (team code hiq1jfr)
The deformed Hermitian-Yang-Mills (dHYM) equation is the mirror equation for the special Lagrangian equation.
The “small radius limit” of the dHYM equation is the J-equation, which is closely related to the constant scalar curvature K"ahler (cscK) metrics.
In this talk, I will explain my recent result that the solvability of the J-equation is equivalent to a notion of stability.
I will also explain my similar result on the supercritical dHYM equation.
Title: Algebra & Combinatorics Seminar: Distinguishing newforms by their Hecke eigenvalues
Speaker: Sanoli Gun (IMSc, Chennai)
Date: Fri, 20 Nov 2020
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, we will discuss some of the existing techniques for distinguishing newforms. We will also report on a recent joint work with Kumar Murty and Biplab Paul.
Title: APRG Seminar: Bilinear multipliers on Banach function spaces
Speaker: Oscar Blasco (Universidad de Valencia, Spain)
Date: Wed, 18 Nov 2020
Time: 4 pm
Venue: Microsoft Teams (online)
Let $X_1,X_2, X_3$ be Banach spaces of measurable functions in $L^0(\mathbb R)$ and let $m(\xi,\eta)$ be a locally integrable function in $\mathbb R^2$. We say that $m\in \mathcal{BM}_{(X_1,X_2,X_3)}(\mathbb R)$ if \begin{equation} B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i \langle\xi+\eta, x\rangle}d\xi d\eta, \end{equation} defined for $f$ and $g$ with compactly supported Fourier transform, extends to a bounded bilinear operator from $X_1 \times X_2$ to $X_3$.
In this talk we present some properties of the class $\mathcal{BM}_{(X_1,X_2,X_3)}(\mathbb R)$ for general spaces which are invariant under translation, modulation and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus in the case $m(\xi,\eta)=M(\xi-\eta)$ and find conditions for these classes to contain non zero multipliers in terms of the Boyd indices for the spaces.
Title: Geometry & Topology Seminar: Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles
Speaker: Jean-Pierre Demailly (Grenoble)
Date: Mon, 16 Nov 2020
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths – and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Another outcome of the approach is a new concept of volume for vector bundles.
Title: Algebra & Combinatorics Seminar: Denominator identities for the periplectic Lie superalgebra p$(n)$
Speaker: Shifra Reif (Bar-Ilan University, Israel)
Date: Fri, 13 Nov 2020
Time: 3 pm
Venue: Microsoft Teams (online)
We present the denominator identities for the periplectic Lie superalgebras and discuss their relations to representations of $\mathbf{p}(n)$ and $\mathbf{gl}(n)$. Joint work with Crystal Hoyt and Mee Seong Im.
Title: APRG Seminar: Fractional Poincaré inequalities and Harmonic Analysis
Speaker: Carlos Pérez (University of the Basque Country and BCAM)
Date: Wed, 11 Nov 2020
Time: 4 pm
Venue: Microsoft Teams (online)
In this expository lecture we will discuss some recent results concerning fractional Poincaré and Poincaré-Sobolev inequalities with weights, the degeneracy. These results improve some well known estimates due to Fabes-Kenig-Serapioni from the 80’s in connection with the local regularity of solutions of degenerate elliptic equations and also some more recent results by Bourgain-Brezis-Minorescu. Our approach is different from the usual ones and is based on methods that come from Harmonic Analysis. This is especially visible when the connection of this theory with the BMO space and its different variants will be shown.
Title: Geometry & Topology Seminar: SYZ mirror symmetry for del Pezzo surfaces and rational elliptic surfaces
Speaker: Tristan Collins (MIT)
Date: Mon, 09 Nov 2020
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
I will discuss some aspects of SYZ mirror symmetry for pairs $(X,D)$ where $X$ is a del Pezzo surface or a rational elliptic surface
and $D$ is an anti-canonical divisor. In particular I will explain the existence of special Lagrangian fibrations, mirror symmetry
for (suitably interpreted) Hodge numbers and, if time permits, I will describe a proof of SYZ mirror symmetry conjecture for del Pezzo surfaces.
This is joint work with Adam Jacob and Yu-Shen Lin.
Title: Algebra & Combinatorics Seminar: A weight-dependent inversion statistic and Catalan numbers
Speaker: Michael Schlosser (University of Vienna, Austria)
Date: Fri, 06 Nov 2020
Time: 3 pm
Venue: Microsoft Teams (online)
We introduce a weight-dependent extension of the inversion statistic, a classical Mahonian statistic on permutations. This immediately gives us a new weight-dependent extension of $n!$. By restricting to $312$-avoiding permutations our extension happens to coincide with the weighted Catalan numbers that were considered by Flajolet in his combinatorial study of continued fractions. We show that for a specific choice of weights the weighted Catalan numbers factorize into a closed form, hereby yielding a new $q$-analogue of the Catalan numbers, different from those considered by MacMahon, by Carlitz, or by Andrews. We further refine the weighted Catalan numbers by introducing an additional statistic, namely a weight-dependent extension of Haglund’s bounce statistic, and obtain a new family of bi-weighted Catalan numbers that generalize Garsia and Haiman’s $q,t$-Catalan numbers and appear to satisfy remarkable properties. This is joint work with Shishuo Fu.
Title: APRG Seminar: Recent developments on $L^p$-$L^q$ boundedness of Fourier multipliers
Speaker: Vishvesh Kumar (Ghent University, Belgium)
Date: Wed, 04 Nov 2020
Time: 4 pm
Venue: Microsoft Teams (online)
In his seminal paper (Acta Math. 1960), H"ormander established the $L^p$-$L^q$ boundedness of Fourier multipliers on $\mathbb{R}^n$ for the range $1<p \leq 2 \leq q<\infty.$ Recently, Ruzhansky and Akylzhanov (JFA, 2020) extended H"ormander’s theorem for general locally compact separable unimodular groups using group von Neumann algebra techniques and as a consequence, they obtained the $L^p$-$L^q$ boundedness of spectral multipliers for general positive unbounded invariant operators on locally compact separable unimodular groups.
In this talk, we will discuss the $L^p$-$L^q$ boundedness of global pseudo-differential operators and Fourier multipliers on smooth manifolds for the range $1<p\leq 2 \leq q<\infty$ using the nonharmonic Fourier analysis developed by Ruzhansky, Tokmagambetov, and Delgado. As an application, we obtain the boundedness of spectral multipliers, embedding theorems, and time asymptotic the heat kernels for the anharmonic oscillator.
This talk is based on my joint works with Duván Cardona (UGent), Marianna Chatzakou (Imperial College London), Michael Ruzhansky (UGent), and Niyaz Tokmagambetov (UGent).
Title: APRG Seminar: Ramanujan's Master theorem
Speaker: Sanjoy Pusti (IIT, Bombay)
Date: Wed, 28 Oct 2020
Time: 4 pm
Venue: Microsoft Teams (online)
Ramanujan’s Master theorem states that (under certain conditions) if a function $f$ on $\mathbb R$ can be expanded around zero in a power series of the form \begin{equation} f(x)=\sum_{k=0}^\infty (-1)^k a(k) x^k, \end{equation} then \begin{equation} \int_0^\infty f(x) x^{-\lambda-1}\,dx=-\frac{\pi}{\sin\pi\lambda} a(\lambda), \, \text{ for }\lambda\in\mathbb C. \end{equation} This theorem can be thought of as an interpolation theorem, which reconstructs the values of $a(\lambda)$ from its given values at $a(k), k\in \mathbb N\cup {0}$. In particular if $a(k)=0$ for all $k\in \mathbb N\cup {0}$, then $a$ is identically $0$. By selecting particular values for the function $a$, Ramanujan applied this theorem to compute several definite integrals and power series. This explains why it is referred to as the “Master Theorem”.
Based on the duality of Riemannian symmetric spaces of compact and noncompact type inside a common complexification, Bertram, Olafsson-Pasquale proved an analogue of this theorem on Riemannian symmetric spaces of noncompact type.
In the first part of the talk, we shall discuss an analogue of this theorem for radial sections of line bundles over Poincare upper half plane. This is a joint work with Swagato K Ray.
In the second half, we shall discuss an analogue of this theorem for Sturm-Liouville operators. This is a joint work with Jotsaroop Kaur.
Title: APRG Seminar: Maximal estimates for bilinear Bochner Riesz means
Speaker: Jotsaroop Kaur (IISER Mohali)
Date: Wed, 21 Oct 2020
Time: 4 pm
Venue: Microsoft Teams (online)
We prove some new $L^p$ estimates for the maximal function associated to bilinear Bochner Riesz means in all dimensions $n\geq 1$. This is a joint work with Saurabh Shrivastava.
Title: Geometry & Topology Seminar: Plateau problem in the pseudo-hyperbolic space
Speaker: Jeremy Toulisse (Nice)
Date: Mon, 19 Oct 2020
Time: 4:00 pm
Venue: MS teams (team code hiq1jfr)
The pseudo-hyperbolic space $H^{2,n}$ is the pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how to solve an asymptotic Plateau problem in this space: given a topological circle in the boundary at infinity of $H^{2,n}$, we construct a unique complete maximal surface bounded by this circle. This construction relies on Gromov’s theory of pseudo-holomorphic curves. This is a joint work with François Labourie and Mike Wolf.
Title: Algebra & Combinatorics Seminar: Fusion product decomposition of g-stable affine Demazure modules
Speaker: R. Venkatesh (IISc Mathematics)
Date: Fri, 16 Oct 2020
Time: 3 pm
Venue: Microsoft Teams (online)
The affine Demazure modules are the Demazure modules that occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We call them $\mathfrak{g}$-stable if they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a $\mathfrak{g}$-stable affine Demazure module is isomorphic to the fusion (tensor) product of smaller $\mathfrak{g}$-stable affine Demazure modules, thus completing the main theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). We obtain a new combinatorial proof for the key fact that was used in Chari et al. (op cit.), to prove the decomposition of $\mathfrak{g}$-stable affine Demazure modules. Our proof for this key fact is uniform, avoids the case-by-case analysis, and works for all finite-dimensional simple Lie algebras.
Title: APRG Seminar: Carleman Estimates and Unique Continuation
Speaker: Ramesh Manna (IISc Mathematics)
Date: Wed, 14 Oct 2020
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, we discuss Carleman estimates for Laplacian, which implies strong unique continuation for $-\Delta u+Vu$ with potential $V \in L^{\infty}.$ We briefly discuss unique continuation in certain critical situations such as when the potential is in $L^{n/2}_{loc}$ assuming Fourier restriction theorems. Then $L^{n/2}_{loc}$ case is a well-known result of Kenig-Jerison. Our proof of unique continuation is based on the Carleman estimate where it is a consequence of the spectral gap of Laplace Beltrami on the sphere.
Title: Geometry & Topology Seminar: Compactness and partial regularity theory of Ricci flows in higher dimensions
Speaker: Richard Bamler (UC Berkeley)
Date: Mon, 12 Oct 2020
Time: 9:00 pm
Venue: MS teams (team code hiq1jfr)
We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
Title: Algebra & Combinatorics Seminar: Regular Bernstein blocks
Speaker: Manish Mishra (IISER Pune)
Date: Fri, 09 Oct 2020
Time: 3 pm
Venue: Microsoft Teams (online)
Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. The category $R(G)$ of smooth representations of $G(F)$ has a decomposition into a product of indecomposable subcategories called Bernstein blocks and to each block is associated a non-negative real number called Moy-Prasad depth. We will begin with recalling all this basic theory. Then we will focus the discussion on ‘regular’ blocks. These are ‘most’ Bernstein blocks when the residue characteristic of $F$ is suitably large. We will then talk about an approach of studying blocks in $R(G)$ by studying a suitably related depth-zero block of certain other groups. In that context, I will explain some results from a joint work with Jeffrey Adler. One of them being that the Bernstein center (i.e., the center of a Bernstein block) of a regular block is isomorphic to the Bernstein center of a depth-zero regular block of some explicitly describable another group. I will give some applications of such results.
Title: APRG Seminar: Characterization of eigenfunctions of the Laplace-Beltrami operator using Fourier multipliers
Speaker: Muna Naik (Harish-Chandra Research Institute, Prayagraj)
Date: Wed, 07 Oct 2020
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, we will try to characterize eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers via Roe-Strichartz type theorems in rank one symmetric spaces of noncompact type. This work has its origin in a simple result of Roe, which says that if all the derivatives and antiderivatives of a given function on the real line are uniformly bounded, then the function is a linear combination of sin(x) and cos(x). We will talk about ramification of this result in context of characterizing eigenfunctions of the Laplace-Beltrami operator. The talk will be based on a joint work with Prof. Rudra P. Sarkar.
Title: APRG Seminar: Bilinear spherical maximal functions of product type
Speaker: Kalachand Shuin (IISER Bhopal)
Date: Wed, 30 Sep 2020
Time: 4 pm
Venue: Microsoft Teams (online)
The spherical averages often make their appearance in partial differential equations. For instance, the solution of the wave equation \begin{equation} u_{tt}=\Delta u,\ \ u(x,0)=0,\ \ u_{t}(x,0)=f(x),\ \ in\ \ \mathbb{R}^{3}\ \ is \end{equation} \begin{equation} u(x,t)=\frac{t}{4\pi}\int_{\mathbb{S}^{2}}f(x-ty)d\sigma(y), \end{equation}
where $d\sigma$ is the rotation invariant, normalized surface measure on the sphere $\mathbb{S}^{2}$. In [Proc. Natl. Acad. Sci. USA (1976)], Stein proved the following result:
Theorem. Let $n \geq 3$. Then \begin{equation}\Vert \sup_{t>0} \int_{\mathbb{S}^{n-1}}f(x-ty)d\sigma(y) \Vert_{L^{p}(\mathbb{R}^{n})} \leq C_{p}\Vert f\Vert_{L^{p}(\mathbb{R}^{n})} \end{equation} if, and only if $\frac{n}{n-1}<p\leq\infty$.
The above result was extended to dimension $n=2$, by Bourgain in [J. d’Anal. Math. (1986)]. Later, in [J. d’Anal. Math. (2019)] Lacey proved sparse domination for both lacunary and full spherical maximal functions.
In this talk, I shall talk about the bilinear spherical maximal functions of product type, which is defined in the spirit of bilinear Hardy–Littlewood maximal function. The lacunary and full bilinear spherical maximal functions are defined by \begin{equation} \mathcal{M}_{lac}(f_1,f_2)(x):= \sup_{j\in\mathbb{Z}} \prod_{i=1,2} \int_{\mathbb{S}^{n-1}} f_i(x-2^{j}y_i)d\sigma(y_i), \end{equation} \begin{equation} \mathcal{M}_{full}(f_1,f_2)(x):= \sup_{r>0} \prod_{i=1,2} \int_{\mathbb{S}^{n-1}} f_i(x-ry_i)d\sigma(y_i), \end{equation} where $f_{1},f_{2}\in\mathcal{S}(\mathbb{R}^{n})$, the Schwartz class. We have investigated the sparse domination and weighted boundedness of both the operators $\mathcal{M}_{lac}$ and $\mathcal{M}_{full}$ with respect to the bilinear Muckenhoupt weights $A_{\vec{p},\vec{r}}$. (Joint with Saurabh Shrivastava and Luz Roncal.)
Title: Algebra & Combinatorics Seminar: Densities on Dedekind domains, completions and Haar measure
Speaker: Ignazio Longhi (IISc Mathematics)
Date: Fri, 25 Sep 2020
Time: 3 pm
Venue: Microsoft Teams (online)
A traditional way of assessing the size of a subset X of the integers is to use some version of density. An alternative approach, independently rediscovered by many authors, is to look at the closure of X in the profinite completion of the integers. This for example gives a quick, intuitive solution to questions like: what is the probability that an integer is square-free? Moreover, in many cases, one finds that the density of X can be recovered as the Haar measure of the closure of X. I will discuss some things that one can learn from this approach in the more general setting of rings of integers in global fields. This is joint work with Luca Demangos.
Title: MS Thesis colloquium: Random graphs with given degree sequence
Speaker: Neeladri Maitra (IISc Mathematics)
Date: Thu, 24 Sep 2020
Time: 2:30 pm
Venue: Microsoft Teams (online)
The first part of this talk deals with identifying and proving the scaling limit of a uniform tree with given child sequence. A non-negative sequence of integers $\mathbf{c}=(c_1, c_2, …, c_l)$ with sum $l-1$ is called a child sequence for a rooted tree $t$ on $l$ nodes, if for some ordering $v_1, v_2,…, v_l$ of the nodes, $v_i$ has exactly $c_i$ many children. Consider for each $n$, a child sequence $\mathbf{c}^n$ with sum $n-1$, and let $\mathbf{t}_n$ be the plane tree with $n$ nodes, which is uniformly distributed over the set of all plane trees having $\mathbf{c}^n$ as their child sequence. Broutin and Marckert (2012) prove that under certain assumptions on $\mathbf{c}^n$, the scaling limit of $\mathbf{t}_n$, suitably normalized, is the Brownian Continuum Random Tree (BCRT). We consider a more general setting, where a finite number of vertices of $\mathbf{t}_n$ are allowed to have large degrees. We prove that the scaling limit of $\mathbf{t}_n$ in this regime is the Inhomogeneous Continuum Random Tree (ICRT), in the Gromov-Hausdorff sense.
In the second part, we look at vacant sets left by random walks on random graphs via simulations. Cerny, Teixeira and Windisch (2011) proved that for random $d$-regular graphs, there is a number $u_{\star}$, such that if a random walk is run up to time $un$ with $u<u_{\star}$, $n$ being the total number of nodes in the graph, a giant component of size $\text{O}(n)$ of the subgraph spanned by the vacant nodes i.e. the nodes that are not visited by the random walk, is seen. Whereas if the random walk is run up to time $un$ with $u>u_{\star}$, the size of the largest component of the subgraph spanned by the vacant nodes becomes $\text{o}(n)$. With the help of simulations, we try to investigate whether there is such a phenomenon for supercritical configuration models with heavy-tailed degrees.
Title: APRG Seminar: Certain quasi-analyticity results on Riemannian symmetric spaces of noncompact type
Speaker: Mithun Bhowmik (IISc Mathematics)
Date: Thu, 24 Sep 2020
Time: 4 pm
Venue: Microsoft Teams (online)
An $L^2$ version of the celebrated Denjoy-Carleman theorem regarding quasi-analytic functions was proved by Chernoff on $\mathbb{R}^d$ using iterates of the Laplacian. In 1934, Ingham used the classical Denjoy-Carleman theorem to relate the decay of Fourier transform and quasi-analyticity of an integrable function on $\mathbb{R}$. In this talk, we discuss analogues of the theorems of Chernoff and Ingham for Riemannian symmetric spaces of noncompact type and show that the theorem of Ingham follows from that of Chernoff.
Title: Algebra & Combinatorics Seminar: Totally positive matrices, Polya frequency sequences, and Schur polynomials (Joint with APRG Seminar)
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 18 Sep 2020
Time: 3 pm
Venue: Microsoft Teams (online)
I will discuss totally positive/non-negative matrices and kernels, including Polya frequency (PF) functions and sequences. This includes examples, history, and basic results on total positivity, variation diminution, sign non-reversal, and generating functions of PF sequences (with some proofs). I will end with applications of total positivity to old and new phenomena involving Schur polynomials.
Title: APRG Seminar: An introduction to total positivity
Speaker: Apoorva Khare (IISc Mathematics)
Date: Wed, 09 Sep 2020
Time: 11 am
Venue: Microsoft Teams (online)
I will give a gentle introduction to total positivity and the theory of Polya frequency (PF) functions. This includes their spectral properties, basic examples including via convolution, and a few proofs to show how the main ingredients fit together. Many classical results (and one Hypothesis) from before 1955 feature in this journey. I will end by describing how PF functions connect to the Laguerre–Polya class and hence Polya–Schur multipliers, and mention 21st century incarnations of the latter.
Title: Algebra & Combinatorics Seminar: Algebraic K-theory of varieties with group actions
Speaker: Charanya Ravi (Universität Regensburg, Germany)
Date: Tue, 08 Sep 2020
Time: 3 pm
Venue: Microsoft Teams (online)
Cohomology theories are one of the most important algebraic invariants of topological spaces and this has inspired the definition of several different cohomology theories in algebraic geometry. In this talk, we focus on algebraic K-theory, which is one such classical cohomological invariant of algebraic varieties. After motivating and introducing this notion, we discuss several fundamental properties of algebraic K-theory of varieties with algebraic group actions. Well-known examples of varieties with group actions include toric varieties and flag varieties.
Title: Algebra & Combinatorics Seminar: Some Bernstein projectors for $SL_2$
Speaker: Sandeep Varma (TIFR, Mumbai)
Date: Fri, 04 Sep 2020
Time: 3 pm
Venue: Microsoft Teams (online)
Let $G$ be the group $SL_2$ over a finite extension $F$ of $\mathbb{Q}_p$, $p$ odd. I will discuss certain distributions on $G(F)$, belonging to what is called its Bernstein center (I will explain what this and many other terms in this abstract mean), supported in a certain explicit subset of $G(F)$ arising from the work of A. Moy and G. Prasad. The assertion is that these distributions form a subring of the Bernstein center, and that convolution with these distributions has very agreeable properties with respect to orbital integrals. These are ‘depth $r$ versions’ of results proved for general reductive groups by J.-F. Dat, R. Bezrukavnikov, A. Braverman and D. Kazhdan.
Title: APRG Seminar: Well-posedness and tamed schemes for McKean-Vlasov Stochastic Differential Equations with Common Noise
Speaker: Chaman Kumar (IIT Roorkee)
Date: Wed, 02 Sep 2020
Time: 3 pm
Venue: Microsoft Teams (online)
We prove well-posedness of McKean–Vlasov stochastic differential equations (McKean–Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. We propose (moment) stable numerical schemes for this class of McKean–Vlasov SDEs, namely tamed Euler and tamed Milstein schemes. Further, their rates of convergence in strong sense are shown to be 1/2 and 1 respectively. We employ the notion of measure derivative introduced by P.-L. Lions in his lectures delivered at the College de France. The strong convergence of the tamed Milstein scheme is established under mild regularity assumptions on the coefficients. To demonstrate our theoretical findings, we perform several numerical simulations on popular models such as mean-field versions of stochastic 3/2 volatility models and stochastic double well dynamics with multiplicative noise.
The talk is based on my recent joint works with Neelima (Delhi University), Christoph Reisinger (Oxford University) and Wolfgang Stockinger (Oxford University).
Title: Algebra & Combinatorics Seminar: Codimension two cycles in Iwasawa theory
Speaker: Bharathwaj Palvannan (National Center for Theoretical Sciences, Taiwan)
Date: Fri, 28 Aug 2020
Time: 3 pm
Venue: Microsoft Teams (online)
In classical Iwasawa theory, one studies a relationship called the Iwasawa main conjecture, between an analytic object (the p-adic L-function) and an algebraic object (the Selmer group). This relationship involves codimension one cycles of an Iwasawa algebra. The topic of higher codimension Iwasawa theory seeks to generalize this relationship. We will describe a result in this topic using codimension two cycles, involving an elliptic curve with supersingular reduction. This is joint work with Antonio Lei.
Title: PhD Thesis defence: Sparse bounds for various maximal functions
Speaker: Sourav Hait (IISc Mathematics)
Date: Fri, 21 Aug 2020
Time: 11 am
Venue: Microsoft Teams (online)
Title: PhD Thesis colloquium: Total variation cutoff for random walks on some finite groups
Speaker: Subhajit Ghosh (IISc Mathematics)
Date: Mon, 10 Aug 2020
Time: 3 pm
Venue: Microsoft Teams (online)
This thesis studies the mixing times for three random walk models. Specifically, these are the random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below:
The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrix of this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $\left(n-\frac{3}{2}\right)\log n$ for this shuffle.
The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n\log n$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i< n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.
The random walk on the complete monomial group: Let $G_1\subseteq\cdots\subseteq G_n \subseteq\cdots $ be a sequence of finite groups with $|G_1|>2$. We study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(e,\dots,e,g;$id$)$ and $(e,\dots,e,g^{-1},e,\dots,e,g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n+\frac{1}{2}n\log (|G_n|-1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n\log n$ if $|G_n|=o(n^{\delta})$ for all $\delta>0$.
Title: PhD Thesis colloquium: Trace Estimate For The Determinant Operators
Speaker: Paramita Pramanick (IISc Mathematics)
Date: Thu, 06 Aug 2020
Time: 4 pm
Venue: Microsoft Teams (online)
Let $\boldsymbol T=(T_1, \ldots , T_d)$ be $d$ -tuple of commuting operators on a Hilbert space $\mathcal{H}$. Assume that $\boldsymbol T$ is hyponormal, that is, $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ] :=\big (\big ( \big [ T_j^*,T_i] \big )\big )$ acting on the $d$-fold direct sum of the Hilbert space $\mathcal{H}$ is non-negative definite. The commutator $[T_j^*,T_i]$, $1\leq i,j \leq d$, of a finitely cyclic and hyponormal $d$-tuple is not necessarily compact and therefore the question of finding trace inequalities for such a $d$-tuple does not arise. A generalization of the Berger-Shaw theorem for commuting tuple $\boldsymbol T$ of hyponormal operators was obtained by Douglas and Yan decades ago. We discuss several examples of this generalization in an attempt to understand if the crucial hypothesis{\rm in their theorem requiring the Krull dimension of the Hilbert module over the polynomial ring defined by the map $p\to p(\boldsymbol T)$, $p\in \mathbb C[\boldsymbol z]$, is optimal or not. Indeed, we find examples $\boldsymbol T$ to show that there a large class operators for which $\text{trace}[T_j^*,T_i]$, $1\leq j,i \leq d$, is finite but the $d$-tuple is not finitely polynomially cyclic, which is one of the hypothesis of the Douglas-Yan theorem. We also introduce the weaker notion of “projectively hyponormal operators” and show that the Douglas-Yan theorem remains valid even under this weaker hypothesis. However, one might look for a function of $ \big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$ which may be in trace class. For this, we define an operator valued determinant of a $d\times d$-block operator $\boldsymbol B := \big (\big ( B_{i j} \big ) \big )$ by the formula
\begin{equation} \text{dEt}\big (\boldsymbol{B}\big ):=\sum_{\sigma, \tau \in \mathfrak S_d} \text{sgn}(\sigma)B_{\tau(1),\sigma(\tau(1))}B_{\tau(2),\sigma(\tau(2))},\ldots, B_{\tau(d),\sigma(\tau(d))}. \end{equation}
It is then natural to investigate the properties of the operator
$\mbox{dEt}\big (\big [\big [ \boldsymbol T^*, \boldsymbol T \big ]\big ] \big ),$
in this case, $B_{i j} = [T_j^*,T_i]$.
Indeed, we show that the operator dEt equals the generalized commutator
$\text{GC} \big (\boldsymbol T^*, \boldsymbol T \big )$ introduced earlier by
Helton and Howe. Among other things, we find a trace inequality for the operator
$\mbox{dEt}\big (\big [\big [ \boldsymbol T^*, \boldsymbol T \big ]\big] \big ),$
after imposing certain growth and cyclicity condition on the operator $\boldsymbol T$, namely,
\begin{equation} \text{trace} \big( {\rm dEt} \big( [[ \boldsymbol{T}^*, \boldsymbol{T} ]] \big) \big) \leq m \vartheta d! \prod_{i=1}^{d} |T_i|^2 \end{equation}
for some $\vartheta \geq 1.$ We give explicit examples illustrating the abstract inequality.
Title: MS Thesis defence: Category $\mathcal{O}$ of $\mathfrak{g}$-modules and BGG reciprocity
Speaker: Swarnalipa Datta (IISc Mathematics)
Date: Fri, 15 May 2020
Time: 11:30 am
Venue: Microsoft Teams (online)
In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$-modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra. We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists a one-to-one correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$ admits a $p$-filtration.
Title: PhD Thesis defence: Geometric invariants for a class of submodules of analytic Hilbert modules
Speaker: Samrat Sen (IISc Mathematics)
Date: Thu, 23 Apr 2020
Time: 4 pm
Venue: Microsoft Teams (online)
Title: Algebra & Combinatorics Seminar: Projective representations of Heisenberg groups
Speaker: Sumana Hatui (IISc Mathematics)
Date: Fri, 13 Mar 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
The theory of projective representations of groups, extensively studied by Schur, involves understanding homomorphisms from a group into the projective linear groups. By definition, every ordinary representation of a group is also projective but the converse need not be true. Therefore understanding the projective representations of a group is a deeper problem and many a times also more difficult in nature. To deal with this, an important role is played by a group called the Schur multiplier.
In this talk, we shall describe the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their $t$-variants. We shall discuss the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional complex projective irreducible representations.
This is a joint work with Pooja Singla.
Title: Geometry & Topology Seminar: The two-systole of stretched positive scalar curvature metric on S^2xS^2
Speaker: Thomas Richard (Université Paris-Est Créteil, France)
Date: Wed, 11 Mar 2020
Time: 4:00 pm
Venue: LH-1, Mathematics Department
Let $(M,g)$ be a Riemannian manifold and ‘$c$’ be some homology class of $M$. The systole of $c$ is the minimum of the $k$-volume over all possible representatives of $c$. We will use combine recent works of Gromov and Zhu to show an upper bound for the systole of $S^2 \times \{*\}$ under the assumption that $S^2 \times \{*\}$ contains two representatives which are far enough from each other.
Title: MS Thesis defence: Determinantal processes and stochastic domination
Speaker: Raghavendra Tripathi (IISc Mathematics and University of Washington)
Date: Tue, 10 Mar 2020
Time: 11 am
Venue: LH-1, Mathematics Department
The systematic study of determinantal processes began with the work of Macchi (1975), and since then it has appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths), and physics (fermions, repulsion arising in quantum physics). The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a separable Hilbert space. Let $H$ and K be two finite-dimensional subspaces of a Hilbert space, and $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons (2003) showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and provides a unified approach of proving the result in discrete as well as continuous case.
As an application of the above result, we will obtain the stochastic domination between the largest eigenvalues of Wishart matrix ensembles $W(N, N)$ and $W(N-1, N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M, N)$ has the same distribution as the directed last-passage time $G(M, N)$ on $\mathbb{Z}^2$ with the i.i.d. exponential weight. This was recently used by Basu and Ganguly to obtain stochastic domination between $G(N, N)$ and $G(N-1, N+1)$. Similar connections are also known between the largest eigenvalue of the Meixner ensemble and the directed last-passage time on $\mathbb{Z}^2$ with the i.i.d. geometric weight. We prove another stochastic domination result, which combined with Lyons’ result, gives the stochastic domination between the eigenvalue processes of Meixner ensembles $M(N, N)$ and $M(N-1, N+1)$.
Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Tue, 10 Mar 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
Lectures 1–4 (Jan 14, 16, 29, Feb 5):
- Automorphic Forms on Semisimple Lie Groups
- Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension
- Automorphic Forms in Berezin Quantization and von Neumann Algebras
Lecture 5. Multivariable Scattering Theory (Tuesday, March 10)
Title: Geometry & Topology Seminar: Approximation of weak G-bundles in high dimensions
Speaker: Swarnendu Sil (ETH Zurich)
Date: Mon, 09 Mar 2020
Time: 4:00 pm
Venue: LH-1, Mathematics Department
The analysis for Yang-Mills functional and in general, problems related to higher dimensional gauge theory, often requires one to work with weak notions of principal G-bundles and connections on them. The bundle transition functions for such bundles are not continuous and thus there is no obvious notion of a topological isomorphism class.
In this talk, we shall discuss a few natural classes of weak bundles with connections which can be approximated in the appropriate norm topology by smooth connections on smooth bundles. We also show how we can associate a topological isomorphism class to such bundle-connection pairs, which is invariant under weak gauge changes. In stark contrast to classical notions, this topological isomorphism class is not independent of the connection.
Title: Algebra & Combinatorics Seminar: Formulas for the Gross-Stark units
Speaker: Matthew Honnor (King's College London, UK)
Date: Fri, 06 Mar 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
In the 1980’s Tate stated the Brumer–Stark conjecture which, for a totally real field $F$ with prime ideal $\mathfrak{p}$, conjectures the existence of a $\mathfrak{p}$-unit called the Gross–Stark unit. This unit has $\mathfrak{P}$ order equal to the value of a partial zeta function at 0, for a prime $\mathfrak{P}$ above $\mathfrak{p}$. In 2008 and 2018 Dasgupta and Dasgupta–Spieß, conjectured formulas for this unit. During this talk I shall explain Tate’s conjecture and then the ideas for the constructions of these formulas. I will finish by explaining the results I have obtained from comparing these formulas.
Title: Eigenfunctions Seminar: On the decimal expansion of log(2020/2019) and $e$
Speaker: Yann Bugeaud (IRMA, Université de Strasbourg, France)
Date: Fri, 28 Feb 2020
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-5, Mathematics Department
It is commonly expected that $e$, $\log 2$, $\sqrt{2}$, $\pi$, among other “classical” numbers, behave, in many respects, like almost all real numbers. For instance, they are expected to be normal to base 10, that is, one believes that their decimal expansion contains every finite block of digits from ${0, \ldots , 9}$. We are very far away from establishing such a strong assertion. However, there has been some small recent progress in that direction. After surveying classical results and problems on normal numbers, we will adopt a point of view from combinatorics on words and show that the decimal expansions of $e$, of any irrational algebraic number, and of $\log (1 + \frac{1}{a})$, for a sufficiently large integer $a$, cannot be ‘too simple’, in a suitable sense.
Title: Algebra & Combinatorics Seminar: Joint extreme values of L functions (Joint with APRG Seminar)
Speaker: Akshaa Vatwani (IIT, Gandhinagar)
Date: Wed, 26 Feb 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
We consider $L$-functions $L_1,\ldots,L_k$ from the Selberg class having polynomial Euler product and satisfying Selberg’s orthonormality condition. We show that on every vertical line $s=\sigma+it$ in the complex plane with $\sigma \in(1/2,1)$, these $L$-functions simultaneously take “large” values inside a small neighborhood.
This is joint work with Kamalakshya Mahatab and Lukasz Pankowski.
Title: Algebra & Combinatorics Seminar: Unit-root $L$-functions and $p$-adic differential equations
Speaker: Wansu Kim (KAIST, Daejeon, South Korea)
Date: Mon, 24 Feb 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
We start with reviewing Dwork’s seminal work on a certain $p$-adic hypergeometric function, which has an application to the unit-root $L$-function of the Legendre family of elliptic curves in characteristic $p>2$. Then I would like to overview what can be said about unit-root $L$-function of the family of abelian varieties over a curve, and discuss its potential applications.
Title: Algebra & Combinatorics Seminar: Twisted conjugacy in linear groups over certain rings
Speaker: Parameswaran Sankaran (CMI, Chennai)
Date: Fri, 21 Feb 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
An endomorphism $\phi: G\to G$ of a group yields an action of $G$ on itself, known as the $\phi$-twisted conjugacy action, where $(g,x)\mapsto gx\phi(g^{-1})$. The group $G$ is said to have the property $R_\infty$ if, for any automorphism $\phi$ of $G$, the orbit space of the $\phi$-twisted conjugacy action is infinite. This notion, and the related notion of Reidemeister number, originated from Nielsen fixed point theory.
It is an interesting problem to decide, given an infinite group $G$ whether or not $G$ has property $R_\infty$. We will consider the problem in the case when $G=GL(n,R), SL(n,R), n\ge 2$, when $R$ is either a polynomial ring or a Laurent polynomial ring over a finite field $\mathbb{F}_q$.
The talk is based on recent joint work with Oorna Mitra.
Title: Algebra & Combinatorics Seminar: Shintani descent and character sheaves on algebraic groups
Speaker: Tanmay Deshpande (TIFR, Mumbai)
Date: Fri, 07 Feb 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
Let $G$ be an algebraic group defined over a finite field $\mathbb{F}_q$ and let $m$ be a positive integer. Shintani descent is a relationship between the character theories of the two finite groups $G(\mathbb{F}_q)$ and $G(\mathbb{F}_{q^m})$ of $\mathbb{F}_q$ and $\mathbb{F}_{q^m}$-valued points of $G$ respectively. This was first studied by Shintani for $G=GL_n$. Later, Shoji studied Shintani descent for connected reductive groups and related it to Lusztig’s theory of character sheaves. In this talk, I will speak on the cases where $G$ is a unipotent or solvable algebraic group. I will also explain the relationship with the theory of character sheaves.
Title: APRG Seminar: Purely sequential estimation of a negative binomial mean with applications in ecology
Speaker: Sudeep Bapat (University of Calfornia at Santa Barbara, USA)
Date: Wed, 05 Feb 2020
Time: 2 pm
Venue: LH-1, Mathematics Department
We discuss a set of purely sequential strategies to estimate an unknown negative binomial mean $\mu$ under different forms of loss functions. We develop point estimation techniques where the thatch parameter $\tau$ may be known or unknown. Both asymptotic first-order efficiency and risk efficiency properties will be elaborated. The results will be supported by an extensive set of data analysis carried out via computer simulations for a wide variety of sample sizes. We observe that all of our purely sequential estimation strategies perform remarkably well under different situations. We also illustrate the implementation of these methodologies using real datasets from ecology, namely, weed count data and data on migrating woodlarks. (This is a Skype talk.)
Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Wed, 05 Feb 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesdays, January 29 and February 5)
Lecture 5. Multivariable Scattering Theory
Title: Geometry & Topology Seminar: Enumerative Geometry of the Quintic Threefold
Speaker: Ritwik Mukherjee (NISER, Bhubaneswar)
Date: Fri, 31 Jan 2020
Time: 10:00 am
Venue: LH-1, Mathematics Department
The quintic threefold (the zero set of a homogeneous degree 5 polynomial on CP^4) is one of the most famous examples of a Calabi Yau manifold. It is one of the most studied in the field of Enumerative Geometry. For example, how many lines are there on a Quintic threefold? In this talk we will explain some approaches to count curves on the Quintic threefold. In particular, we will try to explain the following idea: If Y is a submanifold of X, and we understand the Enumerative Geometry of X, how can we answer questions about the Enumerative Geometry of Y? We will try to explain the idea used by Andreas Gathman to compute all the genus zero Gromov-Witten invariants of the Quintic Threefold.
The talk will be self contained and will not assume any prior knowledge of Enumerative Geometry or Gromov-Witten Invariants.
Title: Algebra & Combinatorics Seminar: An extremal problem related to weighted Davenport constant
Speaker: Eshita Mazumdar (ISI, Bangalore)
Date: Fri, 31 Jan 2020
Time: 3 pm
Venue: LH-3, Mathematics Department
For a finite abelian group $G$ and $A \subset [1, \exp(G) - 1]$, the $A$-weighted Davenport Constant $D_A(G)$ is defined to be the least positive integer $k$ such that any sequence $S$ with length $k$ over $G$ has a non-empty $A$-weighted zero-sum subsequence. The original motivation for studying Davenport Constant was the problem of non-unique factorization in number fields. The precise value of this invariant for the cyclic group for certain sets $A$ is known but the general case is still unknown. Typically an extremal problem deals with the problem of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects that satisfies certain requirements. In a recent work with Prof. Niranjan Balachandran, we introduced an Extremal Problem for a finite abelian group related to Weighted Davenport Constant. In this talk I will talk about the behaviour of it for different groups, specially for cyclic group.
Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Wed, 29 Jan 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesdays, January 29 and February 5)
Lecture 5. Multivariable Scattering Theory
Title: PhD Thesis colloquium: Sparse bound of various spherical maximal functions
Speaker: Sourav Hait (IISc Mathematics)
Date: Wed, 22 Jan 2020
Time: 4 pm
Venue: LH-1, Mathematics Department
In 1976, E.M. Stein proved $L^p$ bounds for spherical maximal function on Euclidean space. The lacunary case was dealt on later by C.P. Calderon in 1979. In a recent paper, M. Lacey has proved sparse bound for these functions and $L^p$ bounds will follow immediately as a result.
In this talk, we will look at various maximal functions corresponding to spherical averages and find sparse bounds for those functions. We will also observe some weighted and unweighted estimates that will follow as a consequences.
First, we will show sparse bound for lacunary spherical maximal function on Heisenberg group . Next we move on to full spherical maximal function. Then we study lacunary maximal function corresponding to the spherical average on product of Heisenberg groups. Finally, we will revisit generalized spherical averages on Euclidean space and prove sparse bounds for the related maximal functions.
Title: Eigenfunctions Seminar: Glimpses of KdV Equation and Soliton theory
Speaker: Phoolan Prasad (IISc Mathematics)
Date: Fri, 17 Jan 2020
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
Solitons are solutions of a special class of nonlinear partial differential equations (soliton equations, the best example is the KdV equation). They are waves but behave like particles. The term “soliton” combining the beginning of the word “solitary” with ending “on” means a concept of a fundamental particle like “proton” or “electron”.
The events: (1) sighting, by chance, of a great wave of translation, “solitary wave”, in 1834 by Scott–Russell, (2) derivation of KdV equation by Korteweg de Vries in 1895, (3) observation of a very special type of wave interactions in numerical experiments by Kruskal and Zabusky in 1965, (4) development of the inverse scattering method for solving initial value problems by Gardener, Greene, Kruskal and Miura in 1967, (5) formulation of a general theory in 1968 by P.D. Lax and (6) contributions to deep theories starting from the work by R. Hirota (1971-74) and David Mumford (1978-79), which also gave simple methods of solutions of soliton equations, led to the development of one of most important areas of mathematics in the 20th century.
This also led to a valuable application of solitons to physics, engineering and technology. There are two aspects of soliton theory arising out of the KdV Equation:
- Applied mathematics – analysis of nonlinear PDE leading to dynamics of waves.
- Pure mathematics – algebraic geometry. It is surprising that each one of these can inform us of the other in the intersection that is soliton theory, an outcome of the KdV equation.
The subject is too big but I shall try to give some glimpses (1) of the history, (2) of the inverse scattering method, and (3) show that an algorithm based on algebraic-geometric approach is much easier to derive soliton solutions.
Title: PhD Thesis defence: On the signs of Hecke eigenvalues of modular forms and differential operators on Jacobi forms
Speaker: Ritwik Pal (IISc Mathematics)
Date: Thu, 16 Jan 2020
Time: 10:30 am
Venue: LH-1, Mathematics Department
This talk broadly has two parts. The first one is about the signs of Hecke eigenvalues of modular forms and the second is about a problem on certain holomorphic differential operators on the space of Jacobi forms.
In the first part we will briefly discuss how the statistics of signs of newforms determine them (work of Matomaki-Soundararajan-Kowalski) and then introduce certain ‘Linnik-type’ problems (the original problem was concerning the size of the smallest prime in an arithmetic progression in terms of the modulus) which ask for the size of the first negative eigenvalue (in terms of the analytic conductor) of various types modular forms, which has seen a lot of recent interest. Also specifically we will discuss the problem in the context of Yoshida lifts (a certain subspace of the Siegel modular forms), where in the thesis, we have improved upon the previously known result on this topic significantly. We will prove that the smallest $n$ with $\lambda(n)<0$ satisfy $n < Q_{F}^{1/2-2\theta+\epsilon}$, where $Q_{F}$ is the analytic conductor of a Yoshida lift $F$ and $0<\theta <1/4$ is some constant. The crucial point is establishing a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level.
We will focus on a similar question concerning the first negative Fourier coefficient of a Hilbert newform. If ${C(\mathfrak{m})}_{\mathfrak{m}}$ denotes the Fourier coefficients of a Hilbert newform $f$, then we show that the smallest among the norms of ideals $\mathfrak{m}$ such that $ C(\mathfrak{m})<0$, is bounded by $Q_{f}^{9/20+\epsilon}$ when the weight vector of $f$ is even and $Q_{f}^{1/2+\epsilon}$ otherwise. This improves the previously known result on this problem significantly. Here we would show how to use certain ‘good’ Hecke relations among the eigenvalues and some standard tools from analytic number theory to achieve our goal.
Finally we would talk about the statistical distribution of the signs of the Fourier coefficients of a Hilbert newform and essentially prove that asymptotically, half of them are positive and half negative. This was a breakthrough result of Matomaki-Radziwill for elliptic modular forms, and our results are inspired by those. The proof hinges on establishing some of their machinery of averages multiplicative functions to the number field setting.
In the second part of the talk we will introduce Jacobi forms and certain differential operators indexed by $\{D_{v}\}_{0}^{2m}$ that maps the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the direct sum of the differential operators $D_{v}$ for $v={1,2,…,2m}$ maps $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. Inspired by certain conjectures of Hashimoto on theta series, S. Bocherer raised the question whether any of the differential operators be removed from that map while preserving the injectivity. In the case of even weights S. Das and B. Ramakrishnan show that it is possible to remove the last operator. In the talk we will discuss the case of the odd weights and prove a similar result. The crucial step (and the main difference from the even weight case) in the proof is to establish that a certain tuple of congruent theta series is a vector valued modular form and finding the automorphy of the Wronskian of this tuple of theta series.
Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Thu, 16 Jan 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
The date and time for the third and fourth lectures will be announced in due course.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras
Lecture 5. Multivariable Scattering Theory
Title: PhD Thesis defence: Phase Transition, Percolation at Criticality, Poisson convergence of isolated vertices and Connectivity in Random Connection Models
Speaker: Sanjoy Kumar Jhawar (IISc Mathematics)
Date: Tue, 14 Jan 2020
Time: 11 am
Venue: LH-1, Mathematics Department
This work has two parts. The first part contains the study of phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogeneous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y\mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid \cdot \mid$ is the Euclidean norm. In the inhomogeneous version of the model, points of $\mathcal{P}_{\lambda}$ is endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability
\begin{equation} \left(1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)\right) \end{equation}
for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.
In the second part we consider an inhomogeneous random connection model on a $d$ -dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of isolated vertices converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.
Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Tue, 14 Jan 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.
The date and time for the third and fourth lectures will be announced in due course.
Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)
Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)
Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras
Lecture 5. Multivariable Scattering Theory
Title: APRG Seminar: Brownian structure in universal KPZ objects
Speaker: Milind Hegde (University of California, Berkeley, USA)
Date: Mon, 13 Jan 2020
Time: 2:15 pm
Venue: LH-1, Mathematics Department
Many models of one dimensional random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For a few such models, the limiting interface profile, after scaling by characteristic KPZ scaling exponents of one-third and two-third, is known to be the Airy_2 process shifted by a parabola. This limiting process is expected to be “locally Brownian”, and a recent result gives a quantified bound on probabilities of events under the Airy_2 process on a unit order interval in terms of probabilities of the same events under Brownian motion (of rate two). This comparison also holds in the prelimit for the particular model of Brownian last passage percolation. In this talk, we will introduce KPZ universality and discuss this result and a number of consequences, using last passage percolation as an expository framework.
Joint work with Jacob Calvert and Alan Hammond.
Title: Algebra & Combinatorics Seminar: The Kneser–Tits conjecture for algebraic groups
Speaker: Maneesh Thakur (ISI, Bangalore)
Date: Fri, 10 Jan 2020
Time: 3 pm
Venue: LH-1, Mathematics Department
We will discuss the celebrated Kneser–Tits conjecture for algebraic groups and report on some recent results. We will keep the technicalities to the minimum.
Title: Geometry & Topology Seminar: Ricci flow and diagonalizing the Ricci tensor
Speaker: Anusha Krishnan (Syracuse University)
Date: Mon, 06 Jan 2020
Time: 4:00 pm
Venue: LH-1, Mathematics Department
We will discuss some work on the Ricci flow on manifolds with symmetries. In particular, cohomogeneity one manifolds, i.e. a Riemannian manifold M with an isometric action by a Lie group G such that the orbit space M/G is one-dimensional. We will also explain how this relates to diagonalizing the Ricci tensor on Lie groups and homogeneous spaces.
Title: Algebra & Combinatorics Seminar: On congruences in Iwasawa theory
Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)
Date: Fri, 20 Dec 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
The talk will focus on congruences modulo a prime $p$ of arithmetic invariants that are associated to the Iwasawa theory of Galois representations arising from elliptic curves. These congruences fit in the framework of some deep conjectures in Iwasawa theory which relate arithmetic and analytic invariants.
Title: APRG Seminar: Local geometry of the rough-smooth interface in the two-periodic Aztec diamond
Speaker: Sunil Chhita (Durham University, UK)
Date: Thu, 19 Dec 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, we found that a certain averaging of the height function at the rough smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level lines of the height function. We show after suitable centering and rescaling that a point process from these paths converge to the extended Airy kernel point process provided that the natural parameter associated to the two-periodic Aztec diamond is small enough. This is joint work with Kurt Johansson and Vincent Beffara.
Title: Eigenfunctions Seminar: The Riemann Hypothesis: History and Recent Work
Speaker: Ken Ono (University of Virginia, Charlottesville, USA)
Date: Wed, 18 Dec 2019
Time: 3:15 – 5:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Lecture 1:
Lecture 2:
The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of one-half. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize. In this lecture, Ken Ono will discuss the mathematical meaning of the Riemann hypothesis and why it matters. Along the way, he will tell tales of mysteries about prime numbers and highlight new advances. He will conclude with a discussion of recent joint work with mathematicians Michael Griffin of Brigham Young University, Larry Rolen of Georgia Tech, and Don Zagier of the Max Planck Institute, which sheds new light on this famous problem.
Title: Algebra & Combinatorics Seminar: Oriented and Colorful Variants of Gyárfás–Sumner Conjecture
Speaker: Sunil Chandran (IISc CSA)
Date: Wed, 11 Dec 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
The Gyárfás–Sumner conjecture states the following: Let $a,b$ be positive integers. Then there exists a function $f$, such that if $G$ is a graph of clique number at most $a$ and chromatic number at least $f(a,b)$, then $G$ contains all trees on at most $b$ vertices as induced subgraphs. This conjecture is still open, though for several special cases it is known to be true. We study the oriented version of this conjecture: Does there exist a function $g$, such that if the chromatic number of an oriented graph $G$ (satisfying certain properties) is at least $g(s)$ then $G$ contains all oriented trees on at most $s$ vertices as its induced subgraphs. In general this statement is not true, not even for triangle free graphs. Therefore, we consider the next natural special class – namely the 4-cycle free graphs – and prove the above statement for that class. We show that $g(s) \leq 4s^2$ in this case.
We also consider the rainbow (colorful) variant of this conjecture. As a special case of our theorem, we significantly improve an earlier result of Gyárfás and Sarkozy regarding the existence of induced rainbow paths in $C_4$ free graphs of high chromatic number. I will also discuss the recent results of Seymour, Scott (and Chudnovsky) on this topic.
Title: PhD Thesis defence: Representations of Special Compact Linear Groups of Order Two: Construction, Representation Growth, Group Algebras, and Branching Laws
Speaker: Hassain M. (IISc Mathematics)
Date: Thu, 05 Dec 2019
Time: 3 pm
Venue: LH-3, Mathematics Department
Let $O$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let $e$ be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.
In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.
Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work for $p=2$. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.
Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.
Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{r})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are NOT isomorphic for $r \geq 2e+2$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{r}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{r}))]$ are NOT isomorphic for $r\geq4$.
Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.
Title: APRG Seminar: Automorphic Forms in Several Complex Variables
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Thu, 05 Dec 2019
Time: 2:30-3:30 pm and 3:45-4:45 pm
Venue: LH-1, Mathematics Department
Automorphic and modular forms in one complex variable (on the upper half-plane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
- Automorphic Forms in Complex Analysis and Algebraic Geometry
- Construction of multi-variable Automorphic Forms on Lie groups and Siegel domains
- Berezin Quantization of Automorphic Forms and type II von Neumann Algebras
- Scattering Theory on Symmetric Spaces and their non-compact Quotients
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
- Introduction: Why multi-variable automorphic forms?
- Quotient Spaces by finite linear groups (Cartan’s Theorem)
- Quotient Spaces by properly discontinuous groups
- Projective embedding of compact quotients of bounded domains
2. Construction and Fourier Analysis of Automorphic Forms
- Theta Functions on Lattices and Jordan Algebras
- Poincaré and Eisenstein Series on Lie Groups
- Siegel Domains and Satake Compactification
- Relations to Number theory and arithmetic lattices
3. Automorphic Forms and Operator Theory
- Reproducing Kernels and Berezin Quantization
- Toeplitz operators with automorphic symbols
- Group von Neumann algebras and intertwiners
4. Automorphic Forms in Scattering Theory
- Scattering in euclidean space and the upper half-plane
- Spherical functions on symmetric spaces
- Scattering operators on symmetric spaces
- Fourier transform and scattering matrix
- The Residues of the scattering matrix
Title: APRG Seminar: Fourier analysis on the infinite-dimensional torus
Speaker: Luz Roncal (BCAM – Basque Center for Applied Mathematics, Spain)
Date: Wed, 04 Dec 2019
Time: 10 am
Venue: LH-1, Mathematics Department
We will show both original and known results on Harmonic Analysis for functions defined on the infinite-dimensional torus, which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the one-dimensional torus, with its corresponding Haar measure. Such results will include:
- Fourier Analysis on $\mathbb{T}^{\omega}$: basic properties, absolutely divergent series, Calder'on-Zygmund decomposition and differentiation of integrals.
- Mixed norm $L^{\bar{p}}(\mathbb{T}^{\omega})$ spaces, $\bar{p}=(p_1,p_2,\ldots)$: definition, properties, duality, interpolation and weak spaces.
- M. Riesz Theorems on $\mathbb{T}^{\omega}$, a.e. convergence, rectangular partial sums and $L^{\bar{p}}$ convergence.
Several open problems and other questions will be considered. Some of the results presented are joint work with Emilio Fernandez (Universidad de La Rioja, Spain).
Title: Algebra & Combinatorics Seminar: Some remarks on Leavitt path algebras
Speaker: John Meakin (University of Nebraska at Lincoln, USA)
Date: Tue, 03 Dec 2019
Time: 11 am
Venue: LH-1, Mathematics Department
The study of Leavitt path algebras has two primary sources, the work of W.G. Leavitt in the early 1960’s on the module type of a ring, and the work by Kumjian, Pask, and Raeburn in the 1990’s on Cuntz-Krieger graph $C^*$-algebras. Given a directed graph $\Gamma$ and a field $F$, the Leavitt path algebra $L_F(\Gamma)$ is an $F$-algebra essentially built from the directed paths in the graph $\Gamma$. Reasonable necessary and sufficient graph-theoretic conditions for two directed graphs to have isomorphic Leavitt path algebras do not seem to be known. In this talk I will discuss a recent construction, due to Zhengpan Wang and myself, of a semigroup $LI(\Gamma)$ associated with a directed graph $\Gamma$, that we call the Leavitt inverse semigroup of $\Gamma$. The semigroup $LI(\Gamma)$ is closely related to the corresponding Leavitt path algebra $L_F(\Gamma)$ and the graph inverse semigroup $I(\Gamma)$ of $\Gamma$. Leavitt inverse semigroups provide a certain amount of structural information about Leavitt path algebras. For example if $LI(\Gamma) \cong LI(\Delta)$, then $L_F(\Gamma) \cong L_F(\Delta)$, but the converse is false. I will discuss some topological aspects of the structure of graph inverse semigroups and Leavitt inverse semigroups: in particular, I will provide necessary and sufficient conditions for two graphs $\Gamma$ and $\Delta$ to have isomorphic Leavitt inverse semigroups.
This is joint work with Zhengpan Wang, Southwest University, Chongqing, China.
Title: Algebra & Combinatorics Seminar: Ideals generated by quadrics defined by determinantal conditions
Speaker: Indranath Sengupta (IIT, Gandhinagar)
Date: Wed, 27 Nov 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
We will present some recent studies on ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are matrices whose entries are polynomials of degree at most 1. We will discuss, how a good Groebner basis for these ideals help us compute primary decompositions and gather various other homological informations.
Title: Algebra & Combinatorics Seminar: Graphs with possible loops
Speaker: Pavol Hell (Simon Fraser University, Canada)
Date: Fri, 22 Nov 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
I will discuss a few examples of concepts that have interesting extensions if loops are allowed (but not required). I will include interval graphs, strongly chordal graphs, and other concepts.
Title: APRG Seminar: Automorphic Forms in Several Complex Variables
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Wed, 20 Nov 2019
Time: 2:30-3:30 pm and 4-5 pm
Venue: LH-1, Mathematics Department
Automorphic and modular forms in one complex variable (on the upper half-plane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
- Automorphic Forms in Complex Analysis and Algebraic Geometry
- Construction of multi-variable Automorphic Forms on Lie groups and Siegel domains
- Berezin Quantization of Automorphic Forms and type II von Neumann Algebras
- Scattering Theory on Symmetric Spaces and their non-compact Quotients
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
- Introduction: Why multi-variable automorphic forms?
- Quotient Spaces by finite linear groups (Cartan’s Theorem)
- Quotient Spaces by properly discontinuous groups
- Projective embedding of compact quotients of bounded domains
2. Construction and Fourier Analysis of Automorphic Forms
- Theta Functions on Lattices and Jordan Algebras
- Poincaré and Eisenstein Series on Lie Groups
- Siegel Domains and Satake Compactification
- Relations to Number theory and arithmetic lattices
3. Automorphic Forms and Operator Theory
- Reproducing Kernels and Berezin Quantization
- Toeplitz operators with automorphic symbols
- Group von Neumann algebras and intertwiners
4. Automorphic Forms in Scattering Theory
- Scattering in euclidean space and the upper half-plane
- Spherical functions on symmetric spaces
- Scattering operators on symmetric spaces
- Fourier transform and scattering matrix
- The Residues of the scattering matrix
Title: Geometry & Topology Seminar: Comparisons of non-abelian theta functions
Speaker: Swarnava Mukhopadhyay (TIFR, Mumbai)
Date: Mon, 18 Nov 2019
Time: 4:00 pm
Venue: LH-1, Mathematics Department
We consider the natural embedding for SO(r) into SL(r) and study the corresponding map between the moduli spaces of principal bundles on smooth projective curves. We compare the spaces of global sections of natural line bundles (non-abelian theta functions) for these moduli spaces and their twisted analogues with the space of theta functions. We will discuss how these results can be applied to obtain an alternate proof of a result of Pauly-Ramanan. If time permits, we will also discuss some applications to the monodromy of the Hitchin/WZW connections. This is a joint work with H. Zelaci.
Title: Algebra & Combinatorics Seminar: The G.-C. Rota approach and the Lehmer conjecture
Speaker: Bernhard Heim (German University of Technology, Oman)
Date: Thu, 14 Nov 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Report on joint work with M. Neuhauser. This includes results with C. Kaiser, F. Luca, F. Rupp, R. Troeger, and A. Weisse.
The Lehmer conjecture and Serre’s lacunary theorem describe the vanishing properties of the Fourier coefficients of even powers of the Dedekind eta function.
G.-C. Rota proposed to translate and study problems in number theory and combinatorics to and via properties of polynomials.
We follow G.-C. Rota’s advice. This leads to several new results and improvement of known results. This includes Kostant’s non-vanishing results attached to simple complex Lie algebras, a new non-vanishing zone of the Nekrasov-Okounkov formula (improving a result of G. Han), a new link between generalized Laguerre and Chebyshev polynomials, strictly sign-changes results of reciprocals of the cubic root of Klein’s absolute $j$- invariant, and hence the $j$-invariant itself. Finally we give an interpretation of the first non-sign change of the Ramanujan $\tau(n)$ function by the root distribution of a certain family of polynomials in the spirit of G.-C. Rota.
Title: APRG Seminar: Automorphic Forms in Several Complex Variables
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Wed, 13 Nov 2019
Time: 2:30-3:30 pm and 4-5 pm
Venue: LH-1, Mathematics Department
Automorphic and modular forms in one complex variable (on the upper half-plane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.
I plan to divide these lectures into four parts:
- Automorphic Forms in Complex Analysis and Algebraic Geometry
- Construction of multi-variable Automorphic Forms on Lie groups and Siegel domains
- Berezin Quantization of Automorphic Forms and type II von Neumann Algebras
- Scattering Theory on Symmetric Spaces and their non-compact Quotients
Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.
1. Automorphic Forms in Complex Analysis and Algebraic Geometry
- Introduction: Why multi-variable automorphic forms?
- Quotient Spaces by finite linear groups (Cartan’s Theorem)
- Quotient Spaces by properly discontinuous groups
- Projective embedding of compact quotients of bounded domains
2. Construction and Fourier Analysis of Automorphic Forms
- Theta Functions on Lattices and Jordan Algebras
- Poincaré and Eisenstein Series on Lie Groups
- Siegel Domains and Satake Compactification
- Relations to Number theory and arithmetic lattices
3. Automorphic Forms and Operator Theory
- Reproducing Kernels and Berezin Quantization
- Toeplitz operators with automorphic symbols
- Group von Neumann algebras and intertwiners
4. Automorphic Forms in Scattering Theory
- Scattering in euclidean space and the upper half-plane
- Spherical functions on symmetric spaces
- Scattering operators on symmetric spaces
- Fourier transform and scattering matrix
- The Residues of the scattering matrix
Title: PhD Thesis defence: On the Kobayashi geometry of domains
Speaker: Anwoy Maitra (IISc Mathematics)
Date: Mon, 11 Nov 2019
Time: 11 am
Venue: LH-1, Mathematics Department
We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:
Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.
Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s well-known big-constant/small-constant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.
The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.
A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newly-introduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.
Wolff–Denjoy-type theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoy-type theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.
Title: Eigenfunctions Seminar: Slides Recent breakthroughs in sphere packing
Speaker: Abhinav Kumar (Stony Brook University, USA)
Date: Fri, 08 Nov 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
The sphere packing problem asks for the densest packing by congruent non-overlapping spheres in n dimensions. It is a famously hard problem, though easy to state, and with many connections to different parts of mathematics and physics. In particular, every dimension seems to have its own idiosyncracies, and until recently no proven answers were known beyond dimension 3, with the 3-dimensional solution being a tour de force of computer-aided mathematics.
Then in 2016, a breakthrough was achieved by Viazovska, solving the sphere packing problem in 8 dimensions. This was followed shortly by joint work of Cohn-Kumar-Miller-Radchenko-Viazovska solving the sphere packing problem in 24 dimensions. The solutions involve linear programming bounds and modular forms. I will attempt to describe the main ideas of the proof.
Title: APRG Seminar: Dirichlet spectral problem in a thin domain with a periodically oscillating boundary
Speaker: Klas Pettersson (UiT – The Arctic University of Norway, Narvik, Norway)
Date: Wed, 06 Nov 2019
Time: 11 am
Venue: LH-1, Mathematics Department
We describe the leading terms in the asymptotic behavior of the eigenvalues and the eigenfunctions to an elliptic Dirichlet spectral problem in a thin finite cylindrical domain with a periodically oscillating boundary by means of homogenization. Under suitable scaling and structure assumptions, the eigenfunctions show oscillatory behavior, and asymptotically localize with a profile solving a diffusion equation with quadratic potential on the real line. Methods for analysis of spectral asymptotics for heterogeneous media will be briefly discussed.
Title: Algebra & Combinatorics Seminar: Special Values of L-functions and period relations for motives
Speaker: Chandrasheel Bhagwat (IISER, Pune)
Date: Wed, 06 Nov 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
We will discuss certain rationality results for the critical values of the degree-$2n$ $L$-functions attached to $GL_1 \times O(n,n)$ over a totally real number field for an even positive integer $n$. We will also discuss some relations for Deligne periods of motives. This is part of a joint work with A. Raghuram.
Title: Algebra & Combinatorics Seminar: Fundamental group of a complex ball quotient
Speaker: Tathagata Basak (Iowa State University, Ames, USA)
Date: Mon, 04 Nov 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Let W be a Weyl group and V be the complexification of its natural reflection representation. Let H be the discriminant divisor in (V/W), that is, the image in (V/W) of the hyperplanes fixed by the reflections in W. It is well known that the fundamental group of the discriminant complement ((V/W) – H) is the Artin group described by the Dynkin diagram of W.
We want to talk about an example for which an analogous result holds. Here W is an arithmetic lattice in PU(13,1) and V is the unit ball in complex thirteen dimensional vector space. Our main result (joint with Daniel Allcock) describes Coxeter type generators for the fundamental group of the discriminant complement ((V/W) – H). This takes a step towards a conjecture of Allcock relating this fundamental group with the Monster simple group.
The example in PU(13,1) is closely related to the Leech lattice. Time permitting, we shall give a second example in PU(9,1) related to the Barnes–Wall lattice for which some similar results hold.
Title: PhD Thesis defence: Fourier coefficients of modular forms and mass of pullbacks of Saito--Kurokawa lifts
Speaker: Pramath Anamby (IISc Mathematics)
Date: Mon, 04 Nov 2019
Time: 4:30 pm
Venue: LH-1, Mathematics Department
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms, having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$ have the same eigenvalues of the $p$-th Hecke operator $T_p$ for almost all primes $p$, then $f_1=f_2$.
The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$.
We expect similar applications. We also discuss few results on the square-free Fourier coefficients of elliptic cusp forms.
In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, \mathbb{Z})$ then we consider its Saito–Kurokawa lift $F_g$ and a certain ‘restricted’ $L^2$-norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.
Pullback of a Siegel modular form $F((\tau,z,z,\tau’))$ ($(\tau,z,z,\tau’)$ in Siegel’s upper half-plane of degree 2) to $H \times H$ is its restriction to $z=0$, which we denote by $F|_{z=0}$. Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$-norms of such pullbacks to central values of $L$-functions for all degrees.
In fact, when a Siegel modular form arises as a Saito–Kurokawa lift (say $F=F_g$), results of Ichino relate the mass of the pullbacks to the central values of certain $GL(3) \times GL(2)$ $L$-functions. Moreover, it has been observed that comparison of the (normalized) norm of $F_g$ with the norm of its pullback provides a measure of concentration of $F_g$ along $z=0$. We recall certain conjectures pertaining to the size of the’mass’. We use the amplification method to improve the currently known bound for $N(F_g)$.
Title: Geometry & Topology Seminar: Invariants of annular links, cobordisms and transverse links from combinatorial link Floer complex
Speaker: Apratim Chakraborty (ISI, Bangalore)
Date: Mon, 04 Nov 2019
Time: 4:15 pm
Venue: LH-2, Mathematics Department
We will define an invariant for annular links using the combinatorial link Floer complex that gives genus bounds for annular cobordisms. The celebrated slice-Bennequin inequality relates slice genus of a knot with its contact geometric invariants. We investigate similar relations in our context. In particular, we will define an invariant of transverse knots that refines the transverse invariant $\theta$ in knot Floer homology.
Title: APRG Seminar: On some local smoothing estimate of Fourier integral operators and Hermite functions
Speaker: Ramesh Manna (TIFR-CAM, Bengaluru)
Date: Wed, 30 Oct 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we discuss the “local smoothing” phenomenon for Fourier integral operators with amplitude function belongs to the “symbol class”. We give an overview of the regularity results which have been proven to date. We use harmonic analysis of Hermite functions in the study of Fourier integral operators. Finally, we give an application of the local smoothing estimate to the wave equation and maximal operators. This is a joint work with Prof. P. K. Ratnakumar.
Title: APRG Seminar: A product formula for homogeneous characteristic functions
Speaker: Bhaskar Bagchi (ISI, Bangalore, retd.)
Date: Wed, 30 Oct 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
If $T$ is a cnu (completely non-unitary) contraction on a Hilbert space, then its Nagy-Foias characteristic function is an operator valued analytic function on the unit disc $\mathbb{D}$ which is a complete invariant for the unitary equivalence class of $T$. $T$ is said to be homogeneous if $\varphi(T)$ is unitarily equivalent to $T$ for all elements $\varphi$ of the group $M$ of biholomorphic maps on $\mathbb{D}$. A stronger notion is of an associator. $T$ is an associator if there is a projective unitary representation $\sigma$ of $M$ such that $\varphi(T) = \sigma(\varphi)^* T \sigma(\varphi)$ for all $\varphi$ in $M$. In this talk we shall discuss the following result from a recent work of the speaker with G. Misra and S. Hazra: A cnu contraction is an associator if and only if its characteristic function $\theta$ has the factorization $\theta(z) = \pi_* (\varphi_z) C \pi(\varphi_z), z \in \mathbb{D}$ for two projective unitary representations $\pi, \pi_∗$ of $M$.
Title: Geometry & Topology Seminar: Rational parallelisms and generalized Cartan geometries on complex manifolds
Speaker: Sorin Dumitrescu (Universite Cote d'Azur, Nice)
Date: Mon, 28 Oct 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
This talk deals with (generalized) holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic parallelisms, holomorphic Riemannian metrics, holomorphic affine connections or holomorphic projective connections. A more flexible notion is that of a generalized Cartan geometry which allows some degeneracy of the geometric structure. This encapsulates for example some interesting rational parallelisms. We discuss classification and uniformization results for compact complex manifolds bearing (generalized) holomorphic Cartan geometries.
Title: Geometry & Topology Seminar: Hot spots conjecture for Euclidean domains
Speaker: Sugata Mondal (TIFR, Mumbai)
Date: Mon, 21 Oct 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
In the first part of this talk I shall recall what the Hot spots conjecture is. Putting it in mathematical terms, I shall provide a brief history of the conjecture. If time permits I shall explain a proof of the conjecture for Euclidean triangles.
Title: Geometry & Topology Seminar: Deformations of the Hitchin Integrable System and Supersymmetric Quantum Field Theories
Speaker: Aswin Balasubramanian (Rutgers University)
Date: Mon, 14 Oct 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
I will begin by reviewing the relationship between Hitchin’s Integrable System and 4d N=2 Supersymmetric Quantum Field Theories. I will then discuss two classes of deformations of the Hitchin system which correspond, in the physical context, to relevant and marginal deformations of a conformal theory. The study of relevant deformations turns out to be related to the theory of sheets in a complex Lie algebra and their classification leads to a surprising duality between sheets in a Lie algebra and Slodowy slices in the Langlands dual Lie algebra (work done with J. Distler) . If there is time, I will discuss marginal deformations which are related to studying the Hitchin system as a family over the moduli space of curves including over nodal curves (ongoing project with J. Distler and R. Donagi) .
Title: Eigenfunctions Seminar: Superimposing theta structure on a generalized modular relation
Speaker: Atul Dixit (IIT, Gandhinagar)
Date: Fri, 11 Oct 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
Modular forms are certain functions defined on the upper half plane that transform nicely under $z\to -1/z$ as well as $z\to z+1$. By a modular relation (or a modular-type transformation) for a certain function $F$, we mean that which is governed by only the first map, i.e., $z\to -1/z$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha \beta = 1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha) = F(iw, \beta)$ or other relations of the form $F(z, \alpha) = F(z, \beta)$ etc. The famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eisenstein series on ${\rm SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha) = F(z, iw, \beta)$ recently obtained by Kesarwani, Moll and the speaker. This implies that one can superimpose the theta structure on the Ramanujan-Guinand formula.
The current work arose from answering a similar question - can we superimpose the theta structure on a recent modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$ which generalizes an important result of Ramanujan? In the course of answering this question in the affirmative, we were led to a surprising new generalization of $\zeta(z, a)$. This new zeta function, $\zeta_w(z, a)$, satisfies interesting properties, albeit they are much more difficult to derive than the corresponding ones for $\zeta(z, a)$. In this talk, I will briefly discuss the theory of the Riemann zeta function $\zeta(z)$, the Hurwitz zeta function $\zeta(z, a)$ and then describe the theory of $\zeta_w(z, a)$ that we have developed. In order to obtain the generalized modular relation (with the theta structure) satisfied by $\zeta_w(z, a)$, one not only needs this theory but also has to develop the theory of reciprocal functions in a certain kernel involving Bessel functions as well as the theory of a generalized modified Bessel function. (Based on joint work with Rahul Kumar.)
Title: Algebra & Combinatorics Seminar: Recent developments in the theory of the restricted partition function $p(n,N)$
Speaker: Atul Dixit (IIT, Gandhinagar)
Date: Wed, 09 Oct 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
A beautiful $q$-series identity found in the unorganized portion of Ramanujan’s second and third notebooks was recently generalized by Maji and I. This identity gives, as a special case, a three-parameter identity which is a rich source of partition-theoretic information allowing us to prove, for example, Andrews’ famous identity on the smallest parts function $\mathrm{spt}(n)$, a recent identity of Garvan, and identities on divisor generating functions, to name a few. Guo and Zeng recently derived a finite analogue of Uchimura’s identity on the generating function for the divisor function $d(n)$. This motivated us to look for a finite analogue of my generalization of Ramanujan’s aforementioned identity with Maji. Upon obtaining such a finite version, our quest to look for a finite version of Andrews’ $\mathrm{spt}$-identity necessitated finding finite analogues of rank, crank and their moments. We could obtain finite versions of rank and crank for vector partitions. We were also able to obtain a finite analogue of a partition identity recently conjectured by George Beck and proven by Shane Chern. I will discuss these and some related results. This is joint work with Pramod Eyyunni, Bibekananda Maji and Garima Sood.
Title: Geometry & Topology Seminar: Geometries of representations
Speaker: Gianluca Faraco (TIFR, Mumbai)
Date: Mon, 07 Oct 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Given a closed orientable surface $S$, a $(G,X)-$structure on $S$ is the datum of a maximal atlas whose charts take values on $X$ and transition functions are restrictions of elements in $G$. Any such structure induces a holonomy representation $\rho:\pi_1(\widetilde{S})\to X$ which encodes geometric data of the structure. Conversely, can we recover a geometric structure from a given representation? Is such a structure unique? In this talk we answer these questions by providing old and new results.
Title: PhD Thesis defence: Risk-Sensitive Stochastic Control and Differential Games
Speaker: Somnath Pradhan (IISc Mathematics)
Date: Fri, 04 Oct 2019
Time: 2:30 pm
Venue: R-15 (Chairman's Room), Mathematics Department
We study risk-sensitive stochastic optimal control and differential game problems. These problems arise in many applications including heavy traffic analysis of queueing networks, communication networks, and manufacturing systems.
First, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in $\mathbb{R}^{d}$. We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddle-point equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationary Markov strategies.
Subsequently, we study risk-sensitive ergodic control problem for controlled reflecting diffusion processes in the non-negative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. This, in turn, leads to the complete characterization of optimal controls in the space of stationary Markov controls.
Then we study risk-sensitive zero-sum/nonzero-sum stochastic differential games on the infinite horizon, where the state is a controlled reflecting diffusion in the non-negative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding Hamilton-Jacobi-Isaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies.
Finally, we study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criteria, where the state space is a controlled diffusion process in $\mathbb{R}^{d}.$ Under certain conditions, we establish the existence of a Nash equilibrium in stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.
Title: Algebra & Combinatorics Seminar: A characterization of the family of secant lines to a hyperbolic quadric in $PG(3,q)$, $q$ odd
Speaker: Bikramaditya Sahu (IISc Mathematics)
Date: Fri, 04 Oct 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
In this talk, we shall discuss a combinatorial characterization of the family of secant lines of the 3-dimensional projective space $PG(3,q)$ which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with points and planes. This is joint work with Puspendu Pradhan.
Title: Eigenfunctions Seminar: Slides Perron–Frobenius eigenfunctions of perturbed stochastic matrices
Speaker: Rajeeva L. Karandikar (CMI, Chennai)
Date: Fri, 27 Sep 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
Consider a stochastic matrix $P$ for which the Perron–Frobenius eigenvalue has multiplicity larger than 1, and for $\epsilon > 0$, let
\begin{equation} P^\epsilon := (1 - \epsilon) P + \epsilon Q, \end{equation}
where $Q$ is a stochastic matrix for which the Perron–Frobenius eigenvalue has multiplicity 1. Let $\pi^\epsilon$ be the Perron–Frobenius eigenfunction for $P^\epsilon$. We will discuss the behavior of $\pi^\epsilon$ as $\epsilon \to 0$.
This was an important ingredient in showing that if two players repeatedly play Prisoner’s Dilemma, without knowing that they are playing a game, and if they play rationally, they end up cooperating. We will discuss this as well in the second half.
The talk will include the required background on Markov chains.
Title: Algebra & Combinatorics Seminar: On the Numbers of the Form $x^2+11y^2$
Speaker: Martin Kreuzer (Universität Passau, Germany)
Date: Wed, 25 Sep 2019
Time: 11 am – 12 pm (LH-5) and 3:30 – 4:30 pm (LH-1)
Venue: Rooms LH-5 and LH-1 (respectively), Mathematics Department
Lecture 1: Modular Class Groups
Lecture 2: Number Rings
A famous result of Leonhard Euler says that his so-called “convenient numbers” $N$ have the property that a positive integer $n$ has a unique representation of the form $n=x^2+Ny^2$ with $\gcd(x^2,Ny^2)=1$ if and only if $n$ is a prime, a prime power, twice one of these, or a power of 2. The set of known 65 convenient numbers is ${ 1,2,3,4,5,6,7,8,9,10,12,13,15,\dots,1848 }$, and it is conjectured that these are all of them. So, when we look at this set, we see that 11 is the first “inconvenient” number, and therefore we consider the natural question which positive integers have a representation of the form $n=x^2+11 y^2$ with $\gcd(x,11y)=1$.
Our approach is split into two parts. First we introduce the modular class group $G_{11}$
of level 11 and give a detailed description of its structure. We show that there are four
conjugacy classes of elliptic elements of order 2, we provide concrete matrices representing these
elliptic elements, and we give an explicit representation of $G_{11}$ using them.
Then we conjugate the first of these matrices, namely $t_1=\binom{0, 1}{-1, 0}$, by the elements
of $G_{11}$ and get matrices whose top right entry is of the form $x^2+11 y^2$.
Conversely, we construct elliptic elements $A_n(\ell)$ of order 2 in $G_{11}$ which are
conjugate to one of the generators. Then the matrices conjugate to $t_1$ are the ones
we are interested in, and we find a set of candidate numbers $C$ such that $C=S_1 \cup
S_2$, where $S_1$ is the set we want to characterise. Thus the task is reduced to distinguishing
between $S_1$ and $S_2$.
This problem is addressed in the second part of the talk using number rings in $K=\mathbb{Q}(\sqrt{-11})$. The ring of integers of this number field is $\mathcal{O}_K=\mathbb{Z}[(-11+\sqrt{-11})/2]$, and the more natural ring $\mathbb{Z}[\sqrt{-11}]$ is its order of conductor 2. By realizing the elements of $S_1$ and $S_2$ as norms of elements in $\mathcal{O}_K$, we get some of their basic properties. The main theorem provides a precise description of the primitive representations $n=x^2+11 y^2$ into four classes, where cubic numbers and prime numbers are two classes which admit separate, detailed descriptions. For the prime numbers in $S_1$, we need to use some consequences of ring class field theory for $\mathbb{Z}[\sqrt{-11}]$, but all other results are largely self-contained.
Title: Geometry & Topology Seminar: Tropical curve counting
Speaker: Sushmita Venugopalan (IMSc, Chennai)
Date: Mon, 23 Sep 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Counting holomorphic curves in a symplectic manifold has been an area of research since Gromov’s work on this subject in the 1980s. Symplectic manifolds naturally allow a ‘cut’ operation. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. An interesting feature of curves in a multiply-cut manifold is that they have an underlying ‘tropical graph’, which is a graph that lives in the polytope associated to the cut.
Title: Algebra & Combinatorics Seminar: Linkage principle and Tame cyclic base change for $GL_n(F)$
Speaker: Santosh Nadimpalli (Radboud University, Nijmegen, Netherlands)
Date: Fri, 20 Sep 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite cyclic Galois extension of $F$. The theory of base change associates to an irreducible smooth $\overline{\mathbb{Q}}_l$-representation $(\pi_F, V)$ of ${\rm GL}_n(F)$ an irreducible $\overline{\mathbb{Q}}_l$-representation $(\pi_E, W)$ of ${\rm GL}_n(E)$. The ${\rm GL}_n(E)$-representation $\pi_E$ extends as a representation of ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$. Assume that the central character of $\pi_F$ takes values in $\overline{\mathbb{Z}}_l^\times$, and $l\neq p$. When $\pi_E$ is cuspidal, for any ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$ stable lattice $\mathcal{L}$ in $\pi_E$, Ronchetti supporting the linkage principle of Treumann and Venkatesh conjectured that the zeroth Tate cohomology of $\mathcal{L}$ with respect to ${\rm Gal}(E/F)$ is the Frobenius twist of mod-$l$ reduction of the representation $\pi_F$, i.e.,
\begin{equation} \widehat{H}^0 ({\rm Gal}(E/F), \mathcal{L}) \simeq \overline{\pi} _F^{(l)}. \end{equation}
This conjecture is verified by Ronchetti when $\pi_F$ is a depth-zero cuspidal representation using compact induction model. We will explain a proof in the case where $n=2$ and $\pi_F$ has arbitrary depth, using Kirillov model. If time permits, we will discuss the general case by local Rankin–Selberg convolutions.
Title: APRG Seminar: Modulation Spaces and Applications to Hartree–Fock equations
Speaker: Divyang Bhimani (TIFR-CAM, Bengaluru)
Date: Fri, 20 Sep 2019
Time: 4:30 pm
Venue: LH-1, Mathematics Department
We discuss some progress in the last decade (and ongoing interest) of modulation spaces from the PDE point of view. We prove some local and global well-posedness for Hartree–Fock equations in modulation spaces. We shall also prove similar results when a harmonic potential is added to these equations. This is a joint work with Manoussos Grillakis and Kasso Okoudjou.
Title: APRG Seminar: The Spectrum of Dense Random Geometric Graphs
Speaker: Kartick Adhikari (Technion, Haifa, Israel)
Date: Wed, 18 Sep 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We study the spectrum of a random geometric graph, in a regime where the graph is dense and highly connected. As opposed to other random graph models (e.g. the Erdos-Renyi random graph), even when the graph is dense, not all the eigenvalues are concentrated around 1. In the case where the vertices are generated uniformly in a unit d-dimensional box, we show that for every $0\le k \le d$ there are $\binom{d}{k}$ eigenvalues at $1-2^{-k}$. The rest of the eigenvalues are indeed close to 1. The spectrum of the graph Laplacian plays a key role in both theory and applications. We also show that the corresponding eigenfunctions are tightly related to the geometric configuration of the points. Aside from the interesting mathematical phenomenon we reveal here, the results of this paper can also be used to analyze the homology of the random Vietoris-Rips complex via spectral methods.
The talk will be based on a joint work with R. Adler, O. Bobrowski, and R. Rosenthal.
Title: Algebra & Combinatorics Seminar: Macdonald polynomials and level two Demazure modules for affine $sl_{n+1}$
Speaker: Rekha Biswal (Max Planck Institut, Bonn, Germany)
Date: Mon, 16 Sep 2019
Time: 3:05 pm
Venue: LH-1, Mathematics Department
Macdonald polynomials are a remarkable family of orthogonal symmetric polynomials in several variables. An enormous amount of combinatorics, group theory, algebraic geometry and representation theory is encoded in these polynomials. It is known that the characters of level one Demazure modules are non-symmetric Macdonald polynomials specialized at t=0. In this talk, I will define a class of polynomials in terms of symmetric Macdonald polynomials and using representation theory we will see that these polynomials are Schur-positive and are equal to the graded character of level two Demazure modules for affine $\mathfrak{sl}_{n+1}$. As an application we will see how this gives rise to an explicit formula for the graded multiplicities of level two Demazure modules in the excellent filtration of Weyl modules. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.
Title: Algebra & Combinatorics Seminar: Counting Independent Sets in Graphs and Hypergraphs and Spectral Stability
Speaker: Prasad Tetali (Regents' Professor, Georgia Tech, Atlanta, USA; and Satish Dhawan Visiting Professor, IISc)
Date: Fri, 13 Sep 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
In 2001, Jeff Kahn showed that a disjoint union of $n/(2d)$ copies of the complete bipartite graph $K_{d,d}$ maximizes the number of independent sets over all $d$-regular bipartite graphs on n vertices, using Shearer’s entropy inequality. In this lecture I will mention several extensions and generalizations of this extremal result (to graphs and hypergraphs) and will describe a stability result (in the spectral sense) to Kahn’s result.
The lecture is based on joint works with Emma Cohen, David Galvin, Will Perkins, Michail Sarantis and Hiep Han.
Title: Algebra & Combinatorics Seminar: Consecutive integers divisible by a power of their largest prime factor
Speaker: Jean-Marie de Koninck (Université Laval, Quebec, Canada)
Date: Wed, 11 Sep 2019
Time: 4:30 pm
Venue: LH-1, Mathematics Department
The connection between the multiplicative and additive structures of an arbitrary integer is one of the most intriguing problems in number theory. It is in this context that we explore the problem of identifying those consecutive integers which are divisible by a power of their largest prime factor. For instance, letting $P(n)$ stand for the largest prime factor of $n$, then the number $n=1294298$ is the smallest integer which is such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2$. No one has yet found an integer $n$ such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2,3$. Why is that? In this talk, we will provide an answer to this question and explore similar problems.
Title: Algebra & Combinatorics Seminar: Two footnotes to Galois's Memoirs
Speaker: Chandan Dalawat (Harish-Chandra Research Institute, Allahabad)
Date: Fri, 06 Sep 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We will review the history of solvability of polynomial equations by radicals, concentrating on the two Memoirs of Evariste Galois. We will show how the first Memoir allows us to determine all equations of prime degree which are solvable by radicals, and the second Memoir similarly leads to the determination of all primitive equations which are solvable by radicals. A finite separable extension $L$ of a field $K$ is called primitive if there are no intermediate extensions, and solvable if the Galois group of its Galois closure is a solvable group. Galois himself proved in his Second Memoir that if $L$ is both primitive and solvable over $K$, then the degree $[L:K]$ has to be the power of a prime. We parametrise the set of all primitive solvable extensions in terms of other more computable things attached to $K$. Thus, when $K$ is a local field with finite residue field of characteristic $p$, we can explicitly write down all primitive extensions! This involves the determination of all irreducible $\mathbb{F}_p$-representations of the absolute Galois group of $K$.
Title: Algebra & Combinatorics Seminar: Stark-Heegner cycles for Bianchi modular forms
Speaker: Guhan Venkat (Université Laval, Quebec, Canada; and Morningside Center of Mathematics, Beijing, China)
Date: Wed, 04 Sep 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
In his seminal paper in 2001, Henri Darmon proposed a systematic construction of p-adic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of p-adic cohomology classes/cycles in the Harris–Soudry–Taylor representation associated to a Bianchi cusp form, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato–Beilinson conjecture as well as Gross–Zagier type results. This is based on a joint work with Chris Williams (Imperial College London).
Title: Eigenfunctions Seminar: Entropic relaxations of Monge–Kantorovich optimal transport
Speaker: Soumik Pal (University of Washington, Seattle, USA)
Date: Fri, 30 Aug 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
Lecture 1: An introduction to Monge–Kantorovich optimal transport and entropic relaxations
Abstract:
The mathematics of Monge–Kantorovich optimal transport (OT) has grown to be a unifying theme in many scientific disciplines, from purely mathematical areas such as analysis, geometry, and probability to revolutionary new methods in economics, statistics, machine learning and artificial intelligence. Much of these are due to impressive leaps in computational methods in OT that hinge on entropy based approximations. For dynamical OT such relaxations correspond to stochastic processes called Schrödinger bridges. This is an introductory talk on OT, entropic relaxations, and Schrödinger bridges that will attempt to give a flavor of this "hot area" by sewing through various recent advances.
Lecture 2: On the difference between entropic cost and the optimal transport cost
Abstract:
Consider a Monge–Kantorovich problem of transporting densities with a strictly convex cost function. The entropic relaxation cost is known to converge to the cost of optimal transport. We are interested in the difference between the two, suitably scaled. We show that this limit is always given by the relative entropy of the target density with respect to a Riemannian volume measure that measures the local sensitivity of the Monge map. In the special case of the quadratic Wasserstein transport, this relative entropy is exactly one half of the difference of entropies of the target and initial densities. The proofs are based of Gaussian approximations to Schrödinger bridges which can be interpreted as a higher order large deviation.
Title: APRG Seminar: The Heisenberg commutation relation and operator algebras
Speaker: Soumyashant Nayak (University of Pennsylvania, Philadelphia, USA)
Date: Wed, 28 Aug 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
The historical development of the study of operator algebras is intimately tied with the quest to develop a mathematical formalism for quantum mechanics. This is what motivated von Neumann to study ‘rings of operators’ in his seminal series of papers with Murray. He was dissatisfied with his original Hilbert space formalism involving the so-called type $I_{\infty}$ algebras and envisaged the type $II_1$ factors as providing the appropriate description of the logic of quantum systems. In this talk, we will briefly trace the history of the field before turning our attention to the question of representing the Heisenberg commutation relation, $QP - PQ = i\hbar I$, using operators on a Hilbert space. In the type $II_1$ case, this obstinately leads towards non-selfadjoint operators and non-selfadjoint operator algebras which have so far been second-class citizens in comparison to their self-adjoint counterparts (namely, C*-algebras, von Neumann algebras). We will conduct some mock drills in the world of matrices before moving our discussion to the algebra of affiliated operators corresponding to a $II_1$ factor (otherwise known as Murray-von Neumann algebras).
Title: Geometry & Topology Seminar: Commensurators of thin groups
Speaker: Mahan Mj (TIFR, Mumbai)
Date: Mon, 26 Aug 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
A celebrated theorem of Margulis characterizes arithmetic lattices in terms of density of their commensurators. A question going back to Shalom asks the analogous question for thin subgroups. We shall report on work during the last decade or so and conclude with a recent development. In recent work with Thomas Koberda, we were able to show that for a large class of normal subgroups of rank one arithmetic lattices, the commensurator is discrete.
Title: Algebra & Combinatorics Seminar: Density: How Zariski helped Schur, Cayley, and Hamilton
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 23 Aug 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Computing the determinant using the Schur complement of an invertible minor is well-known to undergraduates. Perhaps less well-known is why this works even when the minor is not invertible. Using this and the Cayley–Hamilton theorem as illustrative examples, I will gently explain one “practical” usefulness of Zariski density outside commutative algebra.
Title: APRG Seminar: Pairs of Commuting Contractions
Speaker: Srijan Sarkar (IISc Mathematics)
Date: Wed, 21 Aug 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
This talk will be a continuation of my presentation on ‘Factorization of Contractions’, which was held at the in-house symposium this year.
We will start by proving the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries which states that: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exist a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_{\Phi}, M_{\Psi})$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where
\begin{equation} \Phi(z)=(P+zP^{\perp})U^* \quad \text{and} \quad \Psi(z)=U(P^{\perp}+zP) \end{equation}
for $z \in \mathbb{D}$ (the unit disc in $\mathbb{C}$).
We shall then proceed to derive a similar characterization for pure contractions. In particular, let $T$ be a pure contraction on a Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$ be the Sz.-Nagy and Foias representation of $T$ for some canonical $\mathcal{Q} \subseteq H^2_{\mathcal{D}}(\mathbb{D})$, where $\mathcal{D}$ is a closed subspace of $\mathcal{H}$. We will show that $T = T_1 T_2$, for some commuting contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there exist $\mathcal{B}(\mathcal{D})$-valued polynomials $\varphi$ and $\psi$ of degree $\leq 1$ such that $\mathcal{Q}$ is a joint ($M^*_{\varphi}, M^*_{\psi}$)-invariant subspace and
\begin{equation} P_{\mathcal{Q}} M_z\mid_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\varphi \psi}\mid_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\psi \varphi}\mid_{\mathcal{Q}} \quad \mbox{and} \quad (T_1, T_2) \cong (P_{\mathcal{Q}} M_{\varphi}\mid_{\mathcal{Q}}, P_{\mathcal{Q}} M_{\psi}\mid_{\mathcal{Q}}). \end{equation}
Moreover, there exist a Hilbert space $\mathcal{E}$ and an isometry $V \in \mathcal{B}(\mathcal{D}; \mathcal{E})$ such that
\begin{equation} \varphi(z) = V^* \Phi(z) V \quad \mbox{and} \quad \psi(z) = V^* \Psi(z) V \quad \quad (z \in \mathbb{D}), \end{equation}
where the pair $(\Phi, \Psi)$, as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on $H^2_{\mathcal{E}}(\mathbb{D})$. As an application, we shall obtain a sharper von Neumann inequality for commuting pairs of contractions.
If time permits, then we will look at some of the recent work by other authors in this direction and also try to figure out the challenges that lie in extending this result to a tuple of contractions.
Title: A central value formula of degree 6 complex L-series and arithmetic applications
Speaker: Aprameyo Pal (University of Duisburg-Essen, Germany)
Date: Mon, 19 Aug 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We prove an explicit central value formula for a family of complex L-series of degree 6 for GL2 × GL3 which arise as factors of certain Garret–Rankin triple product L-series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito–Kurokawa lifts, and as an application, we prove Deligne’s conjecture about the algebraicity of the central values of the considered L-series up to the relevant periods. I would also include some other arithmetic applications towards the subconvexity problem, construction of associated p-adic L function, etc.
This is joint work with Carlos de Vera Piquero.
Title: Eigenfunctions Seminar: Slides Quantitative Diagonalizability
Speaker: Nikhil Srivastava (UC Berkeley, USA)
Date: Fri, 16 Aug 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
A diagonalizable matrix has linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every matrix is the limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta of a matrix whose eigenvectors have condition number poly(n)/delta, confirming a conjecture of E. B. Davies. The proof is based on regularizing the pseudospectrum with a complex Gaussian perturbation.
Joint work with J. Banks, A. Kulkarni, S. Mukherjee.
Title: APRG Seminar: Deterministic counting using complex zeros (with some help from probability)
Speaker: Piyush Srivastava (TIFR, Mumbai)
Date: Wed, 14 Aug 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Problems associated with the algorithmic counting of combinatorial structures arise naturally in many applications and also constitute an interesting sub-field of computational complexity theory. In this talk, we consider one of the most studied of such problems: that of counting proper colourings of bounded degree graphs. Here, a constant maximum degree $\Delta$ and a set of $q$ colours are fixed. The input is a graph $G$ of maximum degree at most $\Delta$, and a parameter $\varepsilon > 0$. The problem is to output “efficiently”, i.e. in time that grows polynomially in $1/\varepsilon$ and the size $n$ of the graph, and up to a multiplicative error of at most $(1+\varepsilon)$, the number of proper colourings of $G$ with the given set of $q$ colours.
The problem has been attacked using randomized Markov chain methods, and a delightfully simple analysis using the path coupling method gives an efficient randomized algorithm when $q \geq 2\Delta + 1$. The best currently known Markov chain analysis, due to Vigoda, requires $q \geq 11 \Delta / 6$. However, the condition required for efficient deterministic algorithms (i.e., those that do not use any randomness and do not have a probability of error) was until recently $q \geq 2.58 \Delta$, which lagged behind even the simple path coupling analysis. In this work, we improve this to $q \geq 2 \Delta$, thus finally matching at least the path coupling bound for deterministic algorithms. Our method, based on a paradigm proposed by Alexander Barvinok, uses information on the complex zeros of the associated Potts model partition function. Perhaps surprisingly, the information we need about the complex zeros of this partition function is obtained using methods inspired from previous analyses of phenomena related to Markov chain algorithms for the problem.
Part of the talk will also be a general survey of Barvinok’s paradigm and the growing body of work connecting it to more probabilistic notions.
Joint work with Singcheng Liu and Alistair Sinclair.
Title: APRG Seminar: Reconstruction of a Riemannian manifold from noisy intrinsic distances
Speaker: Hariharan Narayanan (TIFR, Mumbai)
Date: Wed, 14 Aug 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian manifold $(M,g)$ is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let ${X_1,X_2,\dots,X_N}$ be a set of $N$ sample points sampled randomly from an unknown Riemannian manifold $M$. We assume that we are given the numbers $D_{jk}=d_M(X_j,X_k)+\eta_{jk}$, where $j,k\in {1,2,\dots,N}$. Here, $d_M(X_j,X_k)$ are geodesic distances, $\eta_{jk}$ are independent, identically distributed random variables such that $\mathbb E e^{|\eta_{jk}|}$ is finite. We show that when $N$ is large enough, it is possible to construct an approximation of the Riemannian manifold $(M,g)$ with a large probability. This problem is a generalization of the geometric Whitney problem with random measurement errors. We consider also the case when the information on noisy distance $D_{jk}$ of points $X_j$ and $X_k$ is missing with some probability. In particular, we consider the case when we have no information on points that are far away.
This is joint work with Charles Fefferman, Sergei Ivanov and Matti Lassas.
Title: Families of Drinfeld modular forms for GL(N)
Speaker: Marc-Hubert Nicole (Université d'Aix-Marseille, France)
Date: Tue, 13 Aug 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Hida once described his theory of families of ordinary p-adic modular eigenforms as obtained from cutting “the clear surface out of the pitch-dark well too deep to see through” of the space of all elliptic modular forms. In this colloquium-style talk, we shall peer into the well of Drinfeld modular forms instead of classical modular forms. More precisely, we shall explain how to construct families of finite slope Drinfeld modular forms over Drinfeld modular varieties of any dimension. In the ordinary case (the “clear surface”), we show that the weight may vary p-adically in families of Drinfeld modular forms (a direct analogue of Hida’s Vertical Control Theorem). In the deeper & murkier waters of positive slope, the situation is more subtle: the weight may indeed vary continuously, but not analytically, thereby contrasting markedly with Coleman’s well-known p-adic theory.
Joint work with G. Rosso (Cambridge University / Concordia University (Montréal)).
Title: Geometry & Topology Seminar: Compactness of the space of singular, minimal hypersurfaces with bounded volume and Jacobi eigenvalue
Speaker: Akashdeep Dey (Princeton University, USA)
Date: Tue, 13 Aug 2019
Time: 10 am
Venue: LH-1, Mathematics Department
Let $\{ M_k \}_{k=1}^{\infty}$ be a sequence of closed, singular, minimal hypersurfaces in a closed Riemannian manifold $(N^{n+1},g), n+1 \geq 3$. Suppose, the volumes of $M_k$ are uniformly bounded from above and the $p$-th Jacobi eigenvalues of $M_k$ are uniformly bounded from below. Then, there exists a closed, singular, minimal hypersurface $M$ in $N$ with the above mentioned volume and eigenvalue bounds such that possibly after passing to a subsequence, $M_k$ weakly converges (in the sense of varifolds) to $M$, possibly with multiplicities. Moreover, the convergence is smooth and graphical over the compact subsets of $reg(M) \setminus Y$ where $Y$ is a finite subset of $reg(M)$ with $|Y|\leq p-1$. This result generalizes the previous results of Sharp and Ambrozio-Carlotto-Sharp in higher dimensions.
Title: Algebra & Combinatorics Seminar: Dimers and the Beauville integrable system
Speaker: Terrence George (Brown University, USA)
Date: Wed, 07 Aug 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
To any convex integral polygon $N$ is associated a cluster integrable system that arises from the dimer model on certain bipartite graphs on a torus. The large scale statistical mechanical properties of the dimer model are largely determined by an algebraic curve, the spectral curve $C$ of its Kasteleyn operator $K(x,y)$. The vanishing locus of the determinant of $K(x,y)$ defines the curve $C$ and coker $K(x,y)$ defines a line bundle on $C$. We show that this spectral data provides a birational isomorphism of the dimer integrable system with the Beauville integrable system related to the toric surface constructed from $N$.
This is joint work with Alexander Goncharov and Richard Kenyon.
Title: APRG Seminar: Inverse problems for the wave equation
Speaker: Rakesh (University of Delaware, Newark, USA)
Date: Fri, 02 Aug 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
A three dimensional acoustic medium (operator is $\partial_t^2 {-} \Delta_x + q(x)$) is probed by plane waves coming from infinity and the medium response is measured at infinity. One aims to recover the acoustic property (the function $q(x)$) from this type of measurement. Two such longstanding open problems are the “fixed angle” scattering problem and the “back-scattering” problem. Both these problems involve studying the injectivity and the inversion of some non-linear map from compactly supported smooth functions on $\mathbb{R}^3$ to distributions on $\mathbb{R}^3$. These maps are defined through non-explicit solutions of the perturbed wave equation - the existence and uniqueness of these solutions is well understood.
We will state these two problems, describe what was known and then state our (with Mikko Salo of University of Jyvaskyla, Finland) injectivity result for the “fixed angle scattering” problem. If time permits we will describe some of the ideas used to prove our result for the fixed angle scattering problem – Carleman estimates for the perturbed wave equation play a big role.
Title: MS Thesis colloquium: Category of $\mathfrak{g}$-modules and BGG reciprocity
Speaker: Swarnalipa Datta (IISc Mathematics)
Date: Tue, 30 Jul 2019
Time: 11:30 am
Venue: LH-1, Mathematics Department
In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$-modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra. We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists a one-to-one correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$ admits a $p$-filtration.
Title: Percolation theory: from classical to dependent models
Speaker: Subhajit Goswami (IHES, Bures-sur-Yvette, France)
Date: Fri, 26 Jul 2019
Time: 3:30 - 4:40 PM
Venue: LH-1, Mathematics Department
Introduced in 1957 by Broadbent and Hammersley as a simple probabilistic model for movement of fluid through porous media, percolation theory has emerged as one of the most active and richest areas of research in modern probability theory. As one the most well-known models demonstrating an order-disorder phase transition, the techniques developed in the context of percolation theory have been widely successful in rigorous understanding of models in classical statistical physics, e.g. Ising and Potts. Furthermore in two dimensions it enjoys a deep connection with conformal field theory. In this talk I will attempt to provide a very brief tour of some horizons of classical percolation theory including some very recent results. I will then move onto percolation models in dependent media which pose new challenges to overcome. In this context I will present some recent results about a model related to the geometry of level sets of a canonical random Gaussian function (Gaussian free field) on lattice graphs. I will also briefly discuss some directions for future research. Some of the recent results are based on joint works with Hugo Duminil-Copin, Aran Raoufi, Pierre-Francois Rodriguez, Franco Severo and Ariel Yadin. No knowledge about percolation theory is assumed on the part of the audience.
Title: PhD Thesis colloquium: Geometric Invariants For a Class of Submodules of Analytic Hilbert Modules
Speaker: Samrat Sen (IISc Mathematics)
Date: Wed, 24 Jul 2019
Time: 11 am
Venue: LH-1, Mathematics Department
Title: PhD Thesis colloquium: Phase Transition, Percolation at Criticality and Connectivity in Random Connection Models
Speaker: Sanjoy Kumar Jhawar (IISc Mathematics)
Date: Mon, 22 Jul 2019
Time: 11 am
Venue: LH-1, Mathematics Department
This work has two parts. The first part contains the study of phase transition and percolation at criticality for three planar random graph models, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y \mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid\cdot\mid$ is the Euclidean norm. In the inhomogenous version of the model, points of $\mathcal{P}_{\lambda}$ are endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability
\begin{equation} \left(1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)\right) \end{equation}
for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.
In the second part we consider an inhomogeneous random connection model on a $d$-dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of vertices of degree $j$ converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.
Title: PhD Thesis colloquium: Representations of Special Compact Linear Groups of Order Two: Construction, Representation Growth, Group Algebras, and Branching Laws
Speaker: Hassain M. (IISc Mathematics)
Date: Mon, 15 Jul 2019
Time: 11:30 am
Venue: LH-1, Mathematics Department
Let $O$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let $e$ be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.
In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.
Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work for $p=2$. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.
Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.
Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{2m})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{2m}))]$ are NOT isomorphic for $m > e$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{2m}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{2m}))]$ are NOT isomorphic for $m>1$.
Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.
Title: PhD Thesis colloquium: Hardy's inequalities for Grushin operator and Hermite multipliers on modulation spaces
Speaker: Rakesh Kumar (IISc Mathematics)
Date: Fri, 05 Jul 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We prove Hardy’s inequalities for the fractional power of Grushin operator $\mathcal{G}$ which is chased via two different approaches. In the first approach, we first prove Hardy’s inequality for the generalized sublaplacian. We first find Cowling–Haagerup type of formula for the fractional sublaplacian and then using the modified heat kernel, we find integral representations of the fractional generalized sublaplacian. Then we derive Hardy’s inequality for generalized sublaplacian. Finally using the spherical harmonics, applying Hardy’s inequality for individual components, we derive Hardy’s inequality for Grushin operator. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\mathbb{R}^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\mathbb{R}^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\mathcal{G}_s f$ in $L^p(\mathbb{R}^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy–Littlewood–Sobolev inequality for the Grushin operator.
Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\mathbb{R}^n)$. We find a relation between the boundedness of sublaplacian multipliers $m(\tilde{\mathcal{L}})$ on polarised Heisenberg group $\mathbb{H}^n_{pol}$ and the boundedness of Hermite multipliers $m(\mathcal{H})$ on modulation spaces $M^{p,q}(\mathbb{R}^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe those conditions on multipliers are more than required restrictive. We improve the results for the special case $p=q$ of the modulation spaces $M^{p,q}(\mathbb{R}^n)$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}(\mathbb{R}^n)$ and the boundedness of Fourier multipliers on torus $\mathbb{T}^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr"odinger equation related to Hermite on modulation spaces.
Title: PhD Thesis colloquium: Signs of Hecke eigenvalues of modular forms and differential operators on Jacobi forms
Speaker: Ritwik Pal (IISc Mathematics)
Date: Tue, 02 Jul 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
This talk would have two parts. In the first part, we will discuss some topics which can be classified as ‘Linnik-type’ problems (the motivation being his original question about locating the first prime in an arithmetic progression) in the context of Hecke eigenvalues of modular forms on various groups, and then talk about the distribution of their signs. In the second part we will discuss differential operators on modular forms, and then talk about their applications to questions about Jacobi forms.
It is well-known that the sequence of Hecke eigenvalues mentioned above are often real, and has infinitely many sign changes. First part of the talk would discuss the problem of estimating the location of the first such sign change in the context of Hecke eigenvalues of Yoshida lifts (a certain subspace of the Siegel modular forms) and Fourier coefficients of Hilbert modular forms. We show how to improve the previously best known results on this topic significantly.
The crucial inputs behind these would be to establish a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level for treating Yoshida lifts; and exploiting Hecke relations along with generalising related results due to K. Soundararajan, K. Matomaki et al. for the case of Hilbert modular forms. In both cases we measure the location of the eigenvalues or Fourier coefficients in terms of an analytic object called the ‘analytic conductor’, which would be introduced during the talk. Moreover in the case of Hilbert modular forms, we will also discuss quantitative results about distribution of positive and negative Hecke eigenvalues. The proof depends on establishing a certain result on a particular types of multiplicative functions on the set of integral ideals of a totally real number field.
In the second part of the talk, we will introduce the space of Jacobi forms and certain results due to J. Kramer and, briefly, a conjecture due to Hashimoto on theta series attached to quaternion algebras to motivate the results to follow. The (partial) solution of this conjecture by Arakawa and B"ocherer transfers the question to one about differential operators on Jacobi forms, and we would report on previously known and new results on this topic.
The heart of the second part of the talk would focus on the question about the differential operators on Jacobi forms. It is well known that certain differential operators ${D_{v}}_{0}^{2m}$ map the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the sum of the differential operators $D_{v}$ for $v={1,2,…2m}$ map $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. The question alluded to above boils down to investigate whether one can omit certain differential operators from the list above, maintaining the injective property. In this regard, we would discuss results of Arakawa–B"ocherer, Das–Ramakrishnan, and finally our results. The main point would be to establish automorphy of the Wronskian of a certain tuple of congruent theta series of weight 3/2.
Title: Algebra & Combinatorics Seminar: Journey to the core of partitions
Speaker: Shubham Sinha (UC San Diego, USA)
Date: Mon, 01 Jul 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Integer partitions have been an interesting combinatorial object for centuries but we still have a lot left to understand. In this talk we will see an uncommon way to view partitions using ‘Abaci’ and use this to illustrate partition division (defining t-cores and t-quotients of a given integer partition). I am planning to go in more detail with a special kind of partition called t-core (partitions) and discuss different ways to enumerate related objects.
Title: Formulae in Operator Theory - Story of Two Projections (Ramanujan Medal Award Lecture, INSA)
Speaker: Kalyan B. Sinha (Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore)
Date: Thu, 27 Jun 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Trace is a kind of “special non-commutative integration” and trace formulae attempts to relate traces of certain expressions, involving operators with associated geometric/topological quantities. A simple example, involving two orthogonal projections in a Hilbert space, will be discussed, in which some of these features appear naturally.
Title: PhD Thesis colloquium: On the Kobayashi geometry of domains
Speaker: Anwoy Maitra (IISc Mathematics)
Date: Thu, 27 Jun 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:
Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.
Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s well-known big-constant/small-constant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.
The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.
A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newly-introduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.
Wolff–Denjoy-type theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoy-type theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.
Title: APRG Seminar: Solvability of an operator equation in B(H)-semigroups
Speaker: Sasmita Patnaik (IIT, Kanpur)
Date: Wed, 26 Jun 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
The study of multiplicative B(H)-semigroups with selfadjoint-ideals property (called SI semigroups, in short) is related to the solvability of a certain operator equation. This introduces a new unitary invariant property for B(H)-semigroups. We characterized singly generated SI semigroups for a normal operator and for a rank one operator. The SI property is closely related to the simplicity of B(H)-semigroups.
Title: PhD Thesis colloquium: Fourier coefficients of modular forms and mass of pullbacks of Saito--Kurokawa lifts
Speaker: Pramath Anamby (IISc Mathematics)
Date: Tue, 25 Jun 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms,having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$ have the same eigenvalues of the $p$-th Hecke operator $T_p$ for almost all primes $p$, then $f_1=f_2$.
The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square–free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$. We expect similar applications. We also discuss few results on the square–free Fourier coefficients of elliptic cusp forms.
In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, Z)$ then we consider its Saito–Kurokawa lift $F_g$ and a certain ‘restricted’ $L^2$-norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.
Pullback of a Siegel modular form $F((\tau,z,z,\tau'))$ ($(\tau,z,z,\tau')$ in Siegel’s upper half-plane of degree 2) to $H \times H$ is its restriction to $z=0$, which we denote by $F\|_{z=0}$. Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$-norms of such pullbacks to central values of $L$-functions for all degrees.
In fact, when a Siegel modular form arises as a Saito–Kurokawa lift (say $F=F_g$), results of Ichino relate the mass of the pullbacks to the central values of certain $GL(3) \times GL(2)$ $L$-functions. Moreover, it has been observed that comparison of the (normalized) norm of $F_g$ with the norm of its pullback provides a measure of concentration of $F_g$ along $z=0$. We recall certain conjectures pertaining to the size of the ‘mass’. We use the amplification method to improve the currently known bound for $N(F_g)$.
Title: PhD Thesis defence: Some results on spectral spaces and spectral sequences
Speaker: Samarpita Ray (IISc Mathematics)
Date: Mon, 10 Jun 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchell, an arbitrary small preadditive category may be understood as a “ring with several objects”. In this respect, for a Hopf algebra H, an H-category will denote an “H-module algebra with several objects” and a co-H-category will denote an “H-comodule algebra with several objects”. Modules over such Hopf categories were first considered by Cibils and Solotar. We study the cohomology in such module categories. In particular, we consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D,H)-Hopf modules over a Hopf comodule category D. These generalize relative (A,H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants. We will develop these cohomology theories in a manner similar to the “H-finite cohomology” obtained by Guedenon and the cohomology of relative Hopf modules studied by Caenepeel and Guedenon respectively. This is one of the two thesis problems which we plan to discuss in detail.
If time permits, we will also give a brief presentation of the other thesis project. In the last twenty years, several notions of what is called the algebraic geometry over the “field with one element” ($\mathbb{F}_1$) has been developed. It is in this context that monoids became topologically and geometrically relevant objects of study. In our work, we abstract out the topological characteristics of the prime spectrum of a commutative monoid, endowed with the Zariski topology is homeomorphic to the spectrum of a ring i.e., it is a spectral space. Spectral spaces, introduced by Hochster, are widely studied in the literature. We use ideals and modules over monoids to present many such spectral spaces. We introduce closure operations on monoids and obtain natural classes of spectral spaces using finite type closure operations. In the process, various closure operations like integral, saturation, Frobenius and tight closures are introduced for monoids. We study their persistence and localization properties in detail. Next, we make a study of valuation on monoids and prove that the collection of all valuation monoids having the same group completion forms a spectral space. We also prove that the valuation spectrum of any monoid gives a spectral space. Finally, we prove that the collection of continuous valuations on a topological monoid whose topology is determined by any finitely generated ideal also gives a spectral space.
Title: APRG Seminar: Macdonald's conjecture on hypergeometric functions associated to Jack polynomials
Speaker: Siddhartha Sahi (Rutgers University, USA)
Date: Thu, 06 Jun 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Title: Algebra & Combinatorics Seminar: Metaplectic Representations of Hecke algebras
Speaker: Siddhartha Sahi (Rutgers University, USA)
Date: Tue, 04 Jun 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
We construct certain representations of affine Hecke algebras and Weyl groups, which depend on several auxiliary parameters. We refer to these as “metaplectic” representations, and as a direct consequence we obtain a family of “metaplectic” polynomials, which generalizes the well-known Macdonald polynomials.
Our terminology is motivated by the fact that if the parameters are specialized to certain Gauss sums, then our construction recovers the Kazhdan-Patterson action on metaplectic forms for GL(n); more generally it recovers the Chinta-Gunnells action on p-parts of Weyl group multiple Dirichlet series.
This is joint work with Jasper Stokman and Vidya Venkateswaran.
Title: PhD Thesis colloquium: Risk-Sensitive Stochastic Control and Differential Games
Speaker: Somnath Pradhan (IISc Mathematics)
Date: Thu, 30 May 2019
Time: 11:30 am
Venue: LH-1, Mathematics Department
We study risk-sensitive stochastic optimal control and differential game problems. These problems arise in many applications including heavy traffic analysis of queueing networks, communication networks, and manufacturing systems.
First, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in $\mathbb{R}^{d}$. We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddle-point equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationary Markov strategies.
Subsequently, we study risk-sensitive ergodic control problem for controlled reflecting diffusion processes in the non-negative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. This, in turn, leads to the complete characterization of optimal controls in the space of stationary Markov controls.
Then we study risk-sensitive zero-sum/nonzero-sum stochastic differential games on the infinite horizon, where the state is a controlled reflecting diffusion in the non-negative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding Hamilton-Jacobi-Isaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies.
Finally, we study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criteria, where the state space is a controlled diffusion process in $\mathbb{R}^{d}.$ Under certain conditions, we establish the existence of a Nash equilibrium in stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.
Title: APRG Seminar: Problem governed by the von Kármán equations Morley finite element for a distributed optimal control
Speaker: Sudipto Chowdhury (IIT, Bombay)
Date: Tue, 21 May 2019
Time: 11 am
Venue: LH-1, Mathematics Department
This talk consists of the finite element analysis of a distributed optimal control problem governed by the von Kármán equation that describe the deflection of very thin plates defined on a polygonal domain of R2 with box constraints on the control variable. In this talk we discuss a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower order norms for the state and adjoint variables are derived. The lower order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Finally we discuss several numerical results to illustrate our theoretical results.
Title: Algebra & Combinatorics Seminar: Alternating Schur Algebras and Koszul Duality
Speaker: Amritanshu Prasad (IMSc, Chennai)
Date: Wed, 08 May 2019
Time: 3 pm
Venue: LH-5, Mathematics Department
Schur classified polynomial representations of GL(n) using the commuting actions of the dth symmetric group and GL(n) on the d-fold tensor power of n-dimensional space. We set out to investigate what happens when the symmetric group in this picture is shrunk to the alternating group. This led to a simple interpretation and clearer understanding of the Koszul duality functor on the category of polynomial representations of GL(n).
This is based on joint work with T. Geetha and Shraddha Srivastava (https://arxiv.org/abs/1902.02465).
Title: Algebra & Combinatorics Seminar: Path models for Kostant-Kumar submodules of a tensor product
Speaker: Sankaran Viswanath (IMSc, Chennai)
Date: Tue, 07 May 2019
Time: 11:30 am
Venue: LH-5, Mathematics Department
Consider the tensor product of two irreducible finite dimensional representations of a simple Lie algebra. The submodules generated by a tensor product of extremal vectors of the two components are called Kostant-Kumar submodules. These are parametrized by a double coset space of the Weyl group.
Littelmann’s path model is a very general combinatorial model for representations, which encompasses many classical constructs such as Young tableaux and Lakshmibai-Seshadri chains. The path model for the full tensor product is simply the set of concatenations of paths of the individual components. We describe a way to associate a Weyl group element (rather, a double coset) to each such concatenated path and thereby obtain a path model for Kostant-Kumar submodules.
Finally, we recall the many descriptions of Demazure modules, which may be viewed as the analog of the above picture for single paths (rather than concatenations). In this case, the Weyl group element associated to a path admits different descriptions in different path models in terms of statistics such as initial direction, minimal standard lifts and Kogan faces of Gelfand-Tsetlin polytopes. Along the way, we mention some relations to jeu-de-taquin and the Schutzenberger involution.
This is based on joint work with Mrigendra Singh Kushwaha and KN Raghavan.
Title: Algebra & Combinatorics Seminar: The Brylinski filtration and W-algebras
Speaker: Sankaran Viswanath (IMSc, Chennai)
Date: Fri, 03 May 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Each finite dimensional irreducible representation V of a simple Lie algebra L admits a filtration induced by a principal nilpotent element of L. This, so-called, Brylinski or Brylinski-Kostant filtration, can be restricted to the dominant weight spaces of V, and the resulting Hilbert series is very interesting q-analogs of weight multiplicity, first defined by Lusztig.
This picture can be extended to certain infinite-dimensional Lie algebras L and to irreducible highest weight, integrable representations V. We focus on the level 1 vacuum modules of special linear affine Lie algebras. In this case, we show how to produce a basis of the dominant weight spaces that is compatible with the Brylinski filtration. Our construction uses the so-called W-algebra, a natural vertex algebra associated to L.
This is joint work with Sachin Sharma (IIT Kanpur) and Suresh Govindarajan (IIT Madras).
Title: Algebra & Combinatorics Seminar: Suslin Matrices vs Clifford Algebras
Speaker: Vineeth Chintala (IIT, Bombay)
Date: Wed, 01 May 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
In this talk, we will introduce the notion of an embedding of a quadratic space (in an associative algebra). Familiar examples of embeddings are given by Complex Numbers, Quaternions, Octonions, Clifford Algebras, and Suslin Matrices.
When there are two embeddings of the same quadratic space, then we can gather information about one embedding using the other. This is the main theme of the talk. To illustrate this, we will see the connection between Suslin Matrices and Clifford Algebras. In one direction, we are able to give a simple description of Clifford Algebras using matrices. Conversely, we can now explain some (seemingly) accidental properties of Suslin matrices in a conceptual way.
Title: MS Thesis colloquium: Determinantal processes and stochastic domination
Speaker: Raghavendra Tripathi (IISc Mathematics)
Date: Tue, 30 Apr 2019
Time: 11:30 am
Venue: LH-1, Mathematics Department
The systematic study of determinantal processes began with the work of Macchi (1975), and since then they have appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths, measures on Young diagrams), and physics (fermions). A particularly interesting and well-known example of a discrete determinantal process is the Uniform spanning tree (UST) on finite graphs. We shall describe UST on complete graphs and complete bipartite graphs—in these cases it is possible to make explicit computations that yield some special cases of Aldous’ result on CRT.
The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a Hilbert space of functions on a given set. Let $H$ and $K$ are two finite dimensional subspaces of a Hilbert space, and let $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and also provides a unified approach of proving the result in discrete as well as continuous case.
As an application of the above result, we will obtain the stochastic domination between the largest eigenvalue of Wishart matrix ensembles $W(N,N)$ and $W(N-1,N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M,N)$ has the same distribution as the directed last-passage time $G(M,N)$ on $Z^2$ with i.i.d. exponential weights. We, thus, obtain stochastic domination between $G(N,N)$ and $G(N-1,N+1)$ - answering a question of Riddhipratim Basu. Similar connections are also known between the largest eigenvalue of Meixner ensemble and directed last-passage time on $Z^2$ with i.i.d. geometric weights. We prove a stochastic domination result which combined with the Lyons’ result gives the stochastic domination between Meixner ensemble $M(N,N)$ and $M(N-1,N+1)$.
Title: Geometry & Topology Seminar: Coarse geometry, Baum-Connes conjecture, and metrics of positive scalar curvature on spin foliations
Speaker: Indrava Roy (IMSc, Chennai)
Date: Mon, 29 Apr 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
We shall present the circle of ideas concerning the coarse geometric approach to the Baum-Connes conjecture, via the analytic surgery sequence of N. Higson and J. Roe. In joint work with M.-T. Benameur, we elicit its relations with some secondary invariants of Dirac operators on co-compact coverings, which generalize classical deep results originally due to N. Keswani. We have now partially extended this program further to the context of non-compact, complete spin foliations, which provide new obstructions to the existence of leafwise metrics of (uniformly) positive scalar curvature via the coarse index, generalizing the results of Connes for smooth foliations of compact manifolds. If time permits, we shall also outline the connections of this framework with the work of Gromov-Lawson.
Title: On derivatives of Modular forms
Speaker: Siegfried Bocherer (Universität Mannheim, Germany)
Date: Mon, 22 Apr 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Modular forms are characterized by certain transformation laws. In general, these properties are not preserved by (holomorphic) differentiation. I will discuss several ways to overcome this trouble: Analytically one can use non-holomorphic differential operators (‘‘Maaß-Shimura operators’’) or use bilinear holomorphic differential operators (‘‘Rankin brackets’’). More arithmetically, one can show that derivatives of modular forms are still congruent to modular forms (‘‘Ramanujan’s $\theta$-operators’’). I will describe the connection between the analytic and the arithmetic aspects with focus on some recent results on $\theta$-operators.
Title: Algebra & Combinatorics Seminar: Spinorial Representations of Symmetric Groups
Speaker: Jyotirmoy Ganguly (IISER Pune)
Date: Tue, 16 Apr 2019
Time: 2 pm
Venue: LH-1, Mathematics Department
Representations of Symmetric groups $S_n$ can be considered as homomorphisms to the orthogonal group $\mathrm{O}(d,\mathbb{R})$, where $d$ is the degree of the representation. If the determinant of the representation is trivial, we call it achiral. In this case, its image lies in the special orthogonal group $\mathrm{SO}(V)$. It is called chiral otherwise. The group $\mathrm{O}(V)$ has a non-trivial topological double cover $\mathrm{Pin}(V)$. We say the representation is spinorial if it lifts to $\mathrm{Pin}(V)$. We obtained a criterion for whether the representation is spinorial in terms of its character. We found similar criteria for orthogonal representations of Alternating groups and products of symmetric groups. One can use these results to count the number of spinorial irreducible representations of $S_n$, which are parametrized by partitions of $n$. We say a partition is spinorial if the corresponding irreducible representation of $S_n$ is spinorial. In this talk, we shall present a summary of these results and count for the number of odd-dimensional, irreducible, achiral, spinorial partitions of $S_n$. We shall also prove that almost all the irreducible representations of $S_n$ are achiral and spinorial. This is joint work with my supervisor Dr. Steven Spallone.
Title: Eigenfunctions Seminar: Slides The Pick-Nevanlinna problem: from metric geometry to matrix positivity
Speaker: Gautam Bharali (IISc Mathematics)
Date: Fri, 12 Apr 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
The Pick–Nevanlinna problem refers to the problem of – given two connected open sets in complex Euclidean spaces and finite sets of distinct points in each – characterizing (in terms of the given point data) the existence of a holomorphic map between the two sets that interpolates the given points. The problem gets its name from Pick – who provided a beautiful characterization for the existence of an interpolant when the domain and the co-domain are the unit disc – and from Nevanlinna, who rediscovered Pick’s result. This characterization is in terms of matrix positivity. I shall begin by presenting an easy argument by Sarason, which he says is already implicit in Nevanlinna’s work, for the necessity of the Pick–Nevanlinna condition. How does one prove the sufficiency of the latter condition? Sarason’s ideas have provided the framework for a long chain of complicated characterizations for the more general problem. The last word on this is still to be written. But in the original set-up of Pick, geometry provides the answer. Surprisingly, Pick did not notice that his approach provides a very clean solution to the interpolation problem where the co-domain is any Euclidean ball. We shall see a proof of the latter. This proof uses a general observation about conditional positivity (which also made an appearance in the previous talk), which is attributed to Schur. If time permits, we shall see what can be said when the co-domain is a bounded symmetric domain.
Title: Eigenfunctions Seminar: Slides Schoenberg: from metric geometry to matrix positivity
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 12 Apr 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
I will present a historical account of some work of Schoenberg in metric geometry: from his metric space embeddings into Euclidean space and into spheres (Ann. of Math. 1935), to his characterization of positive definite functions on spheres (Duke Math. J. 1942). It turns out these results can be viewed alternately in terms of matrix positivity: from appearances of (conditionally) positive matrices in analysis, to the classification of entrywise positivity preservers in all dimensions.
Title: Algebra & Combinatorics Seminar: Crossed products of noncommutative tori by cyclic groups
Speaker: Sayan Chakraborty (ISI, Kolkata)
Date: Mon, 08 Apr 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
In this talk I consider crossed product $C^*$-algebras of higher dimensional noncommutative tori with actions of cyclic groups. I will discuss $K$-theory of those $C^*$-algebras and some applications.
Title: Geometry & Topology Seminar: A chain model for the Goresky-Hingston product
Speaker: Arun Maiti (IISc Mathematics)
Date: Mon, 08 Apr 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
A theorem of J. Jones relates cohomology of free loop space of a simply connected manifold to Hochschild homology of its singular cochain algebra. In this talk I will discuss a potential Hochschild chain model for a product on cohomology of free loop space known as the Goresky-Hingston product.
Title: Algebra & Combinatorics Seminar: Determinants, linear recurrences and some combinatorial identities
Speaker: Sudip Bera (IISc Mathematics)
Date: Fri, 05 Apr 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
In this talk we prove that the solution of a general linear recurrence with constant coefficients can be interpreted as the determinant of some suitable matrix using a purely combinatorial method. As a consequence of our approach, we give combinatorial proofs of some recent identities due to Sury and McLaughlin in a unified way.
Title: Geometry & Topology Seminar: Semi-regular tilings
Speaker: Basudeb Datta (IISc Mathematics)
Date: Mon, 01 Apr 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we consider tilings on surfaces (2-dimensional manifolds without boundary). If the face-cycle at all the vertices in a tiling are of same type then the tiling is called semi-regular. In general, semi-regular tilings on a surface M form a bigger class than vertex-transitive and regular tilings on M. This gives us little freedom to work on and construct semi-regular tilings. We present a combinatorial criterion for tilings to decide whether a tiling is elliptic, flat or hyperbolic. We also present classifications of semi-regular tilings on simply-connected surfaces (i.e., on 2-sphere and plane) in the first two cases. At the beginning, we present some examples and brief history on semi-regular tilings.
Title: Algebra & Combinatorics Seminar: Finiteness results on a certain class of modular forms and applications
Speaker: Soumya Bhattacharya (IISER Kolkata)
Date: Wed, 20 Mar 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
‘Holomorphic eta quotients’ are certain explicit classical modular forms on suitable Hecke subgroups of the full modular group. We call a holomorphic eta quotient $f$ ‘reducible’ if for some holomorphic eta quotient $g$ (other than $1$ and $f$), the eta quotient $f/g$ is holomorphic. An eta quotient or a modular form in general has two parameters: weight and level. We shall show that for any positive integer $N$, there are only finitely many irreducible holomorphic eta quotients of level $N$. In particular, the weights of such eta quotients are bounded above by a function of $N$. We shall provide such an explicit upper bound. This is an analog of a conjecture of Zagier which says that for any positive integer $k$, there are only finitely many irreducible holomorphic eta quotients of weight $k/2$ which are not integral rescalings of some other eta quotients. This conjecture was established in 1991 by Mersmann. We shall sketch a short proof of Mersmann’s theorem and we shall show that these results have their applications in factorizing holomorphic eta quotient. In particular, due to Zagier and Mersmann’s work, holomorphic eta quotients of weight $1/2$ have been completely classified. We shall see some applications of this classification and we shall discuss a few seemingly accessible yet longstanding open problems about eta quotients.
This talk will be suitable also for non-experts: We shall define all the relevant terms and we shall clearly state the classical results which we use.
Title: Algebra & Combinatorics Seminar: The study of projective modules over an affine algebra
Speaker: Ravi A. Rao (TIFR, Bombay)
Date: Fri, 15 Mar 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
The study of projective modules along the lines initiated by J.-P. Serre, H. Bass is outlined; the impetus given by A. Suslin in that direction is described, and recent progress made along the vision of A. Suslin on it by Fasel–Rao–Swan is shared briefly.
Title: APRG Seminar: New monotonicity formulas for degenerate parabolic equations with applications to unique continuation and free boundary problems (part 2)
Speaker: Agnid Banerjee (TIFR-CAM, Bengaluru)
Date: Fri, 15 Mar 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
I will continue my discussion on the strong unique continuation theorem for zero order perturbations of the fractional heat equation based on a monotonicity property of a generalized Almgren frequency in weighted Gauss space. This is joint with Nicola Garofalo. Then I will briefly describe how similar considerations are useful while studying the regularity of the free boundary in the obstacle problem for the fractional heat operator where some other monotonicity formulas are also required. This is a more recent work with Donatella Danielli, Nicola Garofalo and Arshak Petrosyan.
Title: Algebra & Combinatorics Seminar: Suslin completion and set-theoretic complete intersections
Speaker: Ravi A. Rao (TIFR, Bombay)
Date: Wed, 13 Mar 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We describe how a completion of the factorial row by Suslin led to showing some ideals in a polynomial ring are set-theoretic complete intersections.
Title: Geometry & Topology Seminar: Stability of quadratic curvature functionals at product of Einstein manifolds
Speaker: Soma Maity (IISER Mohali)
Date: Mon, 11 Mar 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Consider Riemannian functionals defined by L^2-norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature. I will talk about rigidity, stability and local minimizing properties of Einstein metrics and their products as critical metrics of these quadratic functionals. We prove that the product of a spherical space form and a compact hyperbolic manifold is unstable for certain quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small. We also prove the stability of L^{n/2}-norm of Weyl curvature at compact quotients of Sn × Hm.
Title: Algebra & Combinatorics Seminar: Laguerre heaps of segments for the PASEP
Speaker: Xavier Viennot (CNRS and LaBRI, France)
Date: Fri, 08 Mar 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems strongly related to the moments of some classical orthogonal polynomials (Hermite, Laguerre, Askey–Wilson). The partition function has been interpreted with various combinatorial objects such as permutations, alternative and tree-like tableaux, etc. We introduce a new one called “Laguerre heaps of segments”, which seems to play a central role in the network of bijections relating all these interpretations.
Title: APRG Seminar: New monotonicity formulas for degenerate parabolic equations with applications to unique continuation and free boundary problems
Speaker: Agnid Banerjee (TIFR-CAM, Bengaluru)
Date: Fri, 08 Mar 2019
Time: 4:15 pm
Venue: LH-1, Mathematics Department
I will discuss a strong unique continuation theorem for zero order perturbations of the fractional heat equation
based on a monotonicity property of a generalized Almgren frequency in weighted Gauss space. This is a joint work
with Nicola Garofalo. If time permits, I will also talk on how such a truncated variant of the Almgren type
monotonicity formula combined with new Weiss and Monneau type monotonicity formulas allows to study the structure
of the singular set on the free boundary in the obstacle problem for the fractional heat operator.
This part of the talk ( if at all I get time !) is joint with Donatella Danielli, Nicola Garofalo and Arshak Petrosyan.
Title: Algebra & Combinatorics Seminar: A survey of the combinatorial theory of orthogonal polynomials and continued fractions
Speaker: Xavier Viennot (CNRS and LaBRI, France)
Date: Thu, 07 Mar 2019
Time: 2:15 pm
Venue: LH-1, Mathematics Department
The theory of orthogonal polynomials started with analytic continued fractions going back to Euler, Gauss, Jacobi, Stieltjes… The combinatorial interpretations started in the late 70’s and is an active research domain. I will give the basis of the theory interpreting the moments of general (formal) orthogonal polynomials, Jacobi continued fractions and Hankel determinants with some families of weighted paths. In a second part I will give some examples of interpretations of classical orthogonal polynomials and of their moments (Hermite, Laguerre, Jacobi, …) with their connection to theoretical physics.
Title: Eigenfunctions Seminar: Slides Continued fractions, Stein-Chen method and extreme value theory
Speaker: Parthanil Roy (ISI, Bangalore)
Date: Fri, 01 Mar 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
According to Wikipedia, a continued fraction is “an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.” It has connections to various areas of mathematics including analysis, number theory, dynamical systems and probability theory.
Continued fractions of numbers in the open interval (0,1) naturally gives rise to a dynamical system that dates back to Gauss and raises many interesting questions. We will concentrate on the extreme value theoretic aspects of this dynamical system. The first work in this direction was carried out by the famous French-German mathematician Doeblin (1940), who rightly observed that exceedances of this dynamical system have Poissonian asymptotics. However, his proof had a subtle error, which was corrected much later by Iosifescu (1977). Meanwhile, Galambos (1972) had established that the scaled maxima of this dynamical system converges to the Frechet distribution.
After a detailed review of these results, we will discuss a powerful Stein-Chen method (due to Arratia, Goldstein and Gordon (1989)) of establishing Poisson approximation for dependent Bernoulli random variables. The final part of the talk will be about the application of this Stein-Chen technique to give upper bounds on the rate of convergence in the Doeblin-Iosifescu asymptotics. We will also discuss consequences of our result and its connections to other important dynamical systems. This portion of the talk will be based on a joint work with Maxim Sølund Kirsebom and Anish Ghosh.
Familiarity with standard (discrete and absolutely continuous) probability distributions will be sufficient for this talk.
Title: APRG Seminar: Counting nodal components of ‘boundary-adapted arithmetic random waves’
Speaker: James Cann (University College London, UK)
Date: Thu, 28 Feb 2019
Time: 2:15 pm
Venue: LH-1, Mathematics Department
The ‘nodal sets’ (zero sets) of Dirichlet Laplace eigenfunctions for the two-dimensional unit square have raised many questions over the past century. The nodal domain theorems of Courant (1924), Stern (1926) and Pleijel (1956), give deterministic (upper, liminf, and limsup resp.) bounds for the number of nodal domains which can be exhibited by an n-th eigenfunction, assuming the eigenvalues are ordered increasingly and with multiplicities listed. Prominent amongst contemporary research is the closely related question of the number of ‘nodal components’ (connected components of the zero set) of a ‘typical’ eigenfunction.
The spectral degeneracy for the square means that the nodal count of an n-th eigenfunction could take a wide range of values, and to understand the ‘typical’ behaviour of this number we attribute Gaussian random coefficients to a standard basis of eigenfunctions for each eigenspace, to form the ensemble of ‘boundary-adapted arithmetic random waves’. The number of nodal components —now a random variable— can then be studied, and this talk will draw together tools from various areas of mathematics (random fields, integral geometry, number theory, mathematical physics, …) in order to say a few things about its asymptotic properties.
Title: APRG Seminar: Spectral asymptotics for a singularly perturbed periodic elliptic operator
Speaker: Klas Pettersson (University of Tromsø – The Arctic University of Norway)
Date: Tue, 26 Feb 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale ε. We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter ε tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is non-degenerate at this point.
Title: Geometry & Topology Seminar: A compactness theorem for Einstein metrics on the four-sphere
Speaker: Harish Seshadri (IISc Mathematics)
Date: Mon, 25 Feb 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
It is open question if the standard round metric on $S^4$ is the unique positive Einstein metric, up to isometry and scaling. In this talk, I will discuss a compactness theorem which rules out nonstandard unit-volume Einstein metrics whose scalar curvatures lie within a certain range.
Title: Algebra & Combinatorics Seminar: Cohomology of modules over H-categories and co-H-categories
Speaker: Samarpita Ray (IISc Mathematics)
Date: Fri, 22 Feb 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchel, an arbitrary small preadditive category may be understood as a “ring with several objects”. In this respect, for a Hopf algebra H, an H-category will denote an “H-module algebra with several objects” and a co-H-category will denote an “H-comodule algebra with several objects”. Modules over such Hopf categories were first considered by Cibils and Solotar. In this talk, we present a study of cohomology in such module categories.
In particular, we consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D,H)-Hopf modules over a Hopf comodule category D. These generalize relative (A,H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants.
(Joint work with Abhishek Banerjee and Mamta Balodi.)
Title: Geometry & Topology Seminar: A vector bundle version of the Monge-Ampere equation
Speaker: Vamsi Pingali (IISc Mathematics)
Date: Mon, 18 Feb 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Motivated by mirror symmetry considerations (the deformed Hermitian-Yang-Mills equation due to Jacobs-Yau) and a desire to study stability conditions involving higher Chern forms, a vector bundle version of the usual complex Monge-Ampere equation (studied by Calabi, Aubin, Yau, etc) will be discussed. I shall also discuss a Kobayashi-Hitchin type correspondence for a special case (a dimensional reduction to Riemann surfaces).
Title: In Search of Inequalities (V.V. Narlikar Memorial Lecture, INSA)
Speaker: S. Thangavelu (IISc Mathematics)
Date: Fri, 15 Feb 2019
Time: 4:30 pm
Venue: LH-1, Mathematics Department
Mathematical analysts seem to have an obsession with inequalities and they search for them even within equalities! Then they ask for conditions under which equality holds. They even go to the extent of quantifying the inequalities and ask for sharp constants. Strange as it may seem, we demonstrate that this is the case, with several examples.
Title: Algebra & Combinatorics Seminar: Total Variation Cutoff for the Transpose top-2 with random shuffle
Speaker: Subhajit Ghosh (IISc Mathematics)
Date: Fri, 15 Feb 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrix for this shuffle. We mainly use the representation theory of alternating group. We show that the mixing time is of order $\left(n-\frac{3}{2}\right)\log n$ and prove that there is a total variation cutoff for this shuffle.
Title: Algebra & Combinatorics Seminar: Pfaffian Orientations and Conformal Minors
Speaker: Nishad Kothari (University of Campinas, São Paulo, Brazil)
Date: Wed, 13 Feb 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Valiant (1979) showed that unless P = N P, there is no polynomial-time algorithm to compute the number of perfect matchings of a given graph – even if the input graph is bipartite. Earlier, the physicist Kasteleyn (1963) introduced the notion of a special type of orientation of a graph, and we refer to graphs that admit such an orientation as Pfaffian graphs. Kasteleyn showed that the number of perfect matchings is easy to compute if the input graph is Pfaffian, and he also proved that every planar graph is Pfaffian. The complete bipartite graph $K_{3,3}$ is the smallest graph that is not Pfaffian. In general, the problem of deciding whether a given graph is Pfaffian is not known to be in N P.
Special types of minors, known as conformal minors, play a key role in the theory of Pfaffian orientations. In particular, a graph is Pfaffian if and only if each of its conformal minors is Pfaffian. It was shown by Little (1975) that a bipartite graph $G$ is Pfaffian if and only if $G$ does not contain $K_{3,3}$ as a conformal minor (or, in other words, if and only if $G$ is $K_{3,3}$-free); this places the problem of deciding whether a bipartite graph is Pfaffian in co – N P. Several years later, a structural characterization of $K_{3,3}$-free bipartite graphs was obtained by Robertson, Seymour and Thomas (1999), and independently by McCuaig (2004), and this led to a polynomial-time algorithm for deciding whether a given bipartite graph is Pfaffian.
Norine and Thomas (2008) showed that, unlike the bipartite case, it is not possible to characterize all Pfaffian graphs by excluding a finite number of graphs as conformal minors. In light of everything that has been done so far, it would be interesting to even identify rich classes of Pfaffian graphs (that are nonplanar and nonbipartite).
Inspired by a theorem of Lovasz (1983), we took on the task of characterizing graphs that do not contain $K_4$ as a conformal minor – that is, $K_4$-free graphs. In a joint work with U. S. R. Murty (2016), we provided a structural characterization of planar $K_4$-free graphs. The problem of characterizing nonplanar $K_4$-free graphs is much harder, and we have evidence to believe that it is related to the problem of recognizing Pfaffian graphs. In particular, we conjecture that every graph that is $K_4$-free and $K_{3,3}$-free is also Pfaffian. The talk will be mostly self-contained. I will assume only basic knowledge of graph theory. For more details, see: https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.21882.
Title: PhD Thesis defence: A study of some conformal metrics and conformal invariants on planar domains
Speaker: Amar Deep Sarkar (IISc Mathematics)
Date: Wed, 13 Feb 2019
Time: 11 am
Venue: LH-1, Mathematics Department
The main aim of this thesis is to explain the of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.
To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show the convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Caratheodory rigidity constant converges to 1 near smooth boundary points.
Next on the line is a conformal metric defined using holomorphic quadratic differentials. This was done by T. Sugawa and we will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasi-hyperbolic metric on a smoothly bounded domain.
We also study the Hurwitz metric that was introduced by D. Minda. Its construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner.
Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.
Title: APRG Seminar: Universal models in ergodic theory
Speaker: Nishant Chandgotia (Hebrew University of Jerusalem, Israel)
Date: Tue, 12 Feb 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department
A topological dynamical system is a pair $(X,T)$ where $T$ is a homeomorphism of a compact space $X$. A measure preserving action is a triple $(Y, \mu, S)$ where $Y$ is a standard Borel space, $\mu$ is a probability measure on $X$ and $S$ is a measurable automorphism of $Y$ which preserves the measure $\mu$. We say that $(X,T)$ is universal if it can embed any measure preserving action (under some suitable restrictions).
Krieger’s generator theorem shows that if $X$ is $A^{\mathbb{Z}}$ (bi-infinite sequences in elements of $A$) and $T$ is the transformation on $X$ which shifts its elements one unit to the left then $(X,T)$ is universal. Along with Tom Meyerovitch, we establish very general conditions under which $\mathbb{Z}^d$ (where now we have $d$ commuting transformations on $X$)-dynamical systems are universal. These conditions are general enough to prove that the following models are universal:
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A self-homeomorphism with non uniform specification on a compact metric space (answering a question by Quas and Soo and recovering recent results by David Burguet).
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A generic (in the sense of dense $G_\delta$) self-homeomorphism of the 2-torus preserving Lebesgue measure (extending result by Lind and Thouvenot to infinite entropy).
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Proper colourings of the $\mathbb{Z}^d$ lattice with more than two colours and the domino tilings of the $\mathbb{Z}^2$ lattice (answering a question by Şahin and Robinson).
Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson. The talk will not assume background in ergodic theory and dynamical systems.
Title: Geometry & Topology Seminar: Conjugacy invariant norms on groups
Speaker: Arpan Kabiraj (CMI, Chennai)
Date: Mon, 11 Feb 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Given a group G, we will define conjugacy invariant norm and discuss the boundedness of such norms. We will relate this problem with the existence of homogeneous quasimorphisms on groups. If time permits, we will discuss bounded cohomology and its relation to the above. This talk will be largely self-contained. No background of any of the mentioned objects will be required.
Title: Algebra & Combinatorics Seminar: Leavitt path algebras of weighted Cayley graphs $C_n(S,w)$
Speaker: Mohan R. (ISI, Bangalore)
Date: Fri, 08 Feb 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
We say a unital ring $R$ has the Invariant Basis Number (IBN) property in case, for each pair of positive integers $i,j$ if the left $R$-modules $R^i$ and $R^j$ are isomorphic, then $i=j$. The first examples of non IBN rings were studied by William Leavitt in the 1950s and he defined (what are now known as) Leavitt algebras which are ‘universal’ with non IBN property. In 2004 the algebraic structures arising from directed (multi)graphs known as Leavitt path algebras (LPA for short) were initially developed as algebraic analogues of graph $C^*$ algebras. LPAs generalize a particular class of Leavitt algebras.
During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in $C^*$-algebras, and symbolic dynamicists as well. The goal of this talk is to introduce the notion of Leavitt path algebras and to present some results on LPAs arising from weighted Cayley graphs of finite cyclic groups.
Title: Eigenfunctions Seminar: Positive mass theorem and the Yamabe problem
Speaker: Ved Datar (IISc Mathematics)
Date: Fri, 01 Feb 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
As a natural generalization of the classical uniformization theorem, one could ask if every conformal class of Riemannian metrics contains a metric with constant scalar curvature. In 1960, Yamabe attempted to solve the problem, and gave an incorrect “proof” of an affirmative answer to the above question. Yamabe’s idea was to to use techniques from calculus of variations and partial differential equations to minimize a certain functional, now named after him. Building on some work of Trudger, Aubin solved the Yamabe problem under an additional analytic condition, namely that the Yamabe invariant of the manifold is strictly smaller than the corresponding for the n-dimensional sphere. Aubin was able to verify this analytic condition for all manifolds of dimension strictly greater than 5 that are not locally conformally flat. Most remarkably, the problem was finally solved by Schoen in 1984 using the newly proven positive mass theorem of Schoen and Yau, which in turn has origins in general theory of relativity. A curious feature of Schoen’s proof is that it works precisely in the cases left out by Aubin’s result.
In the first part of the talk, I will provide some background in Riemannian geometry, and then describe the developments surrounding the Yamabe problem until Aubin’s work in 1976. A recurring theme will be the role played by the n-dimensional sphere and its conformal transformations, in all aspects of the problem. In the second part of the talk, I will describe the positive mass theorem, and outline Schoen’s proof of the Yamabe problem in low dimensions and for locally conformally flat manifolds. Most of the Riemannian geometric quantities (including scalar curvature, gradients, Laplacians etc.) will be be defined in the talk, and so the hope is that both the talks will be accessible to everyone.
Title: Algebra & Combinatorics Seminar: Bisections and bicolorings of hypergraphs
Speaker: Niranjan Balachandran (IIT, Bombay)
Date: Wed, 30 Jan 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Given a hypergraph $\mathcal{H}$ with vertex set $[n]:={1,\ldots,n}$, a bisecting family is a family $\mathcal{A}\subseteq\mathcal{P}([n])$ such that for every $B\in\mathcal{H}$, there exists $A\in\mathcal{A}$ with the property $|A\cap B|-|A\cap\overline{B}|\in{-1,0,1}$. Similarly, for a family of bicolorings $\mathcal{B}\subseteq {-1,1}^{[n]}$ of $[n]$ a family $\mathcal{A}\subseteq\mathcal{P}([n])$ is called a System of Unbiased Representatives for $\mathcal{B}$ if for every $b\in\mathcal{B}$ there exists $A\in\mathcal{A}$ such that $\sum_{x\in A} b(x) =0$.
The problem of optimal families of bisections and bicolorings for hypergraphs originates from what is referred to as the problem of Balancing Sets of vectors, and has been the source for a few interesting extremal problems in combinatorial set theory for about 3 decades now. We shall consider certain natural extremal functions that arise from the study of bisections and bicolorings and bounds for these extremal functions. Many of the proofs involve the use of polynomial methods (not the Polynomial Method, though!).
(Joint work with Rogers Mathew, Tapas Mishra, and Sudebkumar Prasant Pal.)
Title: Algebra & Combinatorics Seminar: The Polynomial Method in Combinatorics – Two illustrative theorems
Speaker: Niranjan Balachandran (IIT, Bombay)
Date: Wed, 30 Jan 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Since Dvir proved the Finite Kakeya Conjecture in 2008, the Polynomial Method has become a new and powerful tool and a new paradigm for approaching extremal questions in combinatorics (and other areas too). We shall take a look at the main philosophical principle that underlies this method via two recent (2017) theorems. One is the upper bound for Cap-sets by Ellenberg–Gijswijt, and the other, a function field analogue of a theorem of Sarkozy, due to Ben Green.
Title: Quasiconformal maps and open Riemann surfaces (continued)
Speaker: Hiroshige Shiga (Tokyo Tech, Japan)
Date: Fri, 25 Jan 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In the first part of my talk, we discuss the quasiconformal equivalence for general open Riemann surfaces and give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. In the second part, we consider the quasiconformal equivalence of Riemann surfaces which are the complements of Cantor sets.
Title: Quasiconformal maps and open Riemann surfaces
Speaker: Hiroshige Shiga (Tokyo Tech, Japan)
Date: Thu, 24 Jan 2019
Time: 11 am
Venue: LH-1, Mathematics Department
In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In the first part of my talk, we discuss the quasiconformal equivalence for general open Riemann surfaces and give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. In the second part, we consider the quasiconformal equivalence of Riemann surfaces which are the complements of Cantor sets.
Title: APRG Seminar: Hermitian Vector Bundles over Jordan Manifolds
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Wed, 23 Jan 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: APRG Seminar: Symmetric Spaces and Jordan Triples
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Tue, 22 Jan 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: Geometry & Topology Seminar: Holomorphic differentials and stability in Fukaya categories
Speaker: Pranav Pandit (ICTS, Bangalore)
Date: Mon, 21 Jan 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Spectral networks are certain decorated graphs drawn on a Riemann surface. I will describe a conjectural picture in which spectral networks can be viewed as analogues of Hermitian-Einstein metrics on vector bundles, and in which holomorphic differentials on the Riemann surface arise as stability conditions on certain Fukaya-type categories. This talk is based on various joint projects with Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich, and Carlos Simpson.
Title: Algebra & Combinatorics Seminar: Extensions of partial cyclic orders, boustrophedons and polytopes
Speaker: Sanjay Ramassamy (Ecole Normale Superiure, Paris, France)
Date: Fri, 18 Jan 2019
Time: 11 am
Venue: LH-1, Mathematics Department
While the enumeration of linear extensions of a given poset is a well-studied question, its cyclic counterpart (enumerating extensions to total cyclic orders of a given partial cyclic order) has been subject to very little investigation. In this talk I will introduce some classes of partial cyclic orders for which this enumeration problem is tractable. Some cases require the use of a multidimensional version of the classical boustrophedon construction (a.k.a. Seidel-Entringer-Arnold triangle). The integers arising from these enumerative questions also appear as the normalized volumes of certain polytopes.
This is partly joint work with Arvind Ayyer (Indian Institute of Science) and Matthieu Josuat-Verges (Laboratoire d’Informatique Gaspard Monge / CNRS).
Title: Eigenfunctions Seminar: Some unpublished work of Harish-Chandra
Speaker: Ramesh Gangolli (University of Washington, Seattle, USA)
Date: Fri, 18 Jan 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
When Harish-Chandra died in 1983, he left behind a voluminous pile of handwritten manuscripts on harmonic analysis on semisimple Lie groups over real/complex and p-adic fields. The manuscripts were turned over to the archives of the Institute for Advanced Study at Princeton, and are archived there.
Robert Langlands is the Trustee of the Harish-Chandra archive, and has always been interested in finding a way of salvaging whatever might be valuable in these manuscripts. Some years ago, at a conference in UCLA, he encouraged V. S. Varadarajan and me to look at these manuscripts.
The results of our efforts have resulted in the publication of the Volume 5 (Posthumous) of the Collected works of Harish-Chandra by Springer Verlag, which appeared in July this year. This volume covers only those manuscripts that deal with Real or Complex groups. The manuscripts dealing with p-adic groups remain in the archive.
My talk will be devoted to an outline of the results in this volume, without much detail. However, I will try to describe the main ingredient of the methods that cut across most of this work.
Title: APRG Seminar: Totally nonnegative GCD matrices and kernels
Speaker: Dominique Guillot (University of Delaware, Newark, USA)
Date: Wed, 16 Jan 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Let $X=(x_1, … ,x_n)$ be a vector of distinct positive integers. The $n \times n$ matrix with $(i,j)$ entry equal to gcd$(x_i,x_j)$, the greatest common divisor of $x_i$ and $x_j$, is called the GCD matrix on $X$. By a surprising result of Beslin and Ligh (1989), all GCD matrices are positive definite. In this talk, we will discuss new characterizations of the GCD matrices satisfying the stronger property of being totally nonnegative (i.e., all their minors are nonnegative).
Joint work with Lucas Wu (U. Delaware).
Title: Algebra & Combinatorics Seminar: From word games to a Polymath project
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 11 Jan 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Consider the following three properties of a general group $G$:
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Algebra: $G$ is abelian and torsion-free.
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Analysis: $G$ is a metric space that admits a “norm”, namely, a translation-invariant metric $d(.,.)$ satisfying: $d(1,g^n) = |n| d(1,g)$ for all $g \in G$ and integers $n$.
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Geometry: $G$ admits a length function with “saturated” subadditivity for equal arguments: $\ell(g^2) = 2 \; \ell(g)$ for all $g \in G$.
While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a “norm”.
We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and if time permits, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.
(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)
Title: APRG Seminar: Dynamics of Schwarz reflections: mating rational maps with groups
Speaker: Sabyasachi Mukherjee (State University of New York, Stony Brook, USA)
Date: Mon, 07 Jan 2019
Time: 4:30 pm
Venue: LH-1, Mathematics Department (Skype talk)
A domain in the complex plane is called a quadrature domain if it admits a global Schwarz reflection map. Topology of quadrature domains has important applications to physics, and is intimately related to iteration of Schwarz reflection maps. As dynamical systems, Schwarz reflection maps produce various instances of “matings” of rational maps and groups.
We will introduce this new class of dynamical systems, and illustrate the above-mentioned mating phenomenon with a few concrete examples. We will then describe a specific one-parameter family of Schwarz reflection maps such that “typical” maps in this family arise as unique conformal matings of a quadratic anti-holomorphic polynomial and the ideal triangle group.
Time permitting, we will also mention how Schwarz reflection maps provide us with a framework for constructing correspondences on the Riemann sphere that are “matings” of rational maps and groups.
Title: Geometry & Topology Seminar: On elementary equivalence in Artin groups of finite type
Speaker: TVH Prathamesh (University of Innsbruck, Innsbruck, Austria)
Date: Fri, 04 Jan 2019
Time: 4 pm
Venue: LH-1, Mathematics Department
Two groups are considered ‘elementarily equivalent’, if they have the same ‘elementary’ theory. Classification of various families of groups based on elementary equivalence, has been of long standing interest to both group theorists and model theorists, the most celebrated example of which was the elementary equivalence in free groups posed by Tarski. By studying examples of groups with different elementary theories, we can gain insight into the nature of expressibility of properties of groups. In this talk, I shall elementary equivalence in Artin groups of finite type, which forms a generalization of braid groups and are of interest in geometric group theory. This was a part of joint work with Arpan Kabiraj and Rishi Vyas. The talk should be largely self-contained. No background in logic, model theory or braid groups shall be assumed.
Title: Algebra & Combinatorics Seminar: Representation growth of Special Compact Linear Groups of Order Two
Speaker: Hassain M. (IISc Mathematics)
Date: Fri, 04 Jan 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Let $\mathfrak{O}$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Then the abscissa of convergence of representation zeta function of Special Linear group $\mathrm{SL}_2(\mathfrak{O})$ is $1.$ The case $p\neq 2$ is already known in the literature. For $p=2$ we need more tools to prove the result. In this talk I will discuss the difference between those cases and give an outline of the proof for $p=2.$
Let $\mathfrak{p}$ be the maximal ideal of $\mathfrak{O}$ and $|\mathfrak{O}/\mathfrak{p}|=q.$ It is already shown in literature that for $r \geq 1,$ the group algebras $\mathbb C[\mathrm{GL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb C[\mathrm{GL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. Also for $2\nmid q,$ the group algebras $\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. In this talk I will also show that if $2\mid q$ and $\mathrm{char}(\mathfrak{O})=0$ then, the group algebras, $\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{2\ell})]$ and $\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{2\ell}))]$ are not isomorphic for $\ell > \mathrm{e}$, where $\mathrm{e}$ is the ramification index of $\mathfrak{O}.$
Title: APRG Seminar: Discrete Curvature: a brief overview
Speaker: Prasad Tetali (Georgia Institute of Technology, Atlanta, USA)
Date: Tue, 01 Jan 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Inspired by the recent developments in differential geometry and calculus of variations, there have been several approaches to identifying a suitable notion of local (Ricci) curvature on non-smooth spaces, such as graphs and Markov chains. I will describe some of these approaches and review a few recent developments in this topic on discrete curvature. Some of the consequences include a tight Cheeger inequality in abelian Cayley graphs, and diameter bounds on the spectral gap of the graph Laplacian. Several open questions remain.
Title: APRG Seminar: The path of least resistance - from Euclid to Schwarz and plateau (National Mathematics Day Lecture)
Speaker: Kaushal Verma (IISc Mathematics)
Date: Sat, 22 Dec 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
This will be an elementary introduction to the Calculus of Variations, a mathematical theory that provides the basis for understanding physical phenomena. Starting with Euclid, the talk will attempt to describe some aspects of this theory, with examples.
Title: APRG Seminar: The projective spectrum: 3. Non-Euclidean metrics on the resolvent set
Speaker: Rongwei Yang (State University of New York, Albany, USA)
Date: Fri, 21 Dec 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: APRG Seminar: The projective spectrum: 2. Connections with complex geometry
Speaker: Rongwei Yang (State University of New York, Albany, USA)
Date: Fri, 21 Dec 2018
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Title: Algebra & Combinatorics Seminar: Tensor product decomposition for the general linear supergroup $GL(m|n)$
Speaker: Thorsten Heidersdorf (Max-Planck-Institut fur Mathematik, Bonn, Germany)
Date: Thu, 20 Dec 2018
Time: 11 am
Venue: LH-1, Mathematics Department
Let $\mathfrak{gl}(n)$ denote the Lie algebra of the general linear group $GL(n)$. Given two finite dimensional irreducible representations $L(\lambda), L(\mu)$ of $\mathfrak{gl}(n)$, its tensor product decomposition $L(\lambda) \otimes L(\mu)$ is given by the Littlewood-Richardson rule.
The situation becomes much more complicated when one replaces $\mathfrak{gl}(n)$ by the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. The analogous decomposition $L(\lambda) \otimes L(\mu)$ is largely unknown. Indeed many aspects of the representation theory of $\mathfrak{gl}(m|n)$ are more akin to the study of Lie algebras and their representations in prime characteristic or to the BGG category $\mathcal{O}$. I will give a survey talk about this problem and explain why some approaches don’t work and what can be done about it. This will give me the chance to speak about a) the character formula for an irreducible representation $L(\lambda)$, b) Deligne’s interpolating category $Rep(GL_t)$ for $t \in \mathbb{C}$ and c) the process of semisimplification of a tensor category.
Title: APRG Seminar: Stopping cocyles of the CCR flow
Speaker: Alexander Belton (Lancaster University, Lancaster, UK)
Date: Thu, 20 Dec 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
The non-commutative generalisation of the notion of stopping time was introduced by Hudson. Such quantum stopping times may be used to stop the CCR flow and its isometric cocycles, i.e., left operator Markovian cocycles on Boson Fock space. This stopping of the CCR flow yields a homomorphism from the semigroup of stopping times into the semigroup of unital endomorphisms of the algebra of bounded operators on the ambient Fock space, and this homomorphism is a generalised version of the shift semigroup. Moreover, stopped cocycles satisfy a non-deterministic version of the cocycle relation, which is a significant generalisation of a result of Applebaum. (Joint work with K. B. Sinha)
Title: APRG Seminar: The projective spectrum: 1. Applications to group theory
Speaker: Rongwei Yang (State University of New York, Albany, USA)
Date: Wed, 19 Dec 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: PhD Thesis defence: Decomposition of the tensor product of Hilbert modules via the jet construction and weakly homogeneous operators
Speaker: Soumitra Ghara (IISc Mathematics)
Date: Mon, 17 Dec 2018
Time: 11 am
Venue: LH-1, Mathematics Department
Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ be a sesqui-analytic function. We show that if $\alpha,\beta>0$ be such that the functions $K^{\alpha}$ and $K^{\beta}$, defined on $\Omega\times\Omega$, are non-negative definite kernels, then the $M_m(\mathbb{C})$ valued function $K^{(\alpha,\beta)} := K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^m$ is also a non-negative definite kernel on $\Omega\times\Omega$. Then we find a realization of the Hilbert space $(H,K^{(\alpha,\beta)})$ determined by the kernel $K^{(\alpha, \beta)}$ in terms of the tensor product $(H, K^{\alpha})\otimes (H, K^{\beta})$.
For two reproducing kernel Hilbert modules $(H,K_1)$ and $(H,K_2)$, let $A_n, n\geq 0$, be the submodules of the Hilbert module $(H, K_1)\otimes (H, K_2)$ consisting of functions vanishing to order $n$ on the diagonal set $\Delta:= \{ (z,z):z\in \Omega \}$. Setting $S_0=A_0^\perp, S_n=A_{n-1}\ominus A_{n}, n\geq 1$, leads to a natural decomposition of $(H, K_1)\otimes (H, K_2)$ into an infinite direct sum $\oplus_{n=0}^{\infty} S_n$. A theorem of Aronszajn shows that the module $S_0$ is isometrically isomorphic to the push-forward of the module $(H,K_1K_2)$ under the map $\iota:\Omega\to \Omega\times\Omega$, where $\iota(z)=(z,z), z\in \Omega$. We prove that if $K_1=K^{\alpha}$ and $K_2=K^{\beta}$, then the module $S_1$ is isometrically isomorphic to the push-forward of the module $(H,K^{(\alpha, \beta)})$ under the map $\iota$. We also show that if a scalar valued non-negative kernel $K$ is quasi-invariant, then $K^{(1,1)}$ is also a quasi-invariant kernel.
Title: PhD Thesis colloquium: Some Results on Spectral Spaces and spectral sequences
Speaker: Samarpita Ray (IISc Mathematics)
Date: Thu, 13 Dec 2018
Time: 4 pm
Venue: LH-3, Mathematics Department
In the last twenty years, several notions of what is called the algebraic geometry over the “field with one element” has been developed. One of the simplest approaches to this is via the theory of monoid schemes. The concept of a monoid scheme itself goes back to Kato and was further developed by Deitmar and by Connes, Consani and Marcolli. The idea is to replace prime spectra of commutative rings, which are the building blocks of ordinary schemes, by prime spectra of commutative pointed monoids. In our work, we focus mostly on abstracting out the topological characteristics of the prime spectrum of a commutative pointed monoid. This helps to obtain several classes of topological spaces which are homeomorphic to the the prime spectrum of a monoid. Such spaces are widely studied and are called spectral spaces. They were introduced by M. Hochster. We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations like integral, saturation, Frobenius and tight closures on monoids. We prove that the collection of all continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.
Title: APRG Seminar: The gamma function - an eclectic tour
Speaker: Gopal Srinivasan (IIT, Bombay)
Date: Thu, 13 Dec 2018
Time: 11 am
Venue: LH-1, Mathematics Department
In this talk we shall take an eclectic tour through three centuries since the inception of this function by Euler in 1729. The function has been the object of intense study by every great mathematician of the 19th century and continues to tantalize the modern mathematicians.
We shall dwell upon some of the more important identities, their modern ramifications and some new proofs of old results. We shall also discuss some recent developments that have occured in the last decade.
Title: Algebra & Combinatorics Seminar: Combinatorial positive valuations
Speaker: Katharina Jochemko (KTH, Stockholm, Sweden)
Date: Wed, 12 Dec 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
Valuations are a classical topic in convex geometry. The volume plays an important role in many structural results, such as Hadwiger’s famous characterization of continuous, rigid-motion invariant valuations on convex bodies. Valuations whose domain is restricted to lattice polytopes are less well-studied. The Betke-Kneser Theorem establishes a fascinating discrete analog of Hadwiger’s Theorem for lattice-invariant valuations on lattice polytopes in which the number of lattice points — the discrete volume — plays a fundamental role. In this talk, we explore striking parallels, analogies and also differences between the world of valuations on convex bodies and those on lattice polytopes with a focus on positivity questions and links to Ehrhart theory.
Title: APRG Seminar: Wandering subspace problem for m-isometries
Speaker: Sameer Chavan (IIT, Kanpur)
Date: Tue, 11 Dec 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
The wandering subspace problem for an analytic (norm-increasing) m-isometry T on a Hilbert space H asks whether every T-invariant subspace of H can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 is ascribed to Beurling-Lax-Halmos, while that for m = 2 is due to Richter. In this talk, we discuss present status of this problem including some partial solutions in case m > 2.
Title: APRG Seminar: Geometric Quantization in Complex and Harmonic Analysis
Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)
Date: Fri, 07 Dec 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
Professor H. Upmeier of the Marburg University would be visiting the Department of Mathematics, IISc as the InfoSys Visiting Professor during the period Dec 4 - Feb 28. He will give a series of lectures on the broad theme of “Geometric Quantization in Complex and Harmonic Analysis”. The first set of lectures will take place according to the following schedule
- Lecture 1: Friday, Dec 7
- Lecture 2: Monday, Dec 10
- Lecture 3: Wednesday, Dec 12
- Lecture 4: Friday, Dec 14
- Lecture 5: Monday, Dec 17
- Lecture 6: Wednesday, Dec 19
All lectures will take place in LH-1, Department of Mathematics. The first lecture will be at 3 pm. All subsequent lectures will be at 4 pm.
In this series, the speaker will discuss some basic material (connexions, curvature etc) and then cover, with full proofs, the Borel–Weil–Bott theorem and the Kodaira embedding theorem.
A second set of lectures will be announced subsequently.
Title: On the structure and distances of repeated-root constacyclic codes
Speaker: Anuradha Sharma (IIIT, Delhi)
Date: Mon, 03 Dec 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
The main aim of coding theory is to construct codes that are easier to encode and decode, can correct or at least detect many errors, and contain a sufficiently large number of codewords. To study error-detecting and error-correcting properties of a code with respect to various communication channels, several metrics (e.g. Hamming metric, Lee metric, Rosenbloom-Tsfasman (RT) metric, symbol-pair metric, etc.) have been introduced and studied in coding theory.
In this talk, we will establish algebraic structures of all repeated-root constacyclic codes of prime power lengths over finite commutative chain rings. Using their algebraic structures, we will determine Hamming distances, b-symbol distances, RT distances, and RT weight distributions of these codes. As an application of these results, we will identify MDS (maximum-distance separable) Hamming, MDS b-symbol and MDS RT codes within this particular class of constacyclic codes. We will also present an algorithm to decode these codes with respect to the Hamming, symbol-pair and RT metrics.
Title: APRG Seminar: Polynomially convex embeddings of compact real manifolds
Speaker: Purvi Gupta (Rutgers University, USA)
Date: Mon, 19 Nov 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
A compact subset of $\mathbb{C}^n$ is polynomially convex if it is defined by a (possibly infinite) family of polynomial inequalities. In this talk, we will discuss some questions and recent results regarding the minimum embedding (complex) dimension of abstract compact (real) manifolds subject to such convexity constraints. These embeddings arise naturally in connection with higher-dimensional analogues of the fact that the algebra of continuous functions on a circle can be generated by two smooth functions. We will expand on this connection, discuss the local and global challenges in constructing such embeddings, and discuss two distinct cases in which new bounds have been obtained. This is joint work with R. Shafikov.
Title: Algebra & Combinatorics Seminar: Rings with several objects
Speaker: Abhishek Banerjee (IISc Mathematics)
Date: Fri, 16 Nov 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
An ordinary ring may be seen as a preadditive category with just one object. This leads to the powerful analogy, first formulated explicitly by Mitchell in 1975, that a small preadditive category should be seen as a “ring with several objects”. We will trace the history and development of the category of modules over a preadditive category.
Title: APRG Seminar: A $z^k$-invariant subspace without the wandering property
Speaker: Daniel Seco (ICMAT, Madrid, Spain)
Date: Wed, 14 Nov 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
We study operators of multiplication by $z^k$ in Dirichlet-type spaces $D_\alpha$. We establish the existence of $k$ and $\alpha$ for which some $z^k$-invariant subspaces of $D_\alpha$ do not satisfy the wandering property. As a consequence of the proof, any Dirichlet-type space accepts an equivalent norm under which the wandering property fails for some space for the operator of multiplication by $z^k$ for any $k \geq 6$.
Title: Eigenfunctions Seminar: A refinement of the Littlewood-Richardson rule
Speaker: K.N. Raghavan (IMSc, Chennai)
Date: Fri, 09 Nov 2018
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
Symmetric polynomials are interesting not just in their own right. They appear naturally in various contexts, e.g., as characters in representation theory and as cohomology classes of Grassmannians and other homogeneous spaces in geometry. There are many different bases for the space of symmetric polynomials, perhaps the most interesting of which is the one formed by Schur polynomials. The Littlewood-Richardson (LR) rule expresses the product of two given Schur polynomials as a linear combination of Schur polynomials. The first half of the talk will be a tour of these topics that should be accessible even to undergraduate students.
The second half will be an exposition (which hopefully will continue to be widely accessible!) of recent joint research work with Mrigendra Singh Kushwaha and Sankaran Viswanath, both of IMSc. The LR rule above has a natural interpretation as giving the decomposition as a direct sum of irreducibles of the tensor product of two irreducible representations of the unitary (or general linear) group. A generalised version of it (due to Littelmann, still called the LR rule) gives the analogous decomposition for any reductive group. On the tensor product of two irreducible representations, there is the natural “Kostant-Kumar” filtration indexed by the Weyl group. This consists of the cyclic submodules generated by the highest weight vector tensor an extremal weight vector. We obtain a refined LR rule that gives the decomposition as a direct sum of irreducibles of the Kostant-Kumar submodules (of the tensor product). As an application, we obtain alternative proofs of refinements of “PRV type” results proved by Kumar and others. (PRV = Parthasarathy, Ranga Rao, Varadarajan)
Title: Algebra & Combinatorics Seminar: Curve counting and tropical geometry
Speaker: Dhruv Ranganathan (IAS (Princeton), MIT (Boston), USA; CMI (Chennai); Cambridge, UK)
Date: Fri, 02 Nov 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
The counts of algebraic curves in projective space (and other toric varieties) has been intensely studied for over a century. The subject saw a major advance in the 1990s, due to groundbreaking work of Kontsevich in the 1990s. Shortly after, considerations from high energy physics led to an entirely combinatorial approach to these curve counts, via piecewise linear embeddings of graphs, pioneered by Mikhalkin. I will give an introduction to the surrounding ideas, outlining new results and new proofs that the theory enables. Time permitting I will discuss generalizations, difficulties, and future directions for the subject.
Title: Eigenfunctions Seminar: Tropical geometry of moduli spaces
Speaker: Dhruv Ranganathan (IAS (Princeton), MIT (Boston), USA; CMI (Chennai); Cambridge, UK)
Date: Wed, 31 Oct 2018
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
Tropical geometry studies combinatorial structures that arise as “shadows” or “skeletons” of algebraic varieties. These skeletons were motivated in part by the mirror symmetry conjectures in mathematical string theory, but have now grown to function broadly as a tool for the study of algebraic varieties. Much of the work in the subject in the past decade has focused on the geometry of moduli spaces of abstract and parameterized algebraic curves, their tropical analogues, and the relationship between the two. This has led to new results on the topology of $M_{g,n}$, the geometry of spaces of elliptic curves, and to classical questions about the geometry of Hurwitz spaces. I will give an introduction to these ideas and recent advances.
Title: Algebra & Combinatorics Seminar: Factorization theorems for classical group characters
Speaker: Arvind Ayyer (IISc Mathematics)
Date: Fri, 26 Oct 2018
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Characters of classical groups appear in the enumeration of many interesting combinatorial problems. We show that, for a wide class of partitions, and for an even number of variables of which half are reciprocals of the other half, Schur functions (i.e., characters of the general linear group) factorize into a product of two characters of other classical groups. Time permitting, we will present similar results involving sums of two Schur functions. All the proofs will involve elementary applications of ideas from linear algebra.
This is joint work with Roger Behrend.
Title: On the formality of some contact manifolds
Speaker: Jean Baptiste Gatsinzi (Botswana University of Science and Technology, Botswana)
Date: Thu, 25 Oct 2018
Time: 4 pm
Venue: LH-3, Mathematics Department
Formality is a topological property, defined in terms of Sullivan’s model for a space. In the simply-connected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring.
In 1975, Deligne, Griffiths, Morgan and Sullivan proved that any compact Kaehler manifold is formal. We study the analogue for some contact manifolds. Such spaces are obtained as the total space of some circle and sphere bundles over symplectic manifolds. These include some Sasakian manifolds.
Title: Algebra & Combinatorics Seminar: Gelfand's criterion and multiplicity one results
Speaker: Pooja Singla (IISc Mathematics)
Date: Fri, 12 Oct 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
We will describe Gelfand’s criterion for the commutativity of associative algebras and discuss some of its applications towards the multiplicity one theorems for the representations of finite groups.
Title: APRG Seminar: Two applications of forcing
Speaker: Ashutosh Kumar (National University of Singapore)
Date: Tue, 09 Oct 2018
Time: 3:30 pm
Venue: LH-1, Mathematics Department
We’ll discuss two applications of forcing to analysis. (1) For every partition of a set of reals into countable sets, there is a transversal of the same Lebesgue outer measure. (2) The existence of a continuum-sized family of entire functions which take fewer than continuum values at each complex number is undecidable in ZFC plus the negation of the continuum hypothesis. Both results are joint work with Saharon Shelah.
Title: Algebra & Combinatorics Seminar: Patterns in recurring decimals
Speaker: G.V.K. Teja (IISc Mathematics)
Date: Fri, 05 Oct 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
We will study recurrence patterns in decimal expansions of rational numbers (in any integer base for this talk). After making some initial observations, we will compute the length of the repeating part of any fraction. We conclude by explaining this result over a Euclidean domain.
Title: APRG Seminar: Lugsail lag windows and their application to Markov chain Monte Carlo
Speaker: Dootika Vats (University of Warwick, UK)
Date: Thu, 04 Oct 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
Lag windows are commonly used in the time series, steady state simulation, and Markov chain Monte Carlo (MCMC) literature to estimate the long range variances of ergodic averages. We propose a new lugsail lag window specifically designed for improved finite sample performance. We use this lag window for batch means and spectral variance estimators in MCMC simulations to obtain strongly consistent estimators that are biased from above in finite samples and asymptotically unbiased. This quality is particularly useful when calculating effective sample size and using sequential stopping rules where they help avoid premature termination. Further, we calculate the bias and variance of lugsail estimators and demonstrate that there is little loss compared to other estimators. We also show mean square consistency of these estimators under weak conditions. Our results hold for processes that satisfy a strong invariance principle, providing a wide range of practical applications of the lag windows outside of MCMC. Finally, we study the finite sample properties of lugsail estimators in various examples.
Title: Eigenfunctions Seminar: The words that describe symmetric polynomials
Speaker: Amritanshu Prasad (IMSc, Chennai)
Date: Fri, 28 Sep 2018
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
In the first part of this talk we introduce a classical family of symmetric polynomials called Schur polynomials and discuss some of their properties, and a problem that arises from their study, namely the combinatorial interpretation of Littlewood–Richardson coefficients.
In the second part, we explain how the work of Robinson, Schensted, Knuth, Lascoux, and Schuetzenberger on words in an ordered alphabet led to a solution of this combinatorial problem. We will then mention some relatively recent developments in this subject.
Title: Algebra & Combinatorics Seminar: Expanders: From Graphs to Complexes
Speaker: Bharatram Rangarajan (Hebrew University, Jerusalem, Israel)
Date: Wed, 26 Sep 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
The aim of this talk is to give a high-level overview of the theory of expander graphs and introduce motivations and possible approaches to generalizing it to higher dimensions. I shall begin with three perspectives on expansion in graphs- discrepancy, isoperimetry and mixing time, and show a qualitative equivalence of these notions in defining expansion for graphs. Next I shall briefly discuss upper and lower bounds on expansion, and sketch the Lubotzky-Phillips-Sarnak construction of Ramanujan graphs. Finally, I hope to motivate high-dimensional expanders using two interesting topics- the overlapping problem, and the threshold problem.
Title: From Calculus to Number Theory.
Speaker: A. Raghuram (IISER Pune)
Date: Tue, 25 Sep 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
I will start by recalling some classical formulae that one usually encounters in a first course in Calculus. For example, Euler proved in the 1730’s that the sum of reciprocals of squares of positive integers is one-sixth of the square of \pi. Such formulae are the prototypical examples of an entire of research in modern number theory called special values of L-functions. The idea of an L-function is crucial in the work of Andrew Wiles in his proof of Fermat’s Last Theorem. The aim of this lecture will be to give an appreciation for L-functions and to convey the grandeur of this subject that draws upon several different areas of mathematics such as representation theory, algebraic and differential geometry, and harmonic analysis. Towards the end of the talk, I will present some of my own recent results on the special values of certain automorphic L-functions.
Title: Bangalore Probability Seminar: Differential-Algebraic Equations: Rough Paths and Simulation
Speaker: Soumyendu Raha (IISc Mathematics)
Date: Mon, 24 Sep 2018
Time: 3:30 pm
Venue: LH-1, Mathematics Department
We shall discuss the difficulty in solving and numerically integrating differential-algebraic equation systems of the dx/dt = f(x,u), g(x) = 0 where x is in R^n and u is in R^m and m <= n. In this context we shall introduce a horizontal lift and its exponentiation toward construction of a solution. Especially, the solution and behavior of the algebraic variable is of interest. Cases where u can be rough (belong to fractional Hoelder space) are of interest. A numerical approximation that can produce useful results in computer simulations will be discussed.
Title: Bangalore Probability Seminar: A probabilistic approach to the leader problem in random graphs
Speaker: Sanchayan Sen (IISc Mathematics)
Date: Mon, 24 Sep 2018
Time: 2:15 pm
Venue: LH-1, Mathematics Department
Start with a system of particles with possibly different masses, and consider a process where the particles merge, as time passes, according to some random mechanism. At some point of time the identity of the most massive particle–the leader–becomes fixed. We study the fixation time of the identity of the leader in the general setting of Aldous’s multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of near-critical coalescent processes, including the classical Erdos-Renyi process. In particular, this generalizes a result of Luczak. Based on joint work with Louigi Addario-Berry and Shankar Bhamidi.
Title: Algebra & Combinatorics Seminar: Graphs vs. Lie algebras
Speaker: R. Venkatesh (IISc Mathematics)
Date: Fri, 14 Sep 2018
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$ is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$ modulo the relations $[u, v]=0$ if there is no edge between the vertices $u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$ is naturally graded and the graded dimensions of the Lie algebra $\mathfrak{g}$ of $G$ have some deep connections with the vertex colorings of $G$. In this talk, I will explain how to get the generalized chromatic polynomials of $G$ in terms of graded dimensions of the Lie algebra of $G$. We will use this connection to give a Lie theoretic proof of of Stanley’s reciprocity theorem of chromatic polynomials.
Title: Algebra & Combinatorics Seminar: The Diamond Lemma in ring theory
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 07 Sep 2018
Time: 2:30 pm
Venue: LH-1, Mathematics Department
I will give a gentle introduction to the Diamond Lemma. This is a useful technique to prove that certain “PBW-type” bases exist of algebras given by generators and relations. In particular, we will see the PBW theorem for usual Lie algebras.
Title: Moments of the error term in the Sato-Tate law for elliptic curves
Speaker: Stephan Baier (RKMVU, Belur)
Date: Mon, 27 Aug 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
This is joint work with N. Prabhu (IISER Pune). We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method builds on recent work by N. Prabhu and K. Sinha who derived a Central Limit Theorem on the distribution of the errors in the Sato-Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch’s identities connect moments of coefficients of Hasse-Weil L-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak’s identity is combinatorial in nature.
Title: Geodesics on Platonic Solids
Speaker: Jayadev Athreya (University of Washington, Seattle, USA)
Date: Fri, 10 Aug 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
The five Platonic solids are some of the oldest known objects of mathematical study. In this talk (in joint work with David Aulicino and Pat Hooper) we discuss how understanding the flat geometry of Platonic solids leads to very interesting 20th and 21st century mathematics. In particular, we use the geometry of Riemann surfaces to construct a closed geodesic on the dodecahedron that passes through exactly one vertex, solving a long-standing open question.
Title: APRG Seminar: Limit shapes for groves
Speaker: Terrence George (Brown University, USA)
Date: Wed, 08 Aug 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
A grove is a spanning forest of a triangular portion of the triangular lattice with a prescribed boundary connectivity. A large random grove exhibits a limit shape i.e. there is a non-random algebraic curve outside which the grove is “frozen”. TK Petersen and D Speyer proved that for the uniform measure on groves, the curve is the inscribed circle. I will talk about extensions of their results to probability measures on groves that are periodic in appropriate coordinates. These measures give interesting algebraic curves with higher genus and cusp singularities as limit shapes, as well as new “gaseous” phases.
Title: Bangalore Probability Seminar: Gaussian multiplicative chaos limit of the Brownian loop soup Poisson layering fields
Speaker: Tulasi Ram Reddy (NYU, Abu-Dhabi, UAE)
Date: Mon, 06 Aug 2018
Time: 2:15 pm
Venue: LH-1, Mathematics Department
We consider the primary Brownian loop soup (BLS) layering vertex fields and show the existence of the fields in smooth bounded domains for a suitable range of parameters $\beta$’s. To show this at a fixed cutoff, we use Kahane’s theory of Gaussian multiplicative chaos. On the other hand, when the cutoff is removed, we use Weiner-Ito chaos expansion to establish that the $\lambda-\beta^2$ limit as the intensity $\lambda$ of the BLS diverges and $\beta$ goes to 0 such that $\lambda\beta^2$ is constant, is a complex Gaussian multiplicative Chaos. Based on joint work with F. Camia, A. Gandolfi and G. Peccati.
Title: Bangalore Probability Seminar: Location of near maxima of log-correlated fields
Speaker: Rishideep Roy (IIM, Bangalore)
Date: Mon, 06 Aug 2018
Time: 3:30 pm
Venue: LH-1, Mathematics Department
We consider the collection of near maxima of the discrete log-correlated Gaussian field in the interior of a box. We provide a rough description of the geometry of the set of near maxima. We show that two near maxima can other either simultaneously either at microscopic or at macroscopic level, but not at mesoscopic level.
Title: Eigenfunctions Seminar: A tale of two Fields Medallists: Weyl's Law for eigenfunctions of the hyperbolic Laplacian on noncompact quotients
Speaker: Ravi Raghunathan (IIT, Bombay)
Date: Fri, 03 Aug 2018
Time: 3 – 5 pm (with a 15 minute break at 3:55)
Venue: LH-4, Mathematics Department
Weyl’s law gives an asymptotic formula for the number of eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold. In 2005, Elon Lindenstrauss and Akshay Venkatesh gave a proof of this law for quite general quotients of semisimple Lie groups. The proof crucially uses the fact that solutions of the corresponding wave equation propagate with finite speed. I will try to explain what they did in the simplest setting of the upper half plane. I will also try to explain why such eigenfunctions, also known as automorphic forms, are of central importance in number theory. The first 45 minutes of the talk should be accessible to students who have a knowledge of some basic complex analysis and calculus.
Title: Birch Swinnerton-Dyer Conjecture
Speaker: Shubham Sinha (Graduate Student at UC San Diego, USA)
Date: Fri, 27 Jul 2018
Time: 3:30 pm
Venue: LH-2, Mathematics Department
Proofs of many famous problems in Number Theory (including Fermat’s Last Theorem) rely on understanding some properties about Elliptic Curves, which makes this topic inevitable and very interesting . One of the seven ‘Millennium Prize Problems’ stated by the Clay Mathematics Institute is the Birch Swinnerton-Dyer Conjecture, which is a statement regarding Elliptic Curves.
In this talk I will briefly describe the idea of a proof of the Mordell-Weil Theorem and introduce the n-Selmer group and Tate-Shafarevitch group associated to Elliptic Curves. I will define the L-function and some other arithmetic invariants attached to the Elliptic Curves and state the celebrated Birch-Swinnerton-Dyer Conjecture.
Title: Introduction to Elliptic Curves
Speaker: Shubham Sinha (Graduate Student at UC San Diego, USA)
Date: Fri, 27 Jul 2018
Time: 2 pm
Venue: LH-2, Mathematics Department
Elliptic curves are important objects of study in various areas of research in modern mathematics. In this talk I will develop some algebraic and geometric tools to understand the group structures on elliptic curves and their Isogenies (certain kind of homomorphisms). I will specialise the general study of Elliptic Curves over finite fields, define zeta functions of associated Elliptic Curve and state the Weil Conjectures.
Title: PhD Thesis colloquium: A Study of Conformal Metrics and Invariants on Planar Domains
Speaker: Amar Deep Sarkar (IISc Mathematics)
Date: Fri, 27 Jul 2018
Time: 3 pm
Venue: LH-3, Mathematics Department
The main aim of this thesis is to explain the behaviour of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated to the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.
To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Caratheodory rigidity constant converges to 1 near smooth boundary points.
Next on the line is a conformal metric defined using holomorphic quadratic differentials. This was done by T. Sugawa and we will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasi-hyperbolic metric on a smoothly bounded domain.
We also study the Hurwitz metric that was introduced by D. Minda. It’s construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner.
Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.
Title: On bases for local Weyl modules
Speaker: B. Ravinder (Chennai Mathematical Institute, Chennai)
Date: Thu, 26 Jul 2018
Time: 11 am
Venue: LH-3, Mathematics Department
Let G be a finite-dimensional complex simple Lie algebra and G[t] be its current algebra. The degree grading on the polynomial ring gives a natural grading on G[t] and makes it a graded Lie algebra. Local Weyl modules introduced by Chari and Pressley are interesting finite-dimensional graded G[t]-modules. Corresponding to a dominant integral weight x of G there is a local Weyl module denoted by W(x). The zeroth graded piece of W(x) is the irreducible G-module V(x). In this talk, we discuss how to obtain a basis for W(x) from the basis of V(x) given by Gelfand-Tsetlin patterns, when G is of type A and C.
Title: Four Flavours of Combinatorics (Enumerative, Probabilistic, Extremal, and Geometric)
Speaker: Kunal Dutta (INRIA Sophia-Antipolis, France)
Date: Tue, 24 Jul 2018
Time: 4 pm
Venue: LH-3, Mathematics Department
In this talk we shall see three very different areas of applications of combinatorics in mathematics and computer science, illustrating four different flavours of combinatorial reasoning.
The first problem is on the decomposition, into irreducible representations, of the Weil representation of the full symplectic group associated to a finite module of odd order over a Dedekind domain. We shall discuss how a poset structure defined on the orbits of finite abelian p-groups under automorphisms can be used to show the decomposition of the Weil representation is multiplicity-free, as well as parametrize the irreducible subrepresentations, compute their dimensions in terms of p, etc. Joint works with Amritanshu Prasad (IMSc, Chennai).
Next, we consider lower bounds on the maximum size of an independent set, as well as the number of independent sets, in k-uniform hypergraphs, together with an extension to the maximum size of a subgraph of bounded degeneracy in a hypergraph. Joint works with C. R. Subramanian (IMSc, Chennai), Dhruv Mubayi (UIC, Chicago) and Jeff Cooper (UIC, Chicago) and Arijit Ghosh.
Finally, we shall look at Haussler’s Packing Lemma from Computational Geometry and Machine Learning, for set systems of bounded VC dimension. We shall go through its generalisation to the Shallow Packing Lemma for systems of shallow cell complexity, and see how it can be used to prove the existence of small representations of set systems, such as epsilon nets, M-nets, etc. Joint works with Arijit Ghosh (IMSc, Chennai), Nabil Mustafa (ESIEE Paris), Bruno Jartoux (ESIEE Paris) and Esther Ezra (Georgia Inst. Tech., Atlanta).
Title: The Schur multiplier of central product of groups
Speaker: Sumana Hatui (HRI, Allahabad)
Date: Mon, 23 Jul 2018
Time: 11 am
Venue: LH-3, Mathematics Department
Let G be a central product of two groups H and K. In this talk, I shall discuss about the second cohomology group of G, having coefficients in a divisible abelian group D with trivial G-action, in terms of the second cohomology groups of certain quotients of H and K.
Title: Extremal primes for non-CM elliptic curves
Speaker: Neha Prabhu (Queen's University, Kingston, Canada)
Date: Mon, 23 Jul 2018
Time: 3 pm
Venue: LH-3, Mathematics Department
For an elliptic curve $E$ over $\mathbb{Q}$, the distribution of the number of points on $E$ mod $p$ has been well-studied over the last few decades. A relatively recent study is that of extremal primes for a given curve $E$. These are the primes $p$ of good reduction for which the number of points on $E$ mod $p$ is either maximal or minimal. If $E$ is a curve with CM, an asymptotic for the number of extremal primes was determined by James and Pollack. The talk will discuss the non-CM case and focus on obtaining upper bounds. This is joint work with C. David, A. Gafni, A. Malik and C. Turnage-Butterbaugh.
Title: Shear Coordinates of Teichmüller space and the Nielsen Realisation problem
Speaker: Deeparaj Bhat (undergraduate summer student, from Chennai Mathematical Institute, Chennai)
Date: Tue, 17 Jul 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
The aim of this talk is to answer the Nielsen Realisation problem: Can every finite subgroup of the mapping class group can be realised as a subgroup of the isometry group of some hyperbolic surface? In other words, does every finite subgroup fix a point in the Teichmüller space of the surface?
The usual Fenchel-Nielsen coordinates can be thought of as fixing a pants decomposition and keeping track of the length of the boundary of these together with the amount on ‘twist’ while glueing. Shear coordinates on the other hand, instead of using pair of pants, use ideal triangles as the basic pieces. As ideal triangles are unique up to isometry, only the gluing data needs to be tracked in this case. We shall see a convexity result concerning the length of simple closed curves with respect to these coordinates. This result leads to a positive answer for the Nielsen Realisation problem. I’ll be mainly following the paper by Bestvina, Bromberg, Fujiwara, and Souto (AMJ, 2013). Some technical results will be assumed. Familiarity with Fenchel-Nielsen coordinates will be helpful.
Title: APRG Seminar: Eventually Entanglement breaking maps
Speaker: Mizanur Rahaman (University of Waterloo, Waterloo, Canada)
Date: Tue, 10 Jul 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
Since its introduction, the class of entanglement breaking maps played a crucial role in the study of quantum information science and also in the theory of completely positive maps. In this talk, I will present a certain class of linear maps on matrix algebras that have the property that they become entanglement breaking after composing finite or infinite number of times with themselves. These maps are called eventually entanglement breaking maps. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. It turns out that the set of eventually entanglement breaking maps forms a rich class within the set of all unital completely positive maps. I will relate these maps with irreducible positive linear maps which have been studied a lot in the non-commutative Perron-Frobenius theory. Various spectral properties of a ucp map on finite dimensional C*-algebras will be discussed. The motivation of this work is the “PPT-squared conjecture” made by M. Christandl that says that every PPT channel, when composed with itself, becomes entanglement breaking. In this work, it is proved that every unital PPT-channel becomes entanglement breaking after finite number of iterations. This is a joint work with Sam Jaques and Vern Paulsen.
Title: Perverse sheaves on hyperplane arrangements and gluing
Speaker: Asilata Bapat (Australian National University, Canberra, Australia)
Date: Tue, 03 Jul 2018
Time: 3:30 pm
Venue: LH-1, Mathematics Department
A hyperplane arrangement cuts up a vector space into several pieces. The combinatorics and topology of this subdivision is encoded in the associated abelian category of perverse sheaves. This category has an alternate algebraic description due to Kapranov and Schechtman, in terms of representations of a quiver with relations. I will first explain the background and setup. I will then focus on gluing, or “recollement”, which is a recipe to reconstruct the category of perverse sheaves on a space from an open subset and its complement. The aim of the talk is to describe how recollement on the above category of perverse sheaves translates to the category of quiver representations.
Title: What is a ribbon and what can it tell us about Riemann surfaces?
Speaker: Anand Deopurkar (Australian National University, Canberra, Australia)
Date: Mon, 02 Jul 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
A Riemann surface appears in many different guises in mathematics, for example, as a branched cover of the Riemann sphere, an algebraic subset of a projective space, or a complex analytic 1-manifold. What is the relationship between various representations of the same Riemann surface? In the first part of my talk, I will describe a conjectural answer to one aspect of this question, due to Mark Green. In the second part, I will talk about ribbons. Ribbons are a particular kind of non-reduced schemes—spaces that carry “infinitesimal functions.” I will explain how studying these seemingly strange objects helps us understand properties of regular Riemann surfaces relevant for Green’s conjecture.
Title: PhD Thesis colloquium: Decomposition of the tensor product of Hilbert modules via the jet construction
Speaker: Soumitra Ghara (IISc Mathematics)
Date: Fri, 29 Jun 2018
Time: 11 am
Venue: LH-1, Mathematics Department
Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ be a sesqui-analytic function. We show that if $\alpha,\beta>0$ be such that the functions $K^{\alpha}$ and $K^{\beta}$, defined on $\Omega\times\Omega$, are non-negative definite kernels, then the $M_m(\mathbb{C})$ valued function $K^{(\alpha,\beta)} := K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^m$ is also a non-negative definite kernel on $\Omega\times\Omega$. Then we find a realization of the Hilbert space $(H,K^{(\alpha,\beta)})$ determined by the kernel $K^{(\alpha, \beta)}$ in terms of the tensor product $(H, K^{\alpha})\otimes (H, K^{\beta})$.
For two reproducing kernel Hilbert modules $(H,K_1)$ and $(H,K_2)$, let $A_n, n\geq 0$, be the submodules of the Hilbert module $(H, K_1)\otimes (H, K_2)$ consisting of functions vanishing to order $n$ on the diagonal set $\Delta:= \{ (z,z):z\in \Omega \}$. Setting $S_0=A_0^\perp, S_n=A_{n-1}\ominus A_{n}, n\geq 1$, leads to a natural decomposition of $(H, K_1)\otimes (H, K_2)$ into an infinite direct sum $\oplus_{n=0}^{\infty} S_n$. A theorem of Aronszajn shows that the module $S_0$ is isometrically isomorphic to the push-forward of the module $(H,K_1K_2)$ under the map $\iota:\Omega\to \Omega\times\Omega$, where $\iota(z)=(z,z), z\in \Omega$. We prove that if $K_1=K^{\alpha}$ and $K_2=K^{\beta}$, then the module $S_1$ is isometrically isomorphic to the push-forward of the module $(H,K^{(\alpha, \beta)})$ under the map $\iota$. We also show that if a scalar valued non-negative kernel $K$ is quasi-invariant, then $K^{(1,1)}$ is also a quasi-invariant kernel.
Title: On the Gauss-Bonnet Theorem
Speaker: M. S. Raghunathan (CBS, Mumbai)
Date: Thu, 28 Jun 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
In this talk I will outline a proof of the classical Gauss-Bonnet theorem. The proof uses Chern-Weil theory (which is standard) but more interestingly, Morse theory. The proof has appeared in a recent paper of mine in the Journal of Pure and Applied Mathematics of INSA.
Title: APRG Seminar: Rank Identities and Finite von Neumann Algebras
Speaker: Soumyashant Nayak (University of Pennsylvania, Philadelphia, USA)
Date: Wed, 27 Jun 2018
Time: 2 pm
Venue: LH-1, Mathematics Department
Algebraic identities play a pivotal role in the study of many mathematical structures although once understood, they are subconsciously regarded as being obvious or even tautological. For instance, polarization identity in convexity results in Hilbert space theory, Sylvester’s determinant identity in the study of determinantal processes, rank identities in the proof of Cochran’s theorem, etc. In this talk, the main goal is to discuss a systematic approach towards developing a theory of rank identities and determinant identities. This makes contact with Cohn’s work on free ideal rings, particularly free associative algebras over a field. By taking a universal approach, we will see how these methods translate to the world of finite von Neumann algebras (specifically II1 factors) where there is a natural notion of center-valued rank which measures the degree of non-degeneracy of an operator, and a notion of determinant known as the Fuglede-Kadison determinant. We will also see some applications to the (non-self-adjoint) algebraic structure of finite von Neumann algebras and to certain operator inequalities.
Title: APRG Seminar: On Connections between Partial Differential Equations and Associated Diffusion Processes
Speaker: Rajeeva Karandikar (CMI, Chennai)
Date: Wed, 27 Jun 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk we will describe connections between second order partial differential equations and Markov processes associated with them. This connection had been an active area of research for several decades. The talk is aimed at Analysts and does not assume familiarity with probability theory.
Title: MS Thesis defence: Correlation Functions in the Finite Toom Model
Speaker: Himanshu Gupta (IISc Mathematics)
Date: Wed, 13 Jun 2018
Time: 3 pm
Venue: LH-3, Mathematics Department
We consider a finite version of the one-dimensional Toom model with closed boundaries. Each site is occupied either by a particle of type 0 or of type 1, where the total number of particles of type 0 and type 1 are fixed to be n_0 and n_1 respectively. We call this an (n_0,n_1)-system. The dynamics are as follows: the leftmost particle in a block can exchange its position with the leftmost particle of the block to its right.
In this thesis, we have shown the following. Firstly, we have proven a conjecture about the density of 1’s in a system with arbitrary number of 0’s and 1’s. Secondly, we have made progress on a conjecture for the nonequilibrium partition function. In particular, we have given an alternate proof of the conjecture for the (1, n_1)-system and (n_0, 1)-system, using an enriched two-dimensional model.
Title: An introduction to toric geometry
Speaker: Yifei Chen (Chinese Academy of Sciences, Beijing, China)
Date: Tue, 12 Jun 2018
Time: 3 - 5 pm
Venue: LH-3, Mathematics Department
Title: MS Thesis defence: Duality for spaces of holomorphic functions into a locally convex topological vector space
Speaker: Kriti Sehgal (IISc Mathematics)
Date: Thu, 07 Jun 2018
Time: 2:30 pm
Venue: LH-1, Mathematics Department
In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function $f : O \to E$, where $O$ is an open subset of the complex plane and $E$ a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.
For $\Omega_1$ a subset of the Riemann sphere, we consider the space of locally holomorphic maps of $\Omega_1$ into $E$ vanishing at infinity if infinity belongs to $\Omega_1$, denoted by $P(\Omega_1,E)$. For two complementary subsets $\Omega_1$ and $\Omega_2$ of the Riemann sphere we prove that given two locally convex topological vector spaces $E$ and $F$ in separating duality, under some general conditions, we can define a separating duality between $P(\Omega_1,E)$ and $P(\Omega_2,F)$.
Title: An introduction to birational geometry in positive characteristic fields
Speaker: Yifei Chen (Chinese Academy of Sciences, Beijing, China)
Date: Tue, 05 Jun 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
We will survey recent progress of birational geometry in positive characteristic fields. As well, we will introduce subadditiviy of Kodaira dimension and canonical bundle formula in positive characteristics. These are joint work with Caucher Birkar, Lei Zhang and Yi Gu.
Title: Number of Solutions of Polynomial equations over Finite Fields
Speaker: Sudhir Ghorpade (IIT, Bombay)
Date: Mon, 04 Jun 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
We will consider the following question:
Given a system of a fixed number of linearly independent homogeneous polynomial equations of a fixed degree with coefficients in a finite field F, what is the maximum number of common solutions they can have in the corresponding projective space over F?
The case of a single homogeneous polynomial (i.e., hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. We will outline these developments and report on some recent progress.
An attempt will be made to keep the prerequisites at a minimum. If there is time and interest, connections to coding theory or to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension will also be outlined.
Title: Extensions of the Deligne-Kazhdan philosophy and applications
Speaker: Radhika Ganapathy (TIFR, Bombay)
Date: Thu, 24 May 2018
Time: 5 pm
Venue: LH-1, Mathematics Department
Kazhdan’s theory loosely states that the complex representation theory of the group G($F$), where G is a split connected reductive group over $\mathbb{Z}$ and $F$ is a non-archimedean local field of characterstic $p$, can be viewed as a “limit” of the complex representation theories of the groups G($F’$), where $F’$ varies over non-archimedean local fields of characteristic 0 with residue characteristic $p$. A similar theory for representations of the Galois group Gal($F_s/F$) is due to Deligne. In this talk we will review this theory, discuss some applications of this theory to the local Langlands correspondence, and some ingredients in generalizing the work of Kazhdan and some variants of it to non-split groups.
Title: Exotic smooth structures on complex projective spaces and complex hyperbolic manifolds
Speaker: Ramesh Kasilingam (ISI, Bangalore)
Date: Fri, 04 May 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
We give examples of two inequivalent smooth structures on the complex projective 9-space such that one admits a metric of nonnegative scalar curvature and the other does not. Following this example and the work of Thomas Farrell and Lowell Jones, we also construct examples of closed negatively curved Riemannian 18-manifolds, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds. (Joint work with Samik Basu.)
Title: APRG Seminar: Well-posedness of nonlocal fracture models and a priori error estimates of numerical approximations
Speaker: Prashant Jha (Louisiana State University, Baton Rouge, USA)
Date: Tue, 01 May 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we present our new results on the numerical analysis of nonlocal fracture models. We begin by giving a brief introduction to the Peridynamic theory and the nonlocal potentials considered in our work. We consider a force interaction characterized by a double well potential. Here, one well, near zero strain, corresponds to the linear response of a material, and the other well, for large strain, corresponds to the softening of a material. We show the existence of a regularized model with evolving displacement field in either Hölder space or Sobolev space. Assuming exact solutions in Hölder space, we obtain apriori error estimates due to finite difference approximation. We show that the error converges to zero, uniformly in time, in the mean square norm. The rate depends on the nonlocal length scale and is proportional to 𝐶(Δ𝑡+ℎ𝛾/𝜖2). Here $ℎ$ is the size of mesh, $\epsilon$ is the nonlocal length scale, $\Delta t$ is the size of time step, and $\gamma \in (0,1]$ is the Hölder exponent. $C$ is the constant independent of mesh size and size of time step and may depend on nonlocal length scale through the norm of the exact solution. We also study the finite element approximation and show that the error uniformly converges to zero at the rate C(Δt+h2/ϵ2). We consider piecewise linear continuous elements. Theoretical claims are supported by numerical results. This is a joint work with Dr. Robert Lipton and is funded by the US Army Research Office under grant/award number W911NF1610456.
Title: Embedding of metric graphs on hyperbolic surfaces
Speaker: Bidyut Sanki (IMSc, Chennai)
Date: Fri, 27 Apr 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is called \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, given any metric graph, its metric can be rescaled so that it can be essentially and isometrically embedded on a closed hyperbolic surface. The essential genus $g_e(G)$ of a metric graph $(G, d)$ is the lowest genus of a surface on which such an embedding of the graph is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ can be essentially and isometrically embedded (possibly after a rescaling the metric $d$) on a surface of genus $g$.
Next, we study minimal embeddings, where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_e^{\max}(G)$ a graph $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{\max}(G)$ are realized.
Title: PhD Thesis defence: Computation of sparse representation of high dimensional time series data and experimental applications
Speaker: Sandeep K. Modi (IISc Mathematics)
Date: Thu, 26 Apr 2018
Time: 2:30 pm
Venue: LH-2, Mathematics Department
Obtaining a sparse representation of high dimensional data is often the first step towards its further analysis. Conventional Vector Autoregressive (VAR) modelling methods applied to such data results in noisy, non-sparse solutions with a too many spurious coefficients. Computing auxiliary quantities such as the Power Spectrum, Coherence and Granger Causality (GC) from such non-sparse models is slow and gives wrong results. Thresholding the distorted values of these quantities as per some criterion, statistical or otherwise, does not alleviate the problem.
We propose two sparse Vector Autoregressive (VAR) modelling methods that work well for high dimensional time series data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments using simulated data, our methods show consistently higher accuracy compared to other contemporary methods in recovering the true sparse model. The relative absence of spurious coefficients in our models permits more accurate, stable and efficient evaluation of auxiliary quantities. Our VAR modelling methods are capable of computing Conditional Granger Causality (CGC) in datasets consisting of tens of thousands of variables with a speed and accuracy that far exceeds the capabilities of existing methods.
Using the Conditional Granger Causality computed from our models as a proxy for the weight of the edges in a network, we use community detection algorithms to simultaneously obtain both local and global functional connectivity patterns and community structures in large networks.
We also use our VAR modelling methods to predict time delays in many-variable systems. Using simulated data from non-linear delay differential equations, we compare our methods with commonly used delay prediction techniques and show that our methods yield more accurate results.
We apply the above methods to the following real experimental data:
- Analysis of data from the Human Connectome Project (HCP): fMRI data from the HCP database is used to compute sparse brain functional connectivity networks. The network and community structures obtained are consistent over independent recording sessions and show good spatial correspondence with known functional and anatomical regions of the brain.
- Analysis of ADHD-200 data: fMRI data from children with ADHD (Attention Deficit Hyperactive Disorder) is used to compute sparse brain functional connectivity networks. Analysis of the network measures obtained provide new ways of differentiating between ADHD and typically developing children using global and node-level network measures. They also enable refinement of published results relating to the rFIC-ACC interaction in fMRI resting state data.
- Time-delay prediction from LFP recordings:When applied to Local Field Potential (LFP) recordings from the rat and monkey, our methods predict consistent delays across a range of sampling frequencies.
-
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.
In addition to the above methods, we apply non-parametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed point-process and real time-series data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.
Title: PhD Thesis colloquium: VAR Models, Granger Causality and Applications to Neuroscience
Speaker: Sandeep K. Modi (IISc Mathematics)
Date: Thu, 26 Apr 2018
Time: 2:30 pm
Venue: LH-2, Mathematics Department
Obtaining a sparse representation of high dimensional data is often the first step towards its further analysis. Conventional Vector Autoregressive (VAR) modelling methods applied to such data results in noisy, non-sparse solutions with a too many spurious coefficients. Computing auxiliary quantities such as the Power Spectrum, Coherence and Granger Causality (GC) from such non-sparse models is slow and gives wrong results. Thresholding the distorted values of these quantities as per some criterion, statistical or otherwise, does not alleviate the problem.
We propose two sparse Vector Autoregressive (VAR) modelling methods that work well for high dimensional time series data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments using simulated data, our methods show consistently higher accuracy compared to other contemporary methods in recovering the true sparse model. The relative absence of spurious coefficients in our models permits more accurate, stable and efficient evaluation of auxiliary quantities. Our VAR modelling methods are capable of computing Conditional Granger Causality (CGC) in datasets consisting of tens of thousands of variables with a speed and accuracy that far exceeds the capabilities of existing methods.
Using the Conditional Granger Causality computed from our models as a proxy for the weight of the edges in a network, we use community detection algorithms to simultaneously obtain both local and global functional connectivity patterns and community structures in large networks.
We also use our VAR modelling methods to predict time delays in many-variable systems. Using simulated data from non-linear delay differential equations, we compare our methods with commonly used delay prediction techniques and show that our methods yield more accurate results.
We apply the above methods to the following real experimental data:
- Analysis of data from the Human Connectome Project (HCP): fMRI data from the HCP database is used to compute sparse brain functional connectivity networks. The network and community structures obtained are consistent over independent recording sessions and show good spatial correspondence with known functional and anatomical regions of the brain.
- Analysis of ADHD-200 data: fMRI data from children with ADHD (Attention Deficit Hyperactive Disorder) is used to compute sparse brain functional connectivity networks. Analysis of the network measures obtained provide new ways of differentiating between ADHD and typically developing children using global and node-level network measures. They also enable refinement of published results relating to the rFIC-ACC interaction in fMRI resting state data.
- Time-delay prediction from LFP recordings:When applied to Local Field Potential (LFP) recordings from the rat and monkey, our methods predict consistent delays across a range of sampling frequencies.
-
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.
In addition to the above methods, we apply non-parametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed point-process and real time-series data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.
Title: On the Yang-Mills functional in Noncommutative Geometry
Speaker: Satyajit Guin (IISER Mohali)
Date: Wed, 25 Apr 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
Classical geometric notion of the Yang-Mills functional has been generalized to the noncommutative context by A. Connes. In this talk we will see a suitable formulation of “subadditivity” and “additivity” of this action functional under a natural hypothesis on spectral triples, and show that in general the Yang-Mills functional is always subadditive. An instance of additivity will be discussed for the case of noncommutative torus.
Title: PhD Thesis defence: Numerical methods for elliptic variational inequalities in higher dimensions
Speaker: Gaddam Sharat (IISc Mathematics)
Date: Mon, 23 Apr 2018
Time: 11 am to 12:30 pm
Venue: LH-1, Mathematics Department
The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3-dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogenous boundary data on a 2D domain. Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. In this talk, we first present a quadratic finite element method for three dimensional ellipticobstacle problem which is optimally convergent (with respect to the regularity). We derive a priori error estimates to show the optimal convergence of the method with respect to the regularity, for this we have enriched the finite element space with element-wise bubble functions. Further, aposteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment. Next, we discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem. Using the localized behavior of DG methods, we derive a priori and a posteriori error estimates forlinear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions.We consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. We then proposea new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results here are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a post-processed solution ˜uh of the discrete solution uh which satisfies the exact boundaryconditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyse them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. Finally, we discuss a uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we discuss the ideas for nodes to element connectivity and red-refinement of a tetrahedron. We present conclusions and possible extensions for the future works.
Title: APRG Seminar: Resonances and scattering poles on symmetric spaces
Speaker: Aprameyan Parthasarathy (University of Paderborn, Germany)
Date: Mon, 09 Apr 2018
Time: 2:30 pm
Venue: LH-1, Mathematics Department
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
Title: APRG Seminar: Resonances and scattering poles on symmetric spaces
Speaker: Aprameyan Parthasarathy (University of Paderborn, Germany)
Date: Mon, 09 Apr 2018
Time: 3:30 pm
Venue: LH-1, Mathematics Department
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
Title: Eigenfunctions Seminar: Geodesic conjugacies of surfaces
Speaker: C. S. Aravinda (TIFR-CAM, Bangalore)
Date: Fri, 06 Apr 2018
Time: 3 – 5 pm
Venue: LH-1, Mathematics Department
A geodesic conjugacy between two closed manifolds is a homeomorphism between their unit tangent bundles that takes geodesic flow orbits of one to that of the other in a time-preserving manner. One of the central problems in Riemannian geometry is to understand the extent to which a geodesic conjugacy determines a closed Riemannian manifold itself. While an answer to the question in this generality has yet remained elusive, we give an overeview of results on closed surfaces – the most important illustrative case where a complete picture about questions of geodesic conjugacy rigidity is available.
Title: On Resolutions, Boij-Soderberg Conjectures, and their resolution
Speaker: Ananthnarayan Hariharan (IIT, Bombay)
Date: Fri, 06 Apr 2018
Time: 11 am
Venue: LH-1, Mathematics Department
We begin with a introduction to the notion of a resolution of a module over a Noetherian ring, leading to Betti numbers over local or graded rings, and some problems related to them. Most of the talk will focus on the graded case. One of the recent developments in this area is the resolution of the Boij-Soderberg conjectures by Eisenbud-Schreyer (2009). We discuss the motivation behind the conjectures, with a quick word on the techniques used in their resolution. If time permits, we will see other scenarios where parts of the Boij-Soderberg conjectures hold, and discuss obstacles in extending the Eisenbud-Schreyer techniques in general. This last part is joint work with Rajiv Kumar.
Title: Branching laws for certain covering groups of p-adic groups
Speaker: Shiv Prakash Patel (IISc Mathematics)
Date: Thu, 05 Apr 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
Let G be a group and H a subgroup of G. Let $\pi_1$ and $\pi_2$ be irreducible representations of $G$ and $H$ respectively. By “branching laws” one refers to the rules of describing the vector space $Hom_{H} (\pi_1, \pi_2)$. The well known Langlands’ conjectures predict connections between the representation theory of reductive groups (over local and global fields) and the study of Galois representations. In the nineties, B. Gross and D. Prasad started a systemic investigations into the study of branching laws for the groups of interest to Langlands program, and their predictions are known as Gross-Prasad conjectures. We discuss two basic examples of these predictions. A covering group of a reductive groups is a certain central extensions. We discuss branching laws involving covering groups (namely, a two fold central extension of p-adic $GL_2$) which may be seen as a variation of Gross-Prasad conjectures for covering groups.
Title: Specialization closed subsets, thick subcategories and Cohen-Macaulay rings
Speaker: Sarang Sane (IIT, Chennai)
Date: Fri, 23 Mar 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
In a topological space, a point x is said to be a specialization of another point y if x is in the closure of y. Specialization closed subsets occur naturally when considering the notion of support in the Zariski topology in algebra/algebraic geometry. We will define them and show their classical use in classifying certain subcategories. This will allow us to give a characterization of Cohen-Macaulay local rings. Time permitting, we will also discuss some reductions of K-theoretic invariants (of derived categories with support).
Title: APRG Seminar: Resonances and scattering poles on symmetric spaces
Speaker: Aprameyan Parthasarathy (University of Paderborn, Germany)
Date: Thu, 22 Mar 2018
Time: 3:30 pm
Venue: LH-1, Mathematics Department
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
Title: APRG Seminar: Liouville first-passage percolation and Watabiki's prediction
Speaker: Subhajit Goswami (IHES, Bures-sur-Yvette, France)
Date: Mon, 19 Mar 2018
Time: 2 pm
Venue: LH-1, Mathematics Department
In this talk I will give a brief introduction to Liouville first-passage percolation (LFPP) which is a model for random metric on a finite planar grid graph. It was studied primarily as a way to understand the random metric associated with Liouville quantum gravity (LQG), one of the major open problems in contemporary probability theory. In short the Liouville quantum gravity is a (conjectured) one parameter family of ``canonical’’ random metrics on a Riemann surface. I will discuss some recent results on this metric and the main focus will be on estimates of the typical distance between two points. I will highlight the apparent disagreement of these estimates with a prediction made in the physics literature about the LQG metric. I will also mention some (of many) future problems in this program. Based on a joint work with Jian Ding.
Title: MS Thesis colloquium: Holomorphic functions into a locally convex topological vector space
Speaker: Kriti Sehgal (IISc Mathematics)
Date: Mon, 19 Mar 2018
Time: 11:15 am
Venue: LH-1, Mathematics Department
In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function $f: O \to E$, where $O$ is an open subset of the complex plane and $E$ a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.
For $\Omega_1$ a subset of the Riemann sphere, we consider the space of locally holomorphic maps of $\Omega_1$ into $E$ vanishing at infinity if infinity belongs to $\Omega_1$, denoted by $P(\Omega_1,E)$. For two complementary subsets $\Omega_1$ and $\Omega_2$ of the Riemann sphere we prove that given two locally convex topological vector spaces $E$ and $F$ in separating duality, under some general conditions, we can define a separating duality between $P(\Omega_1,E)$ and $P(\Omega_2,F)$.
Title: MS Thesis colloquium: Correlation Functions in the Finite Toom Model
Speaker: Himanshu Gupta (IISc Mathematics)
Date: Mon, 19 Mar 2018
Time: 10 am
Venue: LH-1, Mathematics Department
We consider a finite version of the one-dimensional Toom model with closed boundaries. Each site is occupied either by a particle of type 0 or of type 1, where the total number of particles of type 0 and type 1 are fixed to be n_0 and n_1 respectively. We call this an (n_0, n_1)-system. The dynamics are as follows: the leftmost particle in a block can exchange its position with the leftmost particle of the block to its right.
In this thesis, we have shown the following. Firstly, we have proven a conjecture about the density of 1’s in a system with arbitrary number of 0’s and 1’s. Secondly, we have made progress on a conjecture for the nonequilibrium partition function. In particular, we have given an alternate proof of the conjecture for the (1, n_1)-system and (n_0, 1)-system, using an enriched two-dimensional model.
Title: APRG Seminar: Self-normalized concentration in infinite dimension and application to online learning problems
Speaker: Aditya Gopalan (IISc ECE)
Date: Mon, 19 Mar 2018
Time: 3:15 pm
Venue: LH-1, Mathematics Department
We will begin by providing a brief introduction to self-normalizing concentration inequalities for scalar and finite-dimensional martingales, which are very useful in measuring the size of a stochastic process in terms of another, growing, process (the 2009 book of de la Pena et al is a good reference). We will then present a self-normalizing concentration inequality for martingales that live in the (potentially infinite-dimensional) Reproducing Kernel Hilbert Space (RKHS) of a p.s.d. kernel. We will conclude by illustrating applications to online kernel least-squares regression and multi-armed bandits with infinite action spaces, a.k.a. sequential noisy function optimization [Joint work with Sayak Ray Chowdhury (IISc)].
Title: APRG Seminar: Resonances and scattering poles on symmetric spaces
Speaker: Aprameyan Parthasarathy (University of Paderborn, Germany)
Date: Thu, 15 Mar 2018
Time: 3:30 pm
Venue: LH-1, Mathematics Department
In these talks, we’ll outline a simple proof of the Helgason conjecture for Riemannian symmetric spaces of rank one, which gives a correspondence between the eigenfunctions of moderate growth of the Laplacian on the symmetric space with distributions on the distinguished boundary. We’ll then see how to use this to compare quantum resonances and scattering poles on these symmetric spaces. We also intend to indicate progress made towards a new proof of the Helgason conjecture in symmetric spaces of higher rank.
Title: Isocrystals associated to arithmetic jet spaces of abelian schemes
Speaker: Arnab Saha (Australian National University, Canberra, Australia)
Date: Thu, 08 Mar 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
The main aim of this talk is to construct a canonical F-isocrystal $H(A)_K$ for an abelian scheme A over a p-adic complete discrete valuation ring of perfect residue field K. This F-isocrystal $H(A)_K$ comes with a filtration and admits a natural map to the usual Hodge sequence of A. Even though $H(A)_K$ admits a map to the crystalline cohomology of A, the F-structure on $H(A)_K$ is fundamentally distinct from the one on the crystalline cohomology. When A is an elliptic curve, we further show that $H(A)$ itself is an F-crystal and that implies a strengthened version of Buium’s result on differential characters. The weak admissibility of $H(A)$ depends on a modular parameter over the points of the moduli of elliptic curves. Hence the Fontaine functor associates a new p-adic Galois representation to every such weakly admissible F-crystal $H(A)$. This is joint work with Jim Borger.
Title: APRG Seminar: Positive harmonic functions and Potential theory: Analytic and Geometric aspects
Speaker: Koushik Ramachandran (Oklahoma State University, Stillwater, USA)
Date: Wed, 07 Mar 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
Let $\Omega\subset\mathbb{R}^d$ be a unbounded domain. A positive harmonic function u in $\Omega$ that vanishes on the boundary $\partial\Omega$ is called a Martin function on $\Omega$. In this talk, we will discuss various analytic and geometric aspects of Martin functions, namely how fast they grow at infinity, maximum on a slice, and convexity properties of their level lines. If time permits, we will also present a inverse balayage problem from Potential theory.
Title: APRG Seminar: Theory and applications of analytic functions on multiply connected domains
Speaker: Vikas Krishnamurthy (Schrodinger Institute, Vienna, Austria)
Date: Tue, 06 Mar 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
In recent years, the theory of complex valued analytic functions defined on multiply connected domains has been recognized to have several applications in applied mathematics. In this talk, we will review the theory of Schottky-Klein prime functions and other allied special functions defined on multiply connected circular domains, and discuss the numerical computation of these special functions. We will also briefly present applications to selected problems in fluid dynamics through conformal mapping methods.
Title: On Gross-Stark conjecture
Speaker: Mahesh Kakde (King's College, London, UK)
Date: Fri, 02 Mar 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
In 1980, Gross conjectured a formula for the expected leading term at s=0 of the p-adic L-function associated to characters of totally real number fields. The conjecture states a precise relation between this leading term and p-adic regulator of p-units in an abelian extension. In the talk I will present a precise formulation of the conjecture and describe its relevance for Hilbert’s 12th problem. I will then sketch proof of this conjecture of Gross. This is a joint work with Samit Dasgupta and Kevin Ventullo.
Title: Eigenfunctions Seminar: Order and fluctuations in self-driven matter
Speaker: Sriram Ramaswamy (IISc Physics)
Date: Mon, 19 Feb 2018
Time: 3 – 5 pm
Venue: LH-1, Mathematics Department
I will present work done with students and colleagues on the collective behaviour of motile organisms, viewed as interacting particles with an autonomous velocity and noise. The talk will include a bit of stochastic processes, some statistical mechanics, and some hydrodynamics. I will discuss experiments, analytical theory and some computation.
Title: The Edelman-Greene bijection and 132-avoiding sorting networks
Speaker: Samu Potka (KTH, Stockholm, Sweden)
Date: Fri, 09 Feb 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
Edelman and Greene constructed a bijective correspondence between the reduced words of the reverse permutation (n, n - 1, …, 2, 1) and standard Young tableaux of the staircase shape (n - 1, …, 1). Recently, motivated by random sorting networks, we studied this bijection and discovered some new properties in joint work with Svante Linusson. In this talk, I will discuss them and, if time permits, also a related project with Linusson and Robin Sulzgruber on random sorting networks where the intermediate permutations avoid the pattern 132.
Title: PhD Thesis defence: Homogeneous operators
Speaker: Somnath Hazra (IISc Mathematics)
Date: Thu, 08 Feb 2018
Time: 2 pm
Venue: LH-1, Mathematics Department
The classification of homogeneous scalar weighted shifts is known. Recently, Koranyi obtained a large class of inequivalent irreducible homogeneous bi-lateral 2-by-2 block shifts. We construct two distinct classes of examples not in the list of Koranyi. It is then shown that these new examples of irreducible homogeneous bi-lateral 2-by-2 block shifts, together with the ones found earlier by Koranyi, account for every unitarily inequivalent irreducible homogeneous bi-lateral 2-by-2 block shift.
Title: PhD Thesis defence: Analytic models, dilations, wandering subspaces and inner functions
Speaker: Monojit Bhattacharjee (IISc Mathematics)
Date: Thu, 08 Feb 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk we will discuss an analytic model theory for pure hyper-contractions (introduced by J. Agler) which is analogous to Sz.Nagy-Foias model theory for contractions. We then proceed to study analytic model theory for doubly commuting n-tuples of operators and analyze the structure of joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over polydisc. In particular, we completely characterize the doubly commuting quotient modules of a large class of reproducing kernel Hilbert Modules, in the sense of Arazy and Englis, over the unit polydisc.
Inspired by Halmos, in the second half of the talk, we will focus on the wandering subspace property of commuting tuples of bounded operators on Hilbert spaces. We prove that for a large class of analytic functional Hilbert spaces $H_k$ on the unit ball in $\mathbb{C}^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1},…,M_{z_n})$ can be described in terms of suitable $H_k$-inner functions. We also prove that $H_k$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogeneous polynomials as an application. Along the way we also prove a refinement of a result of Arveson on the uniqueness of the minimal dilations of pure row contractions.
Title: The ongoing history of PolyMath
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 02 Feb 2018
Time: 3 pm
Venue: LH-1, Mathematics Department
Given the recent successfully concluded Polymath project – and the next Polymath that has already started over the weekend – I will present the origins of Polymath and discuss its workings, using slides of myself and of a Polymath collaborator.
Title: Categorical Kähler Geometry: from derived categories to dynamical systems
Speaker: Pranav Pandit (IHES, France)
Date: Wed, 31 Jan 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
Mirror symmetry is a phenomenon predicted by string theory. It allows one to translate questions in symplectic geometry to questions in complex geometry, and vice versa. The homological mirror symmetry program interprets mirror symmetry within the unifying categorical framework of derived noncommutative geometry. After introducing these ideas, I will describe an approach to a theory of Kähler metrics in derived noncommutative geometry. We will see how this leads to (i) a non-Archimedean categorical analogue of the Donaldson-Uhlenbeck-Yau theorem, inspired by symplectic geometry, and (ii) the discovery of a refinement of the Harder-Narasimhan filtration which controls the asymptotic behavior of certain geometric flows. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.
Title: Eigenfunctions Seminar: Basic law of large numbers and applications; Estimation of integrals with respect to infinite measures via RSMC
Speaker: Krishna B. Athreya (Professor Emeritus, Iowa State University, Ames, USA)
Date: Fri, 19 Jan 2018
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Let X1 , X2 , X3, … Xn be iid random variables. Laws of large numbers roughly state that the average of these variables converges to the expectation value of each of them when n is large. Various forms of these laws have many applications. The strong and weak laws along with the following three applications will be discussed: a) Coin-tossing. b) The Weierstrass approximation theorem. c) The Glivenko–Cantelli theorem.
In the second half of this talk, a law of large numbers is proven for spaces with infinite “volume” (measure) as opposed to the above version for probability measures (“volume” =1).
Title: APRG Seminar: Self-intersections and Poisson line processes on hyperbolic surfaces
Speaker: Jayadev Athreya (University of Washington, Seattle, USA)
Date: Wed, 17 Jan 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
In joint work with Steve Lalley and Jenya Sapir, we study the tessellation of a compact, hyperbolic surface induced by a typical long geodesic segment. We show, that when properly scaled, the local behavior of a typical geodesic is that of a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.
Title: APRG Seminar: Dynamics Of Determinantal Point Processes
Speaker: Alexander Bufetov (CNRS, Marseilles, France; and Steklov Mathematical Institute, IITP RAS, NRU Higher School of Economics, Russia)
Date: Wed, 17 Jan 2018
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Determinantal point processes, which first appeared in Dyson’s work on random matrices, arise in diverse problems of asymptotic combinatorics, ergodic theory, representation theory. They have strong chaotic properties: for example, the sine-process has the Kolmogorov property and satisfies the Central Limit Theorem. A Functional Limit Theorem for the sine-process has been established in joint work with A. Dymov.
A delicate aspect of the behaviour of a determinantal point process is that particles interact at infinite radius. For instance, Ghosh and Peres showed that, under the sine-process, the number of particles in a bounded interval is determined by the configuration in the outside of the interval. For determinantal point processes with so-called integrable kernels, an explicit description is given of conditional measures of the process in a bounded interval with respect to the fixed exterior. These conditional measures are given by orthogonal polynomial ensembles with explicitly found weights. A key step in the argument is that projections inducing our processes satisfy a weaker analogue of the division axiom of de Branges: in fact, this weak division property, as shown in joint work with Roman Romanov, characterizes integrable kernels. Similar results for determinantal point processes governed by orthogonal projections onto Hilbert spaces of holomorphic functions are obtained in joint work with Y. Qiu. The talk is based on the preprints
https://arxiv.org/abs/1707.03463, (joint with R. Romanov), https://arxiv.org/abs/1701.00111 (joint with A. Dymov), and the paper
Alexander I. Bufetov, Yanqi Qiu, “Conditional measures of generalized Ginibre point processes”, J. Funct. Anal., 272:11 (2017), 4671–4708.
Title: Eigenfunctions Seminar: Homogeneous length functions on Groups: A polymath adventure
Speaker: Siddhartha Gadgil (and Apoorva Khare), IISc Mathematics
Date: Mon, 15 Jan 2018
Time: 4 – 5 pm
Venue: LH-1, Mathematics Department
Terence Tao posted on his blog a question of Apoorva Khare, asking whether the free group on two generators has a length function $l: F_2 \to\mathbb{R}$ (i.e., satisfying the triangle inequality) which is homogeneous, i.e., such that $l(g^n) = nl(g)$. A week later, the problem was solved by an active collaboration of several mathematicians (with a little help from a computer) through Tao’s blog. In fact a more general result was obtained, namely that any homogeneous length function on a group $G$ factors through its abelianization $G/[G, G]$.
I will discuss the proof of this result and also the process of discovery (in which I had a minor role).
Title: APRG Seminar: Analogical Models and Homogenization Disagree
Speaker: Augusto Visintin (Universita' degli Studi di Trento, Italy)
Date: Fri, 12 Jan 2018
Time: 11:30 am
Venue: LH-3, Mathematics Department
Title: APRG Seminar: Mathematical models of hysteresis
Speaker: Augusto Visintin (Universita' degli Studi di Trento, Italy)
Date: Fri, 12 Jan 2018
Time: 10 am
Venue: LH-3, Mathematics Department
Title: Hilbert functions of Gorenstein K-algebras
Speaker: Shreedevi K. Masuti (IMSc, Chennai)
Date: Thu, 11 Jan 2018
Time: 4 pm
Venue: LH-5, Mathematics Department
Gorenstein rings are very common and significant in many areas of mathematics. The following are two important and widely open problems in commutative algebra and algebraic geometry:
- characterization of possible Hilbert functions of Gorenstein K-algebras; and
- classification of Gorenstein K-algebras with given Hilbert functions.
Recently, in a joint work with M.E. Rossi, we obtained partial results to these problems in some cases (K-algebras of socle degree 4). In this talk, we will discuss these new developments.
Title: APRG Seminar: Compactness and Structural Stability of Nonlinear Flows
Speaker: Augusto Visintin (Universita' degli Studi di Trento, Italy)
Date: Thu, 11 Jan 2018
Time: 2:30 - 3:30 and 4 - 5 pm
Venue: LH-3, Mathematics Department
Title: PhD Thesis colloquium: Numerical methods for elliptic variational inequalities in higher dimensions
Speaker: Gaddam Sharat (IISc Mathematics)
Date: Wed, 10 Jan 2018
Time: 10 am
Venue: LH-3, Mathematics Department
The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3-dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogenous boundary data on a 2D domain.
Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. In this talk, we first present a quadratic finite element method for three dimensional ellipticobstacle problem which is optimally convergent (with respect to the regularity). We derive a priorierror estimates to show the optimal convergence of the method with respect to the regularity, forthis we have enriched the finite element space with element-wise bubble functions. Further, aposteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment. Next, we discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem.Using the localized behavior of DG methods, we derive a priori and a posteriori error estimates forlinear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions.We consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. We then proposea new and simpler residual based a posteriori error estimator for finite element approximationof the elliptic obstacle problem. The results here are two fold. Firstly, we address the influenceof the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a post-processed solution uh of the discrete solution uh which satisfies the exact boundaryconditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyse them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. Finally, we discuss a uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we discuss the ideas for nodes to element connectivity and red-refinement of a tetrahedron. We present conclusions and possible extensions for the future works.
Title: APRG Seminar: First order logic on Galton-Watson trees
Speaker: Moumanti Podder (Georgia Institute of Technology, Atlanta, USA)
Date: Tue, 09 Jan 2018
Time: 11:45 am
Venue: LH-3, Mathematics Department
This talk will focus on the rooted Galton-Watson (GW) tree. The offspring distribution we consider is Poisson(\lambda), but our results extend to more general distributions. First order properties on rooted trees capture the local, finite structures inside the tree. We analyze the probabilities of first order properties under the GW measure, and obtain these probabilities as fixed points of contracting distributional maps. Moreover, we come up with nice functions that express these probabilities conditioned on survival of the GW tree. This is joint work with Joel Spencer.
Title: Kahler-Einstein metrics on Fano manifolds
Speaker: Ved Datar (University of California, Berkeley, USA)
Date: Wed, 03 Jan 2018
Time: 4 pm
Venue: LH-1, Mathematics Department
A version of the uniformization theorem states that any compact Riemann surface admits a metric of constant curvature. A deep and important problem in complex geometry is to characterize Kahler manifolds admitting constant scalar curvature Kahler (cscK) metrics or extremal Kahler metrics. Even in the special case of Kahler-Einstein metrics, starting with the work of Yau and Aubin in the 1970’s, a complete solution was obtained only very recently by Chen-Donaldson-Sun (and Tian). Their main results says that a Fano manifold admits a Kahler-Einstein metric if and only if it is K-stable. I will survey some of these recent developments, and then focus on a refinement obtained in collaboration with Gabor Szekelyhidi. This has led to the discovery of new Kahler-Einstein manifolds. If time permits, I will also talk about some open problems on constructing cscK and extremal metrics on blow-ups of extremal manifolds, and mention some recent progress.
Title: PhD Thesis defence: Unfolding Operators in Various Oscillatory Domains: Homogenization of Optimal Control Problems
Speaker: Aiyappan S (IISc Mathematics)
Date: Mon, 18 Dec 2017
Time: 10:30 am
Venue: LH-3, Mathematics Department
Homogenization of boundary value problems posed on rough domains has paramount importance in real life problems. Materials with oscillating (rough) boundary are used in many industrial applications like micro strip radiator and nano technologies, biological systems, fractal-type constructions, etc. In this talk, we will be focusing on homogenization of optimal control problems. We will begin with homogenization of a boundary control problem on an oscillating pillar type domain. Then, we will consider a time-dependent control problem posed on a little more general domain called branched structure domain. Asymptotic analysis of this interior control problem will be explained. Next, we will present a generalized unfolding operator that we have developed for a general oscillatory domain. Using this unfolding operator, we study the homogenization of a non-linear elliptic problem on this general highly oscillatory domain. Also, we analyse an optimal control problem on a circular oscillating domain with the assistance of this operator. Finally, we consider a non-linear optimal control problem on the above mentioned general oscillatory domain and study the asymptotic behaviour.
Title: PhD Thesis defence: The Pick--Nevanlinna interpolation problem: complex-analytic methods in special domains
Speaker: Vikramjit Singh Chandel (IISc Mathematics)
Date: Wed, 06 Dec 2017
Time: 2:30 pm
Venue: LH-1, Mathematics Department
The Pick–Nevanlinna interpolation problem in its fullest generality is as follows:
Given domains $D_1$, $D_2$ in complex Euclidean spaces, and a set ${(z_i,w_i): 1\leq i\leq N}\subset D_1\times D_2$, where $z_i$ are distinct and $N$ is a positive integer $\geq 2$, find necessary and sufficient conditions for the existence of a holomorphic map $F$ from $D_1$ into $D_2$ such that $F(z_i) = w_i$, $1\leq N$.
When such a map $F$ exists, we say that $F$ is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem – which we shall study in this thesis – have been of lasting interest:
INTERPOLATION FROM THE POLYDISC TO THE UNIT DISC: This is the case $D_1 = D^n$ and $D_2 = D$, where $D$ denotes the open unit disc in the complex plane and $n$ is a positive integer. The problem itself originates with Georg Pick’s well-known theorem (independently discovered by Nevanlinna) for the case $n=1$. Much later, Sarason gave another proof of Pick’s result using an operator-theoretic approach, which is very influential. Using this approach for $n\geq 2$, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur–Agler class. This is notable because when $n = 2$ the latter result completely solves the problem for the case $D_1 = D^2$, $D_2 = D$. However, Pick’s approach can also be effective for $n\geq 2$. In this thesis, we give an alternative characterization for the existence of a $3$-point interpolant based on Pick’s approach and involving the study of rational inner functions.
Cole, Lewis and Wermer lifted Sarason’s approach to uniform algebras – leading to a characterization for the existence of an interpolant in terms of the positivity of a large, rather abstractly-defined family of $(N\times N)$ matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a $D^n$-to-$D$ interpolant in terms of the positivity of a family of matrices parametrized by a class of polynomials.
INTERPOLATION FORM THE UNIT DISC TO THE SPECTRAL UNIT BALL: This is the case $D_1 = D$ and $D_2$ is the set of all $(n\times n)$ matrices with spectral radius less than $1$. The interest in this arises from problems in Control Theory. Bercovici, Fois and Tannenbaum adapted Sarason’s methods to give a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc – leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any $n$ and $N=2$. We shall present a necessary condition for the existence of a $3$-point interpolant. This we shall achieve by modifying Pick’s approach and applying the aforementioned result due to Bharali.
Title: Derived categories and Ulrich bundles
Speaker: Kyoung-Seog Lee (Center for Geometry and Physics, Institute for Basic Science (IBS) Pohang, Republic of Korea)
Date: Mon, 04 Dec 2017
Time: 4 pm
Venue: LH-3, Mathematics Department
Derived category is an important tool in homological algebra invented by Grothendieck and Verdier. In these days derived categories play important roles in many areas of algebraic geometry. In this talk, I will discuss derived categories and their applications to study Ulrich bundles on some Fano manifolds.
Title: Closed subroot systems of finite and affine root systems
Speaker: Krishanu Roy (IMSc, Chennai)
Date: Fri, 24 Nov 2017
Time: 10:30 am
Venue: LH-1, Mathematics Department
In this talk, I will tell you about the Borel-de-Sibenthal theorem which gives the classification of all maximal closed subroot systems of finite crystallographic root systems. I will start my talk by introducing the notion of finite root systems and it’s closed subroot systems.
The concept of root system is very fundamental in the theory of Lie groups and Lie algebras. Especially they play a vital role in the classification of finite dimensional semi-simple Lie algebras. Closed subroot systems of finite root systems naturally appear in the Borel-de-Sibenthal theory which describes the closed connected subgroups of a compact Lie group that have maximal rank. The classification of closed subroot systems is essential in the classification of semi-simple subalgebras of semi-simple Lie algebras.
Through out this talk, we will try to stay within the theory of root systems and reflection groups. No knowledge of Lie algebras or Lie groups will be assumed. If time permits I will discuss about my joint work with R. Venkatesh which gives explicit descriptions of the maximal closed subroot systems of affine root systems.
Title: Arithmetic geometry of toric varieties
Speaker: Patrice Philippon (CNRS, Paris, France)
Date: Wed, 22 Nov 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
We will present results obtained in collaboration with J. Burgos and M. Sombra. These extend the well known dictionary between the geometric properties of toric varieties and convex geometry. In particular, we give combinatorial descriptions of classical invariants of arithmetic geometry, such as metric, height or essential minimum.
Title: Eigenfunctions Seminar: Unique factorization property of Schur functions
Speaker: R. Venkatesh (IISc Mathematics)
Date: Sat, 11 Nov 2017
Time: 3 – 4:55 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
In this talk, we prove the unique factorization property of Schur functions. This fundamental property of Schur functions was first observed and proved by C. S. Rajan in 2004. I give a different proof of this beautiful fact which I jointly obtained with my adviser S. Viswanath. I begin my talk with introducing the Schur functions and their connections with representation theory of general linear groups. Basic knowledge of elementary algebra will be assumed like group theory and linear algebra. If time permits, I will tell you about the possible generalizations of this result.
Title: APRG Seminar: Irrational rotations, random affine transformations and the central limit theorem
Speaker: Nishant Chandgotia (Tel Aviv University, Israel)
Date: Mon, 09 Oct 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.
Title: Eigenfunctions Seminar: The quest for a polynomial that is hard to compute
Speaker: Neeraj Kayal (Microsoft Research, Bangalore)
Date: Fri, 06 Oct 2017
Time: 3 – 4:55 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
We consider the computation of n-variate polynomials over a field F via a sequence of arithmetic operations such as additions, subtractions, multiplications, divisions, etc. It has been known for at five decades now that a random n-variate polynomial of degree n is hard to compute. Yet not a single explicit polynomial is provably known to be hard to compute (although we have a lot of good candidates). In this talk we will first describe this problem and its relationship to the P vs NP problem. We will then describe several partial results on this problem, both old and new, along with a more general approach/framework that ties together most of these partial results.
Title: Partition of integers :- a treasure trove of open problems : order matching and unimodality
Speaker: N. Narayanan (IIT, Chennai)
Date: Fri, 06 Oct 2017
Time: 10:30 am
Venue: LH-1, Mathematics Department
A partition of integer $n$ is a sequence $\lambda = (\lambda_1, \lambda_2, \cdots, \lambda_k, \cdots)$ of non negative integers such that $\lambda_i \ge \lambda_{i+1}$ and $\sum_i \lambda_i = n$. It follows that there are finitely many non-zero $\lambda_i$’s. One can restrict the number of them and the largest value of $\lambda_i$ and observe that the set of such partitions form a poset under a suitable relation. Several natural questions arise in this setting. Some of these questions have been answered by Proctor, Stanley and Kathy O’Hara among others. We take a look at some techniques as given by Stanley and ask if it is possible to extend it to higher dimensions.
Title: APRG Seminar: Invertibility and condition number of sparse random matrices
Speaker: Anirban Basak (Weizmann Institute of Science, Israel)
Date: Thu, 05 Oct 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
I will describe our work that establishes (akin to) von Neumann’s conjecture on condition number, the ratio of the largest and the smallest singular values, for sparse random matrices. Non-asymptotic bounds on the extreme singular values of large matrices have numerous uses in the geometric functional analysis, compressed sensing, and numerical linear algebra. The condition number often serves as a measure of stability for matrix algorithms. Based on simulations von Neumann and his collaborators conjectured that the condition number of a random square matrix of dimension $n$ is $O(n)$. During the last decade, this conjecture was proved for dense random matrices.
Sparse matrices are abundant in statistics, neural networks, financial modeling, electrical engineering, and wireless communications. Results for sparse random matrices have been unknown and requires completely new ideas due to the presence of a large number of zeros. We consider a sparse random matrix with entries of the form $\xi_{i,j} \delta_{i,j}, \, i,j=1,\ldots,n$, such that $\xi_{i,j}$ are i.i.d. with zero mean and unit variance and $\delta_{i,j}$ are i.i.d. Ber$(p_n)$, where $p_n \downarrow 0$ as $n \to \infty$. For $p_n < \frac{\log n}{n}$, this matrix becomes non-invertible, and hence its condition number equals infinity, with probability tending to one. In this talk, I will describe our work showing that the condition number of such sparse matrices (under certain assumptions on the moments of $\{\xi_{i,j}\}$) is $O(n^{1+o(1)})$ for all $p_n > \frac{\log n}{n}$, with probability tending to one, thereby establishing the optimal analogous version of the von Neumann’s conjecture on condition number for sparse random matrices.
This talk is based on a sequence of joint works with Mark Rudelson.
Title: Counting complex solutions of polynomial equations via Newton polytopes
Speaker: Jugal Verma (IIT, Bombay)
Date: Tue, 05 Sep 2017
Time: 2:30 - 3:45 and 4 - 5 pm
Venue: LH-1, Mathematics Department
We shall discuss a theorem of Bernstein published in 1975 about the number of common solutions of n complex polynomials in n variables in terms of the mixed volumes of their Newton polytopes. This is a far reaching generalisation of the Fundamental Theorem of Algebra and Bezout’s Theorem about intersections of plane algebraic curves. If time permits, we shall sketch a proof of Bernstein’s theorem using Hilbert functions of monomial ideals in polynomial rings.
Title: Eigenfunctions Seminar: The history of matrix positivity preservers
Speaker: Apoorva Khare (IISc Mathematics)
Date: Fri, 18 Aug 2017
Time: 3 – 4 and 4:15 – 5:15 pm
Venue: LH-1, Mathematics Department
I will give a gentle historical (and ongoing) account of matrix positivity and of operations that preserve it. This is a classical question studied for much of the past century, including by Schur, Polya-Szego, Schoenberg, Kahane, Loewner, and Rudin. It continues to be pursued actively, for both theoretical reasons as well as applications to high-dimensional covariance estimation. I will end with some recent joint work with Terence Tao (UCLA).
The entire talk should be accessible given a basic understanding of linear algebra/matrices and one-variable calculus. That said, I will occasionally insert technical details for the more advanced audience. For example: this journey connects many seemingly distant mathematical topics, from Schur (products and complements), to spheres and Gram matrices, to Toeplitz and Hankel matrices, to rank one updates and Rayleigh quotients, to Cauchy-Binet and Jacobi-Trudi identities, back full circle to Schur (polynomials).
Title: Quantum Theory of Dark matter
Speaker: Krishna M (Ashoka University)
Date: Wed, 09 Aug 2017
Time: 3:30 pm
Venue: LH-1, Mathematics Department
In this talk we discuss a formulation of Quantum Theory of Dark matter and discuss some operators on Hilbert spaces of singular measures.
Title: For good measure ... and bad
Speaker: Malabika Pramanik (UBC, Canada)
Date: Wed, 02 Aug 2017
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Title: PhD Thesis colloquium: The Pick--Nevanlinna interpolation problem: complex-analytic methods in special domains
Speaker: Vikramjit Singh Chandel (IISc Mathematics)
Date: Mon, 17 Jul 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
The Pick–Nevanlinna interpolation problem in its fullest generality is as follows:
Given domains $D_1$, $D_2$ in complex Euclidean spaces, and a set ${(z_i,w_i): 1\leq i\leq N}\subset D_1\times D_2$, where $z_i$ are distinct and $N$ is a positive integer $\geq 2$, find necessary and sufficient conditions for the existence of a holomorphic map $F$ from $D_1$ into $D_2$ such that $F(z_i) = w_i$, $1\leq N$.
When such a map $F$ exists, we say that $F$ is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem – which we shall study in this thesis – have been of lasting interest:
INTERPOLATION FROM THE POLYDISC TO THE UNIT DISC: This is the case $D_1 = D^n$ and $D_2 = D$, where $D$ denotes the open unit disc in the complex plane and $n$ is a positive integer. The problem itself originates with Georg Pick’s well-known theorem (independently discovered by Nevanlinna) for the case $n=1$. Much later, Sarason gave another proof of Pick’s result using an operator-theoretic approach, which is very influential. Using this approach for $n\geq 2$, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur–Agler class. This is notable because when $n = 2$ the latter result completely solves the problem for the case $D_1 = D^2$, $D_2 = D$. However, Pick’s approach can also be effective for $n\geq 2$. In this thesis, we give an alternative characterization for the existence of a $3$-point interpolant based on Pick’s approach and involving the study of rational inner functions.
Cole, Lewis and Wermer lifted Sarason’s approach to uniform algebras – leading to a characterization for the existence of an interpolant in terms of the positivity of a large, rather abstractly-defined family of $(N\times N)$ matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a $D^n$-to-$D$ interpolant in terms of the positivity of a family of matrices parametrized by a class of polynomials.
INTERPOLATION FORM THE UNIT DISC TO THE SPECTRAL UNIT BALL: This is the case $D_1 = D$ and $D_2$ is the set of all $(n\times n)$ matrices with spectral radius less than $1$. The interest in this arises from problems in Control Theory. Bercovici, Fois and Tannenbaum adapted Sarason’s methods to give a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc – leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any $n$ and $N=2$. We shall present a necessary condition for the existence of a $3$-point interpolant. This we shall achieve by modifying Pick’s approach and applying the aforementioned result due to Bharali.
Title: The representation homology of spaces
Speaker: Ajay Ramadoss (Indiana University, USA)
Date: Wed, 12 Jul 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: PhD Thesis colloquium: Analytic models, dilations, wandering subspaces and inner functions
Speaker: Monojit Bhattacharjee (IISc Mathematics)
Date: Mon, 03 Jul 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk we will discuss an analytic model theory for pure hyper-contractions (introduced by J. Agler) which is analogous to Sz.Nagy-Foias model theory for contractions. We then proceed to study analytic model theory for doubly commuting n-tuples of operators and analyze the structure of joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over polydisc. In particular, we completely characterize the doubly commuting quotient modules of a large class of reproducing kernel Hilbert Modules, in the sense of Arazy and Englis, over the unit polydisc.
Inspired by Halmos, in the second half of the talk, we will focus on the wandering subspace property of commuting tuples of bounded operators on Hilbert spaces. We prove that for a large class of analytic functional Hilbert spaces H_k on the unit ball in $\mathbb{C}^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1},…,M_{z_n})$ can be described in terms of suitable H_k-inner functions. We also prove that H_k-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogeneous polynomials as an application. Along the way we also prove a refinement of a result of Arveson on the uniqueness of the minimal dilations of pure row contractions.
Title: PhD Thesis colloquium: Homogeneous operators
Speaker: Somnath Hazra (IISc Mathematics)
Date: Mon, 19 Jun 2017
Time: 2 pm
Venue: LH-1, Mathematics Department
It is known that the characteristic function $\theta_T$ of a homogeneous contraction $T$ with an associated representation $\pi$ is of the form \begin{equation} \theta_T(a) = \sigma_{L}(\phi_a)^* \theta(0) \sigma_{R}(\phi_a), \end{equation}
where, $\sigma_{L}$ and $\sigma_{R}$ are projective representation of the
Mobius group Mob with a common multiplier. We give another proof
of the “product formula”.
Also, we prove that the projective representations $\sigma_L$ and
$\sigma_R$ for a class of multiplication operators, the two
representations $\sigma_{R}$ and $\sigma_{L}$ are unitarily equivalent to
certain known pair of representations $\sigma_{\lambda + 1}$ and
$\sigma_{\lambda - 1},$ respectively. These are described explicitly.
Let $G$ be either (i) the direct product of $n$-copies of the
bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic
automorphism group of the polydisc $\mathbb D^n.$
A commuting tuple of bounded operators $\mathsf{T} = (T_1, T_2,\ldots
,T_n)$ is said to be $G$-homogeneous if the joint spectrum of $\mathsf{T}$
lies in $\overline{\mathbb{D}}^n$ and $\varphi(\mathsf{T}),$ defined using
the usual functional calculus, is unitarily equivalent with $\mathsf{T}$
for all $\varphi \in G.$
We show that a commuting tuple $\mathsf{T}$ in the Cowen-Douglas class of
rank $1$ is $G$ - homogeneous if and only if it is unitarily equivalent
to the tuple of the multiplication operators on either the reproducing
kernel Hilbert space with reproducing kernel $\prod_{i = 1}^{n}
\frac{1}{(1 - z_{i}\overline{w}\_{i})^{\lambda_i}}$ or $\prod_{i = 1}^{n}
\frac{1}{(1 - z_{i}\overline{w}\_{i})^{\lambda}},$ where $\lambda,$
$\lambda_i$, $1 \leq i \leq n,$ are positive real numbers, according as
$G$ is as in (i) or (ii).
Let $\mathsf T:=(T_1, \ldots ,T_{n-1})$ be a $G$-homogeneous $(n-1)$-tuple
of rank $1$ Cowen-Douglas class, where $G$ is the the direct product of
$n-1$-copies of the bi-holomorphic automorphism group of the disc. Let
$\hat{T}$ be an irreducible homogeneous (with respect to the
bi-holomorphic group of automorphisms of the disc) operator in the
Cowen-Douglas class on the disc of rank $2$. We show that every
irreducible $G$ - homogeneous operator, $G$ as in (i), of rank $2$ must be
of the form
\begin{equation}
(T_1\otimes I_{\widehat{H}},\ldots , T_{n-1}\otimes
I_{\widehat{H}}, I_H \otimes \hat{T}).
\end{equation}
We also show that if $G$ is chosen to be the group as in (ii), then there are no irreducible $G$- homogeneous operators of rank $2.$
Title: Homological Invariants and Combinatorics of Finite Simple Graphs
Speaker: Arindam Banerjee (Purdue University, USA)
Date: Thu, 15 Jun 2017
In this talk the interplay between the combinatorial structures of finite simple graphs and various homological invariants like regularity, depth etc. of related algebraic objects shall be discussed. Some open problems, recent developments and ongoing projects shall be discussed. In particular some new techniques developed in my thesis to study Castelnuovo-Mumford regularity of algebraic objects related to graphs shall be discussed in some details.
Title: PhD Thesis colloquium: Unfolding Operators in Various Oscillatory Domains: Homogenization of Optimal Control Problems
Speaker: Aiyappan S (IISc Mathematics)
Date: Thu, 15 Jun 2017
Time: 10:30 am
Venue: LH-3, Mathematics Department
Homogenization of boundary value problems posed on rough domains has paramount importance in real life problems. Materials with oscillating (rough) boundary are used in many industrial applications like micro strip radiator and nano technologies, biological systems, fractal-type constructions, etc. In this talk, we will be focusing on homogenization of optimal control problems. We will begin with homogenization of a boundary control problem on an oscillating pillar type domain. Then, we will consider a time-dependent control problem posed on a little more general domain called branched structure domain. Asymptotic analysis of this interior control problem will be explained. Next, we will present a generalized unfolding operator that we have developed for a general oscillatory domain. Using this unfolding operator, we study the homogenization of a non-linear elliptic problem on this general highly oscillatory domain. Also, we analyse an optimal control problem on a circular oscillating domain with the assistance of this operator. Finally, we consider a non-linear optimal control problem on the above mentioned general oscillatory domain and study the asymptotic behaviour.
Title: Boundary values, resonances and scattering poles on symmetric spaces of rank one
Speaker: Aprameyan Parthasarathy (University of Paderborn, Germany)
Date: Wed, 14 Jun 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: Random structures: Phase transitions, scaling limits, and universality
Speaker: Sanchayan Sen (McGill University, Canada)
Date: Thu, 08 Jun 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
The aim of this talk is to give an overview of some recent results in two interconnected areas:
a) Random discrete structures: One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on $n$ vertices and degree exponent $\tau>3$, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}$. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.
More generally, recent research has provided strong evidence to believe that several objects, including (i) components under critical percolation, (ii) the vacant set left by a random walk, and (iii) the minimal spanning tree, constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the Gromov-Hausdorff sense, and these limiting objects are universal under some general assumptions. We will discuss recent developments in a larger program aimed at a complete resolution of these conjectures.
b) Stochastic geometry: In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90’s, the proof of which relies on a variation of Stein’s method and a quantification of a classical argument in percolation theory.
Based on joint work with Louigi Addario-Berry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.
Title: A noncommutative Matlis-Greenlees-May equivalence
Speaker: Rishi Vyas (Ben Gurion University, Israel)
Date: Tue, 06 Jun 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
The notion of a weakly proregular sequence in a commutative ring was first formally introduced by Alonso-Jeremias-Lipman (though the property that it formalizes was already known to Grothendieck), and further studied by Schenzel and Porta-Shaul-Yekutieli: a precise definition of this notion will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. Every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A.
In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.
Title: Dualizing Complexes in the Noncommutative Arithmetic Context
Speaker: Rishi Vyas (Ben Gurion University, Israel)
Date: Tue, 06 Jun 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
Dualizing complexes were first introduced in commutative algebra and algebraic geometry by Grothendieck and play a fundamental role in Serre-Grothendieck duality theory for schemes. The notion of a dualizing complex was extended to noncommutative ring theory by Yekutieli. There are existence theorems for dualizing complexes in the noncommutative context, due to Van den Bergh, Wu, Zhang, and Yekutieli amongst others.
Most considerations of dualizing complexes over noncommutative rings are for algebras defined over fields. There are technical difficulties involved in extending this theory to algebras defined over more general commutative base rings. In this talk, we will describe these challenges and how to get around them. Time permitting, we will end by presenting an existence theorem for dualizing complexes in this more general setting.
The material described in this talk is work in progress, carried out jointly with Amnon Yekutieli.
Title: Explaining the Earth’s surface observations: a Computational Geodynamics Approach
Speaker: Attreyee Ghosh (IISc Center for Earth Sciences)
Date: Thu, 18 May 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
The field of geodynamics deals with the large scale forces shaping the Earth. Computational geodynamics, which uses numerical modeling, is one of the most important tools to understand the mechanisms within the deep Earth. With the help of these numerical models we can address some of the outstanding questions regarding the processes operating within the Earth’s interior and their control on shaping the surface of the planet. Much of Earth’s surface observations such as gravity anomalies, plate motions, dynamic topography, lithosphere stress field, owe their origin to convection within the Earth’s mantle. While we understand the basic nature of such flow in the mantle, a lot remains unexplained, including the complex rheology of the deep mantle and how this density driven convective flow couples with the shallow surface. In this talk I will discuss how my group is using numerical modeling to understand the influence of the deep mantle on surface observations.
Title: Davenport constant and an external problem related to it
Speaker: Eshita Mazumdar (IIT, Bombay)
Date: Tue, 09 May 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
For a finite abelian group $G$ with $|G| = n$, the Davenport Constant $DA(G)$ is defined to be the least integer $k$ such that any sequence $S$ with length $k$ of elements in $G$ has a non-empty $A$ weighted zero-sum subsequence. For certain sets $A$, we already know the precise value of constant corresponding to the cyclic group $\mathbb{Z} / n \mathbb{Z}$. But for different group $G$ and $A$, the precise value of it is still an open question. We try to find out bounds for these combinatorial invariant for random set $A$. We got few results in this connection. In this talk I would like to present those results and discuss about an extremal problem related to this combinatorial invariant.
Title: Weighted norm inequalities for rough singular integral operators
Speaker: Luz Roncal (BCAM - Basque Center for Applied Mathematics, Spain)
Date: Mon, 24 Apr 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
The study of weighted inequalities in Classical Harmonic Analysis started in 70’s, when B. Muckenhoupt characterised in 1972 the weights $w$ for which the Hardy–Littlewood maximal function is bounded in $L^p(w)$. At that time the question about how the operator depended on the constant associated with $w$, which we denote by $[w]_{A_p}$, was not considered (i.e., quantitative estimates) were not investigated.
From the beginning of 2000’s, a great activity has been carried out in order to obtain the sharp dependence for singular integral operators, reaching the solution of the so-called $A_2$ conjecture by T. P. H\“ytonen.
In this talk we consider operators with homogeneous singular kernels, on which we assume smoothness conditions that are weaker than the standard ones (this is why they are called rough). The first qualitative weighted estimates are due to J. Duoandikoetxea and J. L. Rubio de Francia. For the norm of these operators in the space $L^2(w)$ we obtain a quantitative estimate which is quadratic in the constant $[w]_{A_2}$.
The results are based on a classical decomposition of the rough operators as a sum of other operators with a smoother kernel, for which a quantitative reelaboration of a dyadic decomposition proposed by M. T. Lacey is applied.
We will overview as well the most recent advances, mainly associated with quantitative estimates for these rough singular integrals. In particular, Coifman-Fefferman type inequalities (which are new even in their qualitative version), weighted $A_p$-$A_{\infty}$ inequalities and a quantitative version of weak $(1,1)$ estimates will be shown.
Title: Conformal blocks, strange duality and the moduli space of curves
Speaker: Swarnava Mukhopadhyay (University of Maryland, USA)
Date: Fri, 21 Apr 2017
Time: 10 am
Venue: LH-1, Mathematics Department (via Skype)
Conformal blocks are refined invariants of tensor product of representations of a Lie algebra that give a special class of vector bundles on the moduli space of curves. In this talk, I will introduce conformal blocks and explore connections to questions in algebraic geometry and representation theory. I will also focus on some ``strange” dualities in representation theory and how they give equalities of divisor classes on the moduli space of curves.
Title: Eigenfunctions Seminar: Understanding statistics of a network via sampling
Speaker: Siva Athreya (ISI, Bangalore)
Date: Fri, 21 Apr 2017
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
We shall consider sampling procedures to construct subgraphs to infer properties of a network. Specifically, we shall consider sampling procedures in the context of dense and sparse graph limits. We will explore open questions and interesting explorations at the undergraduate level. The talk will be accessible to a general audience.
Title: Eigenfunctions Seminar: Very weakly negatively-curved domains having some potential theory
Speaker: Gautam Bharali (IISc Mathematics)
Date: Fri, 21 Apr 2017
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Given a metric space (X, d), there are several notions of it being negatively curved. In this talk, we single out certain consequences of negative curvature – which are themselves weaker than (X, d) being negatively curved – that turn out to be very useful in proving results about holomorphic maps. We shall illustrate what this means by giving a proof of the Wolff-Denjoy theorem. (This theorem says that given a holomorphic self-map f of the open unit disc, either f has a fixed point in the open unit disc or there exists a point p on the unit circle such that ALL orbits under the successive iterates of f approach p.) For most of this talk, we shall focus on metric spaces or on geometry in one complex variable. Towards the end, we shall briefly point out what can be proved in domains in higher dimensions that have the aforementioned negative-curvature-type properties. This part of the talk is joint work with Andrew Zimmer.
Title: Riemann-Roch, Alexander Duality and Free Resolutions
Speaker: Madhusudan Manjunath (Mathematisches Forschungsinstitut Oberwolfach, Germany)
Date: Thu, 20 Apr 2017
Time: 4 pm
Venue: LH-1, Mathematics Department (via Skype)
The Riemann-Roch theorem is fundamental to algebraic geometry. In 2006, Baker and Norine discovered an analogue of the Riemann-Roch theorem for graphs. In fact, this theorem is not a mere analogue but has concrete relations with its algebro-geometric counterpart. Since its conception this topic has been explored in different directions, two significant directions are i. Connections to topics in discrete geometry and commutative algebra ii. As a tool to studying linear series on algebraic curves. We will provide a glimpse of these developments. Topics in commutative algebra such as Alexander duality and minimal free resolutions will make an appearance. This talk is based on my dissertation and joint work with i. Bernd Sturmfels, ii. Frank-Olaf Schreyer and John Wilmes and iii. an ongoing work with Alex Fink.
Title: A quantitative form of Schoenberg's theorem in fixed dimension
Speaker: Alexander Belton (Lancaster University, UK)
Date: Wed, 05 Apr 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
The Hadamard product of two matrices is formed by multiplying corresponding entries, and the Schur product theorem states that this operation preserves positive semidefiniteness.
It follows immediately that every analytic function with non-negative Maclaurin coefficients, when applied entrywise, preserves positive semidefiniteness for matrices of any order. The converse is due to Schoenberg: a function which preserves positive semidefiniteness for matrices of arbitrary order is necessarily analytic and has non-negative Maclaurin coefficients.
For matrices of fixed order, the situation is more interesting. This talk will present recent work which shows the existence of polynomials with negative leading term which preserve positive semidefiniteness, and characterises precisely how large this term may be. (Joint work with D. Guillot, A. Khare and M. Putinar.)
Title: On commutators of singular integral operators with BMO functions
Speaker: Carlos Perez (University of Basque Country)
Date: Tue, 28 Mar 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
Commutators of singular integral operators with BMO functions were introduced in the seventies by Coifman-Rochberg and Weiss. These operators are very interesting for many reasons, one of them being the fact that they are more singular than Calderon-Zygmund operators. In this lecture we plan to give several reasons showing the “bad” behavior of these operators.
Title: Eigenfunctions Seminar: The zoo of Lyapunov vectors and nonlinear filters
Speaker: Amit Apte (ICTS, Bangalore)
Date: Fri, 17 Mar 2017
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
I will introduce the concepts of backward, forward, and covariant Lyapunov vectors (that are basically “eigenfunctions” of certain operators) for a dynamical system and discuss their relevance in studying various stability results, both for the system itself and for nonlinear filtering problem, which will also be introduced along the way. The “relevance” to nonlinear filters is in the form of open questions whose answers (as a set of conjectures) may be gleaned from numerical results and linear filtering theory.
Title: Eigenfunctions Seminar: Topology of random points
Speaker: Yogeshwaran D. (ISI, Bangalore)
Date: Fri, 17 Mar 2017
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Given n random points on a manifold embedded in a Euclidean space, we wish to understand what topological features of the manifold can be inferred from the geometry of these points. One procedure is to consider union of Euclidean balls of radius r around the points and study the topology of this random set (i.e., the union of balls). A more robust method (known as persistent homology) of inferring the topology of the underlying manifold is to study the evolution of topology of the random set as r varies. What topological information (for example, Betti numbers and some generalizations) about the underlying manifold can we glean for different choices of r ? This question along with some partial answers in the recent years will be the focus of the talk. I shall try to keep the talk mostly self-contained assuming only knowledge of basic probability and point-set topology.
Title: Detecting free splittings
Speaker: Suraj Kumar (Paris Diderot University, France)
Date: Tue, 14 Mar 2017
Time: 9:30 am
Venue: LH-1, Mathematics Department
The goal of this talk is to present an algorithm which takes a compact square complex belonging to a special class as input and decides whether its fundamental group splits as a free product. The special class is built by attaching tubes to finite graphs in such a way that they satisfy a nonpositive curvature condition. This construction gives rise to a rich class of complexes, including, but not limited to, closed surfaces of positive genus. The algorithm can be used to deduce the celebrated Stallings theorem for this special class, as also the well known Grushko decomposition theorem.
Title: When is $R[\theta]$ integrally closed?
Speaker: Sudesh K Khanduja (IISER Mohali)
Date: Fri, 10 Mar 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: A result for boolean interval of finite groups
Speaker: Mamta Balodi (IMSc)
Date: Thu, 09 Mar 2017
Time: 4 pm
Venue: LH-3, Mathematics Department
A group is cyclic iff its subgroup lattice is distributive. Ore’s generalized one direction of this result. We will discuss a dual version of Ore’s result, for any boolean interval of finite groups under the assumption that the dual Euler totient of the interval is nonzero. We conjecture that the dual Euler totient is always nonzero for boolean intervals. We will discuss some techniques which may be helpful in proving it. We first see that dual Euler totient of an interval of finite groups is the Mobius invariant (upto a sign) of its coset poset P. Next in the boolean group complemented case, we prove that P is Cohen-Macaulay, using the existence of an explicit EL-labeling. We then see that nontrivial betti number of the order complex is nonzero, and so is the dual Euler totient.
Title: Diamond lemma for group graded quasi-algebras
Speaker: Mamta Balodi (IMSc)
Date: Thu, 02 Mar 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
Quasi-algebras were introduced as algebras in a monoidal category. Since the associativity constraints in these categories are allowed to be nontrivial, the class of quasi-algebras contains various important examples of non-associative algebras like the octonions and other Cayley algebras. The diamond lemma is a reduction method used in algebra. The original diamond lemma was stated in graph theory by Newman which was later generalized to associative algebras by Bergman. In this talk, we will see the analog of this lemma for the group graded quasi-algebras with some interesting examples like octonion algebra and generalized octonions
Title: Theoretical models for compressible vortex streets
Speaker: Vikas Krishnamurthy (Imperial College, London)
Date: Thu, 23 Feb 2017
Time: 4 pm
Venue: LH-3, Mathematics Department
Vortex streets are a common feature of fluid flows at high Reynolds numbers and their study is now well developed for incompressible fluids. Much less is known, however, about compressible vortex streets. A fundamental reason appears to be the inapplicability of the point vortex model to compressible flows. In this talk, we discuss point vortices in the context of weakly compressible flows and elaborate on the problems involved. We then adopt the hollow vortex model where each vortex is modelled as a finite-area constant pressure region with non-zero circulation. For weakly compressible flows steady hollow vortex solutions are well known to be candidates for the leading order solution in a perturbative Rayleigh-Jansen expansion of a compressible flow. Here we give details of that expansion based on the vortex street solutions of Crowdy & Green (2012). Physical properties of the compressible vortex streets are described. Our approach uses the Imai-Lamla method coupled with analytic function theory and conformal mapping. (Joint work with Darren Crowdy)
Title: Eigenfunctions Seminar: Punctual gluing of t-structures in algebraic geometry
Speaker: Vaibhav Vaish (ISI, Bangalore)
Date: Fri, 17 Feb 2017
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
The formalism of an “abelian category” is meant to axiomatize the operations of linear algebra. From there, the notion of “derived category” as the category of complexes “upto quasi-isomorphisms” is natural, motivated in part by topology. The formalism of t-structures allows one to construct new abelian categories which are quite useful in practice (giving rise to new cohomology theories like intersection cohomology, for example). In this talk we want to discuss a notion of punctual (=”point-wise”) gluing of t-structures which is possible in the context of algebraic geometry. The essence of the construction is classical and well known, but the new language leads to useful constructions in the motivic world.
Title: Flow Invariance Preserving Feedback Controllers for Sabra Shell Model of Turbulence
Speaker: Dharmatti Sheetal (IISER TVM)
Date: Fri, 17 Feb 2017
Time: 11 am
Venue: LH-1, Mathematics Department
In this talk, I would continue dealing with Sabra shell model of Turbulence and study one of the important questions for fluid flow problems namely, finding controls which are capable of preserving the invariant quantities of the flow. Controls are designed in the feedback form such that resultant controlled flow will preserve certain physical properties of the state such as enstrophy, helicity. We use the theory of nonlinear semigroups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space under consideration.
Title: Eigenfunctions Seminar: Graph Partitioning using its Spectra
Speaker: Anand Louis (IISc CSA)
Date: Fri, 17 Feb 2017
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Graph-partitioning problems are a central topic of research in the study of algorithms and complexity theory. They are of interest to theoreticians with connections to error correcting codes, sampling algorithms, metric embeddings, among others, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications. One of the central problems studied in this field is the sparsest cut problem, where we want to compute the cut which has the least ratio of number of edges cut to size of smaller side of the cut. This ratio is known as the expansion of the cut. In this talk, I will talk about higher order variants of expansion (i.e. notions of expansion corresponding to partitioning the graph into more than two pieces, etc.), and how they relate to the graph’s eigenvalues. The proofs will also show how to use the graph’s eigenvectors to compute partitions satisfying these bounds. Based on joint works with Prasad Raghavendra, Prasad Tetali and Santosh Vempala.
Title: Control and analysis of Shell model of Turbulence
Speaker: Dharmatti Sheetal (IISER TVM)
Date: Thu, 16 Feb 2017
Time: 3:30 pm
Venue: LH-1, Mathematics Department
In this lecture I am going to present control problems associated with shell models of Turbulence. Shell models of turbulence are simplified caricatures of equations of fluid mechanics in wave-vector representation. They exhibit anomalous scaling and local non-linear interactions in wave number space. We would like to study control problem related to one such widely accepted shell model of turbulence known as sabra shell model. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the ow near a priori known state. We derive the optimal controls in terms of the solution of adjoint equation for corresponding linearised problems.
Title: Eigenfunctions Seminar: Playing with the nonlinearity of the Navier-Stokes Equation
Speaker: Samriddhi Sankar Ray (ICTS, Bangalore)
Date: Fri, 03 Feb 2017
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
Understanding the origins of intermittency in turbulence remains one of the most fundamental challenges in applied mathematics. In recent years there has been a fresh attempt to understand this problem through the development of the method of Fourier decimation. In this talk, we will review these recent results and analyse the role of precise numerical simulations in understanding the mathematics of the Navier-Stokes and Euler equations.
Title: Eigenfunctions Seminar: What is finiteness?
Speaker: Abhishek Banerjee (IISc Mathematics)
Date: Fri, 03 Feb 2017
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
It is generally well known that there is an innate notion of things like isomorphisms or epimorphisms. This allows us to talk about isomorphisms or epimorphisms of various objects: groups, rings, algebras, etc. In other words, “isomorphism” is really a categorical notion. However, it is not so well known that finiteness itself is alsocategorical. In this talk, we will discuss how finiteness applies to various categories. This will allow usto see finite sets, finite dimensional vector spaces, finitely generated algebras and compact sets as manifestations of the same basic idea.
Title: Hole probabilities for determinantal point processes in the complex plane
Speaker: Kartick Adhikari (IISc Mathematics)
Date: Tue, 31 Jan 2017
Time: 11 am
Venue: LH-1, Mathematics Department
Consider the infinite Ginibre ensemble (the distributional limit of the eigenvalues of nxn random matrices with i.i.d. standard complex Gaussian entries) in the complex plane. For a bounded set $U$, let $H_r(U)$ denote the probability (hole probability) that no points of infinite Ginibre ensemble fall in the region $rU$. We study the asymptotic behavior of $H_r(U)$ as $r \to \infty$. Under certain conditions on $U$ we show that $\log H_r(U)=C_U \cdot r^4 (1+o(1))$ as $r \to \infty$. Using potential theory, we give an explicit formula for $C_U$ in terms of the minimum logarithmic energy of the set with a quadratic external field. We calculate $C_U$ explicitly for some special sets such as the annulus, cardioid, ellipse, equilateral triangle and half disk.
Moreover, we generalize the above hole probability results for a class of determinantal point processes in the complex plane.
Title: The Space of Metric Measure Spaces
Speaker: Sayantan Maitra (IISc Mathematics)
Date: Fri, 20 Jan 2017
Time: 11 am
Venue: LH-1, Mathematics Department
Title: Linking numbers for algebraic cycles
Speaker: Souvik Goswami (ICMAT, Madrid, Spain)
Date: Wed, 18 Jan 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
In algebraic geometry the concept of height pairing (a particular example of linking numbers) of algebraic cycles lies at the confluence of arithmetic, Hodge theory and topology. In a series of two talks, I will explain the notion of Beilinson’s height pairing for cycles homologous to zero. This will bring into picture the notion of Arakelov/arithmetic intersection theory. I will give sufficient background of this theory and provide examples. Finally, I will talk about my recent work with Dr. Jose Ignacio Burgos, about a generalization of Beilinson’s height pairing for higher algebraic cycles.
Title: Approximate controllability of an impulsive sub-diffusion equation
Speaker: Lakshman Mahato (IIIT Dharwad)
Date: Fri, 13 Jan 2017
In this talk, I shall consider an abstract Cauchy problem for a class of impulsive sub-diffusion equation. Existence and regularity of solution of the problem shall be established via eigenfunction expansion. Further, I shall establish the approximate controllability of the problem by applying unique continuation property via internal control acts on a sub-domain.
Title: Limiting measure for TASEP with a slow bond
Speaker: Sourav Sarkar (UC Berkeley, USA)
Date: Wed, 11 Jan 2017
Time: 3 pm
Venue: LH-1, Mathematics Department
It was shown by Basu, Sidoravicius and Sly that a TASEP starting with the step initial condition, i.e., with one particle each at every nonpositive site of $\mathbb{Z}$ and no particle at positive sites, with a slow bond at the origin where a particle jumping from the origin jumps at a smaller rate $r < 1$, has an asympototic current which is strictly less than 1/4. Here we study the limiting measure of the TASEP with a slow bond. The distribution of regular TASEP started with the step initial condition converges to the invariant product Bernoulli measure with density 1/2. The slowdown due to the slow bond implies that there is a long range effect near the origin where the region to the right of origin is sparser and there is a traffic jam to the left of the slow bond with particle density higher than a half. However, the distribution becomes close to a product Bernoulli measure as one moves far away from the origin, albeit with a different density ? < 1/2 to the right of the origin and ?’ > 1/2 to the left of the origin. This answers a question due to Liggett. The proof uses the correspondence between TASEP and directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times, and the geometric properties of the maximal paths there.
Title: Upscaling of a System of Semilinear Diffusion Reaction Equations in a Heterogeneous Medium: MultiScale Modeling and Periodic Homogenization
Speaker: Hari Shankar Mahato (University of Georgia, USA)
Date: Tue, 10 Jan 2017
Time: 11 am
Venue: LH-1, Mathematics Department
A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multiscale medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed at the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods.
In this talk, diffusion and reaction of several mobile chemical species are considered in the pore space of a heterogeneous porous medium. The reactions amongst the species are modelled via mass action kinetics and the modelling leads to a system of multispecies diffusion; reaction equations (coupled semi-linear partial differential equations) at the micro scale where the highly nonlinear reaction rate terms are present at the right hand sides of the system of PDEs, cf. [2]. The existence of a unique positive global weak solution is shown with the help of a Lyapunov functional, Schaefer’s fixed point theorem and maximal Lp-regularity, cf. [2, 3]. Finally, with the help of periodic homogenization and two-scale convergence we upscale the model from the micro scale to the macro scale, e.g. [1, 3]. Some numerical simulations will also be shown in this talk, however for the purpose of illustration, we restrict ourselves to some relatively simple 2- dimensional situations.
As an extension to the previous model, we consider the mixture of two fluids. For such models, a system of Stokes-Cahn-Hilliard equations will be considered at the micro scale in a perforated porous medium. We first explain the periodic setting of the model and the existence results. At the end homogenization of the model will be shown using some extension theorems on Sobolev spaces, two-scale convergence and periodic unfolding.
Title: Finite point configurations
Speaker: Eyvindur A. Palsson (Virginia Tech, USA)
Date: Mon, 09 Jan 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
As big data sets have become more common, there has been significant interest in finding and understanding patterns in them. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will present recent Falconer type theorems, established by myself and my collaborators, for a wide class of finite point configurations in any dimension. The techniques we used come from analysis and geometric measure theory, and the key step was to obtain bounds on multilinear analogues of generalized Radon transforms.
Title: Siegel-Veech transforms are in L2
Speaker: Jayadev Athreya (University of Washington, USA)
Date: Fri, 06 Jan 2017
Time: 11 am
Venue: LH-1, Mathematics Department
Let H denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on R2 is in L2(H,μ), where μ is the Masur-Veech measure on H, and give applications to bounding error terms for counting problems for saddle connections. We will review classical results in the Geometry of Numbers which anticipate this result. This is joint work with Yitwah Cheung and Howard Masur.
Title: On some fourth order equations associated to Physics
Speaker: Sanjiban Santra (CIMAT, Mexico)
Date: Thu, 05 Jan 2017
Time: 4 pm
Venue: LH-1, Mathematics Department
We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated fourth order Liouville theorem to the problem and possible link with the De Giorgi’s conjecture. This is an attempt to prove the McKenna-Walter conjecture which is open for the last two decades.
Title: A glimpse at bi-free probability.
Speaker: Dan-Virgil Voiculescu (UC Berkeley, USA)
Date: Wed, 28 Dec 2016
Time: 3:30 pm
Venue: LH-1
I will provide a glimpse at the recent extension of free probability to systems with left and right variables based on a notion of bi-freeness. This will include the simplest cases of nonlinear convolution operations on non-commutative distributions snd the analogue of extreme values in this setting.
Title: Homological Algebra of Ideals Related to Finite Simple Graphs.
Speaker: Arindam Banerjee (Purdue University, USA)
Date: Tue, 27 Dec 2016
Time: 3 pm
Venue: LH-1
Given any finite simple graph G one can naturally associate two ideals, namely the edge ideal I(G) and the binomial edge ideal J_G in suitable polynomial rings. In this talk we shall discuss the interplay between combinatorics of the graph and depth and regularity of I(G), J_G and their powers. Some recent progress and some open problems will be discussed.
Title: On 3+1 Lorentzian Einstein manifolds with one rotational isometry
Speaker: Nishanth Gudapati (Yale University, USA)
Date: Wed, 21 Dec 2016
Time: 11 am
Venue: LH-1, Mathematics Department
Differentiable manifolds whose Ricci curvature is proportional to the metric are called Einstein manifolds. Such manifolds have been central objects of study in differential geometry and Einstein’s theory for general relativity, with some strong recent results. In this talk, we shall focus on positively curved 3+1 Lorentzian Einstein manifolds with one spacelike rotational isometry. After performing the dimensional reduction to a 2+1 dimensional Einstein’s equations coupled to ‘shifted’ wave maps, we shall prove two explicit positive mass theorems.
Title: Coarse Geometry for Noncommutative Spaces
Speaker: Tathagata Banerjee (IISc Mathematics)
Date: Mon, 19 Dec 2016
Time: 3 pm
Venue: LH-1, Mathematics Department
Given a proper coarse structure on a locally compact Hausdorff space $X$, one can construct the Higson compactification for the coarse structure. In the opposite direction given a compactification of $X$, one can construct a coarse structure. We use unitizations of a non-unital C$^*$-algebra $A$ to define a noncommutative coarse structure on $A$. We also set up a framework to abstract coarse maps to this noncommutative setting. The original motivation for this work comes from Physics where quantum phenomenon when probed at large scales give classical results. We show equivalence of the canonical coarse structure on the classical plane $\mathbb{R}^{2n}$ with a certain noncommutative coarse structure on the Moyal plane which models the hypothetical phase space of Quantum physics. If time permits we shall also discuss other examples of noncommutative coarse equivalences. This is a joint work with Prof. Ralf Meyer.
Title: Adaptive wavelet-based methods for solution of PDEs and signal analysis.
Speaker: Ratikant Behra (IISER-Kolkata)
Date: Fri, 09 Dec 2016
Time: 3 pm
Venue: LH-1
Title: Entrywise functions preserving positivity: Connections between analysis, algebra, and combinatorics
Speaker: Apoorva Khare (Stanford University, USA)
Date: Thu, 08 Dec 2016
Time: 11 am
Venue: LH-1 (via Skype)
What functions preserve positive semidefiniteness (psd) when applied entrywise to psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, and has also recently received renewed attention due to its applications in high-dimensional statistics. However, effective characterizations of entrywise functions preserving positivity in a fixed dimension remain elusive to date.
I will present recent progress on this question, obtained by: (a) imposing rank and sparsity constraints, (b) restricting to structured matrices, and (c) restricting the class of functions to special families such as polynomials or power functions. These constraints arise in theory as well as applications, and provide natural ways to relax the elusive original problem. Moreover, novel connections to symmetric function theory, matrix analysis, and combinatorics emerge out of these refinements.
Title: Some Recent Progress in Quasilinear Hyperbolic Systems: New Local Solvability Methods and Stochastic Analysis.
Speaker: Manil Mohan (Air Force Institute of Tech, Ohio)
Date: Tue, 06 Dec 2016
Time: 11 am
Venue: LH-1
Quasilinear symmetric and symmetrizable hyperbolic system has a wide range of applications in engineering and physics including unsteady Euler and potential equations of gas dynamics, inviscid magnetohydrodynamic (MHD) equations, shallow water equations, non-Newtonian fluid dynamics, and Einstein field equations of general relativity. In the past, the Cauchy problem of smooth solutions for these systems has been studied by several mathematicians using semigroup approach and fixed point arguments. In a recent work of M. T. Mohan and S. S. Sritharan, the local solvability of symmetric hyperbolic system is established using two different methods, viz. local monotonicity method and a frequency truncation method. The local existence and uniqueness of solutions of symmetrizable hyperbolic system is also proved by them using a frequency truncation method. Later they established the local solvability of the stochastic quasilinear symmetric hyperbolic system perturbed by Levy noise using a stochastic generalization of the localized Minty-Browder technique. Under a smallness assumption on the initial data, a global solvability for the multiplicative noise case is also proved. The essence of this talk is to give an overview of these new local solvability methods and their applications.
Title: The Narasimhan Seshadri Theorem revisited
Speaker: M S Raghunathan (National Center for Mathematics, IIT Mumbai)
Date: Fri, 02 Dec 2016
Time: 3 pm
Venue: LH-1
We give a relatively simple proof of the famous theorem of Narasimhan and Seshadri on vector bundles on a compact Riemann surface. The theorem relates the algebraic geometric notion of stability of vector bundles on a compact Riemann surface with a transcendental construct - unitary representations of a suitable Fuchsian group associated to the Riemann surface.
Title: Torelli theorem for the parabolic Deligne-Hitchin moduli space
Speaker: Tomas Gomez (ICMAT, Spain)
Date: Mon, 14 Nov 2016
Time: 3pm
Venue: LH-1
The Deligne-Hitchin moduli space is a partial compactification of the moduli space of $\lambda$-connections. It includes as closed subvarieties the moduli spaces of Hitchin bundles ($\lambda=0$) and of holomorphic connections ($\lambda=1$), exhibiting the later as a deformation of the former. We show a Torelli theorem for a parabolic version of this moduli space (joint work with David Alfaya). I will try to make the talk accessible to a wide mathematical audience.
Title: The Perron-Frobenius theorem and its application to a class of dynamical systems
Speaker: Sandeep Krishna, NCBS.
Date: Fri, 28 Oct 2016
Venue: Simons Centre, Ground Floor.
The Perron-Frobenius theorem is a powerful and useful result about the eigenvalues and eigenvectors of a non-negative matrix. I will not be proving the theorem but will instead focus on its applications. In particular, I will discuss how it can be used to understand the behaviour of a certain class of dynamical systems, namely certain systems of first-order ordinary differential equations where the couplings between variables are specified by a graph. Some familiarity with graph theory will be useful, but I will try to recapitulate all the basic concepts needed for this talk.
Title: Eigenfunctions Seminar: Algorithmic Applications of An Approximate Version of Caratheodory's Theorem
Speaker: Siddharth Barman (IISc CSA)
Date: Fri, 21 Oct 2016
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
In this talk I will present algorithmic applications of an approximate version of Caratheodory’s theorem. The theorem states that given a set of $d$-dimensional vectors $X$, for every vector in the convex hull of $X$ there exists an epsilon-close (under the $p$-norm, for $2 \leq p < \infty$) vector that can be expressed as a convex combination of at most $b$ vectors of $X$, where the bound $b$ depends on epsilon and the norm $p$, and is independent of the ambient dimension $d$. This theorem can be obtained by instantiating Maurey’s lemma (c.f. Pisier 1980/81 and Carl 1985).
I will describe how this approximate version of Caratheodory’s theorem leads novel additive approximation algorithms for finding (i) Nash equilibria in two-player games (ii) dense subgraphs.
Title: Eigenfunctions Seminar: Fibrations, holomorphic motions and negative curvature
Speaker: Harish Seshadri (IISc Mathematics)
Date: Fri, 21 Oct 2016
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
A classical theorem in Riemannian geometry asserts that products of compact manifolds cannot admit Riemannian metrics with negative sectional curvature. A fibration (or a fibre bundle) is a natural generalization of a product and hence one can ask if a fibration can admit such a metric. This question is still open and I will discuss it in the context of Kahler manifolds. This leads to the study of graphs of holomorphic motions, which originally arose in complex dynamics. I will sketch a proof that the graph cannot be biholomorphic to a ball or more generally, a strongly pseudoconvex domain.
Title: Simultaneous Similarity Classes of tuples of Commuting matrices.
Speaker: Uday Bhaskar (IMSc, Chennai)
Date: Fri, 07 Oct 2016
Time: 2 pm
Venue: LH-1
The enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field is the same as the classifi cation of commuting tuples of matrices over a finite field up to simultaneous similarity. Let C_{n,k}(q) denote the number of isomorphism classes of n-dimensional Fq[x1,…,xk]-modules. The generating function in k of the C_{n,k}(q) is a rational function. The computation of this was done explicitly for n <= 4. I shall give a summary of my recently published work on this study of the C_{n,k}(q)s for n <= 2.
Title: Domains of holomorphy for irreducible admissible Banach representations
Speaker: Aprameyan Parthasarathy (Universitat Paderborn, Germany)
Date: Tue, 27 Sep 2016
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we will report on progress towards a conjecture of B. Krötz about the holomorphic extensions of non-zero K-finite vectors of irreducible admissible Banach representations of simple real Lie groups and the relation to a distinguished domain – the so-called crown domain. We will explain some of the main ideas – the Casselman–Wallach smooth globalisation, vanishing of matrix coefficients at infinity etc. Indeed we prove the conjecture with some additional growth conditions on the Banach globalisations. This is joint work with Gang Liu (Univ. Metz).
Title: Epimorphically Preserved Semigroup Identities
Speaker: Wajih Ashraf
Date: Fri, 23 Sep 2016
Time: 4 pm
Venue: LH-1
Title: Eigenfunctions Seminar: Computing the ''best'' map between polygons
Speaker: Vamsi Pritham Pingali (IISc Mathematics)
Date: Fri, 23 Sep 2016
Time: 2:15 – 4 pm (with a 15 minute break at 3)
Venue: LH-1, Mathematics Department
This talk is about computing (approximately) the “best” map between two polygons taking vertices to vertices. It arises out of a real-life problem, namely, surface registration. Our notion of “best” is extremal quasiconformality (least angle-distortion). I will try to keep the talk as self-contained as possible. It is based on a joint-work with M. Goswami, G. Telang, and X. Gu.
Title: Margulis spacetimes and Anosov representations
Speaker: Sourav Ghosh (Universitat Heidelberg, Germany)
Date: Tue, 20 Sep 2016
Time: 4 pm
Venue: LH-1
In this talk I will describe the interrelationship between Margulis spacetimes and Anosov representations. Moreover, I will define the pressure metric on the moduli space of Margulis spacetimes and sketch some of it’s properties.
Title: Tropical Algebraic Geometry: an Introduction.
Speaker: Madhusudan Manjunath (Queen Mary Univ of London, UK)
Date: Tue, 06 Sep 2016
Time: 4 pm
Venue: LH-1
We start with a gentle introduction to tropical algebraic geometry. We then focus on the tropical lifting problem and discuss recent progress. Tropical analogues of graph curves play an important role in this study. This talk will be accessible to the general mathematical audience.
Title: Tropical Algebraic Geometry: an Introduction.
Speaker: Madhusudan Manjunath (Queen Mary Univ of London, UK)
Date: Mon, 05 Sep 2016
Time: 4 pm
Venue: LH-1
We start with a gentle introduction to tropical algebraic geometry. We then focus on the tropical lifting problem and discuss recent progress. Tropical analogues of graph curves play an important role in this study. This talk will be accessible to the general mathematical audience.
Title: Multiobjective optimisation and tropical geometry
Speaker: Benjamin A. Burton (University of Queensland, Brisbane, Australia)
Date: Fri, 02 Sep 2016
Venue: LH-1, Mathematics Department
Multiobjective optimisation involves optimising several quantities, such as time and money, simultaneously. The result is a polyhedral frontier of best possible solutions, which cannot improve one quantity without a trade-off against another. For linear programming, this frontier can be generated using Benson’s outer approximation algorithm, which uses a sequence of scalarisations (single-objective optimisations), combined with classical algorithms from polytope theory.
Title: A hybrid HPC framework with analysis for a class of stochastic models
Speaker: M. Ganesh, Professor, Colorado School of Mines
Date: Fri, 19 Aug 2016
Venue: LH-1
We consider a class of wave propagation models with aleatoric and epistemic uncertainties. Using mathematical analysis-based, shape-independent, a priori parameter estimates, we develop offline/online strategies to compute statistical moments of a key quantity of interest in such models. We present an efficient reduced order model (ROM) and high performance computing (HPC) framework with analysis for quantifying aleatoric and epistemic uncertainties in the propagation of waves through a stochastic media comprising a large number of three dimensional particles. Simulation even for a single deterministic three dimensional configuration is inherently difficult because of the large number of particles. The aleatoric uncertainty in the model leads to a larger dimensional system involving three spatial variables and additional stochastic variables. Accounting for epistemic uncertainty in key parameters of the input probability distributions leads to prohibitive computational complexity. Our hybrid ROM and HPC framework can be used in conjunction with any computational method to simulate a single particle deterministic wave propagation model.
Title: On residues of certain intertwining operators
Speaker: Sandeep Varma (TIFR, Mumbai)
Date: Tue, 16 Aug 2016
Venue: LH-1, Mathematics Department
Let $G$ be a connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. Let $P = MN$ be a Levi decomposition of a maximal parabolic subgroup of $G$, and $\sigma$ an irreducible unitary supercuspidal representation of $M(F)$. One can then consider the representation Ind$_{P(F)}^{G(F)}\sigma$ (normalized parabolic induction). This induced representation is known to be either irreducible or of length two. The question of when it is irreducible turns out to be (conjecturally) related to local $L$-functions, and also to poles of a family of so called intertwining operators.
Title: Resonances for the Laplacian on Riemannian symmetric spaces of the noncompact type: the rank two case
Speaker: Angela Pasquale
Date: Thu, 04 Aug 2016
Time: 4 pm
Venue: LH-1
Let $\Delta$ be the Laplacian on a Riemannian symmetric space $X=G/K$ of the noncompact type and let $\sigma(\Delta)\subseteq \mathbb{C}$ denote its spectrum. The resolvent $(\Delta-z)^{-1}$ is a holomorphic function on $\mathbb{C} \setminus \sigma(\Delta)$, with values in the space of bounded operators on $L^2(X)$. If we view it as a function with values in Hom$(C_c^\infty(X), C_c^\infty(X)^*)$, then it often admits a meromorphic continuation beyond $\mathbb{C} \setminus \sigma(\Delta)$. We study this meromorphic continuation as a map defined on a Riemann surface above $\mathbb{C} \setminus \sigma(\Delta)$. The poles of the meromorphically extended resolvent are called resonances. The image of the residue operator at a resonance is a $G$-module. The main problems are the existence and the localization of the resonances as well as the study of the (spherical) representations of $G$ so obtained. In this talk, based on joint works with Joachim Hilgert and Tomasz Przebinda, we will describe a variety of different situations occurring in the rank two case.
Title: Geometry of geodesics in random spatial processes: Exactly solvable models and beyond
Speaker: Riddhipratim Basu (Stanford University, USA)
Date: Thu, 28 Jul 2016
Time: 4 pm
Venue: LH-1, Mathematics Department
Kardar, Parisi and Zhang introduced a universality class (the so-called KPZ universality class) in 1986 which is believed to explain the universal behaviour in a large class of two dimensional random growth models including first and last passage percolation. A number of breakthroughs has led to an explosion of mathematically rigorous results in this field in recent years. However, these have mostly been restricted to the class of exactly solvable models, where exact formulae are available using powerful tools of random matrices, algebraic combinatorics and representation theory; beyond this class the understanding remains rather limited. I shall talk about a geometric approach to these problems based on studying the geometry of geodesics (optimal paths), and describe some recent progress along these lines.
Title: Representation homology
Speaker: Ajay Ramadoss (Indiana University, USA)
Date: Wed, 27 Jul 2016
Time: 11:15 am
Venue: LH-1, Mathematics Department
The $n$-dimensional matrix representations of a group or an associative algebra $A$ form a space (algebraic variety) Rep$(A,n)$ called the $n$-th representation variety of $A$. This is a classical geometric invariant that plays a role in many areas of mathematics. The construction of Rep$(A,n)$ is natural (functorial) in $A$, but it is not ‘exact’ in the sense of homological algebra. In this talk, we will explain how to refine Rep$(A,n)$ by constructing a derived representation variety DRep$(A,n)$, which is an example of a derived moduli space in algebraic geometry. For an application, we will look at the classical varieties of commuting matrices, and present a series of combinatorial conjectures extending the famous Macdonald conjectures in representation theory.
Title: Representation homology
Speaker: Ajay Ramadoss (Indiana University, USA)
Date: Tue, 26 Jul 2016
Time: 11 a.m.
Venue: LH-1, Mathematics Department
The $n$-dimensional matrix representations of a group or an associative algebra $A$ form a space (algebraic variety) Rep$(A,n)$ called the $n$-th representation variety of $A$. This is a classical geometric invariant that plays a role in many areas of mathematics. The construction of Rep$(A,n)$ is natural (functorial) in $A$, but it is not ‘exact’ in the sense of homological algebra. In this talk, we will explain how to refine Rep$(A,n)$ by constructing a derived representation variety DRep$(A,n)$, which is an example of a derived moduli space in algebraic geometry. For an application, we will look at the classical varieties of commuting matrices, and present a series of combinatorial conjectures extending the famous Macdonald conjectures in representation theory.
Title: Rank-level duality of Conformal Blocks
Speaker: Swarnava Mukhopadhyay (University of Maryland at College Park, USA)
Date: Mon, 25 Jul 2016
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra.
Title: Algebraic K and L-theory of groups acting on trees
Speaker: S K Roushon (TIFR, Mumbai)
Date: Thu, 26 May 2016
Time: 3:30-4:30 pm
Venue: LH-2
The Farrell-Jones Isomorphism conjecture gives a single statement to understand many standard conjectures in Topology and Algebra= . We will discuss understanding K and L-theory of groups acting on trees from the vertex stabilizers of the action, in the context of Isomorphism conjecture.
Title: Relation between orbit spaces of unimodular elements and improved stability bound
Speaker: Pratyusha Chattopadhyay (ISI Bengaluru)
Date: Thu, 05 May 2016
Time: 3:30 pm
Venue: LH-1
This is a topic in classical algebraic K-Theory. I will recall definitions of elementary linear group, elementary symplectic group, linear transvection group, and symplectic transvection group. These group= s have natural action on the set of unimodular elements. I will briefly discuss how bijections between orbit spaces of unimodular elements under different group actions are established. Finally, I will talk about an application of these results, namely improving injective stability bound for the K1 group.
Title: Eigenfunctions Seminar: Some aspects of integral geometry
Speaker: Sreekar Vadlamani (TIFR-CAM, Bangalore)
Date: Fri, 22 Apr 2016
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
One of the earliest instance of integral geometric discussion can be traced back to Buffon through the “needle problem”. The solution, as we know now, is based on a platform of theory of measures on geometrical spaces which are invariant under some group operations. In this talk, we shall walk through this subject by following a specific line of problems/results called “kinematic fundamental formulae” by studying some specific examples.
Title: Eigenfunctions Seminar: Inverse spectral theory
Speaker: Krishna Maddaly (IMSc, Chennai)
Date: Fri, 22 Apr 2016
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Inverse spectral theory in one dimension is to recover a self-adjoint operator given its spectrum and some spectral measure. In general there are lots of self-adjoint operators (of a given form) this collection is called an iso-spectral set. We will give a brief introduction to the questions in this area and also some connections to other areas of mathematics.
Title: Unique factorization of tensor products for finite-dimensional simple Lie algebras
Speaker: R. Venkatesh ( Weizmann Institute of Science, Israel)
Date: Thu, 21 Apr 2016
Time: 2 pm
Venue: LH-1 (via Skype)
Suppose V is a finite dimensional representation of a complex finite dimensional simple Lie algebra that can be written as a tensor product of irreducible representations. A theorem of C.S. Rajan states that the non-trivial irreducible factors that occur in the tensor product factorization of V are uniquely determined, up to reordering, by the isomorphism class of V. I will present an elementary proof of Rajan’s theorem. This is a joint work with S.Viswanath.
Title: Boundary behaviour of the Kobayashi distance and horosphere boundary in complete hyperbolic manifolds.
Speaker: Herve Gaussier Fourier Institute, Grenoble
Date: Wed, 13 Apr 2016
Title: PhD Thesis colloquium: : Optimal Control Problems and Homogenization
Speaker: Mr. Bidhan Chandra Sardar Ph. D. Research Student
Date: Wed, 06 Apr 2016
Time: 11.00 a.m.
Venue: Lecture Hall I,Dept of Mathematics
We study asymptotic analysis (homogenization) of second-order partial differential equations(PDEs) posed on an oscillating domain. In general, the motivation for studying problems defined on oscillating domains, come from the need to understand flow in channels with rough boundary, heat transmission in winglets, jet engins and so on. There are various methods developed to study homogenization problems namely; multi-scale expansion, oscillating test function method, compensated compactness, two-scale convergence, block-wave method, method of unfolding etc.
In this thesis, we consider a two dimensional oscillating domain (comb shape
type) $\Omega_{\epsilon}$ consists of a fixed bottom region $\Omega^-$ and an oscillatory
(rugose) upper region $\Omega_{\epsilon}^{+}$. We introduce an optimal control problems in
$\Omega_{\epsilon}$ for the Laplacian operator. There are mainly two types of optimal
control problems; namely distributed control andboundary control. For distributed control
problems in the oscillatingdomain, one can put control on the oscillating part or on the fixed
part and similarly for boundary control problem (control on the oscillatingboundary or on the
fixed part the boundary). Considering controls on theoscillating part is more interesting and
challenging than putting control on fixed part of the domain. Our main aim is to characterize
the controlsand study the limiting analysis (as $\epsilon \to 0$) of the optimalsolution.
In the thesis, we consider all the four cases, namely distributed and boundary controls both
on the oscilalting part and away from the oscillating part. Since, controls on the oscillating
part is more exciting, in this talk, we present the details of two sections. First we consider
distributed optimal control problem, where the control is supported on the oscillating part
$Omega_{\epsilon}^{+}$ with periodic controls and with Neumann condition on the oscillating
boundary $\gamma_{\epsilon}$. Secondly, we introduce boundary optimal control
problem, control applied through Neumann boundary condition on the oscillating boundary
$\gamma_{\epsilon}$ with suitable scaling parameters. We characterize the optimal control
using unfolding and boundary unfolding operators and study limiting analysis. In the limit, we
obtain two limit problems according to the scaling parameters and we observe that limit
optimal control problem has three control namely; a distributed control, a boundary control
and an interface control.
Title: Chromatic polynomials of graphs from Kac-Moody algebras.
Speaker: R. Venkatesh ( Weizmann Institute of Science, Israel)
Date: Wed, 16 Mar 2016
Time: 2 pm
Venue: LH-1
Given a simple graph G, the Kac-Moody Lie algebra of G is the Kac-Moody algebra whose simply laced Dynkin diagram is G. We give a new interpretation of the chromatic polynomial of G in terms of the Kac-Moody Lie algebra of G. We show that the chromatic polynomial is essentially th= e q-Kostant partition function of the associated Kac-Moody algebra evaluate= d on the sum of the simple roots. As an application, we construct basis of some of the root spaces of the Kac-Moody algebra of G. This is a joint work with Sankaran Viswanath.
Title: Exchangeable and stationary chains of quantum Gaussian states and an integral formula for their entropy rates.
Speaker: K R Parthasarathy
Date: Fri, 11 Mar 2016
Time: 4 pm
Venue: LH-1
We shall introduce the definition of a k-mode Gaussian state and a chain of such states which determine a C* probability space. We present examples of such states exhibiting properties like exchangeability and stationarity. Stationary chains are determined by block Toeplitz matrices. Using the Kac-Murdoch-Szego theorems on asymptotic spectral distributions of Toeplitz matrices we compute the entropy rates of some of these chains. This leaves many natural problems open.
Title: On ideals and associated notions of convergence
Speaker: Prof. Pratulananda Das, Jadavpur University, Kolkata.
Date: Tue, 08 Mar 2016
Time: 4 pm
Venue: LH-1
In this talk we discuss how the notion of usual convergence is extended using the notion of ideals and the importance of P-ideals. We then show how ideals can be generated and in particular how the P ideals can be generated by matrices other than regular summability matrix
Title: Wild Topology, Group Theory and a conjecture in Number Theory
Speaker: Greg Conner (Brigham Young University, USA)
Date: Wed, 02 Mar 2016
Venue: LH-1, Mathematics Department
One of the most useful analogies in mathematics is the fundamental group functor (also known as the Galois Correspondence) which sends a topological space to its fundamental group while at the same time sending continuous maps between spaces to corresponding homomorphisms of groups in such a way that compositions of maps are preserved.
Title: Eigenfunctions Seminar: Some Aspects of Minimal Surfaces
Speaker: Rukmini Dey (ICTS, Bangalore)
Date: Fri, 26 Feb 2016
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
I will introduce minimal surfaces and explain the Weierstrass-Enneper representation of a minimal surface using hodographic coordinates. I will mention an interesting link between minimal surfaces and Born-Infeld solitons. If time permits, I will explain my work (with P. Kumar and R.K. Singh) on interpolation of two real analytic curves by a minimal surface of the Bjorling-Schwartz type.
Title: Eigenfunctions Seminar: Understanding Wolfe’s Heuristic: Submodular Function Minimization and Projection onto Polytopes
Speaker: Deeparnab Chakrabarty (Microsoft Research, Bangalore)
Date: Fri, 26 Feb 2016
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a very important problem. Theoretically, unconstrained SFM is polynomial time solvable, however, these algorithms are not practical. In 1976, Wolfe proposed a heuristic for projection onto a polytope, and in 1980, Fujishige showed how Wolfe’s heuristic can be used for SFM. For general submodular functions, this Fujishige-Wolfe heuristic seems to have the best empirical performance. Despite its good practical performance, very little is known about Wolfe’s projection algorithm theoretically.
In this talk, I will describe the first analysis which proves that the heuristic is polynomial time for bounded submodular functions. Our work involves the first convergence analysis of Wolfe’s projection heuristic, and proving a robust version of Fujishige’s reduction theorem.
Title: Strong pseudoconvexity in Banach spaces and applications
Speaker: Sofia Ortega Castillo
Date: Fri, 19 Feb 2016
Venue: LH-3, Mathematics Department
Having been unclear how to widely define strong (or strict) pseudoconvexity in the infinite-dimensional context, we compared the concept in the smooth-boundary case with strict convexity. As a result, we accomplished establishing definitions of local uniform pseudoconvexity, uniform pseudoconvexity and strict pseudoconvexity for open and bounded subsets of a Banach space. We will see examples of Banach spaces with uniformly pseudoconvex unit ball, as well as examples of Banach spaces whose unit ball is not even strictly pseudoconvex. As an application of the techniques developed, we show that in finite dimension the concept of strict plurisubharmonicity coincides with strict plurisubharmonicity in distribution.
Title: PhD Thesis colloquium: Operator theory on symmetrized bidisc and tetrablock - some explicit constructions
Speaker: Mr. Haripada Sau Ph. D. Research Scholar
Date: Fri, 12 Feb 2016
Time: 11.00 a.m.
Venue: Lecture Hall I, Dept of Mathematics
A pair of commuting bounded operators $(S,P)$ acting on a
Hilbert space, is
called a $\Gamma$-contraction, if it has the symmetrised bidisc
\begin{equation}
\Gamma=\{ (z_1+z_2,z_1z_2):|z_1| \leq 1,|z_2| \leq 1 \}\subseteq \mathbb{C}^2
\end{equation}
as a spectral set. For every $\Gamma$-contraction $(S,P)$, the operator equation
$S-S^*P=D_PFD_P$ has a unique solution $F$ with numerical radius, $w(F)$ no greater than one,
where $D_P$ is the positive square root of $(I-P^*P)$. This unique operator is called the
fundamental operator of $(S,P)$. This thesis constructs an explicit normal boundary dilation for a
$\Gamma$-contraction. A triple of commuting bounded operators $(A,B,P)$
acting on a
Hilbert space with the closure of the tetrablock
\begin{equation}
E=\{(a_{11},a_{22},\det A):
A=\begin{pmatrix} a_{11} & a_{12} \\
a_{21} & a_{22} \end{pmatrix}\text{ with }\lVert A \rVert
<1\}\subseteq\mathbb{C}^3
\end{equation}
as a spectral set, is called a tetrablock contraction. Every tetrablock contraction
possesses two fundamental operators and these are the unique solutions of
\begin{equation}
A-B^*P=D_PF_1D_P, \ \text{ and } \ B-A^*P=D_PF_2D_P.
\end{equation}
Moreover, $w(F_1)$ and $w(F_2)$ are no greater than one. This thesis also constructs an explicit
normal boundary dilation for a tetrablock contraction. In these constructions, the
fundamental operators play a pivotal role. Both the dilations in the symmetrized bidisc and in the
tetrablock are proved to be minimal. But unlike the one variable case, uniqueness of minimal
dilations usually does not hold good in several variables, e.g., Ando’s dilation is not unique.
However, we show that the dilations are unique under a certain natural condition. In view of the
abundance of operators and their complicated structure, a basic problem in operator
theory is to find nice functional models and complete sets of unitary invariants. We develop a
functional model theory for a special class of triples of commuting bounded operators
associated with the tetrablock. We also find a set of complete sort of unitary invariants for
this special class. Along the way, we find a Beurling-Lax-Halmos type of result for a triple
of multiplication operators acting on vector-valuedHardy spaces. In both the model theory
and unitary invariance,fundamental operators play a fundamental role. This thesis
answers the question when two operators $F$ and $G$ with $w(F)$ and $w(G)$ no greater than one,
are admissible as fundamental operators, in other words, when there exists a
$\Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and
$G$ is the fundamental operator of $(S^*,P^*)$. This thesis also answers a
similar question in the tetrablock setting.
Title: PhD Thesis colloquium: Goldman bracket : Centers, Geometric intersection numbers and Length equivalent curves.
Speaker: Mr. Arpan Kabiraj.
Date: Thu, 11 Feb 2016
Time: 10.00 a.m. - 11.00 a.m.
Venue: Lecture Hall I, Department of Mathematics, IISc.
In the 1980s, Goldman introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F. This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra.
In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures.
We use these results to identify the quotient of the Goldman Lie algebra of a non-closed surface by its center as a sub-algebra of the first Hochschild cohomology of the fundamental group.
Using hyperbolic geometry, we prove a special case of a theorem of Chas, namely, the geometric intersection number between two simple closed geodesics is the same as the number of terms (counted with multiplicity) in the Goldman bracket between them.
We also construct infinitely many pairs of length equivalent curves in an= y hyperbolic surface F of finite type. Our construction shows that given a self-intersecting geodesic x of F and any self-intersection point P of x, we get a sequence of such pairs.
Title: Funk and Hilbert geometry: Euclidean, non-Euclidean and timelike
Speaker: Athanase Papadopolous Universite de Strasbourg
Date: Thu, 04 Feb 2016
Venue: LH3 Math Dept IISc
I will present the Funk and Hilbert metrics on convex sets in the setting of Euclidean, non-Euclidean and timelike geometries. I will explain the motivation for studying these metrics and highlight some of their main properties, concerning geodesics, infinitesima structures and isometries.
Title: Eigenfunctions Seminar: Electoral methods and the probability of the Alabama paradox
Speaker: Svante Linusson (KTH, Stockholm, Sweden)
Date: Fri, 29 Jan 2016
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
There exists various possible methods to distribute seats proportionally between states (or parties) in a parliament. In the first half of the talk I will describe some often used methods and discuss their pros and cons (it’s all in the rounding).
One easy method is called Hamilton’s method (also known as the method of largest reminder or Hare’s method). It suffers from a drawback called the Alabama paradox, which e.g. made USA abandon it for the distribution of seats in the house of representatives between states. It is still in use in many other countries including Sweden.
In the second half of the talk I will describe a joint work with Svante Janson (Uppsala Univ.) where we study the probability that the Alabama paradox will happen. We give, under certain assumptions, a closed formula for the probability that the Alabama paradox occurs given the vector $p_1,\dots,p_m$ of relative sizes of the states.
Title: Eigenfunctions Seminar: Eigenvalues, Eigenvectors, Eigenfunctions, Everywhere!
Speaker: Kalyan B. Sinha (JNCSAR, Bangalore and IMI Distinguished Associate)
Date: Fri, 29 Jan 2016
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
Eigenvalues and eigenvectors appear in many physical and engineering problems, beginning with the noted study by Euler in 1751 of the kinematics of rigid bodies. From a mathematical point of view, for an operator of a suitable class, acting in a vector space, its point spectrum and associated subspaces refer to its “eigenvalues and eigenvectors”, while the subspaces associated with the continuous spectrum of the operator is said to consist of “eigenfunctions”. The basic ideas will be discussed mostly through examples, in some of which a natural connection with the representation of appropriate groups lurks behind.
Title: PhD Thesis defence: On critical points of random polynomials and spectrum of certain products of random matrices
Speaker: Tulasi Ram Reddy A
Date: Wed, 27 Jan 2016
Time: 11 am
Venue: LH-3, Mathematics Department
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.
In the second part we deal with the spectrum of products of Ginibre
matrices. Exact eigenvalue density is known for a very few matrix
ensembles. For the known ones they often lead to determinantal point
process. Let $X_1,X_2,...,X_k$ be i.i.d. matrices of size $n \times n$ whose entries
are independent complex Gaussian random variables. We derive the
eigenvalue density for matrices of the form $Y_1.Y_2....Y_n$, where each
$Y_i = X_i$ or $X_i^{-1}$. We show that the eigenvalues form a determinantal
point process. The case where $k=2$, $Y_1=X_1,Y_2=X_2^{-1}$ was derived
earlier by Krishnapur. The case where $Y_i =X_i$ for all $i=1,2,...,n$, was
derived by Akemann and Burda. These two known cases can be obtained as
special cases of our result.
Title: The Batalin-Vilkovisky structure on a model of the loop space homology.
Speaker: Prof. Jean-Baptiste Gatsinzi, University of Namibia, Namibia.
Date: Fri, 22 Jan 2016
Time: 3 pm
Venue: LH-1
Let X be a closed, simply connected and orientable manifold of dimension m and LX the space of free loops on X. We use Rational Homotopy Theory to construct a model for the loop space homology. We further define a BV structure which is equivalent, in some cases, to the Chas-Sullivan BV operator.
Title: Linking numbers for algebraic cycles
Speaker: Souvik Goswami (ICMAT, Madrid, Spain)
Date: Mon, 18 Jan 2016
Time: 4pm
Venue: LH-1
In algebraic geometry the concept of height pairing (a particular example
Title: Upscaling of a System of Semilinear Diffusion‐Reaction Equations in a Heterogeneous Medium: Multi‐Scale Modeling and Periodic Homogenization.
Speaker: Hari Shankar Mahato (University of Georgia, USA)
Date: Sun, 10 Jan 2016
Time: 11 am
Venue: LH-1
A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multi‐scale medium
Title: On some fourth order equations associated to Physics.
Speaker: Sanjiban Santra (CIMAT, Mexico)
Date: Tue, 05 Jan 2016
Time: 4pm
Venue: LH-1
We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated fourth order Liouville theorem to the problem and possible link with the De Giorgi’s conjecture. This is an attempt to prove the McKenna-Walter conjecture which is open for the last two decades.
Title: Twin Primes
Speaker: M. Ram Murty, Queen's University, Kingston, Canada
Date: Mon, 04 Jan 2016
Time: 4pm
Venue: LH-1
There is a folklore conjecture that there are infinitely many primes p such that p+2 is also prime. This conjecture is still open. However, in the last two years, spectacular progress has been made to show that there are infinitely many primes p such that p+h is also prime with 1< h < 247. We will discuss the history of this problem and explain the new advances in sieve theory that have led to these remarkable results. We will also highlight what we may expect in the future.
Title: A motivic formula for the $L$-function of an abelian variety over a function field
Speaker: Bruno Kahn, University of Paris VI
Date: Fri, 01 Jan 2016
Time: 4 pm
Venue: LH-1
Title: PhD Thesis defence: Homogenization of Periodic Optimal Control Problems in a Domain with highly oscillating boundary
Speaker: Ravi Prakash, University of Conception, Chilie
Date: Fri, 18 Dec 2015
Time: 2:30pm
Venue: LH-3
Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the contemporary applications are much far and wide. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic analysis. We plan to start with an illustrative example of limiting analysis in 1-D for a second order elliptic partial differential equation. We will also see some classical results in the case of periodic composite materials and oscillating boundary domain. The emphasis will be on the computational importance of homogenization in numerics by the introduction of correctors. In the second part of the talk, we will see a study on optimal control problems posed in a domain with highly oscillating boundary. We will consider periodic controls in the oscillating part of the domain with a model problem of Laplacian and try to understand their optimality and asymptotic behavior.
Title: Topics in the history of Harmonic Analysis (6 lectures)
Speaker: Gerald B. Folland (University of Washington, Seattle, USA)
Date: Wed, 16 Dec 2015
Time: 4 pm
Venue: LH-1, Mathematics Department
Title: Voevodsky's smash nilpotence conjecture
Speaker: Ronnie Sebastian (IISER Pune)
Date: Thu, 10 Dec 2015
Time: 4pm
Venue: LH-1
Voevodsky’s conjecture states that numerical and smash equivalence coincide for algebraic cycles. I shall explain the conjecture in more detail and talk about some of the examples for which this conjecture is known.
Title: Enumerative Geometry of rational cuspidal curves on del-Pezzo surfaces.
Speaker: Ritwik Mukherjee (TIFR)
Date: Tue, 01 Dec 2015
Time: 4 pm
Venue: LH-1
Enumerative geometry is a branch of mathematics that deals with the following question: How many geometric objects are there that satisfy certain constraints? The simplest example of such a question is How many lines pass through two points?. A more interesting question is How many lines are there in three dimensional space that intersect four generic lines?. An extremely important class of enumerative question is to ask How many rational (genus 0) degree d curves are there in CP^2 that pass through 3d-1 generic points? Although this question was investigated in the nineteenth century, a complete solution to this problem was unknown until the early 90’s, when Kontsevich-Manin and Ruan-Tian announced a formula. In this talk we will discuss some natural generalizations of the above question; in particular we will be looking at rational curves on del-Pezzo surfaces that have a cuspidal singularity. We will describe a topological method to approach such questions. If time permits, we will also explain the idea of how to enumerate genus one curves with a fixed complex structure by comparing it with the Symplectic Invariant of a manifold (which are essentially the number of curves that are solutions to the perturbed d-bar equation).
Title: Functional equation for Selmer groups
Speaker: Somnath Jha (IIT, Kanpur)
Date: Mon, 30 Nov 2015
Venue: LH-1
Interplay between arithmetic and analytic objects are some of the beautiful aspects of number theory. In this talk, we will discuss several examples of this.
Title: Existence of Nash equilibrium for chance-constrained games
Speaker: Vikas Singh (Universite Paris-Sud)
Date: Thu, 26 Nov 2015
Time: 3:00 pm
Venue: LH1
We consider an n-player strategic game with finite action sets. The payoffs of each player are random variables. We assume that each player uses a satisficing payoff criterion defined by a chance-constraint, i.e., players face a chance-constrained game. We consider the cases where payoffs follow normal and elliptically symmetric distributions. For both cases we show that there always exists a mixed strategy Nash equilibrium of corresponding chance-constrained game.
Title: Eigenfunctions Seminar: Teichmüller space, harmonic maps and measured foliations
Speaker: Subhojoy Gupta (IISc Mathematics)
Date: Fri, 20 Nov 2015
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
This talk shall be an introduction to some aspects of Teichmüller theory, which is concerned with the parameter space of marked Riemann surfaces. I shall describe Wolf’s parametrization of Teichmüller space using harmonic maps, and discuss how it extends to the Thurston compactification. If time permits, I will describe some recent joint work with Wolf.
Title: Eigenfunctions Seminar: Endomorphisms of the algebra of all bounded operators on a Hilbert space
Speaker: B. V. Rajarama Bhat (ISI, Bangalore)
Date: Fri, 20 Nov 2015
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
We get useful insight about various algebraic structures by studying their automorphisms and endomorphisms. Here we give a brief introduction to the theory of semigroups of endomorphisms of the algebra of all bounded operators on a Hilbert space.
Title: Hecke algebras and the Langlands program
Speaker: Manish Mishra (Universitat Heidelberg)
Date: Tue, 17 Nov 2015
Time: 4pm
Venue: LH-1
Given an irreducible polynomial f(x) with integer coefficients and a prime number p, one wishes to determine whether f(x) is a product of distinct linear factors modulo p. When f(x) is a solvable polynomial, this question is satisfactorily answered by the Class Field Theory. Attempts to find a non-abelian Class Field Theory lead to the development of an area of mathematics called the Langlands program. The Langlands program, roughly speaking, predicts a natural correspondence between the finite dimensional complex representations of the Galois group of a local or a number field and the infinite dimensional representations of real, p-adic and adelic reductive groups. I will give an outline of the statement of the local Langlands correspondence. I will then briefly talk about two of the main approaches towards the Langlands program - the type theoretic approach relying on the theory of types developed by Bushnell-Kutzko and others; and the endoscopic approach relying on the trace formula and endoscopy. I will then state a couple of my results involving these two approaches.
Title: Algorithms and complexity for Turaev-Viro invariants
Speaker: Jonathan Spreer, The University of Queensland, Australia
Date: Fri, 13 Nov 2015
Time: 4 pm
Venue: LH-1
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. I will discuss this family of invariants, and present an explicit fixed-parameter tractable algorithm for arbitrary r which is practical—and indeed preferable—to the prior state of the art for real computation.
Title: Exploring parameterised complexity in computational topology
Speaker: Benjamin A. Burton (University of Queensland, Brisbane, Australia)
Date: Thu, 12 Nov 2015
Time: 4 pm
Venue: LH-1, Mathematics Department
Exact computation with knots and 3-manifolds is challenging - many fundamental problems are decidable but enormously complex, and many major algorithms have never been implemented. Even simple problems, such as unknot recognition (testing whether a loop of string is knotted), or 3-sphere recognition (testing whether a triangulated 3-manifold is topologically trivial), have best-known algorithms that are worst-case exponential time.
Title: Landstad-Vaes theory for locally compact quantum groups
Speaker: Sutanu Roy (University of Ottawa, Canada)
Date: Fri, 06 Nov 2015
Time: 4 pm
Venue: LH-1, Mathematics Department
In the seventies of last century, Magnus Landstad characterised the coefficient C-algebra inside the multiplier algebra of a given crossed product C-algebra subject to an action of an Abelian locally compact group G. In 2005, Stefaan Vaes extended the Landstad theory for regular locally compact quantum groups. He gave strong indications that this is not possible for non-regular groups. In this talk I shall explain how Landstad-Vaes theory extends for non-regular groups. To this end we have to consider not necessary continuous, but measurable, actions of locally compact quantum groups. For regular locally compact quantum groups any measurable action is continuous, so our theory contains that of Landstad-Vaes. This is a joint work with Stanislaw Lech Woronowicz.
Title: Nonequilibrium Markov processes conditioned on large deviations
Speaker: Raphael Chetrite (Universite Nice Sophia Antipolis, France)
Date: Thu, 05 Nov 2015
Time: 4 pm
Venue: LH-1, Mathematics Department
I will present a general approach for constructing a Markov process that describes the dynamics of a nonequilibrium process when one or more observables of this process are observed to fluctuate in time away from their typical values.
Title: On Risk Concentration
Speaker: Marie Kratz (ESSEC Business School, CREAR - risk research center)
Date: Mon, 26 Oct 2015
Time: 2:15-3:15 pm
Venue: LH-1, Mathematics Department
We study the local behavior of (extreme) quantiles of the sum of heavy-tailed random variables, to infer on risk concentration. Looking at the literature, asymptotic (for high threshold) results have been obtained when assuming (asymptotic) independence and second order regularly varying conditions on the variables. Other asymptotic results have been obtained in the dependent case when considering specific copula structures. Our contribution is to investigate on one hand, the non-asymptotic case (i.e. for any threshold), providing analytical results on the risk concentration for copula models that are used in practice, and comparing them with results obtained via Monte-Carlo method. On the other hand, when looking at extreme quantiles, we assume a multivariate second order regular variation condition on the vectors and provide asymptotic risk concentration results. We show that many models used in practice come under the purview of such an assumption and provide a few examples. Moreover this ties up related results available in the literature under a broad umbrella. This presentation is based on two joint works, one with M. Dacorogna and L. Elbahtouri (SCOR), the other with B. Das (SUTD).
Title: Eigenfunctions Seminar: Homotopy type theory
Speaker: Siddhartha Gadgil (IISc Mathematics)
Date: Fri, 16 Oct 2015
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
The usual foundations of mathematics based on Set theory and Predicate calculus (and extended by category theory), while successful in many ways, are so far removed from everyday mathematics that the possibility of translation of theorems to their formal counterparts is generally purely a matter of faith. Homotopy type theory gives alternative foundations for mathematics. These are based on an extension of type theory (from logic and computer science) using an unexpectedly deep connection of Types with Spaces discovered by Voevodsky and Awodey-Warren. As a consequence of this relation we also obtain a synthetic view of homotopy theory. In this lecture, I will give a brief introduction to this young subject.
Title: Eigenfunctions Seminar: Nonlinear analysis and PDE
Speaker: K. Sandeep (TIFR-CAM, Bangalore)
Date: Fri, 16 Oct 2015
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
It is well known that finding an explicit solution for partial differential equations is almost impossible in most of the cases. In this talk we will give a brief introduction to the analysis of certain nonlinear PDEs using various tools from analysis. We will discuss some concrete examples and some open problems (depending on time). The talk should be accessible to senior undergraduate students.
Title: Iwasawa theory and the Birch and Swinnerton-Dyer conjecture
Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)
Date: Tue, 08 Sep 2015
Time: 2 pm
Venue: LH-2, Mathematics Department
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising front he L-function of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the L-function. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
Title: Eigenfunctions Seminar: Algorithms for independent component analysis
Speaker: Navin Goyal (Microsoft Research, Bangalore)
Date: Fri, 04 Sep 2015
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Independent component analysis (ICA) is a basic problem that arises in several areas including signal processing, statistics, and machine learning. In this problem, we are given linear superpositions of signals. E.g., we could be receiving signals from several sensors but the receivers only get the weighted sums of these signals. The problem is to recover the original signals from the superposed data. In some situations this turns out to be possible: the main assumption being that the signals at different sensors are independent random variables. While independent component analysis is a well-studied problem, one version of it was not well-understood, namely when the original signals are allowed to be heavy-tailed, such as those with a Pareto distribution. Such signals do arise in some applications. In this talk, I will first discuss the previously known algorithms for ICA and then a new algorithm that applies also to for the heavy-tailed case. The techniques used are basic linear algebra and probability.
Title: Eigenfunctions Seminar: Maximal and extremal lattices
Speaker: Siegfried Böcherer (Universität Mannheim)
Date: Fri, 04 Sep 2015
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
We consider integral lattices $L$ in an Euclidean space $V = \mathbb{R}^m$, i.e. $\mathbb{Z}$-submodules of full rank in $V$ such that all vectors in $L$ have integral length. It is impossible to classify such lattices up to isometry, there are just too many of them in general, even if we fix additional invariants such as the discriminant. Therefore one looks for interesting subclasses of lattices, in particular “extremal lattices”, characterized by the property that the smallest length of a non-zero vector in $L$ is “as large as possible”. There are several ways to make this more precise, we will focus on analytic extremality, where modular forms come in. In particular, we will consider extremality for maximal lattices.
Title: Iwasawa theory and the Birch and Swinnerton-Dyer conjecture
Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)
Date: Tue, 01 Sep 2015
Time: 2 pm
Venue: LH-2, Mathematics Department
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising front he L-function of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the L-function. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
Title: Iwasawa theory and the Birch and Swinnerton-Dyer conjecture
Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)
Date: Mon, 31 Aug 2015
Time: 2 pm
Venue: LH-2, Mathematics Department
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising from the L-function of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the L-function. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
Title: Contact Structures and Contact Invariants in Heegaard Floer Homology.
Speaker: Dheeraj Kulkarni (RKMVU, Kolkata)
Date: Mon, 31 Aug 2015
Time: 3:30 pm
Venue: LH-1, Mathematics Department
There is a dichotomy of contact structures– tight vs overtwisted. Classification of overtwisted contact structures up to isotopy, due to Eliashberg, is well understood. However, there have been few results towards classification of tight contact structures. In general, given a contact structure it is difficult to know whether it is tight or not.
Title: Iwasawa theory and the Birch and Swinnerton-Dyer conjecture
Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)
Date: Tue, 25 Aug 2015
Time: 2 pm
Venue: LH-2, Mathematics Department
The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising front he L-function of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the L-function. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.
Title: Localisation theorems for Bochner-Riesz means.
Speaker: Jotsaroop Kaur (University of Milano, Italy)
Date: Fri, 21 Aug 2015
Time: 4 pm
Venue: LH 1
Define $S_R^\alpha f := \int_{\mathbb{R}^d} (1-\frac{|\xi|^2}{R^2})_+^\alpha$ $\hat{f} (\xi) e^{i2\pi (x.\xi)} d\xi$, the Bochner Riesz means of order $\alpha \geq 0$. Let $f\in L^2(\mathbb{R}^d)$ and $f \neq 0$ in an open, bounded set $B.$ It is known that $S_R^\alpha f$ goes to 0 a.e. in $B$ as $R\rightarrow\infty.$ We study the pointwise convergence of Bochner Riesz means $S_{R}^\alpha f, \alpha>0$ as $R \rightarrow \infty$ on sets of positive Hausdorff measure in $\mathbb{R}^d$ by making use of the decay of the spherical means of Fourier Transform of fractal measures. We get an improvement in the range of the Hausdorff dimension of the sets on which it converges. When $0<\alpha<\frac{d-1}{4},$ we get the best possible result in $\mathbb{R}^2$ and in higher dimensions we improve the result by L.Colzani, G. Gigante and A. Vargas. Steins interpolation theorem also gives us the corresponding result for $f\in L^p(\mathbb{R}^d), 1<p<2.$
Title: Gohberg Lemma, Compactness and Essential Spectrum of Operators on Compact Lie Groups
Speaker: Aparajita Dasgupta (EPFL, Lausanne)
Date: Thu, 20 Aug 2015
Time: 11 am
Venue: LH-1
In this talk we prove a version of the Gohberg lemma on compact Lie groups giving an estimate from below for the distance from a given operator to the set of compact operators on compact Lie groups. As a consequence, we prove several results on bounds for the essential spectrum and a criterion for an operator to be compact. The conditions are given in terms of the matrix-valued symbols of operators. (This is a joint work with Professor Michael Ruzhansky.)
Title: AN INTRODUCTION TO ATLAS: BASIC BACKGROUND AND APPLICATIONS
Speaker: JONATHAN FERNANDES
Date: Fri, 14 Aug 2015
Time: 4pm
Venue: LH-1
Title: Koszul duality theory for algebras.
Speaker: Anita Naolekar (ISI, Bengaluru)
Date: Wed, 12 Aug 2015
Time: 11 am
Venue: LH4
Koszul duality theory is a homological method aiming at constructing an explicit quasi-free resolution for quadratic algebras. We introduce the concepts of bar (and cobar) construction for a quadratic algebra (and coalgebra) and provide a quasi-free resolution for quadratic algebras, under certain assumptions.
Title: Dual pairs and intertwining distributions
Speaker: Angela Pasquale (Universite de Lorraine)
Date: Mon, 10 Aug 2015
Time: 4pm
Venue: LH-1
A reductive dual pair in the group Sp(W) of isometries of a symplectic space W, over a local
Title: Reflected Backward SDEs with General Jumps
Speaker: Youssef Oukine (Universite Cadi Ayyad, Maroc)
Date: Fri, 07 Aug 2015
Time: 3-4 pm
Venue: LH-1
In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson random measure. The reflecting process is right continuous with left limits (RCLL for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the general result we use a fixed point argument in an appropriate space. The second part of the paper is related to BSDEs with two reflecting barriers. Once more we prove existence and uniqueness of the solution of the BSDE.
Title: Operads: An Introduction.
Speaker: Anita Naolekar (ISI, Bengaluru)
Date: Wed, 05 Aug 2015
Time: 11 am
Venue: LH4
An operad is an algebraic device which encodes a type of algebra. The classical types of algebras, ie. associative, commutative and Lie algebras give the first examples of algebraic operads. Operadic point of view has several advantages. Firstly, many results known for classical algebras, when written out in operadic language, can be applied to other types of algebras. Secondly, operadic language simplifies the statements and proofs. Thirdly, even for classical cases, operad theory has provided new results. We start with several equivalent definitions, together with examples, and few basic properties.
Title: PhD Thesis colloquium: On critical points of random polynomials and spectrum of certain products of Ginibre matrices.
Speaker: Tulasi Ram Reddy A
Date: Fri, 24 Jul 2015
Time: 11:00 am
Venue: Lecture Hall I, Department of Mathematics
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.
In the second part we deal with the spectrum of products of Ginibre
matrices. Exact eigenvalue density is known for a very few matrix
ensembles. For the known ones they often lead to determinantal point
process. Let $X_1,X_2,...,X_k$ be i.i.d matrices of size nxn whose entries
are independent complex Gaussian random variables. We derive the
eigenvalue density for matrices of the form $Y_1.Y_2....Y_n$, where each
$Y_i = X_i or (X_i)^{-1}$. We show that the eigenvalues form a determinantal
point process. The case where k=2, $Y_1=X_1,Y_2=X_2^{-1}$ was derived
earlier by Krishnapur. The case where $Y_i =X_i$ for all $i=1,2,...,n$, was
derived by Akemann and Burda. These two known cases can be obtained as
special cases of our result.
Title: PhD Thesis defence: Mechanisation of Knot Theory
Speaker: T. V. H. Prathamesh IISc
Date: Fri, 24 Jul 2015
Time: 11:00am - 12:00pm
Venue: Lecture Hall III, Department of Mathematics
Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many way of mechanising are: (1) generating results using Automated Theorem Provers, (2) Interactive theorem proving in a Proof Assistant which involves a combination of user intervention and automation.
In the first part of this thesis, we reformulate the question of equivalence of two Links in First Order Logic using Braid Groups. This is achieved by developing a set of Axioms whose canonical model is the Infinite Braid Group. This renders the problem of distinguishing Knots and Links, amenable to implementation in First Order Logic based Automated Theorem provers. We further state and prove results pertaining to Models of Braid Axioms.
The second part of the thesis deals with formalising Knot Theory in Higher Order Logic using the Isabelle Proof Assistant. We formulate equivalence of Links in Higher Order Logic. We obtain a construction of Kauffman Bracket in the Isabelle Proof Assistant. We further obtain a machine checked proof of invariance of Kauffman Bracket.
Title: Finite configurations in sparse sets
Speaker: Malabika Pramanik (University of British Columbia, Vancouver, Canada)
Date: Wed, 22 Jul 2015
Time: 11:30 am
Venue: LH-1, Mathematics Department
A problem of interest in geometric measure theory both in discrete and continuous settings is the identification of algebraic and geometric patterns in thin sets. The first two lectures will be a survey of the literature on pattern recognition in sparse sets, with a greater emphasis on continuum problems. The second two contain an exposition of recent work, joint in part with Vincent Chan, Kevin Henriot and Izabella Laba on the existence of linear and polynomial configurations in multi-dimensional Lebesgue-null sets satisfying appropriate Hausdorff and Fourier dimensionality conditions.
Title: Some computational and analytic aspects of Chern-Weil forms
Speaker: Vamsi Pingali (Johns Hopkins University)
Date: Thu, 02 Jul 2015
Time: 4 pm
Venue: LH-1, Mathematics Department
This talk will focus on two aspects of vector bundles. One is the calculation and application of certain characteristic and secondary characteristic forms (i.e. Chern, Chern-Simons, and Bott-Chern forms). This part is joint work with Leon Takhtajan and Indranil Biswas. The second is to study certain fully nonlinear PDE akin to the Monge-Ampére equation arising from these differential geometric objects. An existence result or two will be presented along with the difficulties involved in the PDE. Moreover, other areas of geometry and physics from which the same PDE arise will be pointed out.
Title: PhD Thesis defence: Minimal crystallizations of 3- and 4-manifolds
Speaker: Mr. Biplab Basak (IISc Mathematics)
Date: Thu, 25 Jun 2015
Time: 2 p.m. - 3 p.m.
Venue: Lecture Hall III, Department of Mathematics
This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings:
(i) We introduce the weight of a group which has a presentation with
number of relations is at most the number of generators.
We prove that the number of vertices of any crystallization of a connected
closed 3-manifold $M$ is at least the weight of the
fundamental group of $M$. This lower bound is sharp for the 3-manifolds
$\mathbb{R P}^3$, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$,
$S^{\hspace{.2mm}2} \times S^1$, $\TPSS$ and $S^{\hspace{.2mm}3}/Q_8$,
where $Q_8$ is the quaternion group. Moreover,
there is a unique such vertex minimal crystallization in each of these
seven cases. We also construct crystallizations of
$L(kq-1,q)$ with $4(q+k-1)$ vertices for $q \geq 3$, $k \geq 2$ and
$L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent
result of Swartz,
our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.
(ii) We present an algorithm to find certain types of crystallizations of
$3$-manifolds from a given presentation $\langle S \mid R \rangle$ with
$\#S=\#R=2$. We generalize this algorithm for presentations with three
generators and certain class of relations.
This gives us crystallizations of closed connected orientable 3-manifolds
having fundamental groups $\langle x_1,x_2,x_3 \mid
x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$ with $4(m+n+k-3)+ 2\delta_n^2 + 2
\delta_k^2$ vertices for $m\geq 3$ and $m \geq n \geq k \geq 2$, where
$\delta_i^j$ is the Kronecker delta.
If $n=2$ or $k\geq 3$ and $m \geq 4$ then these crystallizations
are vertex-minimal for all the known cases.
(iii) We found a minimal crystallization of the standard pl K3 surface.
This minimal crystallization is a ‘simple crystallization’.
Using this, we present minimal crystallizations of all simply connected pl
$4$-manifolds of “standard” type, i.e., all the connected sums of
$\mathbb{CP}^2$, $S^2 \times S^2$, and the K3 surface. In particular, we
found minimal crystallizations of a pair of 4-manifolds which are
homeomorphic
but not pl-homeomorphic.
Title: Fast algorithms for data analysis and elliptic partial differential equations
Speaker: Sivaram Ambikasaran (NYU)
Date: Wed, 24 Jun 2015
Time: 11-12
Venue: LH1
Large-dense matrices arise in numerous applications: boundary integral formulation for elliptic partial differential equations, covariance matrices in statistics, inverse problems, radial basis function interpolation, multi frontal solvers for sparse linear systems, etc. As the problem size increases, large memory requirements, scaling as O(N^2), and extensive computational time to perform matrix algebra, scaling as O(N^2) or O(N^3), make computations impractical. I will discuss some novel methods for handling these computationally intense problems. In the first half of the talk, I will discuss my contributions to some of the new developments in handling large dense covariance matrices in the context of computational statistics and Bayesian data assimilation. More specifically, I will be discussing how fast dense linear algebra (O(N) algorithms for inversion, determinant computation, symmetric factorisation, etc.) enables us to handle large scale Gaussian processes, thereby providing an attractive approach for big data applications. In the second half of the talk, I will focus on a new algorithm termed Inverse Fast Multipole Method, which permits solving singular integral equations arising out of elliptic PDE’s at a computational cost of O(N).
Title: Electrostatic skeletons
Speaker: Koushik Ramachandran (ISI, Bangalore)
Date: Wed, 24 Jun 2015
Let P be the equilibrium potential of a compact set K in R^n. An electrostatic skeleton of K is a positive measure such that the closed support S of has connected complement and empty interior, and the Newtonian (or logarithmic, when n = 2) potential of is equal to P near infinity. We prove the existence and uniqueness of an electrostatic skeleton for any simplex. – This message has been scanned for viruses and dangerous content by MailScanner, and is believed to be clean.
Title: Small Combination of Slices in Banach Spaces
Speaker: Dr. Sudeshna Basu (George Washington University)
Date: Wed, 10 Jun 2015
Time: 3.00 p.m.
Venue: LH I, Department of Mathematics
Abstract : In this work, we study certain stability results for Ball Separationproperties in Banach Spaces leading to a discussion in the context of operator spaces. In this work, we study certain stability results for Small Combination of Slices Property (SCSP) leading to a discussion on SCSP in the context of operator spaces. SCSpoints were first introduced as a slice generalisation of the PC (i.e. point ofcontinuity points for which the identity mapping from weak topology to normtopology is continuous.) It is known that X is strongly regular respectively Xis w-strongly regular) if and only if every non empty bounded convex set K in X ( respectively K in X) is contained in the norm closure ( respectively w- closure)of SCS(K)( respectively w-SCS(K)) i.e. the SCS points ( w- SCS points) of K. Later, it was proved that a Banach space has Radon- Nikodym Property (RNP) if and only if it is strongly regular and it has the Krein-Milamn Property(KMP). Subsequently, the concepts of SCS points was used to investigate the structure of non-dentable closed bounded convex sets in Banach spaces. The point version of the result was also shown to be true .
Title: Random Matrices, Numerical Computation, and Applications
Speaker: Alan Edelman (MIT)
Date: Wed, 03 Jun 2015
Time: 4-5 pm
Venue: LH1
This talk is about random matrix theory. Linear Algebra and maybe a little probability are the only prerequisites. Random matrix theory is now finding many applications. Many more applications remain to be found. It is truly matrix statistics, when traditional statistics has been primarily scalar and vector statistics. The math is so much richer, and the applications to computational finance, HIV research, the Riemann Zeta Function, and crystal growth, to name a few, show how important this area is. I will show some of these applications, and invite you to find some of your own.
Title: PhD Thesis defence: Shortest length geodesics on closed hyperbolic surfaces
Speaker: Bidyut Sanki IISc
Date: Fri, 29 May 2015
Time: 10:00am - :11:00am
Venue: Lecture Hall III, Department of Mathematics
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
Title: PhD Thesis defence: Shortest length geodesics on closed hyperbolic surfaces
Speaker: Bidyut Sanki IISc
Date: Fri, 29 May 2015
Time: 10:00am - :11:00am
Venue: Lecture Hall III, Department of Mathematics
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
Title: Automated theorem proving using Learning : concepts and code
Speaker: Siddhartha Gadgil
Date: Fri, 15 May 2015
Time: 11:00 a.m.
Venue: LH-III, Department of Mathematics, IISc
In this very informal seminar, I will discuss various aspects of ongoing work to build an automated theorem proving system using, among other things, machine learning.
Title: Equivariant Cobordism Classes of Milnor Manifolds
Speaker: Swagata Sarkar ISI Kolkata
Date: Tue, 28 Apr 2015
Time: 11:00 a.m. - 12:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Let G be a 2-group, and let Z(G) denote the equivariant cobordism algebra of G-manifolds with finite stationary point sets.A cobordism class in Z(G) is said to be indecomposable if it cannot be expressed as the sum of products of lower dimensional cobordism classes.Indecomposable classes generate the cobordism algebra Z(G). We discuss a sufficient criteria for indecomposability of cobordism classes. Using the above mentioned criterion, we show that the classes of Milnor manifolds (i.e., degree 1 hypersurfaces of the product of two real projective spaces) give non-trivial, indecomposable elements in Z(G) in degrees up to 2^n - 5. This talk is based on joint work with Samik Basu and Goutam Mukherjee.
Title: PhD Thesis colloquium: $L^p$-Asymptotics of Fourier transform of fractal measures
Speaker: K. S. Senthil Raani
Date: Thu, 23 Apr 2015
Time: 3:00pm - 4:00pm
Venue: Lecture Hall I, Department of Mathematics
One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\in C_c^{\infty}(d\sigma)$, where $d\sigma$ is the surface measure on the sphere $S^{n-1}\subset\mathbb{R}^n$. Then
It follows that $\widehat{fd\sigma}\in L^p(\mathbb{R}^n)$ for all
$p>2n/(n-1)$. This result can be extended to compactly
supported measure on $(n-1)$-dimensional manifolds with
appropriate assumptions on the curvature. Similar results are
known for measures supported in lower dimensional manifolds in
$\mathbb{R}^n$ under appropriate curvature conditions. However, the
picture for fractal measures is far from complete. This thesis is
a contribution to the study of asymptotic properties of the
Fourier transform of measures supported in sets of fractal
dimension $0<\alpha<n$ for $p\leq 2n/\alpha$.
In 2004, Agranovsky and Narayanan proved that if $\mu$ is a
measure supported in a $C^1$-manifold of dimension $d<n$, then
$\widehat{fd\mu}\notin L^p(\mathbb{R}^n)$ for $1\leq p\leq \frac{2n}{d}$. We
prove that the Fourier transform of a measure $\mu_E$ supported in
a set $E$ of fractal dimension $\alpha$ does not belong to
$L^p(\mathbb{R}^n)$ for $p\leq 2n/\alpha$. We also study $L^p$-asymptotics
of the Fourier transform of fractal measures $\mu_E$ under
appropriate conditions on $E$ and give quantitative versions of
the above statement by obtaining lower and upper bounds for the
following:
\(\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|\xi|\leq L}|\widehat{fd\mu_E}(\xi)|^pd\xi,\) \(\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{L\leq |\xi|\leq 2L}|\widehat{fd\mu_E}(\xi)|^pd\xi.\)
Title: Eigenfunctions Seminar: Linearity in perceptual space
Speaker: S. P. Arun (IISc CNS)
Date: Fri, 10 Apr 2015
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
Our vision is unsurpassed by machines because we use a sophisticated object representation. This representation is unlike the retinal image: on the one hand, two out-of-phase checkerboards, maximally different in image pixels, appear perceptually similar. On the other hand, two faces, similar in their image pixels, appear perceptually distinct. What is then the nature of perceptual space? Are there principles governing its organization? To address these questions, we have been using visual search to measure similarity relations between objects.
I will summarize a line of research from our laboratory indicative of a surprising linear rule governing distances in perceptual space. In the first study, we found that search time is inversely proportional to the feature difference between the target and distracters. The reciprocal of search time is therefore linear and interestingly, it behaved like a mathematical distance metric. It also has a straightforward interpretation as a saliency signal that drives visual search (Arun, 2012). In a second study, complex searches involving multiple distracters were explained by a linear sum of pair-wise dissimilarities measured from simpler searches involving homogeneous distracters (Vighneshvel & Arun, 2013). In a third study, dissimilarities between objects differing in multiple features were found to combine linearly. This was also true for integral features such as the length and width of a rectangle upon including aspect ratio as an additional feature (Pramod & Arun, 2014). Finally, I will describe some recent results extending these findings to more naturalistic objects.
Title: Eigenfunctions Seminar: Visual search as active sequential testing
Speaker: Rajesh Sundaresan (IISc ECE)
Date: Fri, 10 Apr 2015
Time: 3:30 – 5:30 pm
Venue: LH-1, Mathematics Department
Arun and Olson (2010) conducted a visual search experiment where human subjects were asked to identify, as quickly as possible, an oddball image embedded among multiple distractor images. The reciprocal of the search times for identifying the oddball (in humans) and an ad hoc neuronal dissimilarity index, computed from measured neuronal responses to component images (in macaques), showed a remarkable correlation. In this talk, I will describe a model, an active sequential hypothesis testing model, for visual search. The analysis of this model will suggest a natural alternative neuronal dissimilarity index. The correlation between the reciprocal of the search times and the new dissimilarity index continues to be equally high, and has the advantage of being firmly grounded in decision theory. I will end the talk by discussing the many gaps and challenges in our modeling and statistical analysis of visual search. The talk will be based on ongoing work with Nidhin Koshy Vaidhiyan (ECE) and S. P. Arun (CNS).
Title: Achieving positive information velocity in wireless networks
Speaker: Srikanth Iyer (IISc, Bangalore)
Date: Mon, 06 Apr 2015
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In wireless networks, where each node transmits independently of other nodes in the network (the ALOHA protocol), the expected delay experienced by a packet until it is successfully received at any other node is known to be infinite for signal-to-interference-plus-noise-ratio (SINR) model with node locations distributed according to a Poisson point process. Consequently, the information velocity, defined as the limit of the ratio of the distance to the destination and the time taken for a packet to successfully reach the destination over multiple hops, is zero, as the distance tends to infinity. A nearest neighbor distance based power control policy is proposed to show that the expected delay required for a packet to be successfully received at the nearest neighbor can be made finite. Moreover, the information velocity is also shown to be non-zero with the proposed power control policy. The condition under which these results hold does not depend on the intensity of the underlying Poisson point process.
Title: Factorization of holomorphic eta quotients
Speaker: Soumya Bhattacharya CIRM Trento, Italy
Date: Fri, 20 Mar 2015
Time: 2:00 p.m. - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Unlike integer factorization, a reducible holomorphic eta quotient may not factorize uniquely as a product of irreducible holomorphic eta quotients. But whenever such an eta quotient is reducible, the occurrence of a certain type of factor could be observed: We conjecture that if a holomorphic eta quotient f of level M is reducible, then f has a factor of level M. In particular, it implies that rescalings and Atkin-Lehner involutions of irreducible holomorphic eta quotients are irreducible. We prove a number of results towards this conjecture: For example, we show that a reducible holomorphic eta quotient of level M always factorizes nontrivially at some level N which is a multiple of M such that rad(N) = rad(M) and moreover, N is bounded from above by an explicit function of M. This implies a new and much faster algorithm to check the irreducibility of holomorphic eta quotients. In particular, we show that our conjecture holds if M is a prime power. We also show that the level of any factor of a holomorphic eta quotient f of level M and weight k is bounded w.r.t. M and k. Further, we show that there are only finitely many irreducible holomorphic eta quotients of a given level and provide a bound on the weights of such eta quotients. Finally, we give an example of an infinite family of irreducible holomorphic eta quotients of prime power levels.
Title: Arithmetic of Farey-Ford Packings
Speaker: Amita Malik University of Illinois at Urbana-Champaign
Date: Wed, 18 Mar 2015
Time: 3:30 p.m. - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Farey-Ford Packings are a special case of more general circle packings called Apollonian Circle Packings (ACP). These packings have some very interesting properties, for exmaple, if any four mutually tangent circles have integer curvatures, then so do all others in the packing. This has led to many important problems like prime number theorem in this setting. Kontorovich and Oh explore it from dynamics point of view whereas Bourgain, Fuchs and Sarnak look at them more number theoretically. In this talk, our focus will be on the specialized packings Farey-Ford Packings. We consider some basic statistics associated to these circles and answer some questions about their distributions and asymptotic behavior. One can ask similar questions in the general setting for ACP, and if time permits, we will discuss it in this talk. Some of this is joint work with Athreya, Chaubey and Zaharescu.
Title: PhD Thesis colloquium: Minimal crystallizations of 3- and 4-manifolds
Speaker: Mr. Biplab Basak (IISc Mathematics)
Date: Tue, 17 Mar 2015
Time: 2 p.m. - 3 p.m.
Venue: Lecture Hall III, Department of Mathematics
This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings:
(i) We introduce the weight of a group which has a presentation with
number of relations is at most the number of generators.
We prove that the number of vertices of any crystallization of a connected
closed 3-manifold $M$ is at least the weight of the
fundamental group of $M$. This lower bound is sharp for the 3-manifolds
$\mathbb{R P}^3$, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$,
$S^{\hspace{.2mm}2} \times S^1$, $\TPSS$ and $S^{\hspace{.2mm}3}/Q_8$,
where $Q_8$ is the quaternion group. Moreover,
there is a unique such vertex minimal crystallization in each of these
seven cases. We also construct crystallizations of
$L(kq-1,q)$ with $4(q+k-1)$ vertices for $q \geq 3$, $k \geq 2$ and
$L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent
result of Swartz,
our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.
(ii) We present an algorithm to find certain types of crystallizations of
$3$-manifolds from a given presentation $\langle S \mid R \rangle$ with
$\#S=\#R=2$. We generalize this algorithm for presentations with three
generators and certain class of relations.
This gives us crystallizations of closed connected orientable 3-manifolds
having fundamental groups $\langle x_1,x_2,x_3 \mid
x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$ with $4(m+n+k-3)+ 2\delta_n^2 + 2
\delta_k^2$ vertices for $m\geq 3$ and $m \geq n \geq k \geq 2$, where
$\delta_i^j$ is the Kronecker delta.
If $n=2$ or $k\geq 3$ and $m \geq 4$ then these crystallizations
are vertex-minimal for all the known cases.
(iii) We found a minimal crystallization of the standard pl K3 surface.
This minimal crystallization is a ‘simple crystallization’.
Using this, we present minimal crystallizations of all simply connected pl
$4$-manifolds of “standard” type, i.e., all the connected sums of
$\mathbb{CP}^2$, $S^2 \times S^2$, and the K3 surface. In particular, we
found minimal crystallizations of a pair of 4-manifolds which are
homeomorphic
but not pl-homeomorphic.
Title: Index theory and the Arveson Conjecture
Speaker: Ronald G. Douglas Texas A&M University
Date: Fri, 13 Mar 2015
Time: 3:30 p.m. - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
TBA
Title: Composition Operators on Modulation Spaces and Application to Nonlinear Schrodinger Equation
Speaker: P.K. Ratnakumar HRI, Allahabad
Date: Thu, 12 Mar 2015
Time: 3:30 p.m. - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
TBA
Title: Eigenfunctions Seminar: Weyl's Integration Formula for Unitary Groups
Speaker: Steven Spallone (IISER, Pune)
Date: Fri, 06 Mar 2015
Time: 4:00 – 4:50 pm
Venue: LH-1, Mathematics Department
Weyl discovered a decomposition of the invariant probability measure on a connected compact Lie group, according to elements of a maximal torus and their conjugacy classes. In the case of the unitary group, it allowed Diaconis and Shahshahani to access the distribution of the eigenvalues of a random unitary matrix. We explore these ideas in the lecture.
Title: Eigenfunctions Seminar: Generating a random transposition with Random permutations
Speaker: Pooja Singla (IISc Mathematics)
Date: Fri, 06 Mar 2015
Time: 3:00 – 3:50 pm
Venue: LH-1, Mathematics Department
Consider the problem of n cards labelled 1 through n, lying face up on a table. Suppose two integers a and b are chosen independently and uniformly between 1 and n. The cards labelled a and b are switched. If many such transpositions are made the row of cards will tend to appear in random arrangement. Then question is how many steps are required until the deck is well mixed up (i.e. the permutation is close to random)? Diaconis and Shahshahani used tools of representation theory of symmetric groups to prove that at least 1/2(n log n) steps are required before the deck will be well mixed up for large n. I will explain ideas of their proof and few related problems.
Title: Eigenfunctions Seminar: Kronecker Coefficients, Integer Arrays and the RSK Correspondence
Speaker: Amritanshu Prasad (IMSc, Chennai)
Date: Fri, 06 Mar 2015
Time: 2:00 – 2:50 pm
Venue: LH-1, Mathematics Department
A Kronecker coefficient counts the number of times a representation of a symmetric group occurs in the tensor product of two others. Finding a fast algorithm to determine when a Kronecker coefficient is positive is an open problem. There has been an increased interest in this problem over the last few years as it comes up in the geometric approach to the complexity conjecture P = NP due to Mulmuley and Sohoni. I will explain how a higher dimensional analogue of the Robinson–Schensted–Knuth correspondence relates Kronecker coefficients to the problem of counting the number of integer arrays with specified slice sums.
Title: Computational Aspects of Burnside Algebras II
Speaker: Prof. Dr. Martin Kreuzer University of Passau, Germany
Date: Mon, 02 Mar 2015
Time: 2:00 p.m. - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this two-part talk, we reconsider Burnside algebras, a classical tool in the theory of finite groups, from a computational perspective. Using modern computer algebra systems, many of the results about these rings that were proved in the second half of the last century can be transformed into effective algorithms.
Title: Computational Aspects of Burnside Algebras I
Speaker: Prof. Dr. Martin Kreuzer University of Passau, Germany
Date: Wed, 25 Feb 2015
Time: 2:00 p.m. - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this two-part talk, we reconsider Burnside algebras, a classical tool in the theory of finite groups, from a computational perspective. Using modern computer algebra systems, many of the results about these rings that were proved in the second half of the last century can be transformed into effective algorithms.
Title: Prescribing Gauss curvature using mass transport
Speaker: Jerome Bertrand Universit Paul Sabatier, Toulouse, France
Date: Wed, 25 Feb 2015
Time: 3:30 p.m. - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this talk, I will give a proof of Alexandrovs theorem on the Gauss curvature prescription of Euclidean convex body. The proof is mainly based on mass transport. In particular, it doesnt rely on pdes method nor convex polyhedra theory. To proceed, I will discuss generalizations of well-known results for the quadratic cost to the case of a cost function which assumes infinite value.
Title: Axioms of Adaptivity
Speaker: Prof Carsten Carstensen Humboldt University, Berlin
Date: Mon, 23 Feb 2015
Time: 11:00 a.m. - 12:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Four axioms (A1)–(A4) link estimators and distance functions on a set of admissible refinements together and imply optimality of a standard finite element routine on an abstract level with a loop: solve, estimate, mark, and refine. The presentation provides proofs and examples of the recent review due to C. Carstensen, M. Feischl, M. Page, and D. Praetorius: The axioms of adaptivity, Comput. Math. Appl. 67 (2014) 1195 –1253 and so discusses the current literature on the mathematics of adaptive finite element methods. The presentation concludes with an overview over several applications of the set of axioms. If time permits, some recent developments are discussed on ongoing joint work with Hella Rabus on separate marking.
Title: Eigenfunctions Seminar: The dimer model and generalisations
Speaker: Arvind Ayyer (IISc Mathematics)
Date: Fri, 20 Feb 2015
Time: 2:15 – 4:30 pm (with a 15 minute break at 3:15)
Venue: LH-1, Mathematics Department
In the first half of the talk, I will define the dimer model on planar graphs and prove Kasteleyn’s groundbreaking result expressing the partition function (i.e. the generating function) of the model as a Pfaffian. I will then survey various results arising as a consequence, culminating in the beautiful limit shape theorems of Kenyon, Okounkov and coworkers.
In the second half, I will define a variant of the monomer-dimer model on planar graphs and prove that the partition function of this model can be expressed as a determinant. I will use this result to calculate various quantities of interest to statistical physicists and end with some open questions.
Title: What is Quantum Stochastic Calculus?
Speaker: Prof K. R. Parthasarthy ISI Delhi
Date: Fri, 13 Feb 2015
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
TBA
Title: Indeterminate moment problems and growth of associated entire functions
Speaker: Prof Christian Berg University of Copenhagen
Date: Fri, 06 Feb 2015
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: PhD Thesis colloquium: Weighted norm inequalities for Weyl multipliers and Hermite pseudo-multipliers
Speaker: Sayan Bagchi
Date: Fri, 30 Jan 2015
Time: 2:30pm - 3:30pm
Venue: Lecture Hall I, Department of Mathematics
In this talk we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem we deal with a variation of a theorem of Mauceri concerning the L^p boundedness of operators M which are known to be bounded on L^2: We obtain sufficient conditions on the kernel of the operator M so that it satisfies weighted L^p estimates. As an application we prove L^p boundedness of Hermite pseudo-multipliers.
Title: On applying the Deligne-Kazhdan philosophy to classical groups
Speaker: Sandeep Varma TIFR, Mumbai
Date: Thu, 29 Jan 2015
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
I will report on work in progress with Radhika Ganapathy. One wishes to study irreducible smooth' complex representations of symplectic and (split) orthogonal groups over local function fields, i.e., fields of the form F_q((t)). The theory of close local fields’ proposes to do this by studying the representation theory of these groups over (varying) finite extensions of Q_p. We will discuss an approach to using this philosophy to study the local Langlands correspondence for these groups.
Title: Some qualitative uncertainty principles revisited
Speaker: Alladi Sitaram
Date: Fri, 16 Jan 2015
Time: 2:30 - 3:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: Some qualitative uncertainty principles revisited
Speaker: Alladi Sitaram IISc
Date: Fri, 16 Jan 2015
Time: 2:30 - 3:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
TBA
Title: Geometry of algebraic surfaces and holomorphic convexity
Speaker: B. P. Purnaprajna University of Kansas
Date: Fri, 09 Jan 2015
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
TBA
Title: Some qualitative uncertainty principles revisited
Speaker: Alladi Sitaram IISc
Date: Fri, 09 Jan 2015
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
TBA
Title: Lengths of Monotone Subsequences in a Mallows Permutation
Speaker: Nayantara Bhatnagar (University of Delaware, USA)
Date: Mon, 05 Jan 2015
Time: 2 pm
Venue: LH-1, Mathematics Department
The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the Tracy-Widom distribution.
Title: Iwasawa theory
Speaker: Mahesh Kakde King's College London
Date: Mon, 05 Jan 2015
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
I will start with an overview of classical Iwasawa theory and give a formulation of main conjectures in noncommutative Iwasawa theory. If time permits I will say something about proof of main conjectures in non-commutative Iwasawa theory.
Title: Zagier polynomials and modified N\{o}rlund polynomials
Speaker: Atul Dixit Tulane University, LA
Date: Tue, 23 Dec 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
In 1998, Don Zagier studied the ‘modified Bernoulli numbers’ $B_n^{*}$ whose 6-periodicity for odd $n$ naturally arose from his new proof of the Eichler-Selberg trace formula. These numbers satisfy amusing variants of the properties of the ordinary Bernoulli numbers. Recently, Victor H. Moll, Christophe Vignat and I studied an obvious generalization of the modified Bernoulli numbers, which we call ‘Zagier polynomials’. These polynomials are also rich in structure, and we have shown that a theory parallel to that of the ordinary Bernoulli polynomials exists. Zagier showed that his asymptotic formula for $B_{2n}^{*}$ can be replaced by an exact formula.
Title: On splitting of primes and simple extensions of integrally closed domains
Speaker: Sudesh Khanduja IISER Mohali
Date: Mon, 15 Dec 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: Homogenization of a system of multi-species semilinear diff usion - reaction equations and moving boundary problems
Speaker: Dr. Hari Shankar Mahato University of Erlangen-Nrnberg
Date: Mon, 08 Dec 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multiscale medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed by the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods.
Title: Rational loop space homology of homogeneous spaces
Speaker: Jean-Baptiste Gatsinzi University of Namibia
Date: Fri, 05 Dec 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
We recall some facts about Rational Homotopy Theory, from both Sullivan and Quillen points of view. We show how to find a Sullivan model of a homogeneous space G/H . Let M be a closed oriented smooth manifold of dimension d and LM= map(S^1, M) denote the space of free loops on M . Using intersection products, Chass and Sullivan defined a product on \\mathbb{H}*(LM)=H{+d} (LM) that turns to be a graded commutative algebra and defined a bracket on \\mathbb{H}_(LM) making of it a Gerstenhaber algebra. From the work of Jones, Cohen, F\\‘elix, Thomas and others there is an isomorphism of Gerstenhaber algebras between the Hochschild cohomology HH^(C^(M), C^(M)) and \\mathbb{H}_(LM) . Using a Sullivan model (\\land V, d) of M , we show that that the Gerstenhaber bracket can be computed in terms of derivations on (\\land V, d) . Precisely, we show that HH^(\\land V, \\land V) is isomorphic to H_(\\land(V)\\otimes \\land Z , D) , where Z is the dual of V . We will illustrate with computations for homogeneous spaces.
Title: Optimal differentiation on arbitrary grids
Speaker: Divakar Viswanath (University of Michigan, USA)
Date: Mon, 24 Nov 2014
Time: 2 pm
Venue: LH-1, Mathematics Department
Finite difference formulas approximate the derivatives of a function given its values at a discrete set of grid points. Much of the theory for choosing grid points is concerned with discretization errors and rounding errors are ignored completely. However, when the grids become fine the rounding errors dominate. Finding finite difference formulas which optimize the total error leads to a combinatorial optimization problem with a large search space. In this talk, we will describe a random walk based strategy for tackling the combinatorial optimization problem.
Title: Statistics in the Anderson Model
Speaker: Prof. Krishna Maddaly IMSc
Date: Wed, 19 Nov 2014
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: Eigenfunctions Seminar: The GAP package simpcomp - A toolbox for simplicial complexes
Speaker: Jonathan Spreer (University of Queensland, Brisbane, Australia)
Date: Fri, 07 Nov 2014
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
simpcomp is an official extension to the computer algebra system GAP, that is, simpcomp is part of every full GAP installation. The software allows the user to compute numerous properties of abstract simplicial complexes such as the f -, g- and h-vectors, the fundamental group, the automorphism group, (co-)homology, the intersection form, and many more. It provides functions to generate simplicial complexes from facet lists, orbit representatives, or permutation groups. In addition, functions related to slicings and polyhedral Morse theory as well as a combinatorial version of algebraic blowups and the possibility to resolve isolated singularities of 4-manifolds are implemented.
In this talk, I will give an overview over the existing features of the software and also discuss some ongoing projects such as (i) support for Forman’s discrete Morse theory for homology and fundamental group computations, and (ii) an extension of the features to analyse tightness of triangulated manifolds.
Title: Eigenfunctions Seminar: Knots, algorithms and linear programming: the quest to solve unknot recognition in polynomial time
Speaker: Benjamin A. Burton (University of Queensland, Brisbane, Australia)
Date: Fri, 07 Nov 2014
Time: 2:15 – 3:15 pm
Venue: LH-1, Mathematics Department
In this talk we explore new approaches to the old and difficult computational problem of unknot recognition. Although the best known algorithms for this problem run in exponential time, there is increasing evidence that a polynomial time solution might be possible. We outline several promising approaches, in which computational geometry, linear programming and greedy algorithms all play starring roles.
We finish with a new algorithm that combines techniques from topology and combinatorial optimisation, which is the first to exhibit “real world” polynomial time behaviour: although it is still exponential time in theory, exhaustive experimentation shows that this algorithm can solve unknot recognition for “practical” inputs by running just a linear number of linear programs.
Title: Eigenfunctions Seminar: Shining a Hilbertian lamp on the polydisc
Speaker: Tirthankar Bhattacharyya and E.K. Narayanan (IISc Mathematics)
Date: Fri, 24 Oct 2014
Time: 2:15 – 4:30 pm (with a 15 minute break at 3:15)
Venue: LH-1, Mathematics Department
Abstract: See https://drive.google.com/file/d/0Bx6ccM3uL81GNHZqQUtic3g5TndEMmF3Y0JKSDl3LTBTeFU4/view?usp=sharing.
Title: From Calculus to Number Theory (via Cohomology)
Speaker: A. Raghuram (IISER, Pune)
Date: Thu, 16 Oct 2014
Time: 4 pm
Venue: LH-3, Mathematics Department
Title: The BNS invariant and its application to twisted conjugacy
Speaker: Prof. Parameswaran Sankaran IMSc
Date: Tue, 14 Oct 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: PhD Thesis defence: Compactness theorem for the spaces of distance measure spaces and Riemann surface laminations.
Speaker: Mr. Divakaran D.
Date: Tue, 14 Oct 2014
Time: 11.00 a.m. - 12.00 noon
Venue: Lecture Hall I, Department of Mathematics
Gromov’s compactness theorem for metric spaces, a compactness theorem for the
space of compact metric spaces equipped with the Gromov-Hausdorff distance, is
a landmark theorem with many applications. We give a generalisation of this
result - more precisely, we prove a compactness theorem for the space of
distance measure spaces equipped with the generalised
Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple
$(X, d, μ)$, where $(X, d)$ forms a distance space (a generalisation of a metric
space where, we allow the distance between two points to be infinity) and μ is
a finite Borel measure.
Using this result, we prove that the Deligne-Mumford compactifiaction is the completion of the moduli space of Riemann surfaces under generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov’s compactness theorem for pseudo-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov’s compactness theorem for pseudo-holomorphic curves is an important tool in symplectic topology, its applicability is limited due to the non-existence of a general method to construct pseudo-holomorphic curves. Considering a more general class of domains (in place of Riemann surfaces) is likely to be useful. Riemann surface laminations are a natural generalisation of Riemann surfaces. Theorems such as the uniformisation theorem for surface laminations due to Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and the topological classification of “almost all” leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations,as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.
Title: Riesz potentials and Sobolev embedding theorems
Speaker: Rahul Garg (Technion, Haifa, Israel)
Date: Tue, 23 Sep 2014
Time: 4 pm
Venue: LH-1, Mathematics Department
I shall begin with the mapping properties of classical Riesz potentials acting on $L^p$-spaces. After reviewing the literature, I shall present our new “almost” Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential) in the so-called supercritical exponent. Finally, I shall show how one could apply these estimates to deduce Sobolev embedding theorems. This is joint work with Daniel Spector and is available on arXiv:1404.1563.
Title: An interesting result about finite dimensional complex semi-simple algebras
Speaker: Srikanth Tupurani IMSc
Date: Mon, 22 Sep 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
I will prove the following interesting result about finite dimensional complex semi-simple algebra. Let A be a finite-dimensional complex semi-simple algebra without an M_{2}(C) summand and let S be an involutive unital C-algebra anti-automorphism of A. Then there exists an element a in A such that a and S(a) generate A as an algebra. In the proof I will use some basic results in linear algebra
Title: Derived Category of N-complexes
Speaker: Samarpita Ray IISc
Date: Mon, 22 Sep 2014
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
The notion of N-complexes goes back to a 1996 preprint of Kapranov, in which he considered chains of composable morphisms satisfying d^N = 0 (as opposed to the usual $d^2 = 0$ which gives the usual chain complexes). Later on, much of the usual homological algebra for chain complexes (homotopy of morphisms, spectral sequences, etc) was generalized to N-complexes, mostly by Dubois-Violette. Recently in 2014 there has been a burst of interest in this topic with work of Iyama,Kato,Miyachi defining the corresponding N-derived category. We shall begin the talk with simple definitions of $N$ complexes and their homology groups. Then gradually we will move to the paper of Iyama explaining derived category of $N$-complexes.
Title: Eigenfunctions Seminar: Proper holomorphic maps between bounded symmetric domains
Speaker: Gautam Bharali and Jaikrishnan Janardhanan (IISc Mathematics)
Date: Fri, 19 Sep 2014
Time: 2:15 – 4:30 pm (with a 15 minute break at 3:15)
Venue: LH-1, Mathematics Department
The goal of these talks is to discuss the history, the circumstances, and (the main aspects of) the proof of a recent result of ours, which says: a proper holomorphic map between two nonplanar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism.
Results of this character are motivated by a remarkable theorem of H. Alexander about proper holomorphic self-maps of the Euclidean ball. The first part of the talk will focus on the meanings of the terms in the above theorem. We will give an elementary proof that the automorphism group of a bounded symmetric domain acts transitively on it. But the existence of an involutive symmetry at each point is restrictive in many other ways. For instance, it yields a complete classification of the bounded symmetric domains. This follows from the work of Elie Cartan.
Earlier results in the literature that are special cases of our result focused on different classes of domains in the Cartan classification, and have mutually irreconcilable proofs. One aspect of our work is a unified proof: a set of techniques that works across the Cartan classification. One of those techniques is the use of an algebraic gadget: the machinery of Jordan triple systems.
We will present definitions and examples related to this machinery in the first part of the talk. In the second part of the talk, we shall state a result on finite-dimensional Jordan triple systems that is essential to our work. After this, we shall discuss our proof in greater depth.
Title: HARMONIC ANALYSIS ON EXPONENTIAL HOMOGENEOUS SPACES AND DIFFERENTIAL OPERATORS
Speaker: Ali Baklouti University of Sfax, Sfax, Tunisia
Date: Tue, 02 Sep 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
See https://math.iisc.ac.in/~arvind/Baklouti-Abst.Bangalore14.pdf
Title: On a theorem of Titchmarsh
Speaker: Radouan Daher University of Hassan II, Casablanca, Morocco.
Date: Tue, 26 Aug 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this talk we discuss about the Fourier transforms of functions satisfying Lipschitz conditions of certain order. We cover Fourier transforms on Euclidean spaces, non compact symmetric spaces and also certain hyper groups.
Title: Dynamics via quasi-homomorphisms
Speaker: Siddhartha Gadgil IISc
Date: Fri, 22 Aug 2014
Time: 11:00 a.m. - 12:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
In dynamics, one studies properties of a map from a space to itself, up to change of coordinates in the space. Hence it is important to understand invariants of the map under change of coordinates. An important such invariant is Poincare’s rotation number, associated to invertible maps from the circle to itself. Ghys and others have abstracted the rotation number to give many other important invariants of dynamical systems by viewing it in terms of so called quasi-homomorphisms. Quasi-homomorphisms are like homomorphisms, except that a bounded error is allowed in the definition. In this expository lecture I will introduce quasi-homomorphisms and show some interesting properties, constructions and application, including an alternative construction of the real numbers (due to Ross Street). I shall then show how these can be used to construct dynamical invariants, in particular the rotation number. Only basic algebra and analysis are needed as background for this lecture.
Title: Eigenfunctions Seminar: The Littlewood–Offord and related problems
Speaker: Manjunath Krishnapur (IISc Mathematics)
Date: Fri, 22 Aug 2014
Time: 2:15 – 4:30 pm (with a 15 minute break at 3:15)
Venue: LH-1, Mathematics Department
Given n non-zero real numbers v_1,…,v_n, what is the maximum possible number of subsets {i_1,…,i_k} that can have the same subset sum v_{i_1}+…+v_{i_n}? Littlewood and Offord raised the question, showed that the answer is at most 2^nlog(n)/\sqrt{n} and conjectured that the log(n) can be removed. Erdos proved the conjecture and raised more questions that have continued to attract attention, primarily relating the arithmetic structure of v_1,…,v_n and the maximum number of subsets with a given subset-sum.
In the first lecture, we shall review and prove several of these results (due to Littlewood–Offord, Erdos, Moser, Stanley, Halasz, Tao–Vu, Rudelson–Vershynin…) and show an application to random matrices.
In the second part, we start with a question in random polynomials and see how it leads to a variant of the Littlewood–Offord problem that has not been considered before.
Most of the material presented should be accessible to undergraduate students. Much of the lecture is an outcome of my joint study in summer with Sourav Sarkar (ISI, Kolkata).
Title: Computational representation theory of finite monoids
Speaker: Professor Nicolas M Thiery Laboratoire De Research En Informatique, Universite, Paris
Date: Wed, 20 Aug 2014
Time: 3:30 pm to 4:30 pm
Venue: Lecture Hall I, First Floor, Department of Mathematics, IISc
Title: Crystal approach to affine Schubert calculus
Speaker: Professor Anne Schilling Department of Mathematics University of California USA
Date: Wed, 20 Aug 2014
Time: 2:00 pm to 3:00 pm
Venue: Lecture Hall I, First Floor, Department of Mathematics, IISc
Title: Semigroups and Harmonic Analysis
Speaker: Luz Roncal Universidad de La Rioja, Logrono, Spain
Date: Tue, 19 Aug 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
By using the language of diffusion semigroups, it is possible to define and study classical operators in harmonic analysis. We introduce and develop this idea, and show two applications. First, we investigate fractional integrals and Riesz transforms in compact Riemannian spaces of rank one. Secondly, we carry out the study of operators associated with a discrete Laplacian, namely the fractional Laplacian, maximal heat and Poisson semigroups, square functions, Riesz transforms and conjugate harmonic functions.
Title: Asymptotic behavior of positive harmonic functions in some unbounded domains
Speaker: Subhroshekhar Ghosh (Princeton)
Date: Mon, 18 Aug 2014
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
This talk will concern asymptotic estimates at infinity for positive harmonic functions in a large class of non-smooth unbounded domains. These are domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g., various paraboloids and horns. We will also mention a few related results in probability, e.g harmonic measure (distribution of exit position of Brownian motion) estimates.
Title: HARISH-CHANDRA ( 1923 - 1983 )
Speaker: Prof. Alladi Sitaram IISc
Date: Thu, 07 Aug 2014
Time: (1) 2:00 - 2:45 p.m. (2) 3:15 - 4:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Harish-Chandra is a mathematician whose name in the history of mathematics is permanently etched in stone. I will begin by giving a brief account of his life, and in the second half of the lecture, explain some of the technical terms used in the first half. This lecture is aimed mainly at Ph.D. students.
Title: Vector-valued weighted inequalities for one-sided maximal functions
Speaker: Saurabh Kumar Srivastava IISER Bhopal
Date: Tue, 05 Aug 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this talk we shall briefly discuss the theory of one-sided maximal functions. One-sided maximal function is a variant of the classical Hardy-Littlewood maximal function. The theory of one-sided maximal functions is an active research area. Specially, in the past two decades there has been a lot of research activities in this area.
Title: Towards Rigorous Factoring
Speaker: Dr. Ramarathnam Venkatesan Principal Researcher, Microsoft Research
Date: Thu, 31 Jul 2014
Time: 4:00 pm
Venue: CSA Seminar Hall (Room No. 254, First Floor)
Title: Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence
Speaker: Emmanuel Kwame Essel University of Cape Coast, Ghana
Date: Wed, 30 Jul 2014
Time: 11:30a.m. - 12:30 p.m.
Venue: Lecture Hall V, Department of Mathematics
In this talk we consider a linear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’’s method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.
Title: HOMOTOPY AND HOMOLOGY OF NONCOMMUTATIVE SPACES
Speaker: Snigdhayan Mahanta University of Muenster
Date: Mon, 28 Jul 2014
Time: 3:30 - 4:30pm
Venue: Lecture Hall V, Department of Mathematics
In noncommutative geometry (NCG) one typically treats unital C-algebras as noncommutative compact spaces via Gelfand-Naimark duality. In various applications of NCG to problems in geometry or topology it is customary to first reformulate these problems in terms of certain (co)homology theories for noncommutative spaces. The celebrated Baum-Connes conjecture, that reduces the Novikov conjecture to an assertion in bivariant K-theory, is a prime example of this principle. However, the category of C-algebras is well-known to be de ficient from the viewpoint of homotopy theory or index theory. In this talk I am going to first survey certain (co)homology theories for noncommutative spaces, then present my proposed solution to the aforementioned problem, and finally (time permitting) discuss some applications. I will try to keep it non-technical and accessible to a wide range of audience.
Title: A transference result between radial Fourier multipliers and zonal multipliers on the unit sphere.
Speaker: Sriram Alladi
Date: Thu, 24 Jul 2014
Time: 4:00 pm
Venue: LH-1
Title: Non-backtracking spectrum of random graphs
Speaker: Charles Bordenave Universite de Toulouse
Date: Thu, 24 Jul 2014
Time: 11:30a.m. - 12:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
The non-backtracking matrix of a graph is a non-symmetric matrix on the oriented edge of a graph which has interesting algebraic properties and appears notably in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. It has also be proposed recently in the context of community detection. In this talk, we will study the largest eigenvalues of this matrix for the Erdos-Renyi graph G(n,c/n) and other simple inhomogeneous random graphs. This is a joint work with Marc Lelarge and Laurent Maussouli.
Title: PhD Thesis colloquium: Mechanisation of Knot Theory
Speaker: T. V. H. Prathamesh IISc
Date: Tue, 22 Jul 2014
Time: 11:00am - 12:00pm
Venue: Lecture Hall III, Department of Mathematics
Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many way of mechanising are: (1) generating results using Automated Theorem Provers, (2) Interactive theorem proving in a Proof Assistant which involves a combination of user intervention and automation.
In the first part of this thesis, we reformulate the question of equivalence of two Links in First Order Logic using Braid Groups. This is achieved by developing a set of Axioms whose canonical model is the Infinite Braid Group. This renders the problem of distinguishing Knots and Links, amenable to implementation in First Order Logic based Automated Theorem provers. We further state and prove results pertaining to Models of Braid Axioms.
The second part of the thesis deals with formalising Knot Theory in Higher Order Logic using the Isabelle Proof Assistant. We formulate equivalence of Links in Higher Order Logic. We obtain a construction of Kauffman Bracket in the Isabelle Proof Assistant. We further obtain a machine checked proof of invariance of Kauffman Bracket.
Title: Holonomy fibers of complex projective structures
Speaker: Subhojoy Gupta Caltech
Date: Thu, 17 Jul 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
As a sequel to the talk on July 16th, I shall talk about a recent result (joint work with Shinpei Baba) concerning fibers of the holonomy map from P(S) to the PSL(2,C) character variety. The proof involves some three-dimensional hyperbolic geometry, and train-tracks.
Title: On the rank of symmetric random matrices
Speaker: Rahul Roy ISI, Delhi
Date: Thu, 17 Jul 2014
Time: 11:30 - 12:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
We discuss the question of rank of symmetric and non-symmetric matrices when the entries are i.i.d. non-degenerate random variables. In particular we show that for an $n \times n$ symmetric matrix the probability that it is singular is of the order $O(n^{- (1/4) + \epsilon})$. This is joint work with Paulo Manrique and Victor Perez-Abreu.
Title: The Beginnings of Mathematics in India
Speaker: P. P. Divakaran CMI
Date: Thu, 17 Jul 2014
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
As in other ancient civilisations, mathematics in India began in counting and drawing: numbers and plane geometry. The earliest textually recorded geometry is that of the Sulbasutra (around 800 BC onwards) which are manuals for the construction of Vedic altars, its mathematical high point being the `theorem of the diagonal’, Pythagoras’ theorem to you and me. I will state its earliest formulation and touch briefly on some of the ideas around it as found in the texts.
Title: PhD Thesis colloquium: Shortest length geodesics on closed hyperbolic surfaces
Speaker: Bidyut Sanki IISc
Date: Thu, 17 Jul 2014
Time: 10:00am - :11:00am
Venue: Lecture Hall III, Department of Mathematics
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
Title: Thurston-Teichmller theory and the dynamics of grafting
Speaker: Subhojoy Gupta Caltech
Date: Wed, 16 Jul 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
Complex projective structures on a surface S arise in the complex-analytic description of Teichmller space. The operation of grafting parametrizes the space P(S) of such structures, and yields paths in Teichmller space called grafting rays. I shall introduce these, and describe a result concerning their asymptotic behavior. This talk shall be mostly expository.
Title: An introduction to preferential attachment schemes
Speaker: Sunder Sethuraman (University of Arizona, USA)
Date: Tue, 15 Jul 2014
Time: 10 am and 2 pm
Venue: LH-3, Mathematics Department
Informally, a preferential attachment scheme is a dynamic (reinforcement) process where a network is grown by attaching new vertices to old ones selected with probability proportional to their weight, a function of their degrees. In these two talks, we give an introduction to these schemes and discuss a couple of approaches to the study of the large scale degree structure of these graphs. One fruitful approach is to view the scheme in terms of branching processes. Another is to understand it in terms of Markov decompositions and fluid limits.
Title: On shift-like automorphisms of C^k
Speaker: Sayani Bera IISc
Date: Tue, 15 Jul 2014
Time: 4:00 - 5:00pm
Venue: Lecture Hall III, Department of Mathematics
We will use transcendental shift-like automorphisms of C^k ,k>2 to construct two examples of non-degenerate entire mappings with prescribed ranges. The first example generalizes a result of Dixon-Esterle in C^2, i.e., we construct an entire mapping of C^k, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. The second example shows the existence of a Fatou-Bieberbach domain in C^k, k>2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay-Rudin.
Title: PhD Thesis defence: A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities
Speaker: Ms. Kamana Porwal IISc
Date: Mon, 14 Jul 2014
Time: 11:00 a.m. - 12:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Unlike the partial differential equations, the solutions of variational inequalities exhibit singularities even when the data is smooth due to the existence of free boundaries. Therefore the numerical procedure of these problems based on uniform refinement becomes inefficient due to the loss of the order of convergence. A popular remedy to enhance the efficiency of the numerical method is to use adaptive finite element methods based on computable a posteriori error bounds. Discontinuous Galerkin methods play a very important role in the local mesh adaptive refinement techniques.
The main focus in this thesis has been on the derivation of reliable and efficient error bounds for the discontinuous Galerkin methods applied to elliptic variational inequalities. The variational inequalities can be split into two kinds, namely, inequalities of the first kind and the second kind. We study an elliptic obstacle problem and a Signorini contact problem in the category of the first kind, while the frictional plate contact problem in the category of the fourth order variational inequalities of second kind. The mathematical analysis of error estimation in this class of problems crucially depends on a suitable nonlinear smoothing function that enriches the smoothness of the numerical solution. Another remarkable advantage of discontinuous Galerkin methods has been realized in the applications to higher order problems. Numerical experiments support the theoretical results and exhibit optimal convergence.
Title: Morse theory on the space of paths on Homogenous spaces
Speaker: Chaitanya Senapathi TIFR
Date: Fri, 11 Jul 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
1961 Frankel in a novel approach used the curvature of Complex Projective space to show complex submanifolds of complimentary dimension intersect. Based on this approach Scheon and Wolfson reproved the Barth-Lefshetz type theorems using the Morse Theory on the space of paths, in the case the ambient space is a Hermitian Symmetric Space. In this talk we describe how to extend their work to a much larger class of Homogenous spaces.
Title: An Introduction to Minimal Surfaces
Speaker: Mr. Devang S Ram Mohan IISc
Date: Thu, 10 Jul 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
In this presentation we briefly recall some results on harmonic maps. Subsequently, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, certain partial results regarding the uniqueness of such a soap film are discussed.
Title: Dynamics of distal group actions
Speaker: Riddhi Shah (Jawaharlal Nehru University, New Delhi)
Date: Wed, 25 Jun 2014
Time: 3 pm
Venue: LH-1, Mathematics Department
An automorphism $T$ of a locally compact group is said to be distal if the closure of the $T$-orbit of any nontrivial element stays away from the identity. We discuss some properties of distal actions on groups. We will also relate distal groups with behaviour of powers of probability measures on it.
Title: PhD Thesis defence: Some Problems in Multivariable Operator Theory
Speaker: Mr. Santanu Sarkar (IISc)
Date: Mon, 23 Jun 2014
Time: 11.30 a.m.
Venue: Lecture Hall I, Department of Mathematics
This thesis investigates two different types of problems in multi-variable operator theory. The first one deals with the defect sequence for a contractive tuple and the second one deals with wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in (I) and (II) below.
I. We introduce the defect sequence for a contractive tuple of
Hilbert space operators and investigate its properties. We show that there
are upper bounds for the defect dimensions. The upper bounds are
different in the non-commutative and in the commutative case. The tuples
for which these upper bounds are obtained are called maximal contractive
tuples. We show that the creation operator tuple on the full Fock space
and the co-ordinate multipliers on the
Drury-Arveson space are maximal. We also show that if M is an
invariant subspace under the creation operator tuple on the full Fock
space, then the restriction is always maximal. But the situation is
starkly different for co-invariant subspaces. A characterization for a
contractive tuple to be maximal is obtained. We define the notion of
maximality for a submodule of the Drury-Arveson module on the
d-dimensional unit ball. For $d=1$, it is shown that every submodule of the
Hardy module over the unit disc is maximal. But for $d>2$, we prove that any
homogeneous submodule or a submodule generated by polynomials is not
maximal. We obtain a characterization of maximal submodules of the
Drury-Arveson module. We also study pure tuples and see how the defect
dimensions play a role in their irreducibility.
II. We investigate the following question : Let $(T_1, ....., T_n)$ be a
commuting $n$-tuple of bounded linear operators on a Hilbert space $H$. Does
there exist a generating wandering subspace $W$ for $(T_1, ....., T_n)$? We
got some affirmative answers for the doubly commuting
invariant subspaces of the Bergman space and the Dirichlet space over the
unit polydisc. We show that for any doubly commuting invariant subspace
of the Bergman space or the Dirichlet space over polydisc, the tuple
consisting of restrictions of co-ordinate multiplication operators
always possesses a generating wandering subspace.
Title: Analysis based fast algorithms for applied mathematics
Speaker: Prof. Sivaram Ambikasaran (Courant Institute of Mathematical Sciences, NYU)
Date: Thu, 12 Jun 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
The dominant problem in applied mathematics is the application of linear operators and solving linear equations. Dense linear systems arise in numerous applications: Boundary integral equations for elliptic partial differential equations, covariance matrices in statistics, Bayesian inversion, Kalman filtering, inverse problems, radial basis function interpolation, density functional theory, multi-frontal solvers for sparse linear systems, etc. As the problem size increases, large memory requirements scaling as $O(N^2)$ and extensive computational time to perform matrix algebra scaling as $O(N^2)$ or $O(N^3)$ make computations impractical, where $N$ is the underlying number of degrees of freedom. This talk will discuss new fast algorithms that scale as $O(N)$ for a class of problems, given a prescribed tolerance. Applications in the context of Gaussian processes, Integral equations for electromagnetic scattering, Symmetric factorization for Brownian dynamics, Bayesian inversion, Kalman filtering, multi-frontal solvers will also be presented.
Title: Constant Mean Curvature surfaces
Speaker: Prof. Rukmini Dey (HRI)
Date: Thu, 05 Jun 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this expository talk, I will focus on CMC surfaces which are not minimal surfaces. I will talk about the link between CMC surfaces and integrable systems. I will then talk about how CMC surfaces fall out of a constrained optimization problem. I will give examples of rotational and helicoidal CMC surfaces and the isometry between them. If time permits I will talk about the classification of CMC surfaces, namely its Weirstrass representation.
Title: A dynamic model of capital inflow and financial crisis in a two country and multi-country framework over infinite time horizon
Speaker: Gopal Basak (ISI, Kolkata)
Date: Mon, 02 Jun 2014
Time: 2:00-3:00 pm
Venue: LH-I, department of mathematics
We construct a model of capital inflow in a two and multi-country framework. A capital-scarce country, typically a developing country with a high return on capital borrows from a capital-rich country, typically a developed country to finance domestic investment. In the process both the countries gain, raising the world welfare. We formulate the problem in terms of utility maximization with respect to both develop and developing countries’ perspective over infinite time horizon and numerically solve for optimal interest rate, borrowing/lending amount, exchange rate using dynamic programming principle.
Title: Geometric Quantization and Coherent States
Speaker: Prof. Rukmini Dey (HRI)
Date: Fri, 30 May 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this expository talk, I will first review prequantization of symplectic manifolds. I will then talk of polarizations and geometric quantization. I will then focus on geometric quantization of Kahler manifolds and the Rawnsley coherent states. If time permits I will talk of coadjoint orbits and coherent states.
Title: PhD Thesis colloquium: Compactness theorem for the spaces of distance measure spaces and Riemann surface laminations.
Speaker: Mr. Divakaran D.
Date: Fri, 23 May 2014
Time: 11.00 a.m. - 12.00 noon
Venue: Lecture Hall I, Department of Mathematics
Gromov’s compactness theorem for metric spaces, a compactness theorem for the
space of compact metric spaces equipped with the Gromov-Hausdorff distance, is
a landmark theorem with many applications. We give a generalisation of this
result - more precisely, we prove a compactness theorem for the space of
distance measure spaces equipped with the generalised
Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple
$(X, d, μ)$, where $(X, d)$ forms a distance space (a generalisation of a metric
space where, we allow the distance between two points to be infinity) and μ is
a finite Borel measure.
Using this result, we prove that the Deligne-Mumford compactifiaction is the completion of the moduli space of Riemann surfaces under generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov’s compactness theorem for pseudo-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov’s compactness theorem for pseudo-holomorphic curves is an important tool in symplectic topology, its applicability is limited due to the non-existence of a general method to construct pseudo-holomorphic curves. Considering a more general class of domains (in place of Riemann surfaces) is likely to be useful. Riemann surface laminations are a natural generalisation of Riemann surfaces. Theorems such as the uniformisation theorem for surface laminations due to Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and the topological classification of “almost all” leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations,as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.
Title: Roots of Units, or A sum worthy of Gauss
Speaker: Chandan Singh Dalwat HRI
Date: Tue, 06 May 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
One looks at a certain sum G involving the p-th roots of unity (where p is a prime number), called the quadratic gaussian sum. It is easy to see that G^2=p, which means that G itself is either the positive or the negative square root of p. Which one ? It took Gauss many years to find the answer and to prove the result. Since then some other proofs of this result have been given, and it has become the central example of what is called the root number of an L-function. So the result is very important.
Title: PhD Thesis colloquium: Contractivity and Complete contractivity
Speaker: Mr. Avijit Pal (IISc)
Date: Fri, 02 May 2014
Time: 2:00 - 3:00pm
Venue: Lecture Hall I, Department of Mathematics
We study homomorphisms $\rho_{V}$($\rho_{V}(f)=\left (
\begin{smallmatrix}
f(w)I_n& \sum_{i=1}^{m} \partial_if(w)V_{i} \\
0 & f(w)I_n
\end{smallmatrix}\right ), f \in \mathcal O(\Omega_\mathbf A)$) defined on
$\mathcal
O(\Omega_\mathbf A)$, where $\Omega_\mathbf A$ is a bounded
domain of the form:
\(\begin{eqnarray*}
\Omega_\mathbf A & := &\{(z_1 ,z_2, \ldots, z_m) :\|z_1 A_1
+\cdots + z_mA_m \|_{\rm op} < 1\}
\end{eqnarray*}\)
for some choice of a linearly independent set of $n\times n$
matrices $\{A_1, \ldots, A_m\}.$
From the work of V. Paulsen and E. Ricard, it follows that if
$n\geq 3$ and $\mathbb B$ is any ball in $\mathbb C^m$, then there exists
a contractive linear map which is not complete
contractivity. It is known that contractive homomorphisms of the
disc and the bi-disc algebra are completely contractive, thanks
to the dilation theorem of B. Sz.-Nagy and Ando. However, an
example of a contractive homomorphism of the (Euclidean) ball
algebra which is not completely contractive was given by G. Misra. The
characterization of those balls in $\mathbb C^2$ for which
contractive linear maps which are always comletely contractive
remained open. We answer this question.
The class of homomorphism of the form $\rho_V$ arise from
localization of operators in the Cowen-Douglas class of $\Omega.$ The
(complete) contractivity of a homomorphism in this class
naturally produces inequalities for the curvature of the
corresponding Cowen-Douglas bundle. This connection and some of
its very interesting consequences are discussed.
Title: Cryptanalysis of RSA Variants and Implicit Factorization
Speaker: Santanu Sarkar CMI
Date: Thu, 01 May 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
The famous RSA public key cryptosystem is possibly the most studied topic in cryptology research. For efficiency and security purposes, some variants of the basic RSA has been proposed. In this talk, we will first discuss two such RSA variants: Prime Power RSA and Common Prime RSA.
Title: Brownian excursions into an interval
Speaker: Navin Kashyap IISc, Bangalore
Date: Wed, 30 Apr 2014
Time: 2:00pm
Venue: LH-1, Department of Mathematics, IISc
The dynamics of excursions of Brownian motion into a set with more than one boundary point , which no longer have the structure of a Poisson process, requires an extension of Ito’s excursion theory , due to B.Maisonneuve. In this talk we provide a `bare hands’ calculation of the relevant objects - local time, excursion measure - in the simple case of Brownian excursions into an interval (a,b), without using the Maisonneuve theory. We apply these computations to calculate the asymptotic distribution of excursions into an interval, straddling a fixed time t, as t goes to infinity.
Title: An Introduction to Minimal Surfaces
Speaker: Mr. Devang S Ram Mohan IISc
Date: Tue, 29 Apr 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this presentation we briefly recall some results on harmonic maps. Subsequently, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, certain partial results regarding the uniqueness of such a soap film are discussed.
Title: PhD Thesis defence: On Walkup's class of manifolds and tight triangulations
Speaker: Mr. Nitin Singh
Date: Mon, 28 Apr 2014
Venue: Lecture Hall - I, Department of Mathematics, IISc
By a triangulation of a topological space $X$, we mean a simplicial complex $K$ whose geometric carrier is homeomorphic to $X$. The topological properties of the space can be expressed in terms of the combinatorics of its triangulation. Simplicial complexes have gained in prominence after the advent of powerful computers as they are especially suitable for computer processing. In this regard, it is desirable for a triangulation to be as efficient as possible. In this thesis we study different notions of efficiency of triangulations, namely, minimal triangulations, tight triangulations and tight neighborly triangulations.
a) Minimal Triangulations: A triangulation of a space is called minimal if
it contains minimum number of vertices among all triangulations of the
space. In general, it is hard to construct a minimal triangulation, or to
decide if a given triangulation is minimal. In this work, we present
examples of minimal triangulations of connected sums of sphere bundles
over the circle.
b) Tight Triangulations: A simplicial complex (triangulation) is called
tight w.r.t field $\mathbb{F}$ if for any induced subcomplex, the induced
homology maps from the subcomplex to the whole complex are all injective.
We normally take the field to be $\mathbb{Z}_2$. Tight triangulations have
several desirable properties. In particular any simplex-wise linear
embedding of a tight triangulation (of a PL manifold) is “as convex” as
possible. Conjecturally, tight triangulations of manifolds are minimal,
and it is known to be the case for most tight triangulations of manifolds.
Examples of tight triangulations are extremely rare, and in this thesis we
present a construction of an infinite family of tight triangulations,
which is only the second of its kind known in literature.
c) Tight Neighborly Triangulations: For dimensions three or more, Novik and Swartz obtained a lower bound on the number of vertices in a triangulation of a manifold, in terms of its first Betti number. Triangulations that meet this bound are called tight neighborly. The examples of tight triangulations constructed in the thesis are also tight neighborly. In addition, it is proved that there is no tight neighborly triangulation of a manifold with first Betti number equal to two.
Title: HOMOGENEOUS OPERATORS AND SOME IRREDUCIBLE REPRESENTATIONS OF THE MOBIUS GROUP
Speaker: Mr. Chandramouli. K IISc
Date: Mon, 28 Apr 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this report, after recalling the definition of the M\\obius group, we define homogeneous operators, that is, operators $T$ with the property $\\varphi(T)$ is unitarily equivalent to $T$ for all $\\varphi$ in the M\\obius group and prove some properties of homogeneous operators. Following this, (i) we describe isometric operators which are homogeneous. (ii) we describe the homogeneous operators in the Cowen-Douglas class of rank 1. Finally, Multiplier representations which occur in the study of homogeneous operators are discussed. Following the proof of Kobayashi, the multiplier representations are shown to be irreducible.
Title: Eigenfunctions Seminar: Optimal Design Problems
Speaker: M. Vanninathan (TIFR-CAM, Bangalore)
Date: Fri, 25 Apr 2014
Time: 2 – 3 pm
Venue: LH-1, Mathematics Department
Motivated by applications, we introduce a class of optimization problems with constraints. Difficulties in solving these problems are highlighted. Mathematical developments to overcome these difficulties are discussed.
Title: Eigenfunctions Seminar: Gromov hyperbolicity and the Kobayashi metric
Speaker: Harish Seshadri (IISc Mathematics)
Date: Fri, 25 Apr 2014
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
I will discuss the negative curvature properties of certain intrinsic metrics in complex analysis. The talk will be accessible to graduate students.
Title: PhD Thesis colloquium: On the structure of proper holomorphic mappings
Speaker: Mr. Jaikrishnan Janardhanan IISc
Date: Fri, 11 Apr 2014
Time: 2:30 - 3:30pm
Venue: Lecture Hall I, Department of Mathematics
The aim of this thesis is to give explicit descriptions
of the set of proper holomorphic mappings between two complex
manifolds with reasonable restrictions on the domain and target
spaces. Without any restrictions, this problem is intractable
even when posed for domains in $C^n$. We present results for
special classes of manifolds. We study, broadly, two types of
structure results:
I. Descriptive: Our first result is a structure theorem for finite proper holomorphic mappings between products of connected, hyperbolic open subsets of compact Riemann surfaces. A special case of our result follows from the techniques used in a classical result of Remmert and Stein adapted to the above setting. However, the presence of factors that have no boundary, or boundaries that consist of a discrete set of points, necessitates the use of alternative techniques. Specifically: we make use of a finiteness theorem of Imayoshi.
II. Rigidity:
A famous theorem of H. Alexander proves the non-existence of
non-injective proper holomorphic self-maps of the unit ball
in $C^n,\ n > 1$. Several extensions of this result for various
classes of domains have been established since the appearance
of Alexander’s result. Our first rigidity result establishes
the non-existence of non-injective proper holomorphic self-maps
of bounded, balanced pseudoconvex domains of finite type (in
the sense of D’Angelo) in $C^n,\ n > 1$. This generalizes a result
in $C^2$ due to Coupet, Pan and Sukhov to higher dimensions. In
higher dimensions, several aspects of their argument do not work.
Instead, we exploit the circular symmetry and a recent result in
complex dynamics by Opshtein.
Our next rigidity result is for bounded symmetric domains. We prove that a proper holomorphic map between two non-planar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. Furthermore, our proof of this result does not rely on the fine structure (in the sense of Wolf et al.) of bounded symmetric domains. Thus, we are able to apply some of our techniques to more general classes of domains. We illustrate this through a rigidity result for certain convex balanced domains whose automorphism groups are only assumed to be non-compact. For the bounded symmetric domains, our key tool is that of Jordan triple systems.
Title: A Characterisation of the Fourier transform on the Heisenberg group
Speaker: Dr. Lakshmi Lavanya IISc
Date: Fri, 11 Apr 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on R^n as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. Analogously, we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group. This is a joint work with Prof. S. Thangavelu.
Title: A Characterisation of the Fourier transform on the Heisenberg group
Speaker: Dr. Lakshmi IISc
Date: Fri, 11 Apr 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on R^n as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. Analogously, we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group. This is a joint work with Prof. S. Thangavelu.
Title: A formal proof of Feit-Higman theorem in Agda
Speaker: Mr. Balaji Department of Mathematics.
Date: Mon, 07 Apr 2014
Time: 10.00 a.m.
Venue: Lecture Hall I, Department of Mathematics
Generalised polygons are incidence structures that generalise projective planes (generalised triangles) and are closely related to finite groups. The Feit-Higman theorem states that any generalised n-gon is either an ordinary polygon or n = 2, 3, 4, 6, 8 or 12.
Title: Eigenfunctions Seminar: Recent progress towards the twin prime conjecture
Speaker: Satadal Ganguly (ISI, Kolkata)
Date: Fri, 28 Mar 2014
Time: 2 – 3 pm
Venue: LH-1, Mathematics Department
A great progress on the twin prime problem was made last year by Yitang Zhang and it made him famous in mathematical circles almost overnight. He proved the existence of a positive real number M such that there are infinitely many pairs of primes that differ by at most M. The twin prime conjecture predicts that M can be taken to be two. I shall give an overview of the works leading up to Zhang’s spectacular achievement.
Title: Eigenfunctions Seminar: Quadrature domains - an introduction
Speaker: Kaushal Verma (IISc Mathematics)
Date: Fri, 28 Mar 2014
Time: 3 – 4 pm
Venue: LH-1, Mathematics Department
Quadrature domains are those on which holomorphic functions satisfy a generalized mean value equality. The purpose of this talk will be to reflect on the Schwarz reflection principle and to understand how it leads to examples of quadrature domains.
Title: A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities
Speaker: Ms. Kamana Porwal IISc
Date: Tue, 25 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Discontinuous Galerkin methods have received a lot of attention in the past two decades since these are high order accurate and stable methods which can easily handle complex geometries, irregular meshes with hanging nodes and different degree polynomial approximation in different elements. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main tools to steer the adaptive mesh refinement. In this talk, we present a posteriori error analysis of discontinuous Galerkin methods for variational inequalities of the first kind and the second kind. Particularly, we study the obstacle problem and the Signorini problem in the category of variational inequalities of the first kind and the plate frictional contact problem for the variational inequality of the second kind. Numerical examples will be presented to illustrate the theoretical results.
Title: On the global character of the solutions of some nonlinear difference equations
Speaker: Esha Chatterjee Ghosh
Date: Fri, 21 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
Non linear difference equations of order more than one is a relatively new and dynamic area of research in applied mathematics. In particular, the theory of Rational difference equations emerged in the last two decades and is an ongoing challenging field of study. In this talk we will present a brief introduction to nonlinear difference equations with some applications to various fields. Some techniques used in the qualitative analysis of the global character of solutions will be outlined. We will also discuss the results of a recent paper in which the speaker has addressed and generalized, an open problem posed by E. Camouzis and G. Ladas.
Title: Critical Growth Elliptic Problem with Singular Discontinuous Nonlinearity in R^2
Speaker: Dr. Dhanya IISc
Date: Fri, 14 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Elliptic problems with discontinuous nonlinearity has its own difficulties due to the non-differentiability of the associated functional. Hence, a generalized gradient approach developed by Chang has been used to solve such problems if the associated functional is known to be Lipchitz continuous. In this talk, we will consider critical elliptic problem in a bounded domain in $\\mathbb{R}^2$ with the simultaneous presence of a Heaviside type discontinuity and a power-law type singularity and investigate the existence of multiple positive solutions. Here discontinuity coupled with singularity does not fit into any of the known framework and we will discuss our approach employed to obtain positive solutions.
Title: Discontinuous Galerkin Methods for Diffusion-Dominated Radiative Transfer Problems
Speaker: Prof. Dr. Guido Kanschat University of Heidelberg
Date: Tue, 11 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
While discontinuous Galerkin (DG) methods had been developed and analyzed in the 1970s and 80s with applications in radiative transfer and neutron transport in mind, it was pointed out later in the nuclear engineering community, that the upwind DG discretization by Reed and Hill may fail to produce physically relevant approximations, if the scattering mean free path length is smaller than the mesh size. Mathematical analysis reveals, that in this case, convergence is only achieved in a continuous subspace of the finite element space. Furthermore, if boundary conditions are not chosen isotropically, convergence can only be expected in relatively weak topology. While the latter result is a property of the transport model, asymptotic analysis reveals, that the forcing into a continuous subspace can be avoided. By choosing a weighted upwinding, the conditions on the diffusion limit can be weakened. It has been known for long time, that the so called diffusion limit of radiative transfer is the solution to a diffusion equation; it turns out, that by choosing the stabilization carefully, the DG method can yield either the LDG method or the method by Ern and Guermond in its diffusion limit. Finally, we will discuss an efficient and robust multigrid method for the resulting discrete problems.
Title: PhD Thesis colloquium: A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities
Speaker: Ms. Kamana Porwal IISc
Date: Tue, 11 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Discontinuous Galerkin methods have received a lot of attention in the past two decades since these are high order accurate and stable methods which can easily handle complex geometries, irregular meshes with hanging nodes and different degree polynomial approximation in different elements. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main tools to steer the adaptive mesh refinement. In this talk, we present a posteriori error analysis of discontinuous Galerkin methods for variational inequalities of the first kind and the second kind. Particularly, we study the obstacle problem and the Signorini problem in the category of variational inequalities of the first kind and the plate frictional contact problem for the variational inequality of the second kind. Numerical examples will be presented to illustrate the theoretical results.
Title: On the Gromov hyperbolicity of convex domains
Speaker: Herve Gaussier Fourier Institut, Grenoble
Date: Mon, 10 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
We discuss the Gromov hyperbolicity of the Kobayashi metric on smooth convex domains in Euclidean space. We prove that if the boundary contains a holomorphic disc then the domain is not hyperbolic. On the other hand we give examples of convex (but not strongly pseudoconvex) domains which are hyperbolic. This is a joint work with Harish Seshadri.
Title: On open three-manifolds
Speaker: Gerard Besson Fourier Institut, Grenoble
Date: Fri, 07 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
We will describe quite carefully two classes of open three-manifolds. For one of them almost nothing is known about its Riemannian geometry and we will state open questions. For the second class a theorem is available as well as a complete classification.
Title: On the Gromov hyperbolicity of convex domains
Speaker: Herve Gaussier Fourier Institut, Grenoble
Date: Wed, 05 Mar 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
We discuss the Gromov hyperbolicity of the Kobayashi metric on smooth convex domains in Euclidean space. We prove that if the boundary contains a holomorphic disc then the domain is not hyperbolic. On the other hand we give examples of convex (but not strongly pseudoconvex) domains which are hyperbolic. This is a joint work with Harish Seshadri.
Title: A series identity, possibly connected with a divisor problem, in Ramanujan's Lost Notebook
Speaker: Atul Dixit (Tulane University, USA)
Date: Wed, 05 Mar 2014
Time: 5 pm
Venue: LH-2, Mathematics Department
On page 336 in his lost notebook, S. Ramanujan proposes an identity that may have been devised to attack a divisor problem. Unfortunately, the identity is vitiated by a divergent series appearing in it. We prove here a corrected version of Ramanujan’s identity. While finding a plausible explanation for what may have led Ramanujan to consider a series that appears in this identity, we are led to a connection with a generalization of the famous summation formula of Vorono. One of the ramifications stemming from this work allows us to obtain a one-variable generalization of two double Bessel series identities of Ramanujan that were proved only recently. This is work in progress and is joint with Bruce C. Berndt, Arindam Roy and Alexandru Zaharescu.
Title: Support data and Balmer spectrum
Speaker: Dr. Umesh Dubey IISc
Date: Fri, 21 Feb 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Grothendieck and Verdier introduced the notion of triangulated category to extract homological informations. This structure appear in many branches of Mathematics. Balmer introduced the notion of spectrum of tensor triangulated category as classifying support data. This opens a way for geometric study of theses abstract categorical data. Balmer also defined the structure sheaf on this spectrum and as an application reconstructed large class of schemes from the category of perfect complexes associated with them. In this talk we’ll recall Balmer’s construction of spectrum and its application to reconstruction problem. We’ll also discuss computation of Balmer spectrum for some other tensor triangulated categories obtained in a joint work with Vivek Mallick.
Title: Eigenfunctions Seminar: How to integrate entire functions in polar coordinates?
Speaker: S. Thangavelu (IISc Mathematics)
Date: Fri, 14 Feb 2014
Time: 2 – 3 pm
Venue: LH-1, Mathematics Department
In this talk we address the problem of integrating entire functions (of several complex variables) in ‘polar coordinates’!
Title: Eigenfunctions Seminar: Tight triangulated manifolds
Speaker: Bhaskar Bagchi (ISI, Bangalore)
Date: Fri, 14 Feb 2014
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Wolfgang Kuhnel introduced the notion of tightness and conjectured that all tight triangulations of closed manifolds are strongly minimal. In a recent paper with B. Datta (European J. Comb. 36 (2014), 294–313), the speaker obtained a very general combinatorial criterion for tightness and found results in partial confirmation of this conjecture. We shall discuss these results.
Title: Powers in products of terms of Pell's and Pell-Lucas Sequences
Speaker: Prof Shanta Laishram Indian Statistical Institute Delhi
Date: Wed, 12 Feb 2014
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
It is known that there are only finitely many perfect powers in non degenerate binary recurrence sequences. However explicitly finding them is an interesting and a difficult problem for binary recurrence sequences. A recent breakthrough result of Bugeaud, Mignotte and Siksek states that Fibonacci sequences (F_n) given by F_0 = 0; F_1 = 1 and F_{n+2} = F_{n+1} + F_n for n >= 0 are perfect powers only for F_0 = 0; F_1 = 1; F_2 = 1; F_6 = 8 and F_12 = 144.
Title: On weak universality for directed polymers
Speaker: Prof Konstantin Khanin University of Toronto
Date: Mon, 10 Feb 2014
Time: 3:15 - 4:15 p.m.
Venue: Lecture Hall I, Department of Mathematics
The talk is based on a joint paper with T.Alberts and J.Quastel. We consider directed polymers in a random environment. However the inverse temperature is scaled with the length of a polymer. It turn out that with a proper critical scaling one can get a nontrivial universal behaviour for the partition function and the end point distribution. Moreover the limiting partition function distribution asymptotically converges to the Tracy-Widom law as the rescaled inverse temperature tends to infinity.
Title: From random matrices to long range dependence
Speaker: Prof Arijit Chakrabarty ISI Delhi
Date: Mon, 10 Feb 2014
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this talk. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, a phenomenon resembling that in the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues in a completely different way. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.
Title: Linear Algebra 1.5
Speaker: Prof. Dr. Martin Kreuzer University of Passau, Germany
Date: Fri, 07 Feb 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Most algebraist believe they know Linear Algebra. The purpose of this talk is to indicate that this is not necessarily true. We show a substantial amount of little known basic Linear Algebra and its connection to Algebraic Geometry, in particular to the theory of zero-dimensional subschemes of affine spaces, and to Computer Algebra, in particular to the task of solving zero-dimensional polynomial systems. Here are some questions which we will answer in this talk: What are the big kernel and the small image of an endomorphism? What are its eigenspaces and generalized eigenspaces if it has no eigenvalues? What is the kernel of an ideal? What is a commendable endomorphism? And what is a commendable family of endomorphisms? How is this connected to curvilinear and Gorenstein schemes? And how can you use this to solve polynomial systems?
Title: An algorithm for locating the nucleolus of assignment games.
Speaker: Prof T.E.S. Raghavan University of Illinois at Chicago
Date: Wed, 29 Jan 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In cooperative game theory , Shapley value and the nucleolus are two fundamental solution concepts. They associate a unique imputation for the players whose coalitional worths are given a priori.
Title: Policy improvement algorithm for zero sum two person stochastic games of perfect information in Cesaro payoffs.
Speaker: Prof T.E.S. Raghavan University of Illinois at Chicago
Date: Tue, 28 Jan 2014
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
If the data defining a problem and at least one solution to the problem lie in the same Archimedean ordered field induced by the data, we say that the problem has order field property. When this property holds one may hope to do only finitely many arithmetic operations on the data to arrive at one such a solution. For example if we start with rational data, the value and a pair of optimal strategies for the players in a zero sum two person game have rational entries. This was first noticed by Herman Weyl , and it was a precursor to solving them via the simplex method. For bimatrix games while Nash exhibited an equilibrium in mixed strategies, it was Vorobev and Kuhn who checked that the order field property holds for bimatrix games. Later Lemke and Howson gave the so called linear complementarity algorithm to locate an equilibrium pair in the same data field. For three person games, Nash with Shapley constructed simple counter examples for lack of order field property. In general stochastic games fail to have order field property.
Title: Constructions of Boolean Functions that are Significant in Cryptography and Coding Theory
Speaker: Prof Sumanta Sarkar ISI Kolkata
Date: Mon, 20 Jan 2014
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
A Boolean function is a mapping from the set of all binary n-tuples to the set {0, 1}. Boolean functions are important building blocks in designing secure cryptosystems known as stream ciphers. Boolean functions also form an important class of linear codes, known as the Reed-Muller codes. Over the last few decades, a lot of research has been done on Boolean function for its applications in cryptography and coding theory.
Title: Eigenfunctions Seminar: Statistical estimation of integrals over infinite measure spaces
Speaker: Krishna B. Athreya (Iowa State University, Ames, USA and IISc Mathematics)
Date: Fri, 17 Jan 2014
Time: 3:30 – 4:30 pm
Venue: LH-1, Mathematics Department
Given a L1 function f over an infinite measure space S how does one estimate the integral of f using statistical tools? In this talk we propose using regenerative sequences with heavy tails to do this. We obtain a consistent estimator and show the rate of convergence is slower than $1/\sqrt{n}$. When S is countable the SSRW works.
Title: Eigenfunctions Seminar: Approximation Theory and the Design of Fast Algorithms
Speaker: Nisheeth Vishnoi (Microsoft Research, Bangalore)
Date: Fri, 17 Jan 2014
Time: 2 – 3 pm
Venue: LH-1, Mathematics Department
I will talk about techniques to approximate real functions such as $x^s,$ $x^{-1}$ and $e^{-x}$ by simpler functions and how these results can be used in the design of fast algorithms. The key lies in the fact that such approximations imply faster ways to approximate primitives such as $A^sv,$ $A^{-1}v$ and $\exp({-A})v$, and in the computation of matrix eigenvalues and eigenvectors. Indeed, many fast algorithms reduce to the computation of such primitives, which have proved useful for speeding up several fundamental computations, such as random walk simulation, graph partitioning, solving linear systems of equations, and combinatorial approaches for solving semi-definite programs.
Title: PhD Thesis colloquium: Some Problems in Multivariable Operator Theory
Speaker: Mr. Santanu Sarkar (IISc)
Date: Mon, 13 Jan 2014
Time: 3:00 - 4:00pm
Venue: Lecture Hall I, Department of Mathematics
This thesis investigates two different types of problems in multi-variable operator theory. The first one deals with the defect sequence for a contractive tuple and the second one deals with wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in (I) and (II) below.
I. We introduce the defect sequence for a contractive tuple of
Hilbert space operators and investigate its properties. We show that there
are upper bounds for the defect dimensions. The upper bounds are
different in the non-commutative and in the commutative case. The tuples
for which these upper bounds are obtained are called maximal contractive
tuples. We show that the creation operator tuple on the full Fock space
and the co-ordinate multipliers on the
Drury-Arveson space are maximal. We also show that if M is an
invariant subspace under the creation operator tuple on the full Fock
space, then the restriction is always maximal. But the situation is
starkly different for co-invariant subspaces. A characterization for a
contractive tuple to be maximal is obtained. We define the notion of
maximality for a submodule of the Drury-Arveson module on the
d-dimensional unit ball. For $d=1$, it is shown that every submodule of the
Hardy module over the unit disc is maximal. But for $d>2$, we prove that any
homogeneous submodule or a submodule generated by polynomials is not
maximal. We obtain a characterization of maximal submodules of the
Drury-Arveson module. We also study pure tuples and see how the defect
dimensions play a role in their irreducibility.
II. We investigate the following question : Let $(T_1, ....., T_n)$ be a
commuting $n$-tuple of bounded linear operators on a Hilbert space $H$. Does
there exist a generating wandering subspace $W$ for $(T_1, ....., T_n)$? We
got some affirmative answers for the doubly commuting
invariant subspaces of the Bergman space and the Dirichlet space over the
unit polydisc. We show that for any doubly commuting invariant subspace
of the Bergman space or the Dirichlet space over polydisc, the tuple
consisting of restrictions of co-ordinate multiplication operators
always possesses a generating wandering subspace.
Title: L-4 norms of cusp forms in the level aspect
Speaker: Prof Rizwanur Khan Texas A&M university at Qatar
Date: Fri, 03 Jan 2014
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Modular forms, particularly cusp forms, are ubiquitous objects in mathematics. A natural way to understand a cusp form is to study its L-p norms, for in principle the distribution of a function can be recovered from the knowledge of its moments. In this talk I will describe a new bound for the L-4 norm of a cusp form of prime level q, as q tends to infinity. This work is joint with Jack Buttcane.
Title: On some generalizations and applications of Eisenstein-Dumas and Schonemann Criteria
Speaker: Prof Sudesh K. Khanduja IISER, Mohali
Date: Thu, 19 Dec 2013
Time: 11:30 - 12:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: Microlocal Analysis and Tomography
Speaker: Prof Todd Quinto Tufts University
Date: Thu, 12 Dec 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall V, Second floor(new wing), Department of Mathematics
We consider the reconstruction problem for limited angle tomography using filtered backprojection (FBP). We introduce microlocal analysis and use it to explain why the well-known streak artifacts are present at the end of the limited angular range. We explain how to mitigate the streaks and prove that our modified FBP operator is a standard pseudodifferential operator, and so it does not add artifacts. We provide reconstructions to illustrate our mathematical results. This is joint work with Juergen Frikel, Helmholtz Zentrum, Muenchen.
Title: On Frankl's conjecture
Speaker: Aniruddha Venkata Centre for Excellence in Basic Sciences, Mumbai
Date: Thu, 12 Dec 2013
Time: 2:00 - 3:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
A family of sets is said to be union-closed if the union of any two sets from the family remains in the family. Frankl’s conjecture, aka the union-closed sets conjecture, is the remarkable statement that for any finite union-closed family of finite sets, there exists an element that belongs to at least half of the sets in the family. Originally stated in 1979, it is still wide open. This will be an informal discussion on progress towards the conjecture.
Title: On the structure of proper holomorphic mappings
Speaker: Mr. Jaikrishnan Janardhanan IISc
Date: Wed, 11 Dec 2013
Time: 11.00 am - 12.00 noon
Venue: Lecture Hall I, Department of Mathematics
The aim of this thesis is to give explicit descriptions of the set of proper holomorphic mappings between two complex manifolds with reasonable restrictions on the domain and target spaces. Without any restrictions, this problem is intractable even when posed for domains in C^n. We present results for special classes of manifolds. We study, broadly, two types of structure results:
Title: Low dimensional projective groups
Speaker: Prof Mahan Mj RKM Vivekananda University
Date: Tue, 10 Dec 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Fundamental groups of smooth projective varieties are called projective groups. We shall discuss (cohomological) conditions on dimension that force such a group to be the fundamental group of a Riemann surface.
Title: New directions in complex analysis on symmetric domain
Speaker: Professor Harald Upmeier University of Marburg, Germany
Date: Wed, 04 Dec 2013
Venue: LH-I, department of Mathematics.
The Lectures will focus on topics that go beyond the classical framework of Hilbert spaces of holomorphic functions (Bergman spaces, Hardy spaces) on the unit ball or more general bounded symmetric domains. The main point is to include vector-valued functions and even cohomology classes for non-convex domains. The plan of the lectures is as follows:
Title: Preserving positivity for rank-constrained matrices
Speaker: Apoorva Khare (Stanford University, USA)
Date: Thu, 28 Nov 2013
Time: 3:30 pm
Venue: Lecture Hall I, Department of Mathematics
We study the problem of characterizing functions, which when applied entrywise, preserve Loewner positivity on distinguished submanifolds of the cone of positive semidefinite matrices. Following the work of Schoenberg and Rudin (and several others), it is well-known that entrywise functions preserving positivity in all dimensions are necessarily absolutely monotonic. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $n$. Such characterizations are known only in the $n=2$ case.
Title: Incidence geometries and Algebraic Codes
Speaker: Prof N.S. Narasimha Sastry ISI Bangalore
Date: Fri, 22 Nov 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Tits theory of spherical buildings gives a uniform geometric context to study all finite simple groups (except the alternating groups and the sporadic simple groups) and simple algebraic groups. More generally, the theory of buildings is central to the Lie theory associated with infinite root systems. These structures are `built of’ two basic objects: Coxeter complexes and Moufang generalized polygons. The generalized polygons (which includes projective planes) are rank 2 geometries (incidence geometries with 2 kind of objects - points and lines - and an incidence relation among them) whose classification is fundament, difficult and perhaps impossible.
Title: Incidence geometries and Algebraic Codes
Speaker: Prof N.S. Narasimha Sastry ISI Bangalore
Date: Fri, 15 Nov 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
Tits theory of spherical buildings gives a uniform geometric context to study all finite simple groups (except the alternating groups and the sporadic simple groups) and simple algebraic groups. More generally, the theory of buildings is central to the Lie theory associated with infinite root systems. These structures are `built of’ two basic objects: Coxeter complexes and Moufang generalized polygons. The generalized polygons (which includes projective planes) are rank 2 geometries (incidence geometries with 2 kind of objects - points and lines - and an incidence relation among them) whose classification is fundament, difficult and perhaps impossible.
Title: Admission Prices and Welfare in Queues
Speaker: D. Manjunath (IIT, Bombay)
Date: Mon, 11 Nov 2013
Time: 2:00pm
Venue: LH-II, Department of Mathematics, IISc
A multiqueue system serves a multiclass population. Classes differ in their valuation of time. Oblivious routing in which routing is not informed by current queue status or past decisions is assumed. First, we explore the structure of the routing fractions that maximise social welfare. We then analyse the case customer are strategic and the queues have an admission price. We then argue that admission prices can be set to achieve optimal routing.
Title: On Abhyankar's inertia conjecture
Speaker: Prof Manish Kumar ISI Bangalore
Date: Fri, 08 Nov 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
The conjecture is about various subgroup that can occur as the inertia subgroup of an etale Galois cover of an affine line at a point above infinity over an algebraically closed field of positive characteristic. I will explain this conjecture and mention some positive results supporting this conjecture.
Title: Beurling-Lax-Halmos Theorem and Operator Theory
Speaker: Prof Jaydeb Sarkar, ISI Bangalore
Date: Fri, 25 Oct 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
This talk will be an elementary introduction to (Hilbert) module approach to operator theory. We explore the relationship of the classical von Neumann-Wold decomposition theorem and the Beurling-Lax-Halmos theorem for isometries. We will also discuss a unified approach to the invariant subspace problem for bounded linear operators on Hilbert spaces. The talk will be accessible to general audience including graduate students.
Title: Beurling-Lax-Halmos Theorem and Operator Theory
Speaker: Prof Jaydeb Sarkar ISI Bangalore
Date: Fri, 25 Oct 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
This talk will be an elementary introduction to (Hilbert) module approach to operator theory. We explore the relationship of the classical von Neumann-Wold decomposition theorem and the Beurling-Lax-Halmos theorem for isometries. We will also discuss a unified approach to the invariant subspace problem for bounded linear operators on Hilbert spaces. The talk will be accessible to a general audience including graduate students.
Title: Homotopy type and volume of locally symmetric spaces
Speaker: Prof C S Aravinda TIFR CAM Bangalore
Date: Wed, 23 Oct 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
A conjecture of Gelander states that there is an effective triangulation in a compact deformation retract of a given locally symmetric space, giving linear bounds on the full homotopy type. This talk will explain some of this construction.
Title: Homotopy type and volume of locally symmetric spaces
Speaker: Prof C S Aravinda TIFR Bangalore
Date: Wed, 23 Oct 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
A conjecture of Gelander states that there is an effective triangulation in a compact deformation retract of a given locally symmetric space, giving linear bounds on the full homotopy type. This talk will explain some of this construction.
Title: Ramanujan and hypergeometric series
Speaker: Dr.K. Srinivasa Rao, FNASc.,FTNASc., Senior Professor (Retd.), IMSc
Date: Thu, 12 Sep 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Title: Introduction to the mean curvature flow
Speaker: Dr. Franck Djiideme University of Benin
Date: Wed, 11 Sep 2013
Time: 3.30 p.m.
Venue: Lecture Hall III, Department of Mathematics
The mean curvature flow is a process under which a submanifold is deformed in its mean curvature vector’s direction. This process received more attention since it is an efficient way to construct submanifold which minimizes the volume : it is the negative gradient flow of volume functional. In this firs talk. I will discuss about basic tools in the study of mean curvature and describe some examples .
Title: Discrete holomorphicity and quantized affine algebras
Speaker: Paul ZINN-JUSTIN Universite Pierre et Marie-Curie (Paris 6)
Date: Tue, 10 Sep 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
We consider non-local currents in the context of quantized affine algebras. In two special cases, these currents can be identified with configurations in the six-vertex and Izergin–Korepin nineteen-vertex models. Mapping these to their corresponding Temperley–Lieb loop models, we directly identify non-local currents with discretely holomorphic loop observables. In particular, we show that the bulk discrete holomorphicity relation and its recently derived boundary analogue are equivalent to conservation laws for non-local currents. Joint with Y. Ikhlef, R. Weston and M. Wheeler.
Title: Descent and Cohomological Descent
Speaker: Prof Nitin Nitsure, TIFR, Mumbai
Date: Mon, 26 Aug 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Around 1960, Grothendieck developed the theory of descent. The aim of this theory is to construct geometric objects on a base space – in particular bundles, sheaves and their sections – in terms a generalized covering space which is visualized to lie upstairs' over the base space. The objects over the base space are obtained by descending’ similar objects from the covering space. In late 1960s-early 1970s, Deligne addressed the problem of how to understand cohomology of the base space via cohomology of a covering, by this time descending cohomology classes (instead of just descending global sections of sheaves, which is the case of 0th cohomology). The theory developed by Deligne, known as `Cohomological Descent’ has found important applications to Hodge theory and to cohomology of algebraic stacks. In these two expository lectures, I will begin with a quick look at Grothendieck’s theory of descent, and then go on to give a brief introduction to Deligne’s theory of Cohomological Descent.
Title: Descent and Cohomological Descent
Speaker: Prof Nitin Nitsure, TIFR, Mumbai
Date: Fri, 23 Aug 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Around 1960, Grothendieck developed the theory of descent. The aim of this theory is to construct geometric objects on a base space – in particular bundles, sheaves and their sections – in terms a generalized covering space which is visualized to lie upstairs' over the base space. The objects over the base space are obtained by descending’ similar objects from the covering space. In late 1960s-early 1970s, Deligne addressed the problem of how to understand cohomology of the base space via cohomology of a covering, by this time descending cohomology classes (instead of just descending global sections of sheaves, which is the case of 0th cohomology). The theory developed by Deligne, known as `Cohomological Descent’ has found important applications to Hodge theory and to cohomology of algebraic stacks. In these two expository lectures, I will begin with a quick look at Grothendieck’s theory of descent, and then go on to give a brief introduction to Deligne’s theory of Cohomological Descent.
Title: Mixed Characteristic Polynomials and the Kadison-Singer Problem
Speaker: Prof Nikhil Srivastava Microsoft research, India
Date: Fri, 23 Aug 2013
Time: 2:00 - 4:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
The Kadison-Singer problem is a question in operator theory which arose in 1959 while trying to make Dirac’s axioms for quantum mechanics mathematically rigorous. Over the course of several decades, this question was reduced to several equivalent conjectures about finite matrices, and shown to have significant implications in applied mathematics, computer science, and various branches of pure mathematics.
Title: Projective Normality of G.I.T. quotient varieties modulo finite solvable groups and Weyl groups
Speaker: Prof Santosh Pattanayak, Weizmann Inst. of Science, Israel
Date: Wed, 21 Aug 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
We prove that for any finite dimensional vector space $V$ over an algebraically closed field $K$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the order $|G|$ is a unit in $K$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal{O}(1)^{\otimes |G|}$, where $\mathcal{O}(1)$ denotes the ample generator of the Picard group of $\mathbb P(V)$. We also prove that for the standard representation $V$ of the Weyl group $W$ of a semi-simple algebraic group of type $A_n , B_n , C_n , D_n , F_4$ and $G_2$ over $\mathbb{C}$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal{O}(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V.$
Title: Nonlinear Schrodinger equation and the twisted Laplacian
Speaker: Dr. Vijay Kumar Sohani IISc, Bangalore
Date: Fri, 16 Aug 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this lecture we will study the well posedness problem for the nonlinear Schr\\{o}dinger equation for the magnetic Laplacian on $\\R^{2n}$, corresponding to constant magnetic field, namely the twisted Laplacian on $\\C^n$. We establish the well posednes in certain first order Sobolev spaces associated to the twisted Laplacian, and also in $L^2(\\C^n)$.
Title: Needles, bushes, hairbrushes and polynomials
Speaker: Prof Malabika Pramanik UBC, Canada
Date: Wed, 14 Aug 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Points, lines and circles are among the most primitive and fundamental of mathematical concepts, yet few geometric objects have generated more beautiful and nontrivial mathematics. Deceptively simple in their formulation, many classical problems involving sets of lines or circles remain open to this day. I will begin with a sample that has spearheaded much of modern research, and explore connections with analysis, geometry, combinatorics - maybe also parallel parking.
Title: Control of a queuing system under the moderate deviation scaling
Speaker: Prof. Anup Biswas Technion, Israel
Date: Mon, 12 Aug 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In recent times, exponential type cost structure has become popular in control theory. In this talk we formulate and discuss a risk- sensitive type control problem for a multi-class queuing system under the moderate deviation scaling. It is known that the rate function corresponding to the moderate deviation scaling is of Gaussian type. This property of the rate function is often useful for mathematical analysis. We show that the limiting game corresponding to our control problem is solvable. Also the limiting game has a similarity to the well-studied Brownian control problems. This problem is also related to a conjecture of Damon Wischik (2001). The main difficulty in working with G/G/1 queuing network is that the underlying state dynamics is not Markov. Markov property has proven useful for these type of problems (see e.g., Atar-Goswami-Shwartz(2012)). The standard way to solve these problems is to look at the pde associated with the state dynamics and sandwich the limiting value between the upper and lower value of the game. This technique does not work when the state dynamics is not Markov. We will see that the special structure of the rate function and moderate deviation settings will be helpful to overcome such difficulties. Extension to many-server models will also be discussed.
Title: Weak solutions and convergence analysis of numerical methods for coagulation-fragmentation equations
Speaker: Prof. Ankik Kumar Giri Radon Inst. for Computational and Applied Mathematics (RICAM), Austria
Date: Wed, 07 Aug 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this talk I will mainly focus on the existence and uniqueness of weak solutions to the nonlinear continuous coagulation and multiple fragmentation equations. In addition, the convergence analysis of two numerical methods (the fixed pivot technique and the cell average technique) for solving nonlinear coagulation or Smoluchowski equation is introduced. At the end, the convergence rates obtained from both the techniques are compared and mathematical results of the convergence analysis are also demonstrated numerically.
Title: Gap Distributions and Homogeneous Dynamics
Speaker: Prof. Jayadev Athreya University of Illiniois, Urbana-Champaign
Date: Mon, 05 Aug 2013
Time: 3:30- 4:30pm
Venue: Department of Mathematics, LH-I
We discuss the notion of gap distributions of various lists of numbers in [0, 1], in particular focusing on those which are associated to certain low-dimensional dynamical systems. We show how to explicitly compute some examples using techniques of homogeneous dynamics. This works gives some possible notions of `randomness’ of special trajectories of billiards in polygons, and is based partly on joint works with J. Chaika, J. Chaika and S. Leliever, and with Y.Cheung.
Title: The Hopf map, higher linking and friends
Speaker: Prof. Siddhartha Gadgil IISc
Date: Thu, 18 Jul 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Basic Algebraic Topology is built on considering objects (such as differential forms) up to boundary (e.g., exterior derivative). However, there is more structure in chains and boundaries that can be extracted by not immediately passing to quotients. We will discuss elementary examples of this, such as the Hopf invariant and Milnor’s higher linking.
Title: Potential theory on infinite graphs
Speaker: Prof. Victor Anandam IMSc, Chennai
Date: Fri, 12 Jul 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
The classical (as well as axiomatic) potential theory has been developed in the context of the study of analytic functions on Riemann surfaces, Newton potentials and Markov processes. This talk aims to develop a discrete version of potential theory on infinite graphs, based on the examples of electrical networks and infinite graphs.
Title: A-infinity algebras and the homotopy transfer theorem II
Speaker: Dr. Micah Miller, IISc
Date: Wed, 10 Jul 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
We will continue our discussion of the Homotopy Transfer Theorem for A-infinity algebras. We will introduce the notion of a minimal model for a dg algebra. This will allow us to relate quasi-isomorphisms between strict dg algebras and quasi-isomorphisms between A-infinity algebras.
Title: Induction, equality, spaces, types, A_infinity: An introduction to HoTT
Speaker: Prof. Siddhartha Gadgil
Date: Mon, 08 Jul 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Homotopy Type Theory (HoTT), developed recently in a large collaboration centered at IAS, Princeton, combines elements from type theory, logic and topology to give alternative foundations of mathematics which are much closer to mathematical practice (useful for Automated Reasoning). HoTT also gives new insights into topology.
Title: Induction, equality, spaces, types, A_infinity: An introduction to HoTT
Speaker: Prof. Siddhartha Gadgil IISc
Date: Mon, 08 Jul 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Homotopy Type Theory (HoTT), developed recently in a large collaboration centered at IAS, Princeton, combines elements from type theory, logic and topology to give alternative foundations of mathematics which are much closer to mathematical practice (useful for Automated Reasoning). HoTT also gives new insights into topology.
Title: A-infinity algebras and the homotopy transfer theorem
Speaker: Dr. Micah Miller IISc
Date: Thu, 04 Jul 2013
Time: 3:00 - 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
We will introduce the notion of a differential graded algebra that is not strictly associative, but associative up to chain homotopy. Such objects are called A-infinity algebras or strong homotopy associative algebras. The goal is to state and prove the Homotopy Transfer Theorem, which states that any chain complex that is chain homotopic to an A-infinity algebra has an A-infinity structure and the maps in the chain homotopy can be lifted to a map of A-infinity algebras.
Title: The Euler characteristic, projective modules and the Gersten-Witt complex
Speaker: Prof. Sarang Sane University of Kansas
Date: Tue, 02 Jul 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Starting with the Euler characteristic in graph theory/combinatorics, we trace a brief history, first viewing it as an Euler class in topology, then as an obstruction to splitting of vector bundles and finally get to the more recent notion of the Euler class in algebra/geometry and its use as an obstruction to the splitting of projective modules. This recent notion has two approaches, Euler class groups and Chow-Witt groups, the second of which uses the Gersten-Witt complex as mentioned in the title. Time permitting, we hope to state results about both approaches.
Title: Glimpses of HOTT; Logic and Topology make elegant and useful Foundations
Speaker: Prof. Siddhartha Gadgil Indian Institute of Science
Date: Mon, 01 Jul 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
HOTT (homotopy type theory) is logic built on type theory (mostly from Computer Science) and ideas from topology to give foundations of mathematics that are very elegant and much closer to mathematical practice. This makes HOTT very useful for computer proof systems, and also gives a very nice new synthetic treatment of homotopy theory.
Title: Glimpses of HOTT; Logic and Topology make elegant and useful Foundations
Speaker: Prof. Siddhartha Gadgil Indian Institute of Science
Date: Wed, 26 Jun 2013
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
HOTT (homotopy type theory) is logic built on type theory (mostly from Computer Science) and ideas from topology to give foundations of mathematics that are very elegant and much closer to mathematical practice. This makes HOTT very useful for computer proof systems, and also gives a very nice new synthetic treatment of homotopy theory.
Title: Generalized Radon transforms in image reconstruction problems
Speaker: Prof. Gaik Ambartsoumian University of Texas at Arlington
Date: Thu, 20 Jun 2013
Time: 4:00 - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Integral geometry is a field of mathematics that studies inversions and various properties of transforms, which integrate functions along curves, surfaces and hypersurfaces. Such transforms arise naturally in numerous problems of medical imaging, remote sensing, and non-destructive testing. The most typical examples include the Radon transform and its generalizations. The talk will discuss some problems and recent results related to generalized Radon transforms, and their applications to various problems of tomography.
Title:
Speaker: Prof. Siddhartha Gadgil
Date: Tue, 11 Jun 2013
Venue: Lecture Hall III, Department of Mathematics
The informal seminar on mathematical reasoning continues. This meetingwill be self-contained (i.e., not dependent on previous sessions).
Title: Preserving physical quantities of interest in finite elements with rational bubbles
Speaker: Prof. Johnny Guzman Division of Applied Mathematics, Borwn University, Providence.
Date: Wed, 22 May 2013
Time: 4.00pm - 5.00pm
Venue: Department of Mathematics, LH-1
Many commonly used mixed finite elements for the Stokes problem only conserve mass approximately. We show that if we supplement polynomial basis functions with divergence free rational functions in the finite element method we can conserve mass exactly. Similarly, we show how to supplement polynomial basis in the finite element space with rational functions in when approximating linear elasticity problems in order to preserve angular momentum exactly. This is joint work with Michael Neilan (University of Pittsburgh).
Title: Gradient bounds for p-harmonic systems with vanishing Neumann data in a convex domain
Speaker: Agnid Banerjee, Purdue University
Date: Mon, 20 May 2013
Time: 11am
Venue: Department of Mathematics, LH-3
Let u be a weak solution to a p-harmonic system with vanishing Neumann data on a portion of the boundary of a domain which is convex. We show that subsolution type arguments for some uniformly elliptic PDE’s can be used to deduce that the modulus of the gradient is bounded depending on the Lipschitz character of the domain. In this context, I would like to mention that classical results on the boundedness of the gradient require the domain to be C^{1, Dini}. However, in our case, since the domain is convex, one can make use of the fundamental inequality of Grisvard which can be thought of as an analogue of the use of the barriers for Dirichlet problems in convex domains. Our arguments replaces an argument based on level sets in recent important works of Mazya, Cianchi-Mazya and Geng-Shen involving similar problems. I also intend to indicate some open problems in the regularity theory of degenerate elliptic and parabolic systems. This is a joint work with Prof. John L. Lewis.
Title: Renormalization and reverse renormalization in the dynamics of germs of holomorphic diffeomorphisms in C.
Speaker: Kingshook Biswas Vivekananda University
Date: Fri, 17 May 2013
Time: 11-12 am.
We discuss the renormalization and reverse renormalization constructions used in studying the dynamics of germs of holomorphic diffeomorphisms fixing the origin in C with linear part a rotation. Such a germ is said to be linearizable if it is analytically conjugate to its linear part.
Title: Moebius and conformal maps between boundaries of CAT(-1) spaces.
Speaker: Kingshook Biswas Vivekananda University
Date: Thu, 16 May 2013
Time: 11-12 am.
Motivated by rigidity problems for negatively curved manifolds, we study Moebius and conformal maps $f : \\partial X \\to \\partial Y$ between boundaries of CAT(-1) spaces $X, Y$ equipped with visual metrics.
Title: A new class of high order finite volume schemes
Speaker: Prof. Philip Roe University of Michigan
Date: Mon, 13 May 2013
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-I
- A robust, inexpensive, general-purpose CFD algorithm of better than second-order accuracy would have transformative impact on scientific computing. This talk will describes progress toward this objective.
Title: Geometrical treatment of geometric shock waves
Speaker: Prof. Philip Roe University of Michigan
Date: Fri, 10 May 2013
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-I
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Title: Differential Modular forms on Shimura curves over totally real fields
Speaker: Prof. Debargha Benerjee Australian National University
Date: Thu, 02 May 2013
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-III
- In this talk, we will introduce the theory of differential modular forms for compact Shimura curves over totally real fields. These are the modular forms obtained by applying the arithmetic jet space functor (adjoint to the Witt vector functor) to the ring of modular forms (global sections of certain line bundle). We will show that these differential modular forms help us to detect the ordinary quaternionic abelian schemes and schemes with Frobenius lifts (local analogue of CM abelian schemes). These are also useful to understand the categorical quotient of Shimura curves modulo Hecke correspondences/isogeny. We also construct the Serre-Tate expansions of such differential modular forms as a possible alternative to the Fourier expansion maps.
Title: Systolic inequalities
Speaker: Dr. Thomas Richard, IISc, Bangalore
Date: Tue, 16 Apr 2013
Time: 11: 30 a.m.
Venue: LH- II, Department of Mathematics
The systole of a compact Riemannian manifold (M,g) is the length of the shortest non contractible loop of M, it is attained by a periodic geodesic. A systolic inequality is a lower on the volume of any Riemannian metric depending only on the systole. If one consider the systole as a kind of ‘belt size’ of (M,g), a systolic inequality just says ‘the bigger the belt, the bigger the guy’. In this talk, we will discuss systolic inequalities for surfaces. We will prove optimal results for the torus and the projective plane (which go back to Loewner and Pu in the 50’s) and non optimal results for higher genus surfaces (due to Hebda and Gromov in the 80’s). This topic involve a nice mixture of metric geometry and elementary topology. If time permits, we will say a few words about higher dimensions.
Title: Littlewood-Offord problem
Speaker: Prof. Manjunath Krishnapur IISc, Bangalore
Date: Fri, 05 Apr 2013
Time: 2:00-3:30 pm
Venue: Department of Mathematics, LH-1
In the seqries of 4 lectures, we will cover some aspects of the Littlewood-Offord problem. This problem concerns the anti-concentration phenomenon of sums of independent random variables and has applications to invertibility of random matrices and statistics of real zeros of random polynomials. The subject overlaps probability, combinatorics, additive combinatorics and some algebra.
Title: Metric Measure spaces and Random matrices
Speaker: Prof. Siddhartha Gadgil, IISc, Bangalore
Date: Tue, 02 Apr 2013
Time: 11: 30 a.m.
Venue: LH- II, Department of Mathematics
The geometry of Metric spaces equipped with a probability measure is a very dynamic field. One motivation for the study of such spaces is that they are the natural limits of Riemannian manifolds in many contexts. In this talk, I will introduce basic properties of metric measure spaces and the Gromov-Prohorov distance on them. I will also discuss joint work with Manjunath Krishnapur in which we show that independently sampling points according to the given measure gives an asymptotically bi-Lipschitz correspondence between Metric measure spaces and Random matrices. Finally, I will briefly discuss work with Divakaran in which we study the compactification of the Moduli space of Riemann surfaces in terms of metric measure spaces.
Title: Single and Multi-player Stochastic Dynamic Optimization.
Speaker: Subhamay Saha, IISc, Bangalore
Date: Mon, 01 Apr 2013
Time: 4:00 pm
Venue: Department of Mathematics, LH-1
In this thesis we investigate single and multi-player stochastic dynamic optimization problems in both discrete and continuous time. In the multi-player setup we investigate zero-sum games with both complete and partial information. We study partially observable stochastic games with average cost criterion and the state process being discrete time controlled Markov chain. We establish the existence of the value of the game and also obtain optimal strategies for both players. We also study a continuous time zero-sum stochastic game with complete observation. In this case the state is a pure jump Markov process. We investigate the finite horizon total cost criterion. We characterise the value function via appropriate Isaacs equations. This also yields optimal Markov strategies for both players.
Title: On an ODE related to the Ricci flow
Speaker: Atreyee Bhattacharya IISc, Bangalore
Date: Thu, 28 Mar 2013
Time: 4:00 pm- 5:00 pm
Venue: Department of Mathematics, LH-III
We discuss two topics in this talk. First we study compact Ricci-flat 4- manifolds without boundary and obtain pointwise restrictions on curvature (not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricci-flat Khler surfaces with similar but weaker restrictions on holomorphic sectional curvature. Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both.
Title: An exposition of selected tools from additive combinatorics
Speaker: Mokshay Madiman (Yale university)
Date: Fri, 08 Mar 2013
Time: 2:00-3:30 pm
Venue: LH-1, Department of mathematics.
Tools from additive combinatorics are finding their way into numerous areas of mathematics and applied mathematics, and in particular have been central to recent developments in both random matrix theory and harmonic analysis. The goal of this short course is to understand some of these tools. (If time permits and the audience is willing to pitch in with some talks, we may also cover selected applications.)
Title: Geometrical shock dynamics and Shock vortex interaction
Speaker: Dr. Prasanna Varadarajan Schlumberger Gould Research, Cambridge
Date: Thu, 21 Feb 2013
Time: 4:00 pm-5:00 pm
Venue: LH-III, Department of Mathematics
Title: Languages and Generalized Code Chains.
Speaker: Prof. R. D. Giri, Nagpur University.
Date: Wed, 20 Feb 2013
Time: 3:00 pm-4:00 pm
Venue: LH-III, Department of Mathematics
Title: On condition numbers of a basis
Speaker: Professor B. V. Limaye (IIT Bombay)
Date: Tue, 19 Feb 2013
Time: 11:30 am-12:30 pm
Venue: LH-II, Department of Mathematics
Title: Solutions for differential equations in fluid mechanics by different techniques
Speaker: Prof. Dinesh P. A. M. S. Ramaiah Institute of Technology
Date: Fri, 15 Feb 2013
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-I
Any real physical problem arising in fluid mechanics, when it is translated to mathematical approach generally govern a differential equation with some assumptions. The solution for such a study of the system is solved using analytical or numerical techniques. An attempt has been made to understand the characteristics or behavior or analysis of the physical system using different methods like Lighthills method, perturbation methods, rational approximation or Pad approximation, shooting technique and also highlights on the advantages and limitations of each method are discussed.
Title: Normal spectrum of a subnormal operator
Speaker: Mr. Sumit Kumar IISc, Bangalore
Date: Fri, 15 Feb 2013
Time: 3-4pm
Venue: Department of Mathematics, Lecture Hall II
Let ${H}$ be a separable Hilbert space over the complex field. The class $S := \lbrace N|_{M} : N$ is normal on ${H}$ and ${M}$ is an invariant subspace for $N \rbrace$ of operators was introduced by Halmos and consists of subnormal operators. Each subnormal operator possesses a unique minimal normal extension $\hat{N}$ as shown by Halmos. Halmos proved that $\sigma(\hat{N}) \subseteq \sigma(S)$ and then Bram proved that $\sigma(S)$ is obtained by filling certain number of holes in the spectrum $\sigma(\hat{N})$ of the minimal normal extension $\hat{N}$ of a subnormal operator in ${S}$.
Title: Surface diagrams and 4-manifolds (following Jonathan Williams)
Speaker: Siddhartha Gadgil
Date: Tue, 12 Feb 2013
Time: 11:15 a.m.
Venue: LH-III, Department of Mathematics
I will describe work of Jonathan Williams giving a description of smooth 4-manifolds in terms of certain collections of closed curves on surfaces and certain moves on them. This builds on constructions using Lefschetz pencils and their variants, but is a completely elementary description.
Title: About the density of states in Random Operators
Speaker: Prof Krishna Maddaly IMSc, Chennai
Date: Tue, 12 Feb 2013
Time: 3:15 pm- 4:15 pm
Venue: Department of Mathematics, LH-III
Title: Switching in biological networks
Speaker: Prof. Badal Joshi University of Minnesota
Date: Mon, 11 Feb 2013
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-I
(Bio)chemical reaction networks are used in systems biology to model gene regulatory, protein, metabolic, and other cellular networks. Since existence of multiple steady states (MSS) provides the underpinnings for switching in chemical reaction networks, it is a fundamental problem to determine which network structures permit MSS. There exist several criteria which, when satisfied, establish that a network does not permit MSS regardless of the parameter values. On the other hand, results that establish that a network does permit MSS are rare. I will describe our recent work which provides a novel approach towards solving this problem. In the second part of the talk, I will describe stochastic switching which occurs in a network of neurons which is responsible for the distinct brain states of sleep and wake and for the transitions between the two states.
Title: Derived Geometry and Topological String Theory
Speaker: Dr. Pranav Pandit, University of Vienna
Date: Thu, 07 Feb 2013
Time: 3:30 p.m.
Venue: LH-III, Department of Mathematics
Title: Switching in biological networks
Speaker: Prof. Badal Joshi University of Minnesota
Date: Wed, 06 Feb 2013
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-I
(Bio)chemical reaction networks are used in systems biology to model gene regulatory, protein, metabolic, and other cellular networks. Since existence of multiple steady states (MSS) provides the underpinnings for switching in chemical reaction networks, it is a fundamental problem to determine which network structures permit MSS. There exist several criteria which, when satisfied, establish that a network does not permit MSS regardless of the parameter values. On the other hand, results that establish that a network does permit MSS are rare. I will describe our recent work which provides a novel approach towards solving this problem. In the second part of the talk, I will describe stochastic switching which occurs in a network of neurons which is responsible for the distinct brain states of sleep and wake and for the transitions between the two states.
Title: Real analytic maps and stable Hamiltonian structures
Speaker: Prof Ferit ztrk (Bogazii University, Turkey)
Date: Mon, 28 Jan 2013
Time: 3:30-4:30 pm
Venue: LH-I, Department of Mathematics
We consider a real analytic map f from R^4 to R^2 with a singularity at 0. One method to investigate the singularity is to work on its link L. If 0 is an isolated singularity then it is well known that L is a fibered link in the 3-sphere S^3. This describes immediately a contact structure on S^3. In this talk we suggest that even if 0 is not an isolated singularity, we can associate to the singularity a well-defined stable Hamiltonian structure on S^3, provided that f describes a Seifert fibration on S^3, L being a multi-link in this fibration. This condition is satisfied, for example, when f is complex analytic or f is given as g\\bar{h} with g and h being complex analytic. If the link is already fibered, the stable Hamiltonian structure is nothing but the contact structure mentioned above. Our construction is in fact far more general: given a Seifert multi-link (not necessarily associated to a map from R^4 to R^2) in a Seifert fibered 3-manifold, we build a well-defined stable Hamiltonian structure on the 3-manifold.
Title: The complex of HNN-extensions associated to the free group of rank n
Speaker: Prof. Suhas Jaykumar Pandit ICTP, Trieste
Date: Thu, 17 Jan 2013
Time: 3.30 pm -4.30 pm
Venue: Department of Mathematics, LH-111
In this talk, we shall discuss the complex of HNN - extensions associated to the free group F_n of finite rank n. We shall sketch a proof the following result The group of simplicial automorphisms of this complex is isomorphic to the group Out(F_n) of outer automorphisms of the free group F_n of rank n.
Title: Rigidity and Regularity of Holomorphic Mappings
Speaker: Mr. G. P. Balakumar
Date: Mon, 07 Jan 2013
Time: 4 - 5pm
Venue: Department of Mathematics, LH III
This talk will be on two themes that illustrate the rigidity and regularity of holomorphic mappings. The first part will deal with results concerning the smoothness of continuous CR (Cauchy – Riemann) mappings; in particular, that of Lipschitz continuous CR mappings from h-extendible/semi-regular hypersurfaces into certain Levi co-rank one hypersurfaces, in C^n. The second part will deal with the classification of Kobayashi hyperbolic, finite type rigid polynomial domains with abelian automorphism group in C^3.
Title: Diophantine approximation, arithmetic groups, and ergodic theory
Speaker: Prof Amos Nevo Technion, Israel
Date: Wed, 02 Jan 2013
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-III
One of the main goals of classical metric Diophantine approximation is to quantify the denseness of the rational numbers in the real numbers, or more generally, of Q^d in R^d. An equally natural problem is to quantify the denseness of the rational points on the sphere, and more generally, rational points in other compact and non-compact algebraic sub-varieties in R^d. We will describe a solution to this problem for a large class of homogeneous varieties.
Title: Mean ergodic theorems : from amenable to non-amenable groups
Speaker: Prof Amos Nevo Technion, Israel
Date: Mon, 31 Dec 2012
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-I
We will trace the evolution of the mean ergodic theorem, from its original formulation by von-Neumann to some very recent formulations valid in the context of algebraic groups and their lattice subgroups. We will then present a variety of recent applications of mean ergodic theorems, particularly to counting lattice points.
Title: Affine Stratification Number and Moduli Space of Curves
Speaker: Dr. Chitrabhanu Chaudhuri Northwestern university
Date: Wed, 19 Dec 2012
Time: 4:00 pm- 5:00pm
Venue: Department of Mathematics, LH-I
The affine stratification number of a variety is a measure of how far a variety is from being affine and how close it is to being projective. I shall talk about a certain filtration on the moduli space of curves and the affine stratification number of the open sets occurring in the filtration. This will lead to some cohomology computations where we shall make use of the natural operad structure on the homology of the moduli spaces as well as the some mixed hodge theory.
Title: Construction of Jones Polynomial for Knots (via Tangles)
Speaker: T. V. H. Prathamesh
Date: Fri, 14 Dec 2012
Time: 2.30 pm
Venue: LH-I, Department of Mathematics
Jones Polynomial is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^{1/2} with integer coefficients. We shall discuss the relationship between Jones Polynomial and representation of Knots through Tangles.
Title: Construction of Jones Polynomial for Knots (via Tangles)
Speaker: T. V. H. Prathamesh
Date: Thu, 13 Dec 2012
Time: 2.30 pm
Venue: LH-I, Department of Mathematics
Jones Polynomial is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^{1/2} with integer coefficients. We shall discuss the relationship between Jones Polynomial and representation of Knots through Tangles.
Title: Curvature inequalities for operators in the Cowen-Douglas class
Speaker: Mr. Dinesh Kumar Kesari
Date: Mon, 03 Dec 2012
Time: 11:00 a.m.
Venue: Department of Mathematics, LH-III
The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this talk we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class.
Title: Two studies of almost complex manifolds
Speaker: Dr. Herve Gaussier, Insitut Fourier, Grenoble
Date: Thu, 29 Nov 2012
Time: 3:30 p.m.
Venue: Department of Mathematics, LH-III
The talk will consist of two distinct parts. We will firt study the hyperbolicity of some domains in an almost complex manifold (M,J). In the second part we will study the question of the embeddability of compact almost complex manifolds in complex projective spaces.
Title: Construction of Jones Polynomial for Knots (via Braid Groups and Hecke Algebras)
Speaker: T. V. H. Prathamesh
Date: Tue, 27 Nov 2012
Time: 3.30 pm
Venue: Department of Mathematics, LH-III
Jones Polynomial is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^{1/2} with integer coefficients. Though can be computed easily in terms of what are called skein relations, it originally arose from a studying a particular kind of a ‘trace’ of braid representations into an algebra derived as the quotient of a group ring of braid group. We shall discuss this construction of Jones Polynomial in the first talk. In the second part, we shall discuss the construction of Jones Polynomial from the tangle representation of Knots.
Title: Vector bundles over hypersurfaces of projective varieties
Speaker: Mr. Amit Tripathi
Date: Mon, 26 Nov 2012
Time: 11:00 a.m.- 12.00 noon.
Venue: Department of Mathematics, LH-I
Title: GROTHENDIECK INEQUALITY
Speaker: Mr. Samya Kumar Ray
Date: Wed, 21 Nov 2012
Time: 4:10 p.m.- 5:10 p.m.
Venue: Department of Mathematics, LH-I
Grothendieck published an extraordinary paper entitled Resume de la theorie metrique des produits tensoriels topologiques in 1953. The main result of this paper is the inequality which is commonly known as Grothendieck Inequality. Following Kirivine, in this article, we give the proof of Grothendieck Inequality. We reformulate it in different forms. We also investigate the famous Grothendieck constant KG. The Grothendieck constant was achieved by taking supremum over a special class of matrices. But our attempt will be to investigate it, considering a smaller class of matrices, namely only the positive definite matrices in this class.
Title: Ricci flow of non-smooth spaces
Speaker: Dr. Thomas Richard IISc, Bangalore
Date: Thu, 15 Nov 2012
Time: 4:00 pm- 5:00 pm
Venue: Department of Mathematics, LH-III
Ricci flow is a PDE that deform the metric of a Riemannian manifold in the direction of its Ricci curvature. For compact smooth manifolds, there is a well established existence and uniqueness theory. However for some applications it can be useful to consider Ricci flows of non-smooth spaces, or metric spaces whose distance doesn’t come from a Riemannian metric. We will show that existence and uniqueness holds for the Ricci flow of compact singular surfaces whose curvature is bounded from below in the sense of Alexandrov.
Title: The Whitney-Grauert theorem via contact geometry
Speaker: David Farris
Date: Tue, 06 Nov 2012
Time: 2.00 pm
Venue: Department of Mathematics, LH-III
The Whitney-Grauert theorem states that regular curves in R^2 (i.e. immersions of S^1) are classified up to regular homotopy by the winding number of the derivative. I will present Eliashberg and Geiges’s simple proof of this theorem, in which regular plane curves are realized as projections of curves in R^3 satisfying a certain geometric condition (they’ll be Legendrian curves in the standard contact structure). This is one of the simplest examples of the general pattern of lifting a purely topological problem to an equivalent but simpler problem in contact/symplectic geometry. No knowledge of contact geometry is assumed; the only prerequisite is differential forms on R^3.
Title: On triangulating the Moduli space
Speaker: Prof Siddhartha Gadgil
Date: Tue, 30 Oct 2012
Time: 2.00 pm
Venue: Department of Mathematics, LH-III
I will discuss some constructions, results, questions and applications relating to the space of hyperbolic structures on a surface and its natural compactifications. I will assume that the audience understands what `hyperbolic structures on surfaces’ means.
Title: Shift invariant spaces on the real line and on compact groups
Speaker: Prof Radha Ramakrishnan IIT, Chennai
Date: Tue, 30 Oct 2012
Time: 11:00 am- 12:00 noon
Venue: Department of Mathematics, Lecture Hall III
Title: Riesz transforms associated with Heisenberg groups and Grushin operator
Speaker: Sanjay P. K. IISc, Bangalore
Date: Mon, 29 Oct 2012
Time: 11:30 am- 12:30 pm
Venue: Department of Mathematics, Lecture Hall I
We characterize the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions.
Title: Riesz transforms associated with Heisenberg groups and Grushin operator
Speaker: Mr. Sanjay P. K. IISc, Bangalore
Date: Mon, 29 Oct 2012
Time: 11:30 am- 12:30 pm
Venue: Department of Mathematics, Lecture Hall I
We characterize the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions.
Title: Risk-senstive control of continuous time Markov chains.
Speaker: Subhamay Saha (IISc, Mathematics)
Date: Mon, 22 Oct 2012
Time: 2:00pm
Venue: LH-1, Department of Mathematics, IISc
We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterise the value function via HJB equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.
Title: Stable allocations
Speaker: Prof. Manjunath Krishnapur IISc, Bangalore
Date: Wed, 17 Oct 2012
Time: 4.10 pm
Venue: Department of Mathematics, LH-1
This is an informal lecture to celebrate the unprecedented event that I understand some piece of the work of some economists.
Title: Stable allocations
Speaker: Prof. Manjunath Krishnapur IISc, Bangalore
Date: Tue, 16 Oct 2012
Time: 4.00 pm
Venue: Department of Mathematics, LH-1
This is an informal lecture to celebrate the unprecedented event that I understand some piece of the work of some economists.
Title: The String topology loop product through twisting cochains
Speaker: Dr. Micah Miller IISc, Bangalore
Date: Tue, 09 Oct 2012
Time: 3:30 pm- 4:30 pm
Venue: Department of Mathematics, LH-III
String topology is the study of the free loop space of a manifold LM. The loop product, defined on the homology of LM, is described intuitively as a combination of the intersection product on M and loop concatenation in the based loop space of M. However, since the intersection product is well-defined only on transversally intersecting chains, this description is incomplete. Brown’s theory of twisting cochains provides a chain model of a bundle in terms of the chains on the base and chains on the fiber. We extend this theory so that it can be applied to provide a model of the free loop space. We give a precise definition of the loop product defined at the chain level.
Title: A Google Course-builder course - Hyperbolic Spaces and Groups
Speaker: Siddhartha Gadgil
Date: Fri, 05 Oct 2012
Time: 3:00 p.m.
Venue: LH-III, Department of Mathematics
Google has recently released open source software called `course-builder’ (at https://code.google.com/p/course-builder/) to make interactive online courses consisting of videos interleaved with quizzes. In this brief presentation, I will show as an example a course-builder course I made and describe the process of making such courses. My goal is to convince that making online courses is both easy and worthwhile.
Title: Dynamics of additional food provided predator-prey system with applications to biological control
Speaker: Prof B S R V Prasad VIT University, Vellore
Date: Thu, 20 Sep 2012
Time: 4:00 p.m.- 5:00 p.m.
Venue: Department of Mathematics, LH-III
Necessity to understand the role of additional food as a tool in biological control programs is being increasingly felt, particularly due to its Eco-friendly nature. In this present talk, we develop/analyse a variation of standard predator-prey model with Holling type II function response which presents predator-prey dynamics in presence of some additional food to predators. The aim is to study the consequences of providing additional food on the system dynamics. A thorough mathematical analysis reveals that handling times for the available foods play a vital role in determining the eventual state of the system. It is interesting to observe that by varying the quality (characterised by the handling times) and quantity of additional food we can not only control and limit the prey, but also limit and eradicate the predators. In the context of biological pest control, the results caution the manager on the choice of quality and quantity of the additional food used for this purpose. We further study the controllability aspects of the predator-prey system by considering quality of the additional food as the control variable. Control strategies are offered to steer the system from a given initial state to a required terminal state in a minimum time by formulating Mayer problem of optimal control. It is observed that an optimal strategy is a combination of bang-bang controls and could involve multiple switches. Properties of optimal paths are derived using necessary conditions for Mayer problem. In the light of the results evolved in this work it is possible to eradicate the prey from the system in the minimum time by providing the predator with high quality additional food, which is relevant in the pest management. In the perspective of biological conservation this study highlights the possibilities to drive the state to an admissible interior equilibrium (irrespective of its stability nature) of the system in a minimum time.
Title: From Poisson kernels to stationary measures for random recursions
Speaker: Prof Ewa Damek Wroclaw University
Date: Fri, 14 Sep 2012
Time: 4:00 p.m.- 5:00 pm
Venue: Department of Mathematics, LH-I
Title: Pricing Credit Derivatives in a Markov Modulated Market.
Speaker: Srikanth Iyer (IISc)
Date: Mon, 10 Sep 2012
Time: 2:00pm
Venue: LH-II, Department of Mathematics, IISc
Credit risk refers to the potential losses that can arise due to the changes in the credit quality of financial instruments. There are two approaches to pricing credit derivatives, namely the structural and the reduced form or intensity based models. In the structural approach explicit assumptions are made about the dynamics of a firm’s assets, its capital structure, debt and share holders. A firm defaults when its asset value reaches a certain lower threshold, defined endogenously within the model. In the intensity based approach the firm’s asset values and its capital structure are not modelled at all. Instead the dynamics of default are exogenously given through a default rate or intensity.
Title: Asymptotic behavior of Poisson kernels on NA groups -Fifth of her lecture series
Speaker: Prof Ewa Damek University of Wroclaw, Wroclaw, Poland
Date: Thu, 06 Sep 2012
Time: 4:00 p.m.- 5:00 pm
Venue: Department of Mathematics, LH-I
Title: Asymptotic behavior of Poisson kernels on NA groups - Fourth of her lecture series
Speaker: Prof Ewa Damek University of Wroclaw, Wroclaw, Poland
Date: Wed, 05 Sep 2012
Time: 4:00 p.m.- 5:00 pm
Venue: Department of Mathematics, LH-I
Title: Asymptotic behavior of Poisson kernels on NA groups -Third Lecture
Speaker: Prof Ewa Damek University of Wroclaw, Wroclaw, Poland
Date: Tue, 04 Sep 2012
Time: 4:00 p.m.- 5:00 pm
Venue: Department of Mathematics, LH-I
Title: Operator space projective tensor product
Speaker: Prof. Ajay Kumar, Dept. of Mathematics, University of Delhi
Date: Mon, 03 Sep 2012
Time: 3:00-4:00 pm
Venue: Department of Mathematics, LH-II
Title: Asymptotic behavior of Poisson kernels on NA groups - Second lecture
Speaker: Prof Ewa Damek University of Wroclaw, Wroclaw, Poland
Date: Fri, 31 Aug 2012
Time: 4:00 p.m.- 5:00 pm
Venue: Department of Mathematics, LH-I
Title: Asymptotic behavior of Poisson kernels on NA groups
Speaker: Prof Ewa Damek University of Wroclaw, Wroclaw, Poland
Date: Thu, 30 Aug 2012
Time: 4:00 p.m.- 5:00 pm
Venue: Department of Mathematics, LH-I
Title: Matchings between point processes
Speaker: Mr. Indrajit Jana
Date: Fri, 24 Aug 2012
Time: 11:00 a.m.- 12:00 noon
Venue: Department of Mathematics, LH-III
We study several kinds of matching problems between two point processes. First we consider the set of integers $\\mathbb{Z}$. We assign a color red or blue with probability 1/2 to each integer. We match each red integer to a blue integer using some algorithm and analyze the matched edge length of the integer zero. Next we go to $\\mathbb{R}^{d}$. We consider matching between two different point processes and analyze a typical matched edge length $X$. There we see that the results vary significantly in different dimensions. In dimensions one and two (d=1,2), even $E[X^{d/2}]$ does not exist. On the other hand in dimensions more than two (d>2), $E[\\exp(cX^{d})]$ exist, where $c$ is a constant depends on $d$ only.
Title: Rational points on Elliptic curves and congruences between modular forms
Speaker: Dr. Chandrakant Aribam Mathematics Institute, Heidelberg University
Date: Fri, 24 Aug 2012
Time: 4:00 pm- 5:00 pm
Venue: Department of Mathematics, Lecture Hall 1
Rational points on elliptic curves have found applications in cryptography and in the solution to some problems dating back to antiquity. However, we still do not know how to find an elliptic curve with as many points as possible. In this talk, we will see how the theory of modular forms (of Ramanujan) along with a recently developed theory enables one to understand this problem. Along the way, we will see how congruences between coefficients of modular forms shed light on this problem.
Title: Fourier coefficients of Siegel modular forms
Speaker: Dr. Soumya Das
Date: Thu, 09 Aug 2012
Time: 4:00 pm- 5:00 pm
Venue: Department of Mathematics, Lecture Hall 1
We will discuss some questions concerning the existence of families of Siegel modular forms with some prescribed non-zero Fourier coefficients. If time permits, we also plan to discuss the question of characterizing cusp forms by the growth of their Fourier coefficients. Both are recent joint works with Siegfried Boecherer.
Title: Fourier coefficients of Siegel modular forms
Speaker: Dr. Soumya Das Tata Institute of Fundamental research, Mumbai
Date: Thu, 09 Aug 2012
Time: 4:00 pm- 5:00 pm
Venue: Department of Mathematics, Lecture Hall 1
We will discuss some questions concerning the existence of families of Siegel modular forms with some prescribed non-zero Fourier coefficients. If time permits, we also plan to discuss the question of characterizing cusp forms by the growth of their Fourier coefficients. Both are recent joint works with Siegfried Boecherer.
Title: Introduction to Harmonic analysis on NA groups
Speaker: Dr. Pratyoosh Kumar
Date: Wed, 08 Aug 2012
Time: 4:00 pm- 5:00 pm
Venue: Department of Mathematics, Lecture Hall 1
Title: Birth of Integral Value Transformation (IVT) and some observations in Mathematics and in Genomics
Speaker: Prof. Pabitra Pal Choudhury, ISI, Kolkata
Date: Tue, 31 Jul 2012
Time: 4:00 pm- 5.00 pm
Venue: Department of Mathematics, Lecture Hall 1
Firstly, regarding Carry Value Transformation (CVT) some mathematical observations will be discussed. In using mathematical tools in Genomics we adopted two-way path. One is model based, another is issue based. Both these approaches will be discussed on using Human Olfactory receptors.
Title: On the Mathematical Legacy of S. Chandrasekar
Speaker: Professor S. S. Sritharan, Director, Center for Decision, Risk, Controls and Signals Intelligence (DRCSI), Monterey, CA
Date: Mon, 30 Jul 2012
Time: 4:00 pm- 5.00 pm
Venue: Department of Mathematics, Lecture Hall 1
In this talk we will discuss some of the impact of Subramanian Chandrasekhar’s work on modern applied mathematics. One of the highlight of the talk will be a discussion of Chandrasekhar’s radiative transfer theory where he developed a number of breathtaking mathematical structures such as nonlinear integral equations for the Chandrasekhar H functions and X, Y functions as well as infinite dimensional Riccati (integro-partial differential differential ) equations for the scattering matrix long before the infinite dimensional systems theory.
Title: Some Elementary Mathematics of Integral Value Transformations (IVTs) and Its Associated Dynamical Systems.
Speaker: Dr. Sarif Hassan, Institute of Mathematics and Applications, Bhubaneswar
Date: Mon, 30 Jul 2012
Time: 2:00 pm- 3.00 pm
Venue: Department of Mathematics, Lecture Hall 1
Some basic algebraic structures on the set of 1-dimensional IVTs are introduced. Then discrete dynamical systems are defined through IVT maps and a report on convergence of the dynamical systems has been made. Finally, some problems which are unsolved yet in the domain are discussed.
Title: Some applications of polynomial knots.
Speaker: Prof. Rama Mishra, IISER, Pune
Date: Wed, 25 Jul 2012
Time: 4:00 pm- 5.00 pm
Venue: Department of Mathematics, Lecture Hall 1
Polynomial knots were introduced to represent knots in 3 space by simple polynomial equations. In this talk we will discuss how the degrees of these equations can be used in deriving information of some important knot invariants.
Title: Analytic continuation in several complex variables
Speaker: Mr. Chandan Biswas
Date: Tue, 17 Jul 2012
Time: 10:00 - 11:00 a.m.
Venue: Department of Mathematics, LH-1
We wish to study those domains in $\mathbb{C}^n$, for $n\geq 2$, the so-called domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We shall demonstrate that this study is radically different from that of domains in $\mathbb{C}$ by discussing some examples of special types of domains in $\mathbb{C}^n$, $n\geq 2$, such that every function holomorphic on them extends to a strictly larger domain. This leads to Thullen’s construction of a domain (not necessarily in $\mathbb{C}^n$) spread over $\mathbb{C}^n$, the so-called envelope of holomorphy, which fulfills our criteria. With the help of this abstract approach we shall give a characterization of the domains of holomorphy in $\mathbb{C}^n$.The aforementioned characterization (holomorphic convexity) is very difficult to check. This calls for other (equivalent) criteria for a domain in $\mathbb{C}^n$, $n\geq 2$, to be a domain of holomorphy. We shall survey these criteria. We shall sketch those proofs of equivalence that rely on the first part of our survey: namely, on analytic continuation theorems. If a domain $\Omega\subset \mathbb{C}^n$, is not a domain of holomorphy, we would still like to explicitly describe a domain strictly larger than $\Omega$ to which all functions holomorphic on $\Omega$ continue analytically. One tool that is used most often in such constructions is called “Kontinuitaetssatz”. It has been invoked, without any clear statement, in many works on analytic continuation. The basic (unstated) principle that seems to be in use in these works appears to be a folk theorem. We shall provide a precise statement of this folk Kontinuitaetssatz and give a proof of it.
Title: Homogeneous dynamics and number theory
Speaker: Dr. Anish Ghosh University of East Anglia
Date: Tue, 17 Jul 2012
Time: 4:00 pm- 5:00 pm
Venue: Department of Mathematics, Lecture Hall 1
I will discuss some instances of the interplay between dynamics on homogeneous spaces of algebraic groups and Diophantine approximation, with an emphasis on recent developments. The latter includes joint work with Gorodnik and Nevo, and with Einsiedler and Lytle.
Title: Curvature inequalities for operators in the Cowen- Douglas class
Speaker: Mr. Dinesh Kumar Keshari
Date: Fri, 13 Jul 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall 1
The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this talk we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class.
Title: Relative Symplectic Caps, 4-Genus and Fibered Knots.
Speaker: Mr. Dheeraj Kulkarni
Date: Fri, 13 Jul 2012
Time: 11.00 am- 12.00 noon
Venue: Department of Mathematics, Lecture Hall 1
The $4$-genus of a knot is an important measure of complexity, related to the unknotting number. A fundamental result used to study the $4$-genus and related invariants of homology classes is the \emph{Thom Conjecture}, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsvath-Szabo, which say that \textit{closed} symplectic surfaces minimize genus.
Title: Complex projective structures and dynamics in moduli space
Speaker: Dr. Subojoy Gupta California Institute of Technology
Date: Fri, 13 Jul 2012
Time: 11:00 am- 12:00 noon
Venue: Department of Mathematics, Lecture Hall 1
We shall discuss a new result that relates grafting, which are certain deformations of complex projective structures on a surface, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any fixed Fuchsian holonomy, such geometric structures are dense in moduli space.
Title: Complex projective structures and dynamics in moduli space
Speaker: Dr. Subojoy Gupta California Institute of Technology
Date: Thu, 12 Jul 2012
Time: 11:00 am- 12:00 noon
Venue: Department of Mathematics, Lecture Hall 1
We shall discuss a new result that relates grafting, which are certain deformations of complex projective structures on a surface, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any fixed Fuchsian holonomy, such geometric structures are dense in moduli space.
Title: Extension theorems for subvarieties and vector bundles.
Speaker: Prof. G.V. Ravindra University of Missouri, St. Louis
Date: Tue, 10 Jul 2012
Time: 2:00 - 3:00 p.m.
Venue: Department of Mathematics, Lecture Hall 3
Let Y be a complex, projective manifold and X a smooth hyperplane section in Y. Given a submanifold Z in X, under what conditions is it cut out by a submanifold Z’ in Y. An analogous quesion can be asked for vector bundles on X: namely when is a bundle on X, the restriction of a bundle on Y.
Title: Extension theorems for vector bundles over hypersurfaces
Speaker: Mr. Amit Tripathi
Date: Tue, 10 Jul 2012
Time: 3:30 - 4:30 p.m.
Venue: Department of Mathematics, Lecture Hall 3
In this thesis we study some Questions on vector bundles over hypersurfaces. More precisely, for hypersurfaces of dimension $\\geq 2$, we study the extension problem of Vector bundles. We try to find some conditions under which a vector bundle over an ample divisor of non-singular projective variety, extends as a vector bundle to an open set containing that ample divisor.
Title: KPZ equation from certain microscopic interactions
Speaker: Prof. Sunder Sethuraman University of Arizona
Date: Fri, 29 Jun 2012
Time: 2:00 - 3:00 p.m.
Venue: Department of Mathematics, Lecture Hall 3
We derive a form of the KPZ equation, which governs the fluctuations of a class of interface heights, in terms of a martingale problem, as the scaling limit of fluctuation fields with respect to some particle systems such as zero range processes. This is joint work in progress with P. Goncalves and M. Jara.
Title: Properties of tempered stable distributions
Speaker: Prof. Michael Grabchak UNC Charlotte
Date: Fri, 29 Jun 2012
Time: 3:15 - 4:15 p.m.
Venue: Department of Mathematics, Lecture Hall 3
Tempered stable distributions were introduced in Rosinski 2007 as models that look like infinite variance stable distributions in some central region, but they have lighter (i.e. tempered) tails. We introduce a larger class of models that allow for more variety in the tails. While some cases no longer correspond to stable distributions they serve to make the class more flexible, and in certain subclasses they have been shown to provide a good fit to data. To characterize the possible tails we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs. We will also characterize the weak limits of sequences of tempered stable distributions. If time permits, we will motivate why distributions that are stable-like in some central region but with lighter tails may show up in applications.
Title: Masas and Bimodule Decompositions of $\rm{II}_{1}$ factors
Speaker: Dr.Kunal Mukherjee IMSc, Chennai.
Date: Thu, 28 Jun 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
von Neumann algebras are non commutative analogue of measure spaces. The study of maximal abelian subalgebras (masas) in finite von Neumann algebras is classical to the subject from its birth and is closely tied up with Ergodic theory. Dixmier introduced two types of masas, namely, regular (Cartan) and singular. The philosophies of these two kinds of masas until recently were regarded as being different from each other. After an introduction on the subject, we justify that the existing theories can be unified. Using techniques from Free Probability and playing with suitable amenable groups we exhibit: For each subset S of $\\mathbb{N}$ (could be empty), there exist uncountably many pairwise non conjugate (by automorphism) singular masas in the free group factors for each of which $S\\cup {\\infty\\}$ arises as its Pukanzsky invariant (multiplicity function). If time permits, some other issues related to mixing, coarse bimodules, and Banach’s problem on simple Lebesgue spectrum will be addressed.
Title: Masas and Bimodule Decompositions of $\rm{II}_{1}$ factors
Speaker: Dr. Kunal Mukherjee IMSc, Chennai.
Date: Thu, 28 Jun 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
von Neumann algebras are non commutative analogue of measure spaces. The study of maximal abelian subalgebras (masas) in finite von Neumann algebras is classical to the subject from its birth and is closely tied up with Ergodic theory. Dixmier introduced two types of masas, namely, regular (Cartan) and singular. The philosophies of these two kinds of masas until recently were regarded as being different from each other. After an introduction on the subject, we justify that the existing theories can be unified. Using techniques from Free Probability and playing with suitable amenable groups we exhibit: For each subset S of $\\mathbb{N}$ (could be empty), there exist uncountably many pairwise non conjugate (by automorphism) singular masas in the free group factors for each of which $S\\cup {\\infty\\}$ arises as its Pukanzsky invariant (multiplicity function). If time permits, some other issues related to mixing, coarse bimodules, and Banach’s problem on simple Lebesgue spectrum will be addressed.
Title: Rigidity and Regularity of Holomorphic Mappings.
Speaker: Mr. G. P. Balakumar
Date: Fri, 22 Jun 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
This talk will be on two themes that illustrate the rigidity and regularity of holomorphic mappings. The first part will deal with results concerning the smoothness of continuous CR (Cauchy – Riemann) mappings; in particular, that of Lipschitz continuous CR mappings from h-extendible/semi-regular hypersurfaces into certain Levi co-rank one hypersurfaces, in C^n. The second part will deal with the classification of Kobayashi hyperbolic, finite type rigid polynomial domains in C^3.
Title: Face numbers of simplicial cell manifolds
Speaker: Prof. Mikiya Masuda Osaka City University, Japan
Date: Fri, 08 Jun 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
A simplicial cell complex is roughly speaking a CW complex whose cells are all simplices. This notion is equivalent to that of simplicial poset in combinatorics. A simplicial complex is a simplicial cell complex, but two simplices in a simplicial cell complex may be glued together along more than one simplex on their boundaries. In this talk, I will discuss the characterization of face numbers of simplicial cell decompositions of some manifolds.
Title: An introduction to GKM graphs
Speaker: Prof. Mikiya Masuda Osaka City University, Japan
Date: Thu, 07 Jun 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
The notion of a GKM graph was introduced by Guillemin-Zara [1], motivated by a result of Goresky-Kottwitz-MacPherson [2]. A GKM graph is a regular graph with directions assigned to edges satisfying certain compatibility condition. The 1-skeleton of a simple polytope provides an example of a GKM graph. One can associate a GKM graph $\\mathcal{G}_M$ to a closed manifold $M$ with an action of a compact torus satisfying certain conditions (those manifolds are often called GKM manifolds). Many important manifolds such as toric manifolds and flag manifolds are GKM manifolds. The GKM graph $\\mathcal{G}_M$ contains a lot of geometrical information on $M$, e.g. the (equivariant) cohomology of $M$ can be recovered by $\\mathcal{G}_M$. I will present an overview of some facts on GKM graphs.
Title: Classification of the actions of the circle on 3-manifolds
Speaker: Prof. Soumen Sarkar Korea Advanced Institute of Science and Technology
Date: Wed, 06 Jun 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Title: Grushin multipliers and Toeplitz operators
Speaker: Jotsaroop Kaur
Date: Thu, 31 May 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
The Grushin operator is defined as $G:=-\Delta-|x|^2\partial_t^2$ on $\mathbb{R}^{n+1}$. We study the boundedness of the multipliers $m(G)$ of $G$ on $L^p(\mathbb{R}^{n+1})$. We prove the analogue of the Hormander-Mihlin theorem for $m(G)$. We also study the boundedness of the solution of the wave equation corresponding to $G$ on $L^p(\mathbb{R}^{n+1})$. The main tool in studying the above is the operator-valued Fourier multiplier theorem by Lutz Weis.
Title: Matchings between point processes
Speaker: Mr. Indrajit Jana
Date: Thu, 31 May 2012
Time: 11:00 a.m. - 12:00 noon
Venue: Department of Mathematics, Lecture Hall I
We study several kinds of matching problems between two point processes. First we consider the set of integers $\\mathbb{Z}$. We assign a color red or blue with probability 1/2 to each integer. We match each red integer to a blue integer using some algorithm and analyze the matched edge length of the integer zero. Next we go to $\\mathbb{R}^{d}$. We consider matching between two different point processes and analyze a typical matched edge length $X$. There we see that the results vary significantly in different dimensions. In dimensions one and two (d=1,2), even $E[X^{d/2}]$ does not exist. On the other hand in dimensions more than two (d>2), $E[\\exp(cX^{d})]$ exist, where $c$ is a constant depends on $d$ only.
Title: Structure of the entropy solution of a scalar conservation law with strict convex flux
Speaker: Dr. Shyam Sundar Ghoshal TIFR Centre for Applicable Mathematics, Bangalore
Date: Thu, 24 May 2012
Time: 2:30 - 3:30 p.m.
Venue: Department of Mathematics, Lecture Hall I
We prove the Structure Theorem of the entropy solution. Furthermore we obtain the shock regions each of which represents a single shock at infinity. Using the structure Theorem we construct initial data $u_0\\in C_c^\\f$ for which the solution exhibits infinitely many shocks as $t \\rightarrow \\f$. Also we have generalized the asymptotic behavior (the work of Dafermos, Liu, Kim) of the solution and obtain the rate of decay of the solution with respect to the $N$-wave.
Title: Function Theory on non-compact Riemann surfaces
Speaker: Ms. Eliza Philip
Date: Wed, 23 May 2012
Time: 11:00 a.m. - 12:00 noon
Venue: Department of Mathematics, Lecture Hall I
Given a domain D in the complex plane and a compact subset K, Runge’s theorem provides conditions on K which guarantee that a given function that is holomorphic in some neighbourhood of K can be approximated on K by a holomorphic function on D. We look at an analogous theorem on non-compact Riemann surfaces, i.e., Runge’s approximation theorem, stated and proved by Malgrange. We revisit Malgrange’s proof of the theorem, invoking a very basic result in distribution theory: Weyl’s lemma. We look at two main applications of Runge’s theorem. Firstly, every open Riemann surface is Stein and secondly the triviality of holomorphic vector bundles on non-compact Riemann surfaces. Next, we look at the Gunning-Narasimhan theorem which states that every open (connected) Riemann surface can be immersed into $\\mathbb{C}$. We discuss the proof of this theorem as well, which depends on Runge’s theorem too. Finally we contrast the compact case to the non-compact case, by showing that every compact Riemann surface can be embedded into a large enough complex projective space.
Title: The role of potential theory in complex dynamics
Speaker: Ms. Choiti Bandyopadhyay
Date: Wed, 23 May 2012
Time: 4:00 p.m. - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. We wish to gain a deeper understanding of complex dynamics using the tools and techniques of potential theory, and we will restrict ourselves to the iteration of holomorphic polynomials.
Title: On the stability of certain Riemannian functionals
Speaker: Ms. Soma Maity
Date: Tue, 22 May 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Let $M$ be a closed smooth manifold. Consider the space of Riemannian metrics $\\mathcal{M}$ on $M$. A real valued function on $\\mathcal{M}$ is called a Riemannian functional if it remains invariant under the action of the group of diffeomorphisms of $M$ on $\\mathcal{M}$. We will discuss some geometric properties of the critical points of certain natural Riemannian functionals in this lecture.
Title: On the role of the Bargmann transform in uncertainty principles
Speaker: Rahul Garg
Date: Mon, 21 May 2012
Time: 11:00 a.m. - 12:00 noon
Venue: Department of Mathematics, Lecture Hall I
We consider functions $f$ on $\mathbb{R}^n$ for which both $f$ and their Fourier transforms $\hat{f}$ are bounded by the Gaussian $e^{-\frac{a}{2}|x|^2}$ for some $0<a<1$. Using the Bargmann transform, we show that their Fourier-Hermite coefficients have exponential decay. This is an extension of the one dimensional result of M. K. Vemuri, in which sharp estimates were proved. In higher dimensions, we obtain the analogous result for functions $f$ which are $O(n)$-finite. Here by an $O(n)$-finite function we mean a function whose restriction to the unit sphere $S^{n-1}$ has only finitely many terms in its spherical harmonic expansion. Some partial results are proved for general functions. As a corollary to these results, we obtain Hardy’s uncertainty principle. An analogous problem is studied in the case of Beurling’s uncertainty principle.
Title: Projective modules and complete intersection--a brief survey
Speaker: Prof. N. Mohan Kumar Washington University in St. Louis, USA
Date: Fri, 18 May 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
I will discuss some of the results on projective modules and complete intersections spanning a few decades. So, necessarily, the details have to be sketchy. But, I hope to give the flavour and some of the still persistent questions in the field.
Title: Inverse semigroups and the Cuntz-Li algebras
Speaker: Prof. S. Sundar Indian Stastical Institute, Delhi
Date: Tue, 15 May 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Let R be an integral domain such that every non-zero quotient R/mR is finite. Consider the unitaries and isometries on \\ell^{2}(R) induced by the addition and the multiplication operation of the ring R. The C-algebra generated by these unitaries and isometries is called the ring C-algebra and was studied by Cuntz and Li.
Title: Joint eigenfunctions on the Heisenberg group and support theorems on $\mathbb{R}^n$
Speaker: Mr. Amit Samanta
Date: Tue, 15 May 2012
Time: 11:00 a.m. - 12:00 noon
Venue: Department of Mathematics, Lecture Hall I
This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on $\\mathbb{R}^n$, as described in the following two paragraphs respectively.
Title: The L-4 norm of Hecke cusp forms
Speaker: Prof. Rizwanur Khan Georg-August Universitaet Goettingen, Germany
Date: Fri, 04 May 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
This talk is on a topic in number theory, but should be accessible to a general mathematical audience.
Title: Curvature calculations of the operators in the Cowen-Douglas class
Speaker: Mr. Prahllad Deb
Date: Mon, 30 Apr 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
In a foundational paper Operators Possessing an Open Set of Eigenvalues written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space H possessing an open set W (in complex plane) of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that dimension of the eigenspace at each point w in W is 1, then the map f on W, sending w to ker(T-w), admits a non-zero holomorphic section, say S, and therefore defines a line bundle LT on W.It is well known that the curvature KL of a line bundle LT is a complete invariant for the line bundle LT . On the other hand, define
Title: Analytic continuation in several complex variables
Speaker: Mr. Chandan Biswas
Date: Thu, 26 Apr 2012
Time: 11:00 a.m. - 12:00 noon
Venue: Department of Mathematics, Lecture Hall I
We wish to study those domains in $\\mathbb{C}^n$, for $n\\geq 2$, the so-called domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We shall demonstrate that this study is radically different from that of domains in $\\mathbb{C}$ by discussing some examples of special types of domains in $\\mathbb{C}^n$, $n\\geq 2$, such that every function holomorphic on them extends to a strictly larger domain. This leads to Thullen’s construction of a domain (not necessarily in $\\mathbb{C}^n$) spread over $\\mathbb{C}^n$, the so-called envelope of holomorphy, which fulfills our criteria. With the help of this abstract approach we shall give a characterization of the domains of holomorphy in $\\mathbb{C}^n$.
Title: Analyzing credit risk models in a regime-switching market
Speaker: Mr. Tamal Banerjee
Date: Wed, 25 Apr 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Recently, the financial world witnessed a series of major defaults by several institutions and banks. Therefore, it is not at all surprising that credit risk analysis has turned out to be one of the most important aspects of study in the finance community. As credit derivatives are long term instruments, they are affected by the changes in the market conditions. Thus, it is appropriate to take into consideration the cyclical effects of the market. This thesis addresses some of the important issues in credit risk analysis for a regime-switching market.
Title: Discrete series representation of semisimple Lie groups
Speaker: Prof. R. Parthasarathy Bharathiar University
Date: Mon, 16 Apr 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Two key steps devised by Harish Chandra for his construction of the global characters of discrete series for a non-compact real semisimple Lie group involve
Title: Discrete series representation of semisimple Lie groups
Speaker: Prof. K. Parthasarathy Ramanujan Institute for Advanced Study, University of Madras
Date: Mon, 16 Apr 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Two key steps devised by Harish Chandra for his construction of the global characters of discrete series for a non-compact real semisimple Lie group involve
Title: Comparison results for first-order FEMs
Speaker: Dr. Mira Schedensack Humboldt University, Berlin
Date: Wed, 28 Mar 2012
Time: 2:30 - 3:30 p.m.
Venue: Department of Mathematics, Lecture Hall I
Please see the attachment to this e-mail.
Title: Quasi-optimal adaptive pseudostress approximation of the Stokes equations
Speaker: Dr. Dietmar Gallistl Humboldt University, Berlin
Date: Wed, 28 Mar 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Please see the attachment to this e-mail.
Title: Topological rigidity
Speaker: Prof. Thomas Farrell SUNY at Binghamton, U.S.A.
Date: Tue, 27 Mar 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall III
Title: Topological Rigidity
Title: Some representations of the discrete Heisenberg group
Speaker: Prof. Gerald B. Folland University of Washington
Date: Thu, 01 Mar 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
The operators $f(t) \\rightarrow f(t-a)$ and $f(t) \\rightarrow e^{2\\pi bt} f(t)$ on $L^2(\\mathbb R)$ generate unitary representations of the discrete Heisenberg group $H$ with central character $e^{2\\pi abz}$. What are the irreducible representations of $H$ with this central character, and how can one synthesize the representation just described from them ? When $ab$ is rational, the answers are quite straightforward, but when $ab$ is irrational things are much more complicated. We shall describe results in both cases.
Title: Some representations of the discrete Heisenberg group
Speaker: Prof. Gerald B. Folland University of Washingtom
Date: Thu, 01 Mar 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
The operators $f(t) \\rightarrow f(t-a)$ and $f(t) \\rightarrow e^{2\\pi bt} f(t)$ on $L^2(\\mathbb R)$ generate unitary representations of the discrete Heisenberg group $H$ with central character $e^{2\\pi abz}$. What are the irreducible representations of $H$ with this central character, and how can one synthesize the representation just described from them ? When $ab$ is rational, the answers are quite straightforward, but when $ab$ is irrational things are much more complicated. We shall describe results in both cases.
Title: Lifting of model structures to fibred categories
Speaker: Prof. Abhishek Banerjee Ohio State University, U.S.A.
Date: Mon, 27 Feb 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
A fibred category consists of a functor $p:\mathbf N\longrightarrow \mathbf M$ between categories $\mathbf N$ and $\mathbf M$ such that objects of $\mathbf N$ may be pulled back along any arrow of $\mathbf M$. Given a fibred category $p:\mathbf N\longrightarrow \mathbf M$ and a model structure on the base category $\mathbf M$, we show that there exists a lifting of the model structure on $\mathbf M$ to a model structure on $\mathbf N$. We will refer to such a system as a fibred model category and give several examples of such structures. We show that, under certain conditions, right homotopies of maps in the base category $\mathbf M$ may be lifted to right homotopic maps in the fibred category. Further, we show that these lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences. Finally, we show that if $\mathbf N$ and $\mathbf M$ carry compatible closed monoidal structures and the functor $p$ commutes with colimits, then a Quillen pair on $\mathbf M$ lifts to a Quillen pair on $\mathbf N$.
Title: An invitation to large deviation theory
Speaker: Prof. S.S. Sritharan Center for Decision, Risk, Controls & Signals Intelligence Naval Postgraduate School Monterey, California, USA
Date: Thu, 23 Feb 2012
Time: 10:00 - 11:00 a.m.
Venue: Department of Mathematics, Lecture Hall I
Title: Hurwitz equivalence: some approaches
Speaker: Prof. Siddhartha Gadgil IISc
Date: Wed, 22 Feb 2012
Time: 3:00 - 4:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Hurwitz equivalence is a simple algebraic relation on the set of n-tuples in a group G. This and its generalizations are related to important problems in topology. I discuss some approaches to understanding Hurwitz equivalence.
Title: Representations of semisimple Lie groups
Speaker: Prof. Jyoti Sengupta TIFR Mumbai
Date: Mon, 20 Feb 2012
Venue: Department of Mathematics, Lecture Hall I
Title: On the lumped mass finite element method for parabolic problems
Speaker: Prof. Vidar Thomee Chalmers University of Technology Gothenburg, Sweden
Date: Wed, 08 Feb 2012
Time: 3:00 - 4:00 p.m.
Venue: Department of Mathematics, Lecture Hall III
Title: Liouvillian Extensions and the Galois Theory of Linear Differential Equations
Speaker: Prof. Varadharaj Ravi Srinivasan Catholic University of America Washington, D.C.
Date: Mon, 30 Jan 2012
Time: 3:00 - 4:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Title: Commuting differential operators on Lie groups and spectral multipliers
Speaker: Prof. Alessio Martini Universitaet zu Kiel
Date: Fri, 27 Jan 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Let D_1,…D_n be a system of commuting, formally self-adjoint, left invariant operators on a Lie group G. Under suitable hypotheses, we show that D_1,…D_n are essentially self-adjoint on L^2(G) and admit a joint spectral resolution, and we characterize their joint L^2 spectrum as the set of eigenvalues corresponding to a class of generalized joint eigenfunctions. Moreover, in the case G is a homogeneous group and D_1,…D_n are homogeneous, we obtain L^p-boundedness results for operators of the form m(D_1,…D_n), analogous to the Mihlin-Hormander and Marcinkiewicz multiplier theorems.
Title: Joint spectral multipliers on nilpotent Lie groups
Speaker: Prof. Alessio Martini Universitaet zu Kiel
Date: Wed, 25 Jan 2012
Time: 4:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Title: The chiral boson and function theory on the unit disc
Speaker: Prof. T.R. Ramadas ICTP, Trieste
Date: Wed, 04 Jan 2012
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Conformal field theories (CFTs) are related (in Mathematics) to algebraic geometry, infinite-dimensional Lie algebras and probability, and (in Physics) to critical phenomena and string theory. From a mathematical point of view, much of the formalisation has been from the point of view of algebra – in fact using formal power series. I will give a denition of the simplest chiral or holomorphic CFT using elementary function theory. If time permits, I will also explain operator product expansions.
Title: The $Cos^\lambda$ transform as intertwining operator between generalized principal series representations of SL(n+1,K)
Speaker: Prof. Angela Pasquale Universite Paul Verlaine - Metz
Date: Fri, 23 Dec 2011
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
The $Cos^\\lambda$ transform on real Grassmann manifolds was first studied in convex geometry. The definition of this integral transform has been later extended to Grassmann manifolds over $\\mathbb K$, where $\\mathbb K$ denotes the reals, the complex numbers or the quaternions.
Title: An overview of discontinuous Petrov-Galerkin stabilization techniques
Speaker: Prof. Jay Gopalakrishnan Portland State University
Date: Mon, 19 Dec 2011
Time: 4:00 - 5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
We introduce the concept of optimal test functions that guarantee stability of resulting numerical schemes. Petrov-Galerkin methods seek approximate solutions of boundary value problems in a trial space by weakly imposing all equations via a (possibly different) test space. A basic design principle is that while trial spaces must have good approximation properties, the test space must be chosen for stability. The optimal test functions are those that realize discrete stability constants equal to those in the wellposedness estimates for the undiscretized boundary value problem. When such functions are used within an ultra-weak variational formulation, we obtain Discontinuous Petrov-Galerkin (DPG) methods that exhibit remarkable stability properties. We present the first complete theory for the DPG for Laplace’s equation as well as numerical results for other more complex applications.
Title: A nice group structure on certain orbit spaces of unimodular rows
Speaker: Prof. Anuradha Garge Centre for Excellence in Basic Sciences Mumbai
Date: Thu, 15 Dec 2011
Time: 11:00 a.m. - 12:00 noon
Venue: Department of Mathematics, Lecture Hall I
In this talk, we will begin with the definition of a unimodular row and its relation to Serre’s problem on projective modules. We will then see under what conditions group structures exist on orbit spaces of unimodular rows under elementary action.
Title: Roots of Dehn twists
Speaker: Prof. Kashyap Rajeevsarathy IISER Bhopal
Date: Tue, 13 Dec 2011
Time: 11:00 a.m. - 12:00 noon
Venue: Department of Mathematics, Lecture Hall I
Let $F$ be a closed orientable surface of genus $g \\geq 2$ and $C$ be a simple closed curve in $F$. Let $t_C$ denote a left-handed Dehn twist about $C$. When $C$ is a nonseparating curve, D. Margalit and S. Schleimer showed the existence of such roots by finding elegant examples of roots of $t_C$ whose degree is $2g + 1$ on a surface of genus $g + 1$. This motivated an earlier collaborative work with D. McCullough in which we derived conditions for the existence of a root of degree $n$. We also showed that Margalit-Schleimer roots achieve the maximum value of $n$ among all the roots for a given genus. Suppose that $C$ is a separating curve in $F$. First, we derive algebraic conditions for the existence of roots in Mod$(F)$ of the Dehn twist $t_C$ about $C$. Finally, we show that if $n$ is the degree of a root, then $n \\leq 4g^2 + 2g$, and for $g \\geq 10$, $n \\leq \\frac{16}{5}g^2+ 12g + \\frac{45}{4}$.
Title: Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture--Part II
Speaker: Prof. Jean-Pierre Demailly Institut Fourier, Universite de Grenoble France
Date: Thu, 24 Nov 2011
Time: 4:00-5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Title: Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture--Part I
Speaker: Prof. Jean-Pierre Demailly Institut Fourier, Universite de Grenoble France
Date: Wed, 23 Nov 2011
Time: 4:00-5:00 p.m.
Venue: Department of Mathematics, Lecture Hall I
Title: Dilations, functional model and a complete unitary invariant of a $\Gamma$-contraction
Speaker: Sourav Pal Department of Mathematics, IISc
Date: Mon, 21 Nov 2011
Time: 3:30-4:30 p.m.
Venue: Department of Mathematics, Lecture Hall I
In this talk, we shall define $\\Gamma$-contractions, which were introduced by Jim Agler and Nicholas Young. We shall construct an explicit $\\Gamma$-isometric dilation of a $\\Gamma$-contraction and produce a genuine functional model. A crucial operator equation has to be solved for constructing such a dilation. We shall show how the existence of such a solution characterizes a $\\Gamma$-contraction. This solution, which is unique, also provides a complete unitary invariant.
Title: Challenges in high dimensional Bayesian nonparametrics and some possible solutions
Speaker: Prof. Anjishnu Banerjee Duke University, USA
Date: Thu, 20 Oct 2011
Time: 11:30 a.m.-12:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
Large dimensional data presents many challenges for statistical modeling via Bayesian nonparametrics, both with respect to theroetical issues and computational aspects. We discuss some models that can accomodate large dimensional data and have attractive theoretical properties, specially focussing on kernel partition processes, which are a generalization of the well known Dirichlet Processes. We discuss issues of consistency. We then move onto some typical computational problems in Bayesian nonparametrics, focussing initially on Gaussian processes (GPs). GPs are widely used in nonparametric regression, classification and spatio-temporal modeling, motivated in part by a rich literature on theoretical properties. However, a well known drawback of GPs that limits their use is the expensive computation, typically O($n^3$) in performing the necessary matrix inversions with $n$ denoting the number of data points. In large data sets, data storage and processing also lead to computational bottlenecks and numerical stability of the estimates and predicted values degrades with $n$. To address these problems, a rich variety of methods have been proposed, with recent options including predictive processes in spatial data analysis and subset of regressors in machine learning. The underlying idea in these approaches is to use a subset of the data, leading to questions of sensitivity to the subset and limitations in estimating fine scale structure in regions that are not well covered by the subset. Partially motivated by the literature on compressive sensing, we propose an alternative random projection of all the data points onto a lower-dimensional subspace, which also allows for easy parallelizability for further speeding computation. We connect this with a wide class of matrix approximation techniques. We demonstrate the superiority of this approach from a theoretical perspective and through the use of simulated and real data examples. We finally consider extensions of these approaches for dimension reduction in other non parametric models.
Title: Sampling and meta-learning for function reconstruction -- Case study: blood glucose prediction
Speaker: Prof. Sivananthan Sampath Radon Institute for Computational and Applied Mathematics Linz, Austria
Date: Wed, 19 Oct 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
In this talk, we discuss two problems: the sampling problem for a given class of functions on $\mathbb{R}$ (direct problem) and the reconstruction of a function from finite samples (inverse problem). In the sampling problem, we are given a class of functions $V\subset L^2(\mathbb{R})$ and one seeks to find sets of discrete samples such that any $f$ in $V$ can be completely recovered from its values at the sample points. Here we address the sampling problem for the class of functions $V = V(\varphi)$, the (integer) shift invariant space defined by a generator $\varphi$ with some general assumption. In the second problem, we discuss the problem of reconstruction of a real-valued function $f$ on $X\subset \mathbb{R}^d$ from the given data $\{(x_i,y_i)\}_{i=1}^n\subset X\times\mathbb{R}$, where it is assumed that $y_i=f(x_i)+\xi_i$ and $(\xi_1,…,\xi_n)$ is a noise vector. In particular, we are interested in reconstructing the function at points outside the closed convex hull of $\{x_1,…,x_n\}$, which is the so-called extrapolation problem. We consider this problem in the framework of statistical learning theory and regularization networks. In this framework, we address the major issues: how to choose an appropriate hypothesis space and regularized predictor for given data through a meta-learning approach. We employ the proposed method for blood glucose prediction in diabetes patients. Further, using real clinical data, we demonstrate that the proposed method outperforms the state-of-art (time series and neural-network-based models).
Title: Limiting behaviour of sample autocovariance matrix
Speaker: Prof. Arup Bose ISI Kolkata
Date: Tue, 18 Oct 2011
Time: 11:30 a.m.-12:30 p.m.
Venue: Department of Mathematics, Lecture Hall III
The empirical spectral distribution (ESD) of the sample variance covariance matrix of i.i.d. observations under suitable moment conditions converges almost surely as the dimension tends to infinity. The limiting spectral distribution (LSD) is universal and is known in closed form with support [0,4].
Title: Representations of Symmetric Groups via the RSK Correspondence
Speaker: Prof. Amritanshu Prasad IMSc, Chennai
Date: Fri, 23 Sep 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
I will explain how the purely combinatorial Robinson-Schensted-Knuth correspondence can be used to give a simple proof of the classification of irreducible representations of symmetric groups in the semisimple case. It turns out that all the standard results in the representation theory of symmetric groups can be recovered using this approach.
Title: Regularization by 1/2-Laplacian and Vanishing Viscosity Limit for HJB Equations
Speaker: Prof. Imran Biswas TIFR Centre for Applicable Mathematics
Date: Fri, 16 Sep 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
We investigate the regularizing effect of adding small fractional Laplacian, with critical fractional exponent 1/2 , to a general first order HJB equation. Our results include some regularity estimates for the viscosity solutions of such perturbations, making the solutions classically well-defined. Most importantly, we use these regularity estimates to study the vanishing viscosity approximation to first order HJB equations by 1/2-Laplacian and derive an explicit rate of convergence for the vanishing viscosity limit.
Title: Ergodicity of the geodesic flow of the Weil-Petersson metric
Speaker: Prof. Keith Burns Northwestern University
Date: Fri, 19 Aug 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: The Goldman Bracket and Intersection Numbers II
Speaker: Prof. Siddhartha Gadgil IISc
Date: Wed, 17 Aug 2011
Time: 2:00-3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
I shall give more details of the relation of intersections to hyperbolic geometry and give a sketch of the proof of the main theorem. I shall also outline to the relation of intersection numbers to the so-called Hurwitz equivalence, and hence to smooth 4-manifolds.
Title: The Goldman Bracket and Intersection Numbers
Speaker: Prof. Siddhartha Gadgil IISc
Date: Mon, 08 Aug 2011
Time: 2:00-3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
The Goldman bracket associates a Lie Algebra to closed curves on a surface. I shall describe the bracket and its basic properties. I shall also sketch some joint work with Moira Chas, where we show that the Goldman bracket together with the operation of taking powers determines geometric intersection and self-intersection numbers.
Title: Loewner matrices
Speaker: Prof. Rajendra Bhatia ISI Delhi
Date: Fri, 05 Aug 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Let f be a smooth function on the real line. The divided difference matrices of order n, whose, (i,j)-th entries are the divided differences of f at (\\lambda_i,\\lambda_j) – where \\lambda_1,…,\\lambda_n are prescribed real numbers – are called Loewner matrices. In a seminal paper published in 1934 Loewner used properties of these matrices to characterise operator monotone functions. In the same paper he established connections between this matrix problem, complex analytic functions, and harmonic analysis. These elegant connections sent Loewner matrices into the background. Some recent work has brought them back into focus. In particular, characterisation of operator convex functions in terms of Loewner matrices has been obtained. In this talk we describe some of this work.
Title: Clustering, Percolation and directionally convex ordering of point processes
Speaker: Prof. D. Yogeshwaran Ecole Normale Superieure - INRIA
Date: Wed, 13 Jul 2011
Time: 2:30-3:30 p.m.
Venue: Department of Mathematics, Indian Institute of Science, Lecture Hall I
Heuristics indicate that point processes exhibiting clustering of points have larger critical radii for the percolation of their continuum percolation models than spatially homogeneous point processes. I will explain why the dcx ordering of point processes is suitable to compare their clustering tendencies. Hence, it is tempting to conjecture that the critical radius is increasing in dcx order. We will prove the conjecture for some non-standard critical radii; however it is false for the standard critical radii. I will discuss the implications of these results. A powerful implication is that point processes dcx-smaller than a homogeneous Poisson point process admit uniformly non-degenerate lower and upper bounds on their critical radii. In fact, all the above results hold under weaker assumptions of ordering of moment measures and void probabilities of the point processes. Examples of point processes comparable to Poisson point processes in this weaker sense include determinantal and permanental point processes with trace-class integral kernels. Perturbed lattices are the most general examples of dcx sub- and super-Poisson point processes. More generally, we show that point processes dcx-smaller than a homogeneous Poisson point process exhibit phase transitions in certain percolation models based on the level-sets of additive shot-noise fields of these point process. Examples of such models are k-percolation and SINR-percolation. This is a joint work with Bartek Blaszczyszyn.
Title: Model Theory and a few Applications
Speaker: Prof. Koushik Pal University of California, Berkeley
Date: Wed, 22 Jun 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Model Theory, as a subject, has grown tremendously over the last few decades. Starting out as a branch of mathematical logic, it has now wide applications in most branches of mathematics, with algebra, algebraic geometry, number theory, combinatorics and even analysis, to name a few. In this talk, I will give a brief introduction to model theory, talk about the compactness theorem (one of the main tools in model theory) and how it is used, and give one famous application of model theory to algebra, namely, the Ax-Kochen Theorem, the answer to Artin’s Conjecture. Time permitting, I will talk about a few more recent results in this direction.
Title: Clustering, Percolation and directionally convex ordering of point processes
Speaker: Prof. D. Yogeshwaran Ecole Normale Superieure - INRIA
Date: Mon, 13 Jun 2011
Time: 2:30-3:30 p.m.
Venue: Department of Mathematics, Indian Institute of Science, ROOM TO BE ANNOUNCED
Heuristics indicate that point processes exhibiting clustering of points have larger critical radii for the percolation of their continuum percolation models than spatially homogeneous point processes. I will explain why the dcx ordering of point processes is suitable to compare their clustering tendencies. Hence, it is tempting to conjecture that the critical radius is increasing in dcx order. We will prove the conjecture for some non-standard critical radii; however it is false for the standard critical radii. I will discuss the implications of these results. A powerful implication is that point processes dcx-smaller than a homogeneous Poisson point process admit uniformly non-degenerate lower and upper bounds on their critical radii. In fact, all the above results hold under weaker assumptions of ordering of moment measures and void probabilities of the point processes. Examples of point processes comparable to Poisson point processes in this weaker sense include determinantal and permanental point processes with trace-class integral kernels. Perturbed lattices are the most general examples of dcx sub- and super-Poisson point processes. More generally, we show that point processes dcx-smaller than a homogeneous Poisson point process exhibit phase transitions in certain percolation models based on the level-sets of additive shot-noise fields of these point process. Examples of such models are k-percolation and SINR-percolation. This is a joint work with Bartek Blaszczyszyn.
Title: On a small quotient of a huge absolute Galois group
Speaker: Prof. Sunil K. Chebolu Illinois State University, U.S.A.
Date: Fri, 20 May 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Let G be the absolute Galois group of a field that contains a primitive p-th root of unity. This is a profinite group which is a central object of study in arithmetic algebraic geometry. In joint work with Ido Efrat and Jan Minac, we have shown that a remarkably small quotient of this big group determines the entire Galois cohomology of G. As application of this result, we give new examples of profinite groups that are not realisable as absolute Galois groups of fields. I will present an overview of this work. ALL ARE CORDIALLY INVITED
Title: Noncommutative analogues of the Fej\'er-Riesz Theorem
Speaker: Michael Dritschel
Date: Mon, 25 Apr 2011
Venue: LH 1, Department of Mathematics, IISc
The classical Fej\\‘er-Riesz Theorem states that a nonnegative trigonometric polynomial can be factored as the absolute square of an analytic polynomial. Indeed, the factorization can be done with an outer polynomial. Various generalizations of this result have been considered. For example, Rosenblum showed that the result remained true for operator valued trigonometric polynomials. If one instead considers operator valued polynomials in several variables, one obtains factorization results for strictly positive polynomials, though outer factorizations become much more problematic. In another direction, Scott McCullough proved a factorization result for so-called hereditary trigonometric polynomials in freely noncommuting variables (strict positivity not needed). In this talk we consider an analogue of (hereditary) trigonometric polynomials over discrete groups, and give a result which includes a strict form of McCullough’s theorem as well as the multivariable version of Rosenblum’s theorem.
Title: Noncommutative analogues of the Fej\'er-Riesz Theorem
Speaker: Michael Dritschel
Date: Mon, 18 Apr 2011
Venue: LH 1, Department of Mathematics, IISc
The classical Fej\\‘er-Riesz Theorem states that a nonnegative trigonometric polynomial can be factored as the absolute square of an analytic polynomial. Indeed, the factorization can be done with an outer polynomial. Various generalizations of this result have been considered. For example, Rosenblum showed that the result remained true for operator valued trigonometric polynomials. If one instead considers operator valued polynomials in several variables, one obtains factorization results for strictly positive polynomials, though outer factorizations become much more problematic. In another direction, Scott McCullough proved a factorization result for so-called hereditary trigonometric polynomials in freely noncommuting variables (strict positivity not needed). In this talk we consider an analogue of (hereditary) trigonometric polynomials over discrete groups, and give a result which includes a strict form of McCullough’s theorem as well as the multivariable version of Rosenblum’s theorem.
Title: Quantum Field Theory -- the Mathematics in it
Speaker: Prof. Kalyan B. Sinha JNCASR and IISc
Date: Fri, 01 Apr 2011
Time: 11:00 a.m. to 12:30 p.m. (first lecture)
Venue: Lecture Hall I, Department of Mathematics
Beginning with the attempts of Heisenberg and Pauli in the 1920’s, the subject grewat an astonishing speed, culminating in the remarkable predictive successof Quantum Electrodynamics.Different attemptsto bring the subjectto a sound mathematical footing (comparable to that of Quantum Mechanics)– whetheranalytic, operator-algebraic or geometric – have tastedonly partial success. These talks will try to give a bird’s-eye view of the mathematical areas (ideas)spawned by these attempts, keeping in view the recent book of Folland on this subject.
Title: Calculus on Fractal Curves in R^n and Physical Applications
Speaker: Dr. Seema University of Pune, Pune
Date: Thu, 10 Mar 2011
Time: lh-i, department of mathematics indian institute of science, bangalore
Venue: 4:00-5:00
ALL ARE INVITED
Title: SRB-measure leaks
Speaker: Prof. Shrihari Sridharan Chennai Mathematical Institute
Date: Mon, 07 Mar 2011
Time: 4:00 p.m. - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In this talk, we shall study about the leaking rate of the Sinai-Ruelle-Bowen (SRB) measure through holes of positive measure constructed in the Julia set of hyperbolic rational maps (open dynamics). The dependence of this rate on the size and position of the hole shall be explained. For an easier and better understanding, the simple quadratic map restricted on the unit circle will be analysed thoroughly. Time permitting, we will also compute the Hausdorff dimension of the survivor set.
Title: On the distribution of the eigenvalues of non-selfadjoint operators
Speaker: Prof. Michael Demuth University of Clausthal Germany
Date: Tue, 01 Mar 2011
Time: 4:00 p.m. - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Let A be a selfadjoint operator. We are interested in the discrete spectrum of B = A+M where B is non-selfadjoint. If the resolvent difference is in the Schatten class S_p, then we have an estimate on the distribution of the eigenvalues of B. By means of this estimate we can give qualitative estimates for the number of eigenvalues of B or their moments. That can be applied to Schodinger operators with complex potentials.
Title: Feedback controls of minimal norms
Speaker: Prof. Jean-Pierre Raymond Universite Paul Sabatier, Toulouse France
Date: Wed, 23 Feb 2011
Time: 4:00 p.m. - 5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
In finite-dimensional control theory, feedback control using controllability Gramian goes back to results by Kleinman and Lukes in 1968 and 1970. Some years before, R. Bass characterized controls of minimal norms also using controllability Gramians. The extension of these results to infinite-dimensional systems has a long history. Surprisingly, only reversible infinite-dimensional systems have been considered in those results. We shall present existing results in the literature and we shall characterize stabilizing controls of minimal norms for parabolic systems. This is a joint work with S. Kesavan. Application to the stabilization of the Navier-Stokes equations will be given.
Title: Stochastic Navier-Stokes Equation with Levy noise - where Harmonic Analysis and Stochastic Analysis Meet
Speaker: Prof. S. S. Sritharan Naval Postgraduate School, Monterey, USA
Date: Fri, 11 Feb 2011
Time: 11:15 a.m. - 12:15 p.m.
Venue: Lecture Hall I, Department of Mathematics
Certain classes of non-local pseudo-differential operators can be associated with Markov processes and this result has an infinite dimensional counterpart too. The best known example is the Levy process with its generator an integro-differential operator. In this talk we will give an introduction to stochastic Navier-Stokes equation with jump (Levy) noise and point out opportunities for harmonic as well as stochastic analysis to gain understanding in solvability theory and applications such as control and filtering theory.
Title: On the connections between Stochastic PDE and PDE
Speaker: Prof. B. Rajeev ISI, Bangalore Centre
Date: Fri, 28 Jan 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Stochastic partial differential equations (SPDE) are partial differential equations (PDE) with a `noise term’. One can think of these as a semi-martingale in a function space or a space of distributions with a drift (a bounded variation process involving a second order elliptic partial differential operator ) and a noise term which is a martingale. When the martingale term is suitably structured, the solutions of these SPDE’s are closely related to certain finite dimensional diffusion processes and may be viewed as generalized solutions of the classical stochastic differential equations of Ito, Stroock-Varadhan and others. In this talk [based on Rajeev and Thangavelu (2008) and Rajeev (2010)], we describe how the expected values of the solutions give rise to solutions of PDE’s associated with the diffusion.
Title: Pseudoholomorphic curves and applications
Speaker: Prof. Herve Gaussier Institut Fourier Grenoble, France
Date: Fri, 21 Jan 2011
Time: 2:00-3:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: Cofinal types of ultrafilters
Speaker: Prof. Dilip Raghavan University of Toronto
Date: Fri, 21 Jan 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
I will describe some joint work with Todorcevic on the Tukey theory of ultrafilters on the natural numbers. The notion of Tukey equivalence tries to capture the idea that two directed posets look cofinally the same, or have the same cofinal type. As such, it provides a device for a rough classification of directed sets based upon their cofinal type, as opposed to an exact classification based on their isomorphism type. This notion has recently received a lot of attention in various contexts in set theory. As background, I will illustrate the idea of rough classification with several examples, and explain how rough classification based on Tukey equivalence fits in with other work in set theory. The talk will be based on the paper Cofinal types of ultrafilters. A preprint of the paper is available on my website: http://www.math.toronto.edu/raghavan .
Title: Extremal discs in almost complex manifolds
Speaker: Prof. Herve Gaussier Institut Fourier Grenoble, France
Date: Thu, 20 Jan 2011
Time: 2:30-3:30 p.m.
Venue: Lecture Hall III, Department of Mathematics
Title: Serre's mass formula in prime degree
Speaker: Prof. Chandan Singh Dalawat HRI, Allahabad
Date: Wed, 19 Jan 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Serre’s mass formula counts the number of totally ramified degree-$n$ extensions $E$ of a local field $F$, each extension being assigned some weight depending upon how ramified it is. We will present an elementary proof of this formula when the degree $n$ is prime.The background material will be covered, so that the talk should be accessible to a broad mathematical audience.
Title: Serre's mass formula in prime degree
Speaker: Prof. Chandan Singh Dalawat HRI, Allahabad
Date: Tue, 18 Jan 2011
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Serre’s mass formula counts the number of totally ramified degree-$n$ extensions $E$ of a local field $F$, each extension being assigned some weight depending upon how ramified it is. We will present an elementary proof of this formula when the degree $n$ is prime.The background material will be covered, so that the talk should be accessible to a broad mathematical audience.
Title: The Brylinski Beta function
Speaker: Murali K. Vemuri (Chennai Mathematical Institute)
Date: Mon, 10 Jan 2011
Time: 4 pm
Venue: LH-3, Mathematics Department
An analogue of Brylinski’s knot beta function is defined for a submanifold of $d$-dimensional Euclidean space. This is a meromorphic function on the complex plane. The first few residues are computed for a surface in three dimensional space.
Title: The tyger phenomenon for the Galerkin-truncated Burgers and Euler equations
Speaker: Prof. Uriel Frisch Observatoire de la Cote d'Azur Nice, France
Date: Fri, 24 Dec 2010
Time: lh 1, department of mathematics indian institute of science, bangalore
Venue: 4:00
It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wavenumbers in excess of a threshold $\kg$ exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. At large $\kg$, for smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation which we call a “tyger”, is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features (shocks, layers with strong vorticity gradients, etc). These tygers appear when complex-space singularities come within one Galerkin wavelength $\lambdag = 2\pi/\kg$ from the real domain and typically arise far away from preexisting small-scale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first - in the Burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to $\kg ^{-2/3}$ and $\kg ^{-1/3}$ respectively - but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T.D. Lee in 1952. The sudden dissipative anomaly-the presence of a finite dissipation in the limit of vanishing viscosity after a finite time $\ts$-, which is well known for the Burgers equation and sometimes conjectured for the 3D Euler equation, has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit $\kg\to\infty$. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and thus prevents the flow from converging to the inviscid-limit solution. There are indications that it may be possible to purge the tygers and thereby to recover the correct inviscid-limit behaviour.
Title: Forbidden configurations, extremal set systems, and Steiner designs
Speaker: Prof. Niranjan Balachandran Caltech U.S.A.
Date: Tue, 14 Dec 2010
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
One of the fundamental problems in Extremal Combinatorics concerns maximal set systems with forbidden subconfigurations. One such open problem concerns the conjecture due to Anstee and Sali on the order of maximal configurations with certain forbidden subconfigurations. I shall talk about some well known results, talk about the Anstee-Sali conjecture, and finally talk about some of my recent work concerning Steiner designs occuring as maximal forbidden configurations for certain natural choices of subconfigrations. This generalizes a result of Anstee and Barekat.
Title: Hilbert spaces based on complex Hermite polynomials, and related quantizations
Speaker: Prof. Jean-Pierre Gazeau Universit Paris-7 Denis Diderot France
Date: Mon, 13 Dec 2010
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
I will report on the existence and properties of Hilbert spaces based on different families of complex, holomorphic or not, functions, like Hermite polynomials. The resulting coherent state quantizations of the complex plane will be presented. Some interesting issues will be examined, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules.
Title: An iterated observer method for recovering the initial state for a class of PDE's
Speaker: Prof. Marius Tucsnak Institut Elie Cartan Universite Nancy-1 France
Date: Wed, 24 Nov 2010
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
Title: Distributed Processing and Temporal Codes in Neuronal Networks
Speaker: Wolf Singer (Director, Max Planck Institute for Brain Research Frankfurt/Main Frankfurt Institute for Advanced Studies)
Date: Thu, 18 Nov 2010
Time: LH-1, Mathematics Department
Venue: 11 am
Higher cognitive functions require the coordination of large assemblies of spatially distributed neurons in ever changing constellations. It is proposed that this coordination is achieved through temporal coordination of oscillatory activity in specific frequency bands. Since there is no supra-ordinate command centre in the brain, the respective patterns of synchronous activity self-organize, which has important implications on concepts of intentionality and top down causation. Evidence will be provided that synchronisation supports response selection by attention, feature binding, subsystem integration, short-term memory, flexible routing of signals across cortical networks and access to the work-space of consciousness. The precision of synchronisation is in the millisecond range, suggesting the possibility that information is encoded not only in the co-variation of discharge rates but also in the precise timing of individual action potentials. This could account for the high speed with which cortical circuits can encode and process information. Recent studies in schizophrenic patients indicate that this disorder is associated with abnormal synchronisation of oscillatory activity in the high frequency range (beta and gamma). This suggests that some of the cognitive deficits characteristic for this disease result from deficient binding and subsystem integration.
Title: The Cauchy-Riemann equations in a product domain
Speaker: Prof. Debraj Chakrabarti Department of Mathermatics IIT Bombay
Date: Tue, 16 Nov 2010
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
An important question in complex analysis is to solve the inhomogenous Cauchy-Riemann equations (also called the d-bar equation) in a domain in C^n. The question of boundary smoothness in the d-bar problem is classically dealt with by solving the associated d-bar Neumann problem and showing that the solution operator, the $\\overline{\\partial}$-Neumann operator is compact. For many domains of interest, in particular the product domains, this approach fails. We discuss in this talk some new results on the regularity of the d-bar problem in product domains. This work is joint with Mei-Chi Shaw.
Title: Projective Geometry and Holomorphic curves
Speaker: Prof. Siddhartha Gadgil Indian Institute of Science
Date: Wed, 22 Sep 2010
Time: 4:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
A geometry is a collection of lines and points satisfying the usual incidence axioms. By a theorem of Gromov, given an almost complex structure (which is `tame’) on the complex projective plane CP^2, we obtain a geometry by declaring appropriate holomorphic curves to be the lines. Ghys asked whether Desargues’s theorem (a Euclidean geometry result related to symmetry) characterises the standard complex structure. We show that this is indeed the case.
Title: Distinguishing knots
Speaker: Prof. Siddhartha Gadgil Indian Institute of Science
Date: Fri, 17 Sep 2010
Time: 2:00 p.m.
Venue: Lecture Hall III, Department of Mathematics
Title: Analytic continuation of the resolvent, resonances, and Huygens' principle for Riemannian symmetric spaces of noncompact type
Speaker: Prof. Angela Pasquale Universite Paul Verlaine--Metz France
Date: Mon, 13 Sep 2010
Time: 4:00-5:00 p.m.
Venue: Lecture Hall I, Department of Mathematics
The abstract of this talk has been posted at:
Title: Triangulating the Deligne-Mumford Compactification of Riemann Surfaces
Speaker: Dr. Siddhartha Gadgil, IISc.
Date: Tue, 17 Aug 2010
Time: 2:00 p.m.
Venue: LH-1, Department of Mathematics
The space of all complex structures on a surface, and the Deligne-Mumford compactification of this space, play an important role in many areas of mathematics. We give a combinatorial description of a space that is homotopy equivalent to the Deligne-Mumford compactification, in the case of surfaces with at least one puncture.
Title: The mysterious geometry of soap bubbles
Speaker: Prof. Michael Hutchings, U.C.Berkeley.
Date: Fri, 13 Aug 2010
Time: 4:00 p.m.
Venue: Faculty Hall, IISc.
Why are soap bubbles spherical? Why do double soap bubbles have the shape that they do (three spherical caps meeting along a circle at 120 degree angles)? The single bubble problem was solved in the 19th century, and the double bubble problem was solved a few years ago. The analogous problem for triple soap bubbles remains a mystery. We will give an introduction to these problems and their solutions (when known).
Title: Exploring polynomial convexity of certain classes of sets
Speaker: Mr. Sushil Gorai, IISc.
Date: Wed, 14 Jul 2010
Time: 4:00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
It is very difficult in general to determine when a given compact in C^n, n>1, is polynomially convex. In this talk, we shall discuss polynomial convexity of some classes of sets. First, we shall consider two totally-real surfaces in C^2 that contain the origin and have distinct tangent planes there. We shall discuss how the local polynomial convexity of the union of the tangent planes at (0,0) influences local polynomial convexity of the union of the surfaces at (0,0). Secondly, we will present a condition for local polynomial convexity of unions of more than two totally-real planes in C^2 containing the origin. Next, we shall talk about pluri- subharmonicity. Using this notion we shall give a new proof of an approximation theorem of Axler and Shields and also generalize it. Polynomial convexity plays a very central role in our proof. Finally we shall discuss a characterization for (large) compact patches of smooth totally-real graphs in C^{2n} to be polynomially convex.
Title: A study of the metric induced by the Robin function
Speaker: Mr. Diganta Borah, IISc.
Date: Fri, 09 Jul 2010
Time: 4:00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Let D be a smoothly bounded pseudoconvex domain in C^n, n > 1. Using G(z, p), the Green function for D with pole at p in D, associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kahler metric (the so-called Lambda–metric) using the Robin function arising from G(z, p). The purpose of this thesis is to study this metric by deriving its boundary asymptotics and using them to calculate the holomorphic sectional curvature along normal directions. It is also shown that the Lambda–metric is comparable to the Kobayashi (and hence to the Bergman and Caratheodory metrics) when D is strongly pseudoconvex. The unit ball in C^n is also characterized among all smoothly bounded strongly convex domains on which the Lambda–metric has constant negative holomorphic sectional curvature. This may be regarded as a version of Lu-Qi Keng’s theorem for the Bergman metric.
Title: Topological T - Duality
Speaker: Dr. Ashwin Pande Australian National University Australia.
Date: Wed, 07 Jul 2010
Time: 4.00 p.m.
Venue: Department of Mathematics, LH - III
Title: Certain convolution inequalities on rank one symmetric spaces of noncompact type
Speaker: Prof Swagato Ray IIT, Kanpur
Date: Mon, 14 Jun 2010
Time: 4:00 pm
Venue: LH-I, Department of Mathematics
Kunze Stein inequality can be thought of as a semisimple version of Young’s inequality. A remarkable observation of M.G.Cowling and S.Meda shows that these inequalities can be naturally extended to Lorentz spaces. The final result in this direction was proved by A. Ionescu. In this talk we will try to explain the central ideas behind Kunze Stein type convolution inequalities for Lorentz spaces.
Title: Some generalizations of Hartogs' lemma on analytic continuation
Speaker: Ms. Purvi Gupta
Date: Fri, 04 Jun 2010
Time: 3:30 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
It is well known that there exist domains in C^n, n > 1, such that all functions holomorphic therein extend holomorphically past the boundary. In this talk, we shall examine this surprising phenomenon by discussing refinements of the fundamental example of Hartogs. We shall look at a generalization of Hartogs’ construction discovered by Chirka. Finally, we shall provide a partial answer to a related question raised by Chirka.
Title: Arithmeticity of complex hyperbolic lattices.
Speaker: Martin Deraux Fourier Institute, Grenoble
Date: Mon, 24 May 2010
Time: monday at 2:00 p.m. wednesday at 4:00 p.m.
Venue: LH-1 IISc Mathematics Department
Title: Kinematical conservation laws and propagation of nonlinear waves in three dimensions
Speaker: K. R. Arun, Research student, Department of Mathematics, IISc
Date: Tue, 11 May 2010
We derive the conservation form of equations of evolution of a front propagating in three dimensions. We obtain a system of six conservation laws, known as 3-D kinematical conservation laws (KCL) in a ray coordinate system. The conservative variables of 3-D KCL are also constrained by a stationary vector constraint, known as geometric solenoidal constraint, which consists of three divergence-free type conditions. The 3-D KCL is an under-determined system, and therefore, additional closure relations are required to get a complete set of equations. We consider two closure relations for 3-D KCL: (1) energy transport equation of a weakly nonlinear ray theory (WNLRT) to study the propagation of a nonlinear wavefront, (2) transport equations of a shock ray theory (SRT) to study the propagation of a curved weak shock front. In both the cases we obtain a weakly hyperbolic system of balance laws. For the numerical simulation we use a high-resolution semi-discrete central scheme. The second order accuracy of the scheme is based on MUSCL type reconstructions and TVD Runge-Kutta time stepping procedures. A constrained transport technique is used to enforce the geometric solenoidal constraint and in all the test problems considered, the constraint is satisfied up to very high accuracy. We present the results of extensive numerical experiments, which confirm the efficiency and robustness of the method and also its ability to capture many physically realistic features of the fronts.
Title: Some aspects of the Kobayashi and Caratheodary metrics on pseudoconvex domains.
Speaker: Ms. Prachi, IISc.
Date: Wed, 21 Apr 2010
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
This thesis considers two themes, both of which emanate from and involve the Kobayashi and the Carath\\‘{e}odory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\\mathbf C^2$ and on convex finite type domains in $\\mathbf C^n$ using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in $ \\mathbf C^2$ and convex finite type domains in $ \\mathbf C^n$ in terms of Euclidean parameters. Second a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for $C^1$-isometries of the Kobayashi and Carath\\‘{e}odory metrics on a smoothly bounded strongly pseudoconvex domain.
Title: Test function realizations and Agler-Herglotz representations
Speaker: Prof. James Pickering, Newcastle University, UK
Date: Mon, 19 Apr 2010
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Brody hyperbolicity and homotopy theory
Speaker: Dr. Simone Borghesi
Date: Fri, 16 Apr 2010
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
The most immediate way to use classical homotopy theory to study topological spaces endowed with extra structures, such as complex spaces, is to forget these structures and study the underlying topological space. Usually this procedure is inadequate since what we are forgetting is not homotopy invariant nor topological invariant. There are more sophisticated approaches to use homotopy theory to try to detect different complex structures on the same topological space which prove to be effective, for instance if the topological space is a complex variety and the complex structures endow the tangent bundle of different Chern classes/numbers. A much more thorough approach can be achieved by using model category theory to provide the category of complex spaces with holomorphic maps of an homotopy theory which realizes to the ordinary topological one, it is biholomorphic but not topological invariant. These techniques have been previously implemented by Morel and Voevodsky to create the so called A^1 homotopy theory of algebraic varieties few years ago.
Title: Generalized Calabi-Eckmann manifolds
Speaker: Prof. Parameswaran Sankaran, IMSc, Chennai.
Date: Tue, 13 Apr 2010
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Intersection numbers, embedded spheres and Geosphere laminations for free groups
Speaker: Dr. Suhas Pandit, IISER, Pune
Date: Tue, 13 Apr 2010
Time: 2:00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Free groups and the group of their outer automorphisms have been extensively studied in analogy with (fundamental groups of) surfaces and the mapping class groups of surfaces. We study the analogue of intersection numbers of simple curves, namely the Scott-Swarup algebraic intersection number of splittings of a free group and we also study embedded spheres in $3$- manifold of the form $ M =\\sharp_n S^2 \\times S^1 $. The fundamental group of $M$ is a free group of rank $n$. This $3$-manifold will be our model for free groups. We construct geosphere laminations in free group which are analogues of geodesic laminations on a surface.
Title: Fuglede's conjecture, Tilings and Spectra
Speaker: Prof. Shobha Madan, IIT Kanpur.
Date: Mon, 15 Mar 2010
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: On concave univalent functions
Speaker: Dr. Bappaditya Bhowmik
Date: Thu, 25 Feb 2010
Time: 11:00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
One of the most important subclasses of the class of normalized analytic univalent functions on the open unit disc D is the class of convex functions. In this talk we will focus on meromorphic analogues of the results known for this class. I.e. we consider functions that map D conformally onto a set whose complement is a bounded convex set. We shall begin with a brief history of Livingston’s conjecture which concerns the exact set of variability of the Taylor coefficients for concave functions. Thereafter, we shall discuss some new results concerning the closed convex hull of concave functions and extreme points of it.
Title: Can Computers do Mathematics? Ventures in Topology/Geometry
Speaker: Dr. Siddhartha Gadgil, IISc.
Date: Mon, 11 Jan 2010
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We discuss applications of computers to prove mathematical theorems. In particular we discuss possible future applications in Topology/Geometry. This will also be the introductory lecture to the new course `Computer Assisted Topology/Geometry’
Title: Effective Models and Algebraic Topology II.
Speaker: Prof. Dennis Sullivan, Stony Brook.
Date: Tue, 29 Dec 2009
Time: 2:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Minimal intersections of curves on surfaces.
Speaker: Dr. Moria Chas, Stony Brook.
Date: Tue, 29 Dec 2009
Time: 11:00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Negative sectional curvature and product complex structures.
Speaker: Dr. Harish Seshadri, IISc.
Date: Tue, 29 Dec 2009
Time: 10:00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: String Topology
Speaker: Prof. Dennis Sullivan, Stony Brook.
Date: Mon, 21 Dec 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Can Computers do Mathematics? Ventures in Topology/Geometry
Speaker: Dr. Siddhartha Gadgil, IISc.
Date: Mon, 21 Dec 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We discuss applications of computers to prove mathematical theorems. In particular we discuss possible future applications in Topology/Geometry. This will also be the introductory lecture to the new course `Computer Assisted Topology/Geometry’
Title: Algebraic structures related to the intersection of curves on surfaces
Speaker: Dr. Moria Chas, Stony Brook.
Date: Mon, 21 Dec 2009
Time: 2:30 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Given a surface, one can consider the set of free homotopy classes of oriented closed curves (this is the set of equivalence classes of maps from the circle into the surface, where two such maps are equivalent if the corresponding curves can be deformed one into the other.) Given a free homotopy class one can ask what is the minimum number of times (counted with multiplicity) in which a curve in that class intersects itself. This is the minimal self-intersection number of the free homotopy class. Analogously, given two classes, one can ask what is the minimum number of times representatives of these classes intersect. This is the minimal intersection number of these two classes.
Title: Hyperbolicity of cycle spaces and Automorphism groups of flag domains
Speaker: Prof. Alan Huckleberry, Ruhr University, Bochum, Germany
Date: Fri, 11 Dec 2009
Time: 2:30 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
The appropriate context for algebraic-geometric realizations of holomorphic representations of a complex semisimple group is that of a compact homogeneous projective flag manifold Z = G/Q. One topic of more current interest is the study of the possibility of realizing (infinite-dimensional) unitary representations of a real form G0 of G on function- and/or cohomology-spaces of open G_0-orbits D in Z (flag domains) and their cycle spaces. After an introduction for nonspecialists, we will indicate a proof by Schubert incidence geometry of the Kobayashi hyperbolicity of the relevant cycle spaces. This will then be applied to give an exact description of the group Aut_O(D) of holomorphic automorphisms.
Title: Realization and interpolation in the Schur class of the polydisk
Speaker: Prof. Michael Dritschel, Newcastle University, UK.
Date: Tue, 01 Dec 2009
Time: 4:00 p.m.
Venue: CHEP Lecture Hall
The classical realization theorem gives various characterizations of functions in the unit ball of $H^\\infty(\\mathbb D)$, the bounded analytic functions on the unit disk $\\mathbb D$, which happens to also correspond to the multipliers of Hardy space $H^2(\\mathbb D)$. This realization theorem yields an elegant way of solving the Nevanlinna-Pick interpolation problem. In the mid 80s, Jim Agler discovered an analogue of the realization theorem over the polydisk. While for $d=2$, it once again gives a characterization of the elements of the unit ball of $H^\\infty(\\mathbb D^d)$ (and so allows one to solve interpolation problems in $H^\\infty(\\mathbb D^2)$), for $d>2$, the class of functions which are realized is a proper subset of the unit ball of $H^\\infty(\\mathbb D^d)$ — the so-called Schur-Agler class.
Title: The connection between Clifford analysis and operator theory
Speaker: Prof. Brian Jefferies, University of New South Wales, Sydney
Date: Mon, 30 Nov 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Clifford analysis is a higher dimensional analogue of single variable complex analysis. Although functions take values in a finite dimensional Clifford algebra, the representation formula for Clifford regular functions is simpler and more powerful than for holomorphic functions of several complex variables. The talk shows how Clifford analysis techniques can be employed in operator theory for a functional calculus of $n$-tuples of operators.
Title: Limiting Spectral Distribution of Large Dimensional Random Matrices: Another Look at the Moment Method
Speaker: Prof. Arup Bose, ISI, Kolkata
Date: Wed, 21 Oct 2009
Time: 2:00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices include the moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample variance covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalence classes and relating the limits of the counts to certain volume calculations. We develop this method further and provide a general framework to deal with symmetric patterned matrices with entries coming from an independent sequence. This approach can be extended to cover matrices of the form Ap =XX’/n where X is a pxn matrix with p going infinity and n = n(p) going to infinity and p/n going to y between 0 and infinity. The method can also be used to cover some situations where the input sequence is a suitable linear process. Several new classes of limit distributions arise and many interesting questions remain to be answered.
Title: Buseman functions in a harmonic manifold
Speaker: Dr. Hemangi Shah, IISc.
Date: Mon, 19 Oct 2009
Time: 4:00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functions on M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally we prove that the total Busemann function is continuous in C^\\infty topology. As a consequence of it we show that the uniform divergence of geodesics holds in these spaces.
Title: Mean-field Limits of Interacting Particle Systems
Speaker: Dr. D. Yogeshwaran, Ecole Normale Superieure / INRIA, Paris.
Date: Wed, 02 Sep 2009
Time: 11.30 - 12.45
Venue: ECE 1.07
Consider a system of N interacting particles evolving over time with Markovian dynamics. The interaction occurs only due to a shared resource. I shall briefly sketch the classical tightness-existence-uniqueness approach used to prove weak convergence of the system to a limiting system as N tends to infinity. I shall illustrate another approach in more detail with the example of a nonlinear Markov chain. In the case of exchangeable particles, weak convergence of the system implies propogation of chaos i.e, in the limiting system particles evolve independently due to a deterministically shared resource.
Title: Weighted zero-sum problems: recent progress
Speaker: Prof. S. D. Adhikari, HRI, Allahabad.
Date: Tue, 01 Sep 2009
Time: 4:00 p.m.
Venue: Lecture Hall - II, Dept. of Mathematics
After a brief introduction to some classical group invariants, we proceed to consider their generalizations with weights. We discuss some recent results on these weighted generalizations.
Title: Comparison of critical percolation radii of directionally convex orderered point processes.
Speaker: Dr. D. Yogeshwaran, Ecole Normale Superieure / INRIA, Paris.
Date: Tue, 01 Sep 2009
Time: 4:00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Heuristics indicate that more the clustering in a point process, worse the percolation will be. We shall in this talk see a first step towards a formal proof of this heuristic. Here is a more precise abstract of the results we shall see in the talk.
Title: The d-th polarization constant of Rd
Speaker: Prof. Rakesh, University of Delaware
Date: Fri, 28 Aug 2009
Time: 4:00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
The polarization conjecture for R^d is that
Title: A complex hyperbolic reflection group and the bimonster
Speaker: Dr. Tathagatha Basak, IPMU, Japan
Date: Fri, 21 Aug 2009
Time: 11:00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Let R be the reflection group of the complex leech lattice plus a hyperbolic cell. Let D be the incidence graph of the projective plane over the finite field with 3 elements. Let A(D) be the Artin group of D: generators of A(D) correspond to vertices of D. Two generators braid if there is an edge between them, otherwise they commute.
Title: Gibbs measure asymptotics
Speaker: Prof. K. B. Athreya
Date: Fri, 14 Aug 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Let H :R –> R. with multiple minima on R.For T> 0 consider the probability measure on R with pdf proportional to exp(–H/T). In this talk we discuss the problem of weak limits of this measure as T goes to infinity. It depends on the behaviour of H near its minima locations. Both Gaussian and stable limit laws arise as weak limits.
Title: Interacting diffusion models and the effect of size
Speaker: Dr.Soumik Pal, University of Washington, Seattle
Date: Tue, 04 Aug 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Consider a multidimensional diffusion model where the drift and the diffusion coefficients for individual coordinates are functions of the relative sizes of their current value compared to the others. Two such models were introduced by Fernholz and Karatzas as models for equity markets to reflect some well-known empirically observed facts. In the first model, called ‘Rank-based’, the time-dynamics is determined by the ordering in which the coordinate values can be arranged at any point of time. In the other, named the ‘Volatility-stabilized’, the parameters are functions of the ratio of the current value to the total sum over all coordinates. We show some remarkable properties of these models, in particular, phase transitions and infinite divisibility. Relationships with existing models of queueing, dynamic spin glasses, and statistical genetics will be discussed. Part of the material is based on separate joint work with Sourav Chatterjee and Jim Pitman.
Title: Gaussian Minkowski functionals: an overview of infinite dimensional geometry in Wiener space
Speaker: Dr. V. Sreekar
Date: Fri, 24 Jul 2009
Time: 4 - 5 pm.
Venue: LH-III, Department of Mathematics
Gaussian Minkowski functionals (GMFs) for reasonably smooth subsets of Euclidean spaces, are defined as coefficients appearing in the the Gaussian-tube-formula in finite dimensional Euclidean spaces. The fact that the measure in consideration here is Gaussian, itself makes the whole analysis an infinite dimensional one. Therefore, one might want to generalize the definition of Gaussian Minkowski functionals to the subsets of Wiener space which arise from reasonably smooth (in Malliavin sense) Wiener functionals. As in the finite dimensional case, we shall identify the GMFs in the infinite dimensional case, as the coefficients appearing in the tube formula in Wiener space. Finally, we shall try to apply this infinite dimensional generalization to get results about the geometry of excursion sets of a reasonably large class of random fields defined on a “smooth” manifold.
Title: Wave Propagation on Networks
Speaker: Prof. E. Zuazua Scientific Director BCAM - Basque Center for Applied Mathematics, Spain
Date: Fri, 24 Jul 2009
Time: lh 1, department of mathematics indian institute of science, bangalore
Venue: 4:00
Title: The ubiquity of the symplectic Hamiltonian equations in mechanics
Speaker: Prof. Manuel de Len Instituto de Ciencias Matemticas Consejo Superior de Investigaciones Cientificas
Date: Tue, 21 Jul 2009
Time: lh 1, department of mathematics indian institute of science, bangalore
Venue: 4:00
ALL ARE INVITED
Coffee/Tea: 3:45 pm
Title: Horocycles on the modular surface and diophantine approximation
Speaker: Dr. Jayadev Athreya, Yale University
Date: Mon, 20 Jul 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We describe a relationship between cusp excursions of horocycles on the modular surfaces SL(2, R)/SL(2, Z) and diophantine approximation. Some of the work we discuss will be joint with G. Margulis, and some joint with Y. Cheung
Title: Large deviations for a tagged particle in one dimensional simple exclusion.
Speaker: Prof. Sunder Sethuraman, Iowa State Univ.
Date: Tue, 07 Jul 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Quotient Varieties modulo Finite Groups
Speaker: Prof. S. S. Kannan, CMI
Date: Wed, 01 Jul 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Let $V$ be finite dimensional vector space over the field of complex numbers. Let $G$ be a finite subgroup of $GL(V)$, group of all $\mathbb{C}$- linear automorphisms of $V$. Then, the apmple generator of the Picard group of the projective space $\mathbb P(V)$ descends to the quotient variety $\mathbb{P}(V)/G$. Let $L$ denote the descent. We prove that the polarised variety $\mathbb P(V)/G, L$ is projectively normal when $G$ is solvable or $G$ is generated by pseudo reflections.
Title: Operator Theory and Algebraic Topology
Speaker: R. G. Douglas Texas A&M University
Date: Fri, 05 Jun 2009
Time: l h 1, department of mathematics indian institute of science, bangalore
Venue: 4.00
About thirty-five years ago, several problems in operator theory concerning almost normal operators led L. G. Brown, P. A. fillmore and me to introduce methods and point of view from algebraic topology to solve them. By the time we were done concrete realization of K-homology was introduced as well as new insight obtained for the index theorem of Atiyah-Singer.
Title: Systems Biology Approaches for the Environmental Biotechnology Applications
Speaker: Prof. Radhakrishnan Mahadevan Department of Chemical Engineering and Applied Chemistry University of Toronto
Date: Wed, 06 May 2009
Time: 2:00 pm
Venue: Lecture Hall I, Department of Mathematics
Title: On spectrum and arithmetic
Speaker: Prof. C. S. Rajan, TIFR, Mumbai
Date: Mon, 20 Apr 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We will discuss the notion of spectrum and arithmetic of spaces, and expound on the expectation that they should mutually determine each other for the class of locally symmetric spaces associated to congruent arithmetic lattices.
Title: The homogeneous shifts via inductive algebras
Speaker: Prof. M. K. Vemuri Chennai Mathematical Institute Chennai
Date: Mon, 20 Apr 2009
Time: lh -1, department of mathematics indian institute of science, bangalore
Venue: 11.30
Title: Size biasing with applications to Markov chains and branching processes.
Speaker: K. B. Athreya, Iowa State university
Date: Sun, 19 Apr 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
If X is a positive random variable with a finite mean then the probability distribution with density proprtional to X is called its size biased version. For Markov chains admitting a positive eigen function one can construct a size biased version of this chain which is also Markov.. In this talk we derive conditions for the two chains to be dominated by each other over the full trajectory space.. We then apply this to derive a LLOGL result for supercritical branching processes with arbitrary type space.
Title: Godunov-type schemes for hyperbolic systems with parameter dependent source; the case of Euler system with friction.
Speaker: Prof. Edwige Godlewski, Laboratoire Jacques-Louis Lions, Universit Pierre et Marie Curie Paris 6
Date: Mon, 23 Mar 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We are interested in deriving schemes having some ‘well-balanced’ and ‘asymptotic preserving’ properties for the approximation of a nonlinear hyperbolic system with source term. In the case of Euler system with friction, the scheme is derived from simple Riemann solvers or equivalently using a relaxation scheme for the enlarged Euler system with ‘potential’. All interested are Welcome
Title: On a conjecture of Tits and Weiss
Speaker: Prof. Maneesh Thakur, ISI, Delhi
Date: Fri, 20 Mar 2009
Time: 10:00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Tits and Weiss in their book Moufang Polygons have conjectured that the groups of rational points of certain forms of E_6 are generated by some inner transformations. These groups occur as groups of similitudes of certain cubic forms in 27 variables. We will explain this conjecture and report on some results and reductions towards a positive answer.
Title: On spectrum and arithmetic
Speaker: Prof. C. S. Rajan, TIFR, Mumbai
Date: Fri, 20 Mar 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We will discuss the notion of spectrum and arithmetic of spaces, and expound on the expectation that they should mutually determine each other for the class of locally symmetric spaces associated to congruent arithmetic lattices.
Title: Size biasing with applications to Markov chains and branching processes.
Speaker: K. B. Athreya, Iowa State university
Date: Thu, 19 Mar 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
If X is a positive random variable with a finite mean then the probability distribution with density proprtional to X is called its size biased version. For Markov chains admitting a positive eigen function one can construct a size biased version of this chain which is also Markov.. In this talk we derive conditions for the two chains to be dominated by each other over the full trajectory space.. We then apply this to derive a LLOGL result for supercritical branching processes with arbitrary type space.
Title: Assessing copy number variation using genome-wide alignments
Speaker: Dr. Deepayan Sarkar, Fred Hutchinson Cancer Research Institute USA
Date: Mon, 02 Mar 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Polynomial and Rational Hulls
Speaker: Prof. Nikolay Shcherbina Department of Mathematics University of Wuppertal, Germany
Date: Fri, 20 Feb 2009
Time: department of mathematics indian institute of science, bangalore
Venue: 4PM
We will discuss the notions of polynomial and rational convexity. In particular, the question of existence of analytic structure on the additional part of the hull will be considered. Applications of polynomial and rational convexity to other problems of complex analysis will also be given.
Title: An unusual problem in large deviations
Speaker: S.R.S. Varadhan, Courant Institute
Date: Thu, 12 Feb 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We will investigate the large deviation rates for sums of the form $\sum_i f(x_i) g(x_{2i})$ where $\{x_i\}$ is a nice Markov process. In other words calculate
$\lim_{n\to\infty}{1\over n}\log E[\exp \sum_{i=1}^n f(x_i) g(x_{2i})]$
where $\{x_i\}$ is Markov Chain with transition probability $\pi(x, y)$.
Title: Recent Trends in Invariant Theory
Speaker: Prof. Roger Howe Mathematics Department, Yale University
Date: Tue, 10 Feb 2009
Time: l h 1, department of mathematics indian institute of science, bangalore
Venue: 04:00pm
Title: Modulation Spaces and Banach Gelfand Triples
Speaker: Prof. Hans G. Feichtinger University of Vienna, Austria
Date: Mon, 19 Jan 2009
Time: l h i, department of mathematics indian institute of science, bangalore
Venue: 11:15
http://www.univie.ac.at/nuhag-php/scheduler/index.php
Title: Analysis of the Wright{Fisher Equation
Speaker: Prof. D. W. Stroock M.I.T, USA
Date: Fri, 16 Jan 2009
Time: l h 1, department of mathematics indian institute of science, bangalore
Venue: 4:00
Title: Homogeneous Vector bundles in Operator theory.
Speaker: Prof. A. Koranyi, Lehman College, CUNY
Date: Thu, 15 Jan 2009
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Support Vector Machines in Machine Learning
Speaker: Hans D. Mittelmann (Department of Math & Stats, Arizona State University)
Date: Tue, 06 Jan 2009
Time: department of mathematics indian institute of science, bangalore
Venue: 02:30
Very large datasets occur in the area of machine learning (ML). The tasks are having a computer “learn” to read handwriting, to understand speech, to recognize faces, to filter spam e-mail etc. Mathematically, these problems lead to huge optimization problems, of an, however not unfavorable type, namely convex quadratic programs (QP).
Title: Optimization Software for Financial Mathematics
Speaker: Hans D. Mittelmann (Department of Math & Stats, Arizona State University)
Date: Mon, 05 Jan 2009
Time: department of mathematics indian institute of science, bangalore
Venue: 02:30
Information about available software to solve a large variety of optimization problems is provided at plato.asu.edu/guide.html while some of this software is evaluated at plato.asu.edu/bench.html. Starting with these sources an overview will be given on codes that are particularly useful for applications in mathematical finance.
Title: Stochastic Lagrangian Particle systems for the Navier-Stokes and Burgers equations.
Speaker: Dr. Gautam Iyer, Stanford University
Date: Fri, 02 Jan 2009
Time: 2:30 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
I will introduce an exact stochastic representation for certain non-linear transport equations (e.g. 3D-Navier-Stokes, Burgers) based on noisy Lagrangian paths, and use this to construct a (stochastic) particle system for the Navier-Stokes equations. On any fixed time interval, this particle system converges to the Navier-Stokes equations as the number of particles goes to infinity.
Title: Comparitive description of Circuits in Classical and Quantum Computers
Speaker: Prof. K. R. Parthasarathy Indian Statistical Institute, Delhi
Date: Tue, 30 Dec 2008
Time: l h 1, department of mathematics indian institute of science, bangalore
Venue: 11:15
Title: KRS bases for rings of invariants
Speaker: Dr. K. N. Raghavan, I.M.Sc.
Date: Fri, 26 Dec 2008
Time: 2:30 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
The talk will be a report on an ongoing research project (joint with Preena Samuel and K.V.Subrahmanyam). Representation theoretic consequences will be worked out of the combinatorial characterizations of left, right, and two-sided Kazhdan-Lusztig cells of the symmetric group. Applications to invariant theory and both ordinary and modular representation theory of the symmetric group will be given.
Title: Complex Interpolation for Banach Spaces of Operators
Speaker: Prof. G. Pisier University Of Paris 6 and Texas A & M University
Date: Wed, 24 Dec 2008
Time: department of mathematics indian institute of science, bangalore
Venue: 11:15
https://math.iisc.ac.in/~imi/downloads/gpisier.pdf
Title: Approximation of currents and applications to complex geometry
Speaker: Prof. J. Demailly Institnt Fourier France
Date: Tue, 23 Dec 2008
Time:
Venue:
Title: Infinitesimal Hecke algebras
Speaker: Apoorva Khare (UC Riverside, USA)
Date: Wed, 17 Dec 2008
Time: 11 am
Venue: Lecture Hall - I, Dept. of Mathematics
We study families of infinite-dimensional algebras that are similar to semisimple Lie algebras as well as symplectic reflection algebras. Infinitesimal Hecke algebras are deformations of semidirect product Lie algebras, and we study two families over $\mathfrak{sl}(2)$ and $\mathfrak{gl}(2)$. Both of them have a triangular decomposition and a nontrivial center, which allows us to define and study the BGG Category $\mathcal{O}$ over them - including a (central character) block decomposition, and an analog of Duflo’s Theorem about primitive ideals. We then discuss certain related setups.
Title: Dirichlet Processes and Reflected Diffusions
Speaker: Dr. Kavita Ramanan, Carnegie Mellon University
Date: Tue, 16 Dec 2008
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Reflected diffusions arise in many contexts. We identify a rather general condition under which these diffusions belong to the class of so-called Dirichlet processes, which are a generalization of continuous semimartingales that admit many nice properties, including an Ito formula. We also provide an example arising from applications, in which the reflected diffusion fails to be a semimartingale, but belongs to the class of Dirichlet processes. This is partly based on joint work with Weining Kang.
Title: The Embedded Contact Homology of Circle Bundles over Riemann Surfaces
Speaker: Dr. David Farris, U.C. Berkeley
Date: Tue, 16 Dec 2008
Time: 11.30 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Embedded contact homology (ECH) is an invariant of three-manifolds due to Hutchings, Sullivan, and Taubes. It uses a contact structure on a three-manifold to produce an invariant of the underlying topological manifold. The invariant is the homology of a chain complex generated by certain closed orbits of the Reeb vector field (of interest in classical dynamics), whose differential counts certain holomorphic curves in the symplectization of the contact three-manifold. Few nontrivial examples of ECH have been computed. In this talk, I will give some background and context on ECH and then describe the computation of the ECH of circle bundles over Riemann surfaces, in which the relevant holomorphic curves are actually meromorphic sections of complex line bundles.
Title: Temporal spike pattern learning in biological systems
Speaker: Dr. Sachin S. Talathi (University of Florida, Gainesville, USA)
Date: Fri, 14 Nov 2008
Time: department of mathematics indian institute of science, bangalore
Venue: 4pm
The central question will be focused around the design and implementation of neuronal circuitry involved in the task of decoding sensory information. Sensory information is encoded in the form of sequence of action potential spikes by the peripheral nervous system. This information is then passed onto the central nervous system (CNS) to generate an appropriate response. The action potential spikes are identical in shape and therefore it is assumed that all the information about the environment is embedded in the timing of occurrences of these spikes. The question then is, what neural architecture exists in the CNS to decode this information?
Title: Complex dimensions, attached analytic discs and parametric argument principle
Speaker: Mark Agranovsky, Bar-Ilan University
Date: Fri, 19 Sep 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Complex dimensions, attached analytic discs and parametric argument principle
Speaker: Mark Agranovsky, Bar-Ilan University
Date: Fri, 12 Sep 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Conservation laws admitting delta shock wave type solutions
Speaker: Prof. Prof. G. D. Veerappa Gowda TIFR-CAM, Bangalore
Date: Fri, 29 Aug 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Noncommutative geometry and mirror symmetry
Speaker: Dr. Pranav Pandit University of Pennsylvania
Date: Mon, 25 Aug 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
I will outline an approach to noncommutative geometry, due largely to M. Kontsevich, where certain A-infinity categories play the role of spaces. This noncommutative geometry program draws much of its inspiration from homological mirror symmetry. The talk will be purely expository: it will not contain any new results.
Title: Some uncertainty principles on solvable Lie groups
Speaker: Prof. Ali Baklouti Department of Mathematics, University of Sfax, Sfax, Tunisia
Date: Fri, 22 Aug 2008
Time: department of mathematics indian institute of science, bangalore
Venue: L
The aim of this series of lectures is to seek analogies of some known uncertainty principles, proved in the case of the real line to certain solvable Lie groups. We will speak about uncertainty principles of Hardy, Cowling Price, Morgan and Beurling and present some recent results on their non commutative analogues. Furthermore, we will also talk about sharpness of the decay condition in Hardys principle and give some new result in that context.
Title: Inconsistency of the Bootstrap in Problems Exhibiting Cube Root Asymptotics
Speaker: Dr. Moulinath Banerjee University of Michigan
Date: Wed, 13 Aug 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1\\over 3}$. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown non-increasing density function $f$ on $[0,\\infty)$, is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for $f(t_0)$, where $t_0 \\in (0,\\infty)$ is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function or its least concave majorant, does not have any weak limit in probability. Bootstrapping from a smoothed version of the least concave majorant, however, leads to strongly consistent estimators and the $m$ out of $n$ bootstrap method is also consistent. Our results cast serious doubt on some previous claims about bootstrap consistency (in the class of cube root problems) in the published literature.
Title: Estimation of function thresholds using multistage adaptive procedures
Speaker: Dr. Moulinath Banerjee University of Michigan
Date: Tue, 12 Aug 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
In this talk, I will discuss threshold estimation for a regression function in some different settings. The threshold can either be a change–point, i.e. a point of jump discontinuity in an otherwise smooth curve, or the first time that a regression function crosses a certain level. Both problems have numerous applications in a variety of spheres, like biology (pharmacology, dose-response experiments) and engineering. Our goal is to estimate thresholds of this type given a fixed budget of points to sample from, but with the flexibility that batch sampling can be done in several stages, so that adaptive strategies are possible. Our strategy is to use multistage zoom-in procedures to estimate the threshold: an initial fraction of the sample is invested top come up with a first guess, an adequate neighborhood of the first guess is chosen, more points are sampled from this neighborhood and the initial estimate id updated. The procedure continues thus, ending in a finite number of stages. Such zoom-in procedures result in accelerated convergence rates over any one–stage method. Approximations to relative efficiencies are computed and optimal allocation strategies, as well as recipes for construction of confidence sets discussed.
Title: Preferential Attachment Random graphs with general weight function and general input sequence.
Speaker: Dr. K. B. Athreya Iowa State University
Date: Mon, 04 Aug 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear.We also discuss the case of arbitrary input sequence.
Title: Conformal invariance in probability and statistical physics
Speaker: Dr. Manjunath Krishnapur University of Toronto
Date: Tue, 29 Jul 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
This is an expository talk whose aim will be to give an introduction to SLE (Schramm-Loewner evolution), discovered by Oded Schramm in 2000 to describe many critical statistical mechanical systems.
Title: Modular Forms, Elliptic Curves and Galois Representations.
Speaker: Dr. Aftab Pande Cornell University
Date: Wed, 23 Jul 2008
Time: 4.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
I will give a brief overview of Wiles’ proof of Fermat’s Last Theorem, and explain the connection between modular forms and elliptic curves via Galois representations e.g. the Taniyama-Shimura conjecture. In the second half, I’ll explain some recent results on p-adic modular forms and deformations of Galois representations. If time permits, I’ll outline a future project on ranks of elliptic curves.
Title: Lattice Point Asymptotics and Volume Growth on Teichmuller space
Speaker: Dr. Jayadev Athreya Department of Mathematics, Yale University
Date: Wed, 16 Jul 2008
Time: 11.30 a.m.
Venue: Lecture Hall - III, Dept. of Mathematics
We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of B_R(x), and also for for the cardinality of the intersection of B_R(x) with an orbit of the mapping class group.
Title: Lattice Point Asymptotics and Volume Growth on Teichmuller space
Speaker: Dr. Jayadev Athreya Department of Mathematics, Yale University
Date: Fri, 11 Jul 2008
Time: 4.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of B_R(x), and also for for the cardinality of the intersection of B_R(x) with an orbit of the mapping class group.
Title: The Gorenstein Colength of an Artinian Local Ring
Speaker: Ananth Hariharan
Date: Tue, 24 Jun 2008
Time: 4:00 p.m.
Venue: LH 1, Mathematics Department
The question we are interested in is the following: Given an Artinian local ring, how ‘close’ can we get to it by an Artinian Gorenstein local ring. In this talk I will make the notion of being ‘close’ precise numerically. We will exhibit some natural bounds on this number and discuss some (old and new) results. In particular, if R is a quotient of a power series ring over a field of characteristic zero by a power of the maximal ideal, we will see how one can compute this number.
Title: Eigencurve methods for some generalised eigenvalue problems
Speaker: Professor Paul Binding, Department of Mathematics, University of Calgary, Canada
Date: Fri, 23 May 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Many applied problems give rise to generalised eigenvalue problems of the form Ay = s By, where A and B are operators of various possible types, and s is an eigenparameter. Two parameter embeddings Ay = s By - t Cy generate useful methods of attack, and can be traced back about a century, but new applications are still appearing. They provide ways to visualise simply some quite complicated phenomena, and in this talk I will discuss some older and newer ones that I have found interesting.
Title: The MOR cryptoystem and special linear groups over finite fields
Speaker: Dr. Ayan Mahalanobis, Stevens institute, New Jersey
Date: Wed, 21 May 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
The ElGamal cryptosystem is in the heart of public key cryptography. It is known that the MOR cryptosysetm generalizes it from the cyclic group to the automorphism group of a (non-abelian) group. I will start by describing the MOR cryptosystem and then we will use the special linear group over a finite field as the platform group. It seems likely that this project is competitive with the elliptic curves over finite fields in terms of security. I’ll explain why I think so. Then we can talk about challenges in implementation of this cryptosystem. All interested are Welcome
Title: Null controllability of a fluid-structure model
Speaker: Professor Jean-Pierre Raymond
Date: Mon, 25 Feb 2008
Time: 4.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
We consider a system coupling the Stokes system with an elastic structure modelled by a finite dimensional system. We prove that this system can be driven to zero by a control action only in the fluid equation. The proof is based on a global Carleman inequality. Because of the coupling between the fluid equation and the structure new boundary terms appear in the Carleman inequality, and estimating these terms requires new techniques.
Title: 3-D Kinematical Conservation Laws: equations of evolution of a surface
Speaker: Prof. Phoolan Prasad
Date: Fri, 22 Feb 2008
Venue: Lecture Hall 3, Department of Mathematics
Title: MATHEMATICAL MODELLING IN HEMODYNAMICS I
Speaker: Dr. Anna Zau=9Akov=E1 Institute of Numerical Simulation, Hamburg University of Technology, Germany
Date: Fri, 15 Feb 2008
Venue: LH-1 , Math Dept
2D Mathematical model an numerical simulations of non-Newtonian shear depen= dent=20 flow with fluid-structure interaction
Title: Watson-Crick pairing for RNA and Milnor's link invariants
Speaker: Dr. Siddhartha Gadgil, IISc
Date: Wed, 06 Feb 2008
Time: 4:00 p. m.
Venue: Lecture Hall III, Dept. of Mathematics
We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which can be interpreted in terms of the Heisenberg group as well as lattice paths, which we call the Heisenberg invariant.
Title: Lorenz knots and modular knots.
Speaker: P. Dehornoy.
Date: Mon, 04 Feb 2008
Time: 4-5 pm
Venue: LH 1, IISc Maths Dept
Title: Watson-Crick pairing for RNA and Milnor's link invariants
Speaker: Dr. Siddhartha Gadgil, IISc
Date: Tue, 22 Jan 2008
Time: 4:00 p. m.
Venue: Lecture Hall III, Dept. of Mathematics
We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which can be interpreted in terms of the Heisenberg group as well as lattice paths, which we call the Heisenberg invariant.
Title: Mathematical Aspects of 2-dimensional Yang-Mills Theory
Speaker: Prof. Ambar Sengupta Louisiana State University.
Date: Fri, 18 Jan 2008
Time: 4.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Yang-Mills gauge theory, classical as well as quantum, in two dimensions is both mathematically tractable and poses interesting questions. The quantum theory for the field has a mathematically precise formulation in terms of a Gaussian measure. Classical holonomies become group-valued random variables in this setting. This talk will present an overview of some mathematical problems, solutions, and ideas arising from two-dimensional Yang-Mills theory
Title: Graph Spectra: A Tool for Analyzing Networks
Speaker: Dr. D. Anirban Banerjee, Max-Planck Institute, Leipzig
Date: Thu, 17 Jan 2008
Time: 4.15 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
The existing graph invariants can retrieve certain structural informations, but they are not sufficient to capture all qualitative aspects of a graph. One of the aim of graph theory to identify on one hand the unique and special feature for the network from a particular class and on the other hand the universal qualities that are shared by other network structures. It is a challenge to specify the domain of a given a network structure, on the basis of certain unique qualitative features. We develop theoretical scheme and apply the general method, based on the spectral plot of the normalized graph Laplacian, that is easily visually analyzed and can be considered as excellent diagnostic to categorize the networks from different sources. Construction with different graph operation related to evolution of a network produce specific eigenvalue, describe certain processes of graph formation that leave characteristic traces in the spectrum. We show how useful plausible hypothesis about evolutionary process can be made by investigating the spectra of a graph constructed from actual data. Based on this idea we have reconstructed protein-protein interaction network which is structurally more close to real protein-protein interaction networks than the networks constructed by other models. We also introduced a tentative classification scheme for empirical networks based on global qualitative properties detected through the spectrum of the Laplacian of the graph underlying the network.This method identifies several distinct types of networks across different domains of applications, which is rather difficult by other existing tool and parameters. Thus we infer that spectral distribution is complete qualitative characterization of a graph.
Title: Hochschild homology of rings of differential operators and integration over complex manifolds.
Speaker: Dr. D. Ajay Ramadoss, University of Oklahoma.
Date: Tue, 08 Jan 2008
Time: 4.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Let X be a compact complex manifold and let E be a holomorphic vector bundle on X. Any global holomorphic differential operator D on E induces an endomorphism of $\\text{H}^{\\bullet}(X,E)$. The super-trace of this endomorphism is the super-trace of D. This is a linear functional on the 0-th Hochschild homology of Diff(E), the algebra of global holomorphic differential operators on E. While the Hochschild homology in the usual sense of Diff(E) is too big for explicit computation, there is a notion of completed Hochschild homology of Diff(E) with a very nice property: If HH_i(Diff(E)) denotes the i-th completed Hochschild homology, then HH_i(Diff(E)) is isomorphic to \\text{H}^{2n-i}(X), the 2n-i th cohomology of X with complex coefficients. We shall attempt to outline how the supertrace mentioned above extends to a linear functional on the 0-th completed Hochschild homology of Diff(E), and thus, on H^{2n}(X). A priori, this linear functional depends on E. It however, can be shown that it is precisely the integral over X. This fact also helps one connect the local Riemann-Roch theorems of Nest-Tsygan to the Hirzebruch Riemann-Roch theorem. Analogous results about similar constructions using cyclic homology instead of Hochschild homology are also available.
Title: Directionally convex ordering of random measures, shot-noise fields and some applications to wireless networks.
Speaker: Dr. D. Yogeshwaran, Ecole Normale Superieure, Paris.
Date: Fri, 04 Jan 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
In the standard stochastic geometric setting, wireless networks can be modeled as point processes and their performances as certain mean functionals of the point process. Obtaining closed form expressions of such functionals is not easy for a general class of point processes. This motivates a comparative study. We study comparison of one such class of mean functionals - the additive and extremal shot-noise fields - which arise naturally in modeling of wireless networks, as ingredients of the so called Signal-to-Interference-Noise-Ratio.
Title: Holomorphic extension of CR functions from hypersurfaces with singularities
Speaker: Dr. Debraj Chakrabarti, University of Western Ontario
Date: Thu, 20 Dec 2007
Time: 11.30 a.m.
Venue: Lecture Hall - III, Dept. of Mathematics
We show that Trepreau’s theorem (minimality of a hypersurface at a point implies one sided extension of all CR functions) does not hold if the hypersurface is allowed to have singularities. We formulate a geometric condition called two sided support which is the obstruction to such extension.
Title: The Hodge Conjecture for certain abelian four-folds: a tentative gauge theoretic approach.
Speaker: Prof. T. R. Ramadas, ICTP, Trieste
Date: Tue, 18 Dec 2007
Time: 2.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Title: The Newton Polygon
Speaker: Prof. Sriram Abhyankar, Purdue Universtiy
Date: Fri, 14 Dec 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Ergodic theory, abelian groups, and point processes associated with stable random fields
Speaker: Dr. Parthanil Roy, ETH Zurich and Michigan State University
Date: Mon, 10 Dec 2007
Time: 10.00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We consider a point process sequence induced by a stationary symmetric -stable (0 < < 2) discrete parameter random eld. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), that if the random eld is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specic class of stable random elds generated by conservative actions whose eective dimensions can be computed using the structure theorem of nitely generated abelian groups. The corresponding point processes sequence is not tight and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques. (This talk is based on a joint work with Gennady Samorodnitsky) All interested are Welcome
Title: Some queer diffusions
Speaker: Prof. Daniel Stroock, MIT
Date: Fri, 07 Dec 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Least area surfaces are incompressible
Speaker: Dr. Siddhartha Gadgil, IISc
Date: Thu, 06 Dec 2007
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Least area surfaces are incompressible
Speaker: Dr. Siddhartha Gadgil, IISc
Date: Wed, 05 Dec 2007
Time: 4:00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Domains with non-compact automorphism group
Speaker: Dr. Kaushal Verma, IISc
Date: Fri, 30 Nov 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
The purpose of the talk will be to discuss the problem about classifying domains in C^n that have a non-compact group of holomorphic automorphisms. Examples and a flavour of some of the techniques that have been successful so far will be provided.
Title: Linear and integer programming: The state of the art
Speaker: Prof. Martin Groetschel (Institut fur Mathematik, Technische Universitt Berlin)
Date: Wed, 28 Nov 2007
Time: 4:00 pm
Venue: Faculty Hall
In this lecture I will survey - aimed at a broad mathematical audience - the development of linear and integer programming. The history of these subjects began about one and a half centuries ago but their “boom” started in the 1950s only. Theory, algorithm design and analysis dominated the first years of development. Computational progress was particularly significant in the last twenty years. In fact, the advances in linear and integer programming software are on at least the same level as those in computing machinery.
These achievements combined with successful efforts to model applications make it possible to solve today real world problems of breath taking size and diversity. I will report about some of these success stories in my talk.
Title: Galois action on L-values
Speaker: Dr. N. Tejaswi, IISc
Date: Fri, 23 Nov 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
To an elliptic curve E, one can associate various twists L(E,\\chi,s) of L-functions. The values of these L-functions at integers are expected to behave well under the action of a Galois group. In this talk, we will explain how a Galois group acts on some special L-values and indicate some approaches in studying the behaviour under such actions.
Title: Spectral Pick interpolation from a complex-geometric viewpoint
Speaker: Dr. Gautam Bharali
Date: Fri, 16 Nov 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
The spectral Pick-interpolation problem, i.e. to determine when there exists a holomorphic map from the unit disc to the class of complex matrices of spectral radius less than one that interpolates prescribed data, has a complicated solution using operator-theory and control-theory methods. The difficulty in implementing this solution motivated a new approach pioneered by Agler and Young. Their methods led to a checkable necessary condition for Pick interpolation. But, from a complex-geometric viewpoint, it was unclear why the latter condition should be sufficient. In this talk, we will demonstrate that this condition is not sufficient. We will also present an inequality – largely linear-algebraic in flavour – that provides a necessary condition for matricial data for which the Agler-Young-type test provides no conclusions.
Title: Chern-Cheeger-Simons theory of secondary classes
Speaker: Dr. Jaya Iyer, IMSc.
Date: Tue, 13 Nov 2007
Time: 2.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Title: Cycles on complete intersections
Speaker: Dr. Jaya Iyer, IMSc.
Date: Mon, 12 Nov 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We discuss some questions on the triviality of rational Chow groups and give some examples in this direction.
Title: Ergodic theory, abelian groups, and point processes associated with stable random fields
Speaker: Dr. Parthanil Roy, ETH Zurich and Michigan State University
Date: Sat, 10 Nov 2007
Time: 10.00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We consider a point process sequence induced by a stationary symmetric -stable (0 < < 2) discrete parameter random eld. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), that if the random eld is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specic class of stable random elds generated by conservative actions whose eective dimensions can be computed using the structure theorem of nitely generated abelian groups. The corresponding point processes sequence is not tight and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques. (This talk is based on a joint work with Gennady Samorodnitsky) All interested are Welcome
Title: History of irrational and transcendental numbers
Speaker: Prof. Michel Waldschmidt, Institut de Mathmatiques de Jussieu, Paris
Date: Tue, 06 Nov 2007
Time: 2.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Title: Negatively curved Kahler manifolds
Speaker: Martin Deraux (Institut Fourier, Grenoble)
Date: Wed, 31 Oct 2007
Lecture Series on “ Negatively curved Kahler manifolds” (4 Lectures)
Speaker: Dr. Martin Deraux, Institut Fourier, Grenoble
Time and place: LH 2, IISc Math Dept
- Wednesdays (31st Oct and 7th Nov) 11-12.30 PM
- Thursdays (1st Nov and 8th Nov) 3-4.30 PM
Abstract:
- Basics of Kahler geometry, uniformization.
- Intro to complex hyperbolic geometry, review about symmetric spaces, constructions of discrete groups and lattices, Calabi conjecture and Yau uniformization.
- Overview of Deligne-Mostow theory.
- Branched covering constructions and non-locally symmetric examples.
Title: Non-negative isotropic curvature
Speaker: Dr. Harish Seshadri, IISc
Date: Fri, 26 Oct 2007
Time: 4.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Title: Quantum Cryptography, Mutually Unbiased Basis and what is wrong with Classical Cryptography
Speaker: Professor Asha Rao, RMIT, Melbourne, Australia
Date: Fri, 28 Sep 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Starting with a brief description of classical cryptography and the need for quantum cryptography, I will elaborate on some of the mathematical questions within quantum cryptography, ending with the details on a a particular problem - the construction of mutually unbiased basis.
Title: Quantum field theory as an aid to geometers Finite Element Methods
Speaker: Professor Dr. Katrin Wendland Univ. Augsburg, Germany
Date: Mon, 24 Sep 2007
Time: 10.00 a.m.
Venue: Lecture Hall - II, Dept. of Mathematics
String theorists believe that elementary particles can be described by one dimensional string-like objects that propagate in certain geometries. An appropriate mathematical treatment combines the study of quantum field theory and geometry. Whether or not one believes in the physical (or even mathematical) reality of string theory as a whole, the idea to use one-dimensional test particles to investigate the properties of interesting geometries allows far reaching insights- both in quantum field theory and in geometry.
Title: Some Remarks on the Convergence of Adaptive Finite Element Methods
Speaker: Professor Dr. Carsten Carstensen Humboldt Univ., Germany
Date: Mon, 24 Sep 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Typical adaptive mesh-refining algorithms for first-oder (conforming) finite element methods consist of a sequence of the following steps:
Title: Combinatorial approach to the conjugacy-class algebra of the symmetric group
Speaker: Prof. Jacob Katriel, Technion - Israel Institute of Technology, Haifa
Date: Fri, 14 Sep 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
For A=(1)^{l_1}(2)^{l_2}…(n)^{l_n} a conjugacy class in S_n, let
Supp(A)=\\Sum_{i=1}^n i\\cdot l_i=n-l_1
Evaluation of the coefficient of the calss-sum C in the product of the class sums A and B is reduced to a combinatorial problem in S_k, where k=min{Supp(A),Supp(B),Supp(C)}
Title: Multivariate Distributions, Quantiles and their Monotonicity Properties.
Speaker: Prof. Probal Chaudhari, ISI, Kolkata.
Date: Fri, 07 Sep 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Some notions of multivariate distribution transform and related quantile transform will be introduced and discussed. Some application of these statistical concepts and tools in social and natural sciences will be described.
Title: Some Combinatorial Group Invariants and their genaralisations with weights
Speaker: Prof. S.D. Adhikari, HRI, Allahabad,
Date: Wed, 05 Sep 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Overconvergent modular symbols
Speaker: Dr. B. Baskar, Brandeis University,
Date: Fri, 31 Aug 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
The notion of modular symbols was first introduced by Manin. We state some basic results which relate modular symbols to special values of L-functions and p-adic L-functions. We give Greenberg and Stevens construction of an overconvergent modular symbols which is a measure valued cohomology class attached to a Hida family of modular forms with certain interpolation properties. We will construct the Mazur-Kitagawa two variable p-adic L-function from the overconvergent modular symbol.
Title: On the numerical approximation of nonconservative hyperbolic balance laws
Speaker: Dr. Marcus Kraft Hamburg University of Technology, Germany
Date: Fri, 10 Aug 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Hyperbolic balance laws with source terms allow non-unique entropy solutions. By taking the source term as a variable and stating its time derivative to be zero, a balance law can be rewritten in a quasi-linear form. This then reveals the loss of strict hyperbolicity at critical states, the so called resonance phenomenon. Numerical schemes are sensitive to this phenomenon and it is uncertain, which entropy solution (if it is non-unique) will be created by an appropriate numerical scheme.
Title: Classification problems in topology - some approaches
Speaker: Prof. Himadri Mukherjee North-Eastern Hill University, Shillong
Date: Thu, 02 Aug 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: The arithmetic of elliptic surfaces
Speaker: Dr. Ritabrata Munshi, Rutgers University, USA
Date: Wed, 01 Aug 2007
Time: 2.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
A family of elliptic curves (or an elliptic surface) is given by a Weierstrass equation
Title: The arithmetic of elliptic surfaces
Speaker: Dr. Ritabrata Munshi, Rutgers University, USA
Date: Tue, 31 Jul 2007
Time: 11.30 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
A family of elliptic curves (or an elliptic surface) is given by a Weierstrass equation
Title: On holomorphic Hermitian Bundles
Speaker: Prof. Indranil Biswas, TIFR, Mumbai
Date: Fri, 27 Jul 2007
Time: 4.00 p.m.
Venue: Lecture Hall - III, Dept. of Mathematics
Title: Spectral Theory and Root bases associated with Multiparameter Eigenvalue Problems
Speaker: Mr. J. P. Mohan Das Ph. D. Research Scholar Department of Mathematics
Date: Thu, 26 Jul 2007
Time: 11.00 a.m.
Venue: Lecture Hall III, Department of Mathematics
Title: On a scaling limit for a tagged particle in some zero-range interacting systems
Speaker: Dr. Sunder Sethuraman, Iowa State University
Date: Fri, 29 Jun 2007
Time: 3.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Informally, the zero-range'' particle system follows a collection of dependent random walks on a lattice, each of which interacts infinitesimally only with those particles already present at its location. In this talk, we consider the asymptotics of a distinguished, or tagged particle in this interacting particle system. In particular, we discuss anonequilibrium’’ invariance principle, in one dimension when the transition rates are mean-zero, with respect to a diffusion whose coefficients depend on the ``hydrodynamic’’ density.
Title: Multiplicity Conjecures
Speaker: Dr. Manoj Kummini, University of Kansas
Date: Thu, 28 Jun 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We discuss a conjecture for the Hilbert-Samuel multiplicity (degree) of a homogeneous ideal in a polynomial ring over a field. Herzog-Huneke-Srinivasan conjectured that the multiplicity can be bounded by a function of the maximum twists occuring at different homological degrees in a minimal graded free resolution of the ideal. We will discuss some examples and some cases where the conjecture is known to hold.
Title: Lipschitz isomorphisms between Banach spaces
Speaker: Prof. G. Godefroy University of Paris VI France
Date: Tue, 19 Jun 2007
Time: 4:00 p.m.
Venue: Lecture Hall -I, Dept. of Mathematics
Title: Harmonic analysis approach to the Navier-Stokes equation
Speaker: Prof. S. S. Sritharan, University of Wyoming
Date: Fri, 15 Jun 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: An overview of random matrices
Speaker: Dr. Manjunath Krishnapur, University of Toronto
Date: Thu, 14 Jun 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
We consider the simplest models of Hermitian and non Hermitian random matrices with independent and identically distributed entries. We present basic results on limiting spectral distributions (Wigner’s semicircle law in the Hermitian case and Girko’s circular law in the non Hermitian case).
Title: Higher linking of Knots: Stallings theorem
Speaker: Ms. Geetanjali Kachari, IISc
Date: Mon, 11 Jun 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
A central result used in studying higher linking of knots is a theorem of Stalling on lower central series. Stallings’s proof was based on spectral sequences, but he indicated that the result can be proved directly using Hopf’s description of homology of groups. In this talk Stallings theorem will be explained and a direct proof will be presented.
Title: Adaptive Numerical Solution of Intracellular Calcium Dynamics
Speaker: Prof. Gerald Warnecke, Magdeburg, Germany
Date: Mon, 21 May 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Calcium waves are an important means of intrcellular signaling. Intracellular calcium release at the endoplasmatic reticulum is a prime example of the role of stochastic effects in cellular systems. Realistic models consist of deterministic systems of reaction-diffusion equations in three dimensional space coupled to stochastic transitions of calcium channels at the domain boundary. The resulting dynamics has multiple time and space scales, which complicates computer simulations. In this talk we focus on the PDE aspect of the numerical computations. We use adaptive linear finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. Further, parallel computing is needed for realistic simulations. We describe the algorithmic approach and we demonstrate its efficiency by computational examples. Our single channel model matches experimental data by Mak et al. (PNAS 95, 1998) and results in intriguing dynamics if calcium is used as a carrier. Random openings of the channel accumulate in bursts of calcium blips that may prove central for the understanding of cellular calcium dynamics. We plan to extend our computations to more realistic domain geometries and to use local time stepping methods.
Title: Numerical approximation of population balance equations in process engineering
Speaker: Prof. Gerald Warnecke, Magdeburg, Germany
Date: Mon, 14 May 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Population balance equations are widely used in many chemical and particle process engineering problems involving crystallization, fluidized bed granulation, aerosols etc. Analytical solutions are available only for a limited number of simplified problems and therefore numerical solutions are frequently needed to solve a population balance problem. A general population balance equation for simultaneous aggregation, breakage, growth and nucleation in a well mixed system is given as an integro-partial differential equation for a particle property distribution function. Sectional methods are well known for their simplicity and conservation properties. Therefore numerical techniques belonging to this category are the most commonly used. In these methods, all particles within a computational cell, which in some papers is called a class, section or interval, are supposed to be of the same size. These methods divide the size range into small cells and then apply a balance equation for each cell. The continuous population balance equation is then reduced to a set of ordinary differential equations. However, it is well known that the numerical results by previous sectional methods were rather inaccurate. Furthermore, there is a lack of numerical schemes in the literature which can be used to solve growth, nucleation, aggregation, and breakage processes, i.e. differential and integral terms, simultaneously. We present a new numerical scheme for solving a general population balance equation which assigns particles within the cells more precisely. The technique follows a two step strategy. The first is to calculate the average size of newborn particles in a cell and the other to assign them to neighboring nodes such that important properties of interest are exactly preserved. The new technique preserves all the advantages of conventional discretized methods and provides a significant improvement in predicting the particle size distributions. The technique allows the convenience of using geometric- or equal-size cells. The numerical results show the ability of the new technique to predict very well the time evolution of the second moment as well as the complete particle size distribution. Moreover, a special way of coupling the different processes has been described. It has been demonstrated that the new coupling makes the technique more useful by being not only more accurate but also computationally less expensive. Furthermore, a new idea that considers the growth process as aggregation of existing particle with new small nuclei has been presented. In that way the resulting discretization of the growth process becomes very simple and consistent with first two moments. Additionally, it becomes easy to combine the growth discretization with other processes. Moreover all discretizations including the growth have been made consistent with first two moments. The new discretization of growth is a little diffusive but it predicts the first two moments exactly without any computational difficulties like appearance of negative values or instability etc. The accuracy of the scheme has been assessed partially by numerical analysis and by comparing analytical and numerical solutions of test problems. The numerical results are in excellent agreement with the analytical results and show the ability to predict higher moments very precisely. Additionally, an extension of the proposed technique to higher dimensional problems is discussed
Title: Higher linking of knots.
Speaker: Dr. Siddhartha Gadgil, IISc
Date: Wed, 09 May 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
This will be an informal expository talk on higher linking of knots in three-dimensional space.
Title: Local polynomial convexity of real surfaces in C^n near complex tangencies
Speaker: Dr. Gautam Bharali, IISc
Date: Fri, 13 Apr 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
More than 30 years after the Harvey-Wells paper on complex approximation theory, we still do not know the answers to most of the relevant approximation-theoretic questions on, say, a smooth, compact 2-manifold (call it M) in C^2. Answering many of these questions boils down to examining the local polynomial convexity of M near those points where the (extrinsic) tangent space of M is a complex subspace of the ambient C^2. We shall quickly survey what is currently known, and then look at some recent progress based on examining the Maslov index.
Title: Projective modules over affine algebras
Speaker: Dr. Mrinal Kanti Das, Indian Statistical Institute, Kolkata
Date: Mon, 02 Apr 2007
Time: 10.00 a.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: How interesting are orbifolds?
Speaker: Mainak Poddar ISI Kolkata
Date: Fri, 23 Mar 2007
Time: 4:00 p.m.
Venue: Lecture Hall - I, Department of Mathematics
Orbifolds are generally regarded as generalizations of manifolds. On the other hand, finite groups are also orbifolds. In this survey talk, I will focus on how orbifolds appear naturally in some important areas of Mathematics and Physics. I will also describe some features of the geometry of orbifolds.
Title: Numerical Modeling of Hyperbolic Balance Laws using Bicharacteristics
Speaker: Prof. M. Lukacova, Hamburg University of Technology
Date: Wed, 14 Mar 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Numerical Modeling of Hyperbolic Balance Laws using Bicharacteristics
Speaker: Prof. M. Lukacova, Hamburg University of Technology
Date: Tue, 13 Mar 2007
Time: 4.00 p.m.
Venue: Lecture Hall - II, Dept. of Mathematics
Title: Zariski-Lipman Conjecture
Speaker: Prof. Dilip Patil, IISc
Date: Thu, 08 Mar 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
In this lecture I will give a brief survey on the well-known Zariski-Lipman Conjecture which is still open in general. However, there are some partial results have been proved. I will state these results by giving necessary definitions and concepts required.
Title: Mathematical issues for some control problems in fluid mechanics
Speaker: Prof. Jean-Pierre Raymond Universit Paul Sabatier, Toulouse, France HRI, Allahabad
Date: Wed, 28 Feb 2007
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics
Title: Real vs. Strongly Real Elements in Algebraic Groups
Speaker: Dr. Anupam Kumar Singh, TIFR, Mumbai
Date: Tue, 27 Feb 2007
Time: 4.00 p.m.
Venue: Lecture Hall - II, Dept. of Mathematics
Let $k$ be a field of characteristic not 2 and $G$ be an algebraic group defined over $k$. An element $t$ in $G(k)$ is called real if there exists $g \in G(k)$ such that $gtg^{-1}=t^{-1}$. An element $t\in G(k)$ is called strongly real if $t=\tau_1\tau_2$ where $\tau_i\in G(k)$ and $\tau_i^2=1$. We discuss when a semisimple real element is strongly real in $G(k)$. We investigate this question for classical groups and the groups of type $G_2$ in detail.
Title: Entropy Power inequalities and monotonicity in central limit theorems
Speaker: Prof. Mokshay Madiman University of Yale, USA
Date: Fri, 23 Feb 2007
Venue: Lecture Hall I, Department of Mathematics, IISc
Title: The spherical mean value operator for functions supported in a ball
Speaker: Prof. Rakesh (University of Delaware)
Date: Thu, 01 Feb 2007
Time: 4:00 pm
Venue: LH I