Seminars
By Year
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: 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: Geometry of random geodesics - 3
Speaker: Arjun Krishnan (University of Rochester)
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: Geometry of random geodesics - 2
Speaker: Arjun Krishnan (University of Rochester)
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:
-
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.
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Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.
Title: Geometry of random geodesics - 1
Speaker: Arjun Krishnan (University of Rochester)
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.
The video of this talk is available on the IISc Math Department channel.
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.
The video of this talk is available on the IISc Math Department channel.
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: 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.
The video of this talk is available on the IISc Math Department channel.
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.
The video of this talk is available on the IISc Math Department channel.
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.
The video of this talk is available on the IISc Math Department channel.
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.
The video of this talk is available on the IISc Math Department channel.
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.
The video of this talk is available on the IISc Math Department channel.
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.