Seminars: Mathematics@IISc

Seminars

By Year

Title: Algebra & Combinatorics Seminar: Equivariant Cartan Homotopy formula

Speaker: Safdar Quddus (IISc Mathematics)

Date: Fri, 20 Mar 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

The Cartan homotopy formula was introduced for differentiable manifolds relating a vector field and its Lie derivative. Reinhart gave an algebraic analogue over a commutative algebra for $D$ a derivation. Getzler formulated the same for $A_\infty$ algebras and for $D$ an arbitrary cocycle.

We talk about the equivariant Cartan homotopy formula for cross product $DG$-algebras.

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Title: Algebra & Combinatorics Seminar: Projective representations of Heisenberg groups

Speaker: Sumana Hatui (IISc Mathematics)

Date: Fri, 13 Mar 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

The theory of projective representations of groups, extensively studied by Schur, involves understanding homomorphisms from a group into the projective linear groups. By definition, every ordinary representation of a group is also projective but the converse need not be true. Therefore understanding the projective representations of a group is a deeper problem and many a times also more difficult in nature. To deal with this, an important role is played by a group called the Schur multiplier.

In this talk, we shall describe the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their $t$-variants. We shall discuss the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional complex projective irreducible representations.

Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

Lectures 1–4 (Jan 14, 16, 29, Feb 5):

Automorphic Forms on Semisimple Lie Groups
Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension
Automorphic Forms in Berezin Quantization and von Neumann Algebras

Lecture 5. Multivariable Scattering Theory (Tuesday, March 10)

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Title: Geometry & Topology Seminar: Approximation of weak G-bundles in high dimensions

Speaker: Swarnendu Sil (ETH Zurich)

Date: Mon, 09 Mar 2020

Time: 4:00 pm

Venue: LH-1, Mathematics Department

The analysis for Yang-Mills functional and in general, problems related to higher dimensional gauge theory, often requires one to work with weak notions of principal G-bundles and connections on them. The bundle transition functions for such bundles are not continuous and thus there is no obvious notion of a topological isomorphism class.

In this talk, we shall discuss a few natural classes of weak bundles with connections which can be approximated in the appropriate norm topology by smooth connections on smooth bundles. We also show how we can associate a topological isomorphism class to such bundle-connection pairs, which is invariant under weak gauge changes. In stark contrast to classical notions, this topological isomorphism class is not independent of the connection.

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Title: Algebra & Combinatorics Seminar: Formulas for the Gross-Stark units

Speaker: Matthew Honnor (King's College London, UK)

Date: Fri, 06 Mar 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

In the 1980’s Tate stated the Brumer–Stark conjecture which, for a totally real field $F$ with prime ideal $\mathfrak{p}$, conjectures the existence of a $\mathfrak{p}$-unit called the Gross–Stark unit. This unit has $\mathfrak{P}$ order equal to the value of a partial zeta function at 0, for a prime $\mathfrak{P}$ above $\mathfrak{p}$. In 2008 and 2018 Dasgupta and Dasgupta–Spieß, conjectured formulas for this unit. During this talk I shall explain Tate’s conjecture and then the ideas for the constructions of these formulas. I will finish by explaining the results I have obtained from comparing these formulas.

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Title: Eigenfunctions Seminar: On the decimal expansion of log(2020/2019) and $e$

Speaker: Yann Bugeaud (IRMA, Université de Strasbourg, France)

Date: Fri, 28 Feb 2020

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-5, Mathematics Department

It is commonly expected that $e$, $\log 2$, $\sqrt{2}$, $\pi$, among other “classical” numbers, behave, in many respects, like almost all real numbers. For instance, they are expected to be normal to base 10, that is, one believes that their decimal expansion contains every finite block of digits from ${0, \ldots , 9}$. We are very far away from establishing such a strong assertion. However, there has been some small recent progress in that direction. After surveying classical results and problems on normal numbers, we will adopt a point of view from combinatorics on words and show that the decimal expansions of $e$, of any irrational algebraic number, and of $\log (1 + \frac{1}{a})$, for a sufficiently large integer $a$, cannot be ‘too simple’, in a suitable sense.

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Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesdays, January 29 and February 5)

Lecture 5. Multivariable Scattering Theory

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Title: Geometry & Topology Seminar: Enumerative Geometry of the Quintic Threefold

Speaker: Ritwik Mukherjee (NISER, Bhubaneswar)

Date: Fri, 31 Jan 2020

Time: 10:00 am

Venue: LH-1, Mathematics Department

The quintic threefold (the zero set of a homogeneous degree 5 polynomial on CP^4) is one of the most famous examples of a Calabi Yau manifold. It is one of the most studied in the field of Enumerative Geometry. For example, how many lines are there on a Quintic threefold? In this talk we will explain some approaches to count curves on the Quintic threefold. In particular, we will try to explain the following idea: If Y is a submanifold of X, and we understand the Enumerative Geometry of X, how can we answer questions about the Enumerative Geometry of Y? We will try to explain the idea used by Andreas Gathman to compute all the genus zero Gromov-Witten invariants of the Quintic Threefold.

The talk will be self contained and will not assume any prior knowledge of Enumerative Geometry or Gromov-Witten Invariants.

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Title: Algebra & Combinatorics Seminar: An extremal problem related to weighted Davenport constant

Speaker: Eshita Mazumdar (ISI, Bangalore)

Date: Fri, 31 Jan 2020

Time: 3 pm

Venue: LH-3, Mathematics Department

For a finite abelian group $G$ and $A \subset [1, \exp(G) - 1]$, the $A$-weighted Davenport Constant $D_A(G)$ is defined to be the least positive integer $k$ such that any sequence $S$ with length $k$ over $G$ has a non-empty $A$-weighted zero-sum subsequence. The original motivation for studying Davenport Constant was the problem of non-unique factorization in number fields. The precise value of this invariant for the cyclic group for certain sets $A$ is known but the general case is still unknown. Typically an extremal problem deals with the problem of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects that satisfies certain requirements. In a recent work with Prof. Niranjan Balachandran, we introduced an Extremal Problem for a finite abelian group related to Weighted Davenport Constant. In this talk I will talk about the behaviour of it for different groups, specially for cyclic group.

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Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Wed, 29 Jan 2020

Time: 3 pm

Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesdays, January 29 and February 5)

Lecture 5. Multivariable Scattering Theory

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Title: PhD Thesis colloquium: Sparse bound of various spherical maximal functions

Speaker: Sourav Hait (IISc Mathematics)

Date: Wed, 22 Jan 2020

Time: 4 pm

Venue: LH-1, Mathematics Department

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras

Lecture 5. Multivariable Scattering Theory

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Title: PhD Thesis defence: Phase Transition, Percolation at Criticality, Poisson convergence of isolated vertices and Connectivity in Random Connection Models

Speaker: Sanjoy Kumar Jhawar (IISc Mathematics)

Date: Tue, 14 Jan 2020

Time: 11 am

Venue: LH-1, Mathematics Department

This work has two parts. The first part contains the study of phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogeneous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y\mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid \cdot \mid$ is the Euclidean norm. In the inhomogeneous version of the model, points of $\mathcal{P}_{\lambda}$ is endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability

$\left(1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)\right)$

for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.

In the second part we consider an inhomogeneous random connection model on a $d$ -dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of isolated vertices converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.

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Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Tue, 14 Jan 2020

Time: 3 pm

Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

The date and time for the third and fourth lectures will be announced in due course.

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3, 4. Automorphic Forms in Berezin Quantization and von Neumann Algebras

Lecture 5. Multivariable Scattering Theory

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Title: APRG Seminar: Brownian structure in universal KPZ objects

Title: Eigenfunctions Seminar: The Riemann Hypothesis: History and Recent Work

Speaker: Ken Ono (University of Virginia, Charlottesville, USA)

Date: Wed, 18 Dec 2019

Time: 3:15 – 5:15 pm (with a 15 minute break)

Venue: LH-1, Mathematics Department

Slides

Lecture 1:

Slides

Lecture 2:

The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of one-half. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize. In this lecture, Ken Ono will discuss the mathematical meaning of the Riemann hypothesis and why it matters. Along the way, he will tell tales of mysteries about prime numbers and highlight new advances. He will conclude with a discussion of recent joint work with mathematicians Michael Griffin of Brigham Young University, Larry Rolen of Georgia Tech, and Don Zagier of the Max Planck Institute, which sheds new light on this famous problem.

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Title: Algebra & Combinatorics Seminar: Oriented and Colorful Variants of Gyárfás–Sumner Conjecture

Speaker: Sunil Chandran (IISc CSA)

Date: Wed, 11 Dec 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

The Gyárfás–Sumner conjecture states the following: Let $a,b$ be positive integers. Then there exists a function $f$, such that if $G$ is a graph of clique number at most $a$ and chromatic number at least $f(a,b)$, then $G$ contains all trees on at most $b$ vertices as induced subgraphs. This conjecture is still open, though for several special cases it is known to be true. We study the oriented version of this conjecture: Does there exist a function $g$, such that if the chromatic number of an oriented graph $G$ (satisfying certain properties) is at least $g(s)$ then $G$ contains all oriented trees on at most $s$ vertices as its induced subgraphs. In general this statement is not true, not even for triangle free graphs. Therefore, we consider the next natural special class – namely the 4-cycle free graphs – and prove the above statement for that class. We show that $g(s) \leq 4s^2$ in this case.

We also consider the rainbow (colorful) variant of this conjecture. As a special case of our theorem, we significantly improve an earlier result of Gyárfás and Sarkozy regarding the existence of induced rainbow paths in $C_4$ free graphs of high chromatic number. I will also discuss the recent results of Seymour, Scott (and Chudnovsky) on this topic.

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Title: PhD Thesis defence: Representations of Special Compact Linear Groups of Order Two: Construction, Representation Growth, Group Algebras, and Branching Laws

Speaker: Hassain M. (IISc Mathematics)

Date: Thu, 05 Dec 2019

Time: 3 pm

Venue: LH-3, Mathematics Department

Let $O$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let e be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.

In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.

Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work for p=2. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.

Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.

Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{r})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are NOT isomorphic for $r \geq 2e+2$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{r}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{r}))]$ are NOT isomorphic for $r\geq4$.

Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.

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Title: APRG Seminar: Automorphic Forms in Several Complex Variables

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Thu, 05 Dec 2019

Time: 2:30-3:30 pm and 3:45-4:45 pm

Venue: LH-1, Department of Mathematics

Automorphic and modular forms in one complex variable (on the upper half-plane) belong to the most important topics in complex analysis, with close connections to number theory. In several variables, automorphic forms arise in the study of moduli spaces, with important theorems in algebraic and analytic geometry concerning quotient spaces by properly discontinuous groups of holomorphic transformations and their compactification. In addition, they lead to new and largely unexplored connections to analysis and operator theory, giving rise to von Neumann algebras which are not of type I. Finally, in the multivariable setting the Fourier analysis of automorphic forms (theta functions) introduce new arithmetic structures such as divisibility in Jordan algebras of higher rank.

I plan to divide these lectures into four parts:

Automorphic Forms in Complex Analysis and Algebraic Geometry
Construction of multi-variable Automorphic Forms on Lie groups and Siegel domains
Berezin Quantization of Automorphic Forms and type II von Neumann Algebras
Scattering Theory on Symmetric Spaces and their non-compact Quotients

Recently, in connection with string theory and topological quantum field theory (in dimension up to 8) there has emerged the concept of topological automorphic forms. These connections to topology and mathematical physics may be explored at the end. Another important avenue is the connection to the Langlands program, where automorphic representations are more generally related to (discrete) series representations of algebraic groups.

1. Automorphic Forms in Complex Analysis and Algebraic Geometry

Introduction: Why multi-variable automorphic forms?
Quotient Spaces by finite linear groups (Cartan’s Theorem)
Quotient Spaces by properly discontinuous groups
Projective embedding of compact quotients of bounded domains

2. Construction and Fourier Analysis of Automorphic Forms

Theta Functions on Lattices and Jordan Algebras
Poincaré and Eisenstein Series on Lie Groups
Siegel Domains and Satake Compactification
Relations to Number theory and arithmetic lattices

3. Automorphic Forms and Operator Theory

Reproducing Kernels and Berezin Quantization
Toeplitz operators with automorphic symbols
Group von Neumann algebras and intertwiners

4. Automorphic Forms in Scattering Theory

Scattering in euclidean space and the upper half-plane
Spherical functions on symmetric spaces
Scattering operators on symmetric spaces
Fourier transform and scattering matrix
The Residues of the scattering matrix

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Title: APRG Seminar: Fourier analysis on the infinite-dimensional torus

Speaker: Luz Roncal (BCAM – Basque Center for Applied Mathematics, Spain)

Date: Wed, 04 Dec 2019

Time: 10 am

Venue: LH-1, Mathematics Department

We will show both original and known results on Harmonic Analysis for functions defined on the infinite-dimensional torus, which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the one-dimensional torus, with its corresponding Haar measure. Such results will include:

Fourier Analysis on $\mathbb{T}^{\omega}$: basic properties, absolutely divergent series, Calder'on-Zygmund decomposition and differentiation of integrals.
Mixed norm $L^{\bar{p}}(\mathbb{T}^{\omega})$ spaces, $\bar{p}=(p_1,p_2,\ldots)$: definition, properties, duality, interpolation and weak spaces.
M. Riesz Theorems on $\mathbb{T}^{\omega}$, a.e. convergence, rectangular partial sums and $L^{\bar{p}}$ convergence.

Several open problems and other questions will be considered. Some of the results presented are joint work with Emilio Fernandez (Universidad de La Rioja, Spain).

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Title: Algebra & Combinatorics Seminar: Some remarks on Leavitt path algebras

Title: APRG Seminar: Automorphic Forms in Several Complex Variables

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Wed, 20 Nov 2019

Time: 2:30-3:30 pm and 4-5 pm

Venue: LH-1, Department of Mathematics

I plan to divide these lectures into four parts:

Automorphic Forms in Complex Analysis and Algebraic Geometry
Construction of multi-variable Automorphic Forms on Lie groups and Siegel domains
Berezin Quantization of Automorphic Forms and type II von Neumann Algebras
Scattering Theory on Symmetric Spaces and their non-compact Quotients

1. Automorphic Forms in Complex Analysis and Algebraic Geometry

Introduction: Why multi-variable automorphic forms?
Quotient Spaces by finite linear groups (Cartan’s Theorem)
Quotient Spaces by properly discontinuous groups
Projective embedding of compact quotients of bounded domains

2. Construction and Fourier Analysis of Automorphic Forms

Theta Functions on Lattices and Jordan Algebras
Poincaré and Eisenstein Series on Lie Groups
Siegel Domains and Satake Compactification
Relations to Number theory and arithmetic lattices

3. Automorphic Forms and Operator Theory

Reproducing Kernels and Berezin Quantization
Toeplitz operators with automorphic symbols
Group von Neumann algebras and intertwiners

4. Automorphic Forms in Scattering Theory

Scattering in euclidean space and the upper half-plane
Spherical functions on symmetric spaces
Scattering operators on symmetric spaces
Fourier transform and scattering matrix
The Residues of the scattering matrix

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Title: Geometry & Topology Seminar: Comparisons of non-abelian theta functions

Speaker: Swarnava Mukhopadhyay (TIFR, Mumbai)

Date: Mon, 18 Nov 2019

Time: 4:00 pm

Venue: LH-1, Mathematics Department

We consider the natural embedding for SO(r) into SL(r) and study the corresponding map between the moduli spaces of principal bundles on smooth projective curves. We compare the spaces of global sections of natural line bundles (non-abelian theta functions) for these moduli spaces and their twisted analogues with the space of theta functions. We will discuss how these results can be applied to obtain an alternate proof of a result of Pauly-Ramanan. If time permits, we will also discuss some applications to the monodromy of the Hitchin/WZW connections. This is a joint work with H. Zelaci.

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Title: Algebra & Combinatorics Seminar: The G.-C. Rota approach and the Lehmer conjecture

Speaker: Bernhard Heim (German University of Technology, Oman)

Date: Thu, 14 Nov 2019

Time: 2:30 pm

Venue: LH-1, Mathematics Department

Report on joint work with M. Neuhauser. This includes results with C. Kaiser, F. Luca, F. Rupp, R. Troeger, and A. Weisse.

The Lehmer conjecture and Serre’s lacunary theorem describe the vanishing properties of the Fourier coefficients of even powers of the Dedekind eta function.

G.-C. Rota proposed to translate and study problems in number theory and combinatorics to and via properties of polynomials.

We follow G.-C. Rota’s advice. This leads to several new results and improvement of known results. This includes Kostant’s non-vanishing results attached to simple complex Lie algebras, a new non-vanishing zone of the Nekrasov-Okounkov formula (improving a result of G. Han), a new link between generalized Laguerre and Chebyshev polynomials, strictly sign-changes results of reciprocals of the cubic root of Klein’s absolute $j$- invariant, and hence the $j$-invariant itself. Finally we give an interpretation of the first non-sign change of the Ramanujan $\tau(n)$ function by the root distribution of a certain family of polynomials in the spirit of G.-C. Rota.

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Title: APRG Seminar: Automorphic Forms in Several Complex Variables

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Wed, 13 Nov 2019

Time: 2:30-3:30 pm and 4-5 pm

Venue: LH-1, Department of Mathematics

I plan to divide these lectures into four parts:

Automorphic Forms in Complex Analysis and Algebraic Geometry
Construction of multi-variable Automorphic Forms on Lie groups and Siegel domains
Berezin Quantization of Automorphic Forms and type II von Neumann Algebras
Scattering Theory on Symmetric Spaces and their non-compact Quotients

1. Automorphic Forms in Complex Analysis and Algebraic Geometry

Introduction: Why multi-variable automorphic forms?
Quotient Spaces by finite linear groups (Cartan’s Theorem)
Quotient Spaces by properly discontinuous groups
Projective embedding of compact quotients of bounded domains

2. Construction and Fourier Analysis of Automorphic Forms

Theta Functions on Lattices and Jordan Algebras
Poincaré and Eisenstein Series on Lie Groups
Siegel Domains and Satake Compactification
Relations to Number theory and arithmetic lattices

3. Automorphic Forms and Operator Theory

Reproducing Kernels and Berezin Quantization
Toeplitz operators with automorphic symbols
Group von Neumann algebras and intertwiners

4. Automorphic Forms in Scattering Theory

Scattering in euclidean space and the upper half-plane
Spherical functions on symmetric spaces
Scattering operators on symmetric spaces
Fourier transform and scattering matrix
The Residues of the scattering matrix

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Title: PhD Thesis defence: On the Kobayashi geometry of domains

Speaker: Anwoy Maitra (IISc Mathematics)

Date: Mon, 11 Nov 2019

Time: 11 am

Venue: LH-1, Mathematics Department

We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:

Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.

Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s well-known big-constant/small-constant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.

The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.

A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newly-introduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.

Wolff–Denjoy-type theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoy-type theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.

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Title: Eigenfunctions Seminar: Slides Recent breakthroughs in sphere packing

Speaker: Abhinav Kumar (Stony Brook University, USA)

Date: Fri, 08 Nov 2019

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-1, Mathematics Department

The sphere packing problem asks for the densest packing by congruent non-overlapping spheres in n dimensions. It is a famously hard problem, though easy to state, and with many connections to different parts of mathematics and physics. In particular, every dimension seems to have its own idiosyncracies, and until recently no proven answers were known beyond dimension 3, with the 3-dimensional solution being a tour de force of computer-aided mathematics.

Then in 2016, a breakthrough was achieved by Viazovska, solving the sphere packing problem in 8 dimensions. This was followed shortly by joint work of Cohn-Kumar-Miller-Radchenko-Viazovska solving the sphere packing problem in 24 dimensions. The solutions involve linear programming bounds and modular forms. I will attempt to describe the main ideas of the proof.

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Title: APRG Seminar: Dirichlet spectral problem in a thin domain with a periodically oscillating boundary

Speaker: Klas Pettersson (UiT – The Arctic University of Norway, Narvik, Norway)

Date: Wed, 06 Nov 2019

Time: 11 am

Venue: LH-1, Mathematics Department

We describe the leading terms in the asymptotic behavior of the eigenvalues and the eigenfunctions to an elliptic Dirichlet spectral problem in a thin finite cylindrical domain with a periodically oscillating boundary by means of homogenization. Under suitable scaling and structure assumptions, the eigenfunctions show oscillatory behavior, and asymptotically localize with a profile solving a diffusion equation with quadratic potential on the real line. Methods for analysis of spectral asymptotics for heterogeneous media will be briefly discussed.

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Title: Algebra & Combinatorics Seminar: Special Values of L-functions and period relations for motives

Speaker: Chandrasheel Bhagwat (IISER, Pune)

Date: Wed, 06 Nov 2019

Time: 4 pm

Venue: LH-1, Mathematics Department

We will discuss certain rationality results for the critical values of the degree-$2n$ $L$-functions attached to $GL_1 \times O(n,n)$ over a totally real number field for an even positive integer $n$. We will also discuss some relations for Deligne periods of motives. This is part of a joint work with A. Raghuram.

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Title: Algebra & Combinatorics Seminar: Fundamental group of a complex ball quotient

Speaker: Tathagata Basak (Iowa State University, Ames, USA)

Date: Mon, 04 Nov 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

Let W be a Weyl group and V be the complexification of its natural reflection representation. Let H be the discriminant divisor in (V/W), that is, the image in (V/W) of the hyperplanes fixed by the reflections in W. It is well known that the fundamental group of the discriminant complement ((V/W) – H) is the Artin group described by the Dynkin diagram of W.

We want to talk about an example for which an analogous result holds. Here W is an arithmetic lattice in PU(13,1) and V is the unit ball in complex thirteen dimensional vector space. Our main result (joint with Daniel Allcock) describes Coxeter type generators for the fundamental group of the discriminant complement ((V/W) – H). This takes a step towards a conjecture of Allcock relating this fundamental group with the Monster simple group.

The example in PU(13,1) is closely related to the Leech lattice. Time permitting, we shall give a second example in PU(9,1) related to the Barnes–Wall lattice for which some similar results hold.

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Title: PhD Thesis defence: Fourier coefficients of modular forms and mass of pullbacks of Saito--Kurokawa lifts

Speaker: Pramath Anamby (IISc Mathematics)

Date: Mon, 04 Nov 2019

Time: 4:30 pm

Venue: LH-1, Mathematics Department

In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms, having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$ have the same eigenvalues of the $p$-th Hecke operator $T_p$ for almost all primes $p$, then $f_1=f_2$.

The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$.

We expect similar applications. We also discuss few results on the square-free Fourier coefficients of elliptic cusp forms.

In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, \mathbb{Z})$ then we consider its Saito–Kurokawa lift $F_g$ and a certain ‘restricted’ $L^2$-norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.

Time: 2:30 pm

Venue: R-15 (Chairman's Room), Mathematics Department

We study risk-sensitive stochastic optimal control and differential game problems. These problems arise in many applications including heavy traffic analysis of queueing networks, communication networks, and manufacturing systems.

First, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in $\mathbb{R}^{d}$. We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddle-point equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationary Markov strategies.

Subsequently, we study risk-sensitive ergodic control problem for controlled reflecting diffusion processes in the non-negative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. This, in turn, leads to the complete characterization of optimal controls in the space of stationary Markov controls.

Then we study risk-sensitive zero-sum/nonzero-sum stochastic differential games on the infinite horizon, where the state is a controlled reflecting diffusion in the non-negative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding Hamilton-Jacobi-Isaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies.

Finally, we study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criteria, where the state space is a controlled diffusion process in $\mathbb{R}^{d}.$ Under certain conditions, we establish the existence of a Nash equilibrium in stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.

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Title: Algebra & Combinatorics Seminar: A characterization of the family of secant lines to a hyperbolic quadric in $PG(3,q)$, $q$ odd

Speaker: Bikramaditya Sahu (IISc Mathematics)

Date: Fri, 04 Oct 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

In this talk, we shall discuss a combinatorial characterization of the family of secant lines of the 3-dimensional projective space $PG(3,q)$ which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with points and planes. This is joint work with Puspendu Pradhan.

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Title: Eigenfunctions Seminar: Slides Perron–Frobenius eigenfunctions of perturbed stochastic matrices

Speaker: Rajeeva L. Karandikar (CMI, Chennai)

Date: Fri, 27 Sep 2019

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-1, Mathematics Department

Consider a stochastic matrix $P$ for which the Perron–Frobenius eigenvalue has multiplicity larger than 1, and for $\epsilon > 0$, let

$P^\epsilon := (1 - \epsilon) P + \epsilon Q,$

where $Q$ is a stochastic matrix for which the Perron–Frobenius eigenvalue has multiplicity 1. Let $\pi^\epsilon$ be the Perron–Frobenius eigenfunction for $P^\epsilon$. We will discuss the behavior of $\pi^\epsilon$ as $\epsilon \to 0$.

This was an important ingredient in showing that if two players repeatedly play Prisoner’s Dilemma, without knowing that they are playing a game, and if they play rationally, they end up cooperating. We will discuss this as well in the second half.

The talk will include the required background on Markov chains.

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Title: Algebra & Combinatorics Seminar: On the Numbers of the Form $x^2+11y^2$

Speaker: Martin Kreuzer (Universität Passau, Germany)

Date: Wed, 25 Sep 2019

Time: 11 am – 12 pm (LH-5) and 3:30 – 4:30 pm (LH-1)

Venue: Rooms LH-5 and LH-1 (respectively), Mathematics Department

Lecture 1: Modular Class Groups

Lecture 2: Number Rings

A famous result of Leonhard Euler says that his so-called “convenient numbers” $N$ have the property that a positive integer $n$ has a unique representation of the form $n=x^2+Ny^2$ with $\gcd(x^2,Ny^2)=1$ if and only if $n$ is a prime, a prime power, twice one of these, or a power of 2. The set of known 65 convenient numbers is ${ 1,2,3,4,5,6,7,8,9,10,12,13,15,\dots,1848 }$, and it is conjectured that these are all of them. So, when we look at this set, we see that 11 is the first “inconvenient” number, and therefore we consider the natural question which positive integers have a representation of the form $n=x^2+11 y^2$ with $\gcd(x,11y)=1$.

Our approach is split into two parts. First we introduce the modular class group $G_{11}$ of level 11 and give a detailed description of its structure. We show that there are four conjugacy classes of elliptic elements of order 2, we provide concrete matrices representing these elliptic elements, and we give an explicit representation of $G_{11}$ using them. Then we conjugate the first of these matrices, namely $t_1=\binom{0, 1}{-1, 0}$, by the elements of $G_{11}$ and get matrices whose top right entry is of the form $x^2+11 y^2$. Conversely, we construct elliptic elements $A_n(\ell)$ of order 2 in $G_{11}$ which are conjugate to one of the generators. Then the matrices conjugate to $t_1$ are the ones we are interested in, and we find a set of candidate numbers $C$ such that $C=S_1 \cup S_2$ , where $S_1$ is the set we want to characterise. Thus the task is reduced to distinguishing between $S_1$ and $S_2$.

This problem is addressed in the second part of the talk using number rings in $K=\mathbb{Q}(\sqrt{-11})$. The ring of integers of this number field is $\mathcal{O}_K=\mathbb{Z}[(-11+\sqrt{-11})/2]$, and the more natural ring $\mathbb{Z}[\sqrt{-11}]$ is its order of conductor 2. By realizing the elements of $S_1$ and $S_2$ as norms of elements in $\mathcal{O}_K$, we get some of their basic properties. The main theorem provides a precise description of the primitive representations $n=x^2+11 y^2$ into four classes, where cubic numbers and prime numbers are two classes which admit separate, detailed descriptions. For the prime numbers in $S_1$, we need to use some consequences of ring class field theory for $\mathbb{Z}[\sqrt{-11}]$, but all other results are largely self-contained.

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Title: Geometry & Topology Seminar: Tropical curve counting

Speaker: Sushmita Venugopalan (IMSc, Chennai)

Date: Mon, 23 Sep 2019

Time: 3:30 pm

Venue: LH-1, Mathematics Department

Counting holomorphic curves in a symplectic manifold has been an area of research since Gromov’s work on this subject in the 1980s. Symplectic manifolds naturally allow a ‘cut’ operation. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. An interesting feature of curves in a multiply-cut manifold is that they have an underlying ‘tropical graph’, which is a graph that lives in the polytope associated to the cut.

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Title: Algebra & Combinatorics Seminar: Linkage principle and Tame cyclic base change for $GL_n(F)$

Speaker: Santosh Nadimpalli (Radboud University, Nijmegen, Netherlands)

Date: Fri, 20 Sep 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite cyclic Galois extension of $F$. The theory of base change associates to an irreducible smooth $\overline{\mathbb{Q}}_l$-representation $(\pi_F, V)$ of ${\rm GL}_n(F)$ an irreducible $\overline{\mathbb{Q}}_l$-representation $(\pi_E, W)$ of ${\rm GL}_n(E)$. The ${\rm GL}_n(E)$-representation $\pi_E$ extends as a representation of ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$. Assume that the central character of $\pi_F$ takes values in $\overline{\mathbb{Z}}_l^\times$, and $l\neq p$. When $\pi_E$ is cuspidal, for any ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$ stable lattice $\mathcal{L}$ in $\pi_E$, Ronchetti supporting the linkage principle of Treumann and Venkatesh conjectured that the zeroth Tate cohomology of $\mathcal{L}$ with respect to ${\rm Gal}(E/F)$ is the Frobenius twist of mod-$l$ reduction of the representation $\pi_F$, i.e.,

$\widehat{H}^0 ({\rm Gal}(E/F), \mathcal{L}) \simeq \overline{\pi} \_F^{(l)}.$

Venue: LH-1, Mathematics Department

Macdonald polynomials are a remarkable family of orthogonal symmetric polynomials in several variables. An enormous amount of combinatorics, group theory, algebraic geometry and representation theory is encoded in these polynomials. It is known that the characters of level one Demazure modules are non-symmetric Macdonald polynomials specialized at t=0. In this talk, I will define a class of polynomials in terms of symmetric Macdonald polynomials and using representation theory we will see that these polynomials are Schur-positive and are equal to the graded character of level two Demazure modules for affine $\mathfrak{sl}_{n+1}$. As an application we will see how this gives rise to an explicit formula for the graded multiplicities of level two Demazure modules in the excellent filtration of Weyl modules. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.

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Title: Algebra & Combinatorics Seminar: Counting Independent Sets in Graphs and Hypergraphs and Spectral Stability

Title: Eigenfunctions Seminar: Entropic relaxations of Monge–Kantorovich optimal transport

Speaker: Soumik Pal (University of Washington, Seattle, USA)

Date: Fri, 30 Aug 2019

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-1, Mathematics Department

Slides

Lecture 1: An introduction to Monge–Kantorovich optimal transport and entropic relaxations

Abstract:

The mathematics of Monge–Kantorovich optimal transport (OT) has grown to be a unifying theme in many scientific disciplines, from purely mathematical areas such as analysis, geometry, and probability to revolutionary new methods in economics, statistics, machine learning and artificial intelligence. Much of these are due to impressive leaps in computational methods in OT that hinge on entropy based approximations. For dynamical OT such relaxations correspond to stochastic processes called Schrödinger bridges. This is an introductory talk on OT, entropic relaxations, and Schrödinger bridges that will attempt to give a flavor of this "hot area" by sewing through various recent advances.

Date: Wed, 21 Aug 2019

Time: 3:30 pm

Venue: LH-1, Mathematics Department

This talk will be a continuation of my presentation on ‘Factorization of Contractions’, which was held at the in-house symposium this year.

We will start by proving the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries which states that: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exist a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_{\Phi}, M_{\Psi})$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where

$\Phi(z)=(P+zP^{\perp})U^* \quad \text{and} \quad \Psi(z)=U(P^{\perp}+zP)$

for $z \in \mathbb{D}$ (the unit disc in $\mathbb{C}$).

We shall then proceed to derive a similar characterization for pure contractions. In particular, let $T$ be a pure contraction on a Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$ be the Sz.-Nagy and Foias representation of $T$ for some canonical $\mathcal{Q} \subseteq H^2_{\mathcal{D}}(\mathbb{D})$, where $\mathcal{D}$ is a closed subspace of $\mathcal{H}$. We will show that $T = T_1 T_2$, for some commuting contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there exist $\mathcal{B}(\mathcal{D})$-valued polynomials $\varphi$ and $\psi$ of degree $\leq 1$ such that $\mathcal{Q}$ is a joint ($M^*_{\varphi}, M^*_{\psi}$)-invariant subspace and

$P_{\mathcal{Q}} M_z\mid_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\varphi \psi}\mid_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\psi \varphi}\mid_{\mathcal{Q}} \quad \mbox{and} \quad (T_1, T_2) \cong (P_{\mathcal{Q}} M_{\varphi}\mid_{\mathcal{Q}}, P_{\mathcal{Q}} M_{\psi}\mid_{\mathcal{Q}}).$

Moreover, there exist a Hilbert space $\mathcal{E}$ and an isometry $V \in \mathcal{B}(\mathcal{D}; \mathcal{E})$ such that

Venue: LH-1, Mathematics Department

Problems associated with the algorithmic counting of combinatorial structures arise naturally in many applications and also constitute an interesting sub-field of computational complexity theory. In this talk, we consider one of the most studied of such problems: that of counting proper colourings of bounded degree graphs. Here, a constant maximum degree $\Delta$ and a set of $q$ colours are fixed. The input is a graph $G$ of maximum degree at most $\Delta$, and a parameter $\varepsilon > 0$. The problem is to output “efficiently”, i.e. in time that grows polynomially in $1/\varepsilon$ and the size $n$ of the graph, and up to a multiplicative error of at most $(1+\varepsilon)$, the number of proper colourings of $G$ with the given set of $q$ colours.

In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$-modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra. We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists a one-to-one correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$ admits a $p$-filtration.

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Title: Percolation theory: from classical to dependent models

Speaker: Subhajit Goswami (IHES, Bures-sur-Yvette, France)

Date: Fri, 26 Jul 2019

Time: 3:30 - 4:40 PM

Venue: LH-1, Mathematics Department

Introduced in 1957 by Broadbent and Hammersley as a simple probabilistic model for movement of fluid through porous media, percolation theory has emerged as one of the most active and richest areas of research in modern probability theory. As one the most well-known models demonstrating an order-disorder phase transition, the techniques developed in the context of percolation theory have been widely successful in rigorous understanding of models in classical statistical physics, e.g. Ising and Potts. Furthermore in two dimensions it enjoys a deep connection with conformal field theory. In this talk I will attempt to provide a very brief tour of some horizons of classical percolation theory including some very recent results. I will then move onto percolation models in dependent media which pose new challenges to overcome. In this context I will present some recent results about a model related to the geometry of level sets of a canonical random Gaussian function (Gaussian free field) on lattice graphs. I will also briefly discuss some directions for future research. Some of the recent results are based on joint works with Hugo Duminil-Copin, Aran Raoufi, Pierre-Francois Rodriguez, Franco Severo and Ariel Yadin. No knowledge about percolation theory is assumed on the part of the audience.

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Title: PhD Thesis colloquium: Geometric Invariants For a Class of Submodules of Analytic Hilbert Modules

Speaker: Samrat Sen (IISc Mathematics)

Date: Wed, 24 Jul 2019

Time: 11 am

Venue: LH-1, Mathematics Department

Link to the abstract

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Title: PhD Thesis colloquium: Phase Transition, Percolation at Criticality and Connectivity in Random Connection Models

Speaker: Sanjoy Kumar Jhawar (IISc Mathematics)

Date: Mon, 22 Jul 2019

Time: 11 am

Venue: LH-1, Mathematics Department

This work has two parts. The first part contains the study of phase transition and percolation at criticality for three planar random graph models, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y \mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid\cdot\mid$ is the Euclidean norm. In the inhomogenous version of the model, points of $\mathcal{P}_{\lambda}$ are endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability

$\left(1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)\right)$

Venue: LH-1, Mathematics Department

This talk would have two parts. In the first part, we will discuss some topics which can be classified as ‘Linnik-type’ problems (the motivation being his original question about locating the first prime in an arithmetic progression) in the context of Hecke eigenvalues of modular forms on various groups, and then talk about the distribution of their signs. In the second part we will discuss differential operators on modular forms, and then talk about their applications to questions about Jacobi forms.

It is well-known that the sequence of Hecke eigenvalues mentioned above are often real, and has infinitely many sign changes. First part of the talk would discuss the problem of estimating the location of the first such sign change in the context of Hecke eigenvalues of Yoshida lifts (a certain subspace of the Siegel modular forms) and Fourier coefficients of Hilbert modular forms. We show how to improve the previously best known results on this topic significantly.

The crucial inputs behind these would be to establish a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level for treating Yoshida lifts; and exploiting Hecke relations along with generalising related results due to K. Soundararajan, K. Matomaki et al. for the case of Hilbert modular forms. In both cases we measure the location of the eigenvalues or Fourier coefficients in terms of an analytic object called the ‘analytic conductor’, which would be introduced during the talk. Moreover in the case of Hilbert modular forms, we will also discuss quantitative results about distribution of positive and negative Hecke eigenvalues. The proof depends on establishing a certain result on a particular types of multiplicative functions on the set of integral ideals of a totally real number field.

Venue: LH-1, Mathematics Department

In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms,having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$ have the same eigenvalues of the $p$-th Hecke operator $T_p$ for almost all primes $p$ , then $f_1=f_2$ .

The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square–free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$. We expect similar applications. We also discuss few results on the square–free Fourier coefficients of elliptic cusp forms.

In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, Z)$ then we consider its Saito–Kurokawa lift $F_g$ and a certain ‘restricted’ $L^2$ -norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.

Pullback of a Siegel modular form $F((\tau,z,z,\tau'))$ ( $(\tau,z,z,\tau')$ in Siegel’s upper half-plane of degree 2) to $H \times H$ is its restriction to $z=0$ , which we denote by $F\|_{z=0}$ . Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$ -norms of such pullbacks to central values of $L$-functions for all degrees.

We construct certain representations of affine Hecke algebras and Weyl groups, which depend on several auxiliary parameters. We refer to these as “metaplectic” representations, and as a direct consequence we obtain a family of “metaplectic” polynomials, which generalizes the well-known Macdonald polynomials.

Our terminology is motivated by the fact that if the parameters are specialized to certain Gauss sums, then our construction recovers the Kazhdan-Patterson action on metaplectic forms for GL(n); more generally it recovers the Chinta-Gunnells action on p-parts of Weyl group multiple Dirichlet series.

This is joint work with Jasper Stokman and Vidya Venkateswaran.

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Title: PhD Thesis colloquium: Risk-Sensitive Stochastic Control and Differential Games

Speaker: Somnath Pradhan (IISc Mathematics)

Date: Thu, 30 May 2019

Time: 11:30 am

Venue: LH-1, Mathematics Department

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Title: APRG Seminar: Problem governed by the von Kármán equations Morley finite element for a distributed optimal control

Speaker: Sudipto Chowdhury (IIT, Bombay)

Date: Tue, 21 May 2019

Time: 11 am

Venue: LH-1, Mathematics Department

This talk consists of the finite element analysis of a distributed optimal control problem governed by the von Kármán equation that describe the deflection of very thin plates defined on a polygonal domain of R2 with box constraints on the control variable. In this talk we discuss a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower order norms for the state and adjoint variables are derived. The lower order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Finally we discuss several numerical results to illustrate our theoretical results.

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Title: Algebra & Combinatorics Seminar: Alternating Schur Algebras and Koszul Duality

Speaker: Amritanshu Prasad (IMSc, Chennai)

Date: Wed, 08 May 2019

Time: 3 pm

Venue: LH-5, Mathematics Department

Schur classified polynomial representations of GL(n) using the commuting actions of the dth symmetric group and GL(n) on the d-fold tensor power of n-dimensional space. We set out to investigate what happens when the symmetric group in this picture is shrunk to the alternating group. This led to a simple interpretation and clearer understanding of the Koszul duality functor on the category of polynomial representations of GL(n).

This is based on joint work with T. Geetha and Shraddha Srivastava (https://arxiv.org/abs/1902.02465).

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Title: Algebra & Combinatorics Seminar: Path models for Kostant-Kumar submodules of a tensor product

Speaker: Sankaran Viswanath (IMSc, Chennai)

Date: Tue, 07 May 2019

Time: 11:30 am

Venue: LH-5, Mathematics Department

Consider the tensor product of two irreducible finite dimensional representations of a simple Lie algebra. The submodules generated by a tensor product of extremal vectors of the two components are called Kostant-Kumar submodules. These are parametrized by a double coset space of the Weyl group.

Littelmann’s path model is a very general combinatorial model for representations, which encompasses many classical constructs such as Young tableaux and Lakshmibai-Seshadri chains. The path model for the full tensor product is simply the set of concatenations of paths of the individual components. We describe a way to associate a Weyl group element (rather, a double coset) to each such concatenated path and thereby obtain a path model for Kostant-Kumar submodules.

Finally, we recall the many descriptions of Demazure modules, which may be viewed as the analog of the above picture for single paths (rather than concatenations). In this case, the Weyl group element associated to a path admits different descriptions in different path models in terms of statistics such as initial direction, minimal standard lifts and Kogan faces of Gelfand-Tsetlin polytopes. Along the way, we mention some relations to jeu-de-taquin and the Schutzenberger involution.

This is based on joint work with Mrigendra Singh Kushwaha and KN Raghavan.

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Title: Algebra & Combinatorics Seminar: The Brylinski filtration and W-algebras

Speaker: Sankaran Viswanath (IMSc, Chennai)

Date: Fri, 03 May 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

Each finite dimensional irreducible representation V of a simple Lie algebra L admits a filtration induced by a principal nilpotent element of L. This, so-called, Brylinski or Brylinski-Kostant filtration, can be restricted to the dominant weight spaces of V, and the resulting Hilbert series is very interesting q-analogs of weight multiplicity, first defined by Lusztig.

This picture can be extended to certain infinite-dimensional Lie algebras L and to irreducible highest weight, integrable representations V. We focus on the level 1 vacuum modules of special linear affine Lie algebras. In this case, we show how to produce a basis of the dominant weight spaces that is compatible with the Brylinski filtration. Our construction uses the so-called W-algebra, a natural vertex algebra associated to L.

This is joint work with Sachin Sharma (IIT Kanpur) and Suresh Govindarajan (IIT Madras).

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Title: Algebra & Combinatorics Seminar: Suslin Matrices vs Clifford Algebras

Speaker: Vineeth Chintala (IIT, Bombay)

Date: Wed, 01 May 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

In this talk, we will introduce the notion of an embedding of a quadratic space (in an associative algebra). Familiar examples of embeddings are given by Complex Numbers, Quaternions, Octonions, Clifford Algebras, and Suslin Matrices.

When there are two embeddings of the same quadratic space, then we can gather information about one embedding using the other. This is the main theme of the talk. To illustrate this, we will see the connection between Suslin Matrices and Clifford Algebras. In one direction, we are able to give a simple description of Clifford Algebras using matrices. Conversely, we can now explain some (seemingly) accidental properties of Suslin matrices in a conceptual way.

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Title: MS Thesis colloquium: Determinantal processes and stochastic domination

Speaker: Raghavendra Tripathi (IISc Mathematics)

Date: Tue, 30 Apr 2019

Time: 11:30 am

Venue: LH-1, Mathematics Department

The systematic study of determinantal processes began with the work of Macchi (1975), and since then they have appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths, measures on Young diagrams), and physics (fermions). A particularly interesting and well-known example of a discrete determinantal process is the Uniform spanning tree (UST) on finite graphs. We shall describe UST on complete graphs and complete bipartite graphs—in these cases it is possible to make explicit computations that yield some special cases of Aldous’ result on CRT.

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-1, Mathematics Department

According to Wikipedia, a continued fraction is “an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.” It has connections to various areas of mathematics including analysis, number theory, dynamical systems and probability theory.

Continued fractions of numbers in the open interval (0,1) naturally gives rise to a dynamical system that dates back to Gauss and raises many interesting questions. We will concentrate on the extreme value theoretic aspects of this dynamical system. The first work in this direction was carried out by the famous French-German mathematician Doeblin (1940), who rightly observed that exceedances of this dynamical system have Poissonian asymptotics. However, his proof had a subtle error, which was corrected much later by Iosifescu (1977). Meanwhile, Galambos (1972) had established that the scaled maxima of this dynamical system converges to the Frechet distribution.

After a detailed review of these results, we will discuss a powerful Stein-Chen method (due to Arratia, Goldstein and Gordon (1989)) of establishing Poisson approximation for dependent Bernoulli random variables. The final part of the talk will be about the application of this Stein-Chen technique to give upper bounds on the rate of convergence in the Doeblin-Iosifescu asymptotics. We will also discuss consequences of our result and its connections to other important dynamical systems. This portion of the talk will be based on a joint work with Maxim Sølund Kirsebom and Anish Ghosh.

Familiarity with standard (discrete and absolutely continuous) probability distributions will be sufficient for this talk.

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Title: APRG Seminar: Counting nodal components of ‘boundary-adapted arithmetic random waves’

Speaker: James Cann (University College London, UK)

Date: Thu, 28 Feb 2019

Time: 2:15 pm

Venue: LH-1, Mathematics Department

The ‘nodal sets’ (zero sets) of Dirichlet Laplace eigenfunctions for the two-dimensional unit square have raised many questions over the past century. The nodal domain theorems of Courant (1924), Stern (1926) and Pleijel (1956), give deterministic (upper, liminf, and limsup resp.) bounds for the number of nodal domains which can be exhibited by an n-th eigenfunction, assuming the eigenvalues are ordered increasingly and with multiplicities listed. Prominent amongst contemporary research is the closely related question of the number of ‘nodal components’ (connected components of the zero set) of a ‘typical’ eigenfunction.

The spectral degeneracy for the square means that the nodal count of an n-th eigenfunction could take a wide range of values, and to understand the ‘typical’ behaviour of this number we attribute Gaussian random coefficients to a standard basis of eigenfunctions for each eigenspace, to form the ensemble of ‘boundary-adapted arithmetic random waves’. The number of nodal components —now a random variable— can then be studied, and this talk will draw together tools from various areas of mathematics (random fields, integral geometry, number theory, mathematical physics, …) in order to say a few things about its asymptotic properties.

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Title: APRG Seminar: Spectral asymptotics for a singularly perturbed periodic elliptic operator

Speaker: Klas Pettersson (University of Tromsø – The Arctic University of Norway)

Date: Tue, 26 Feb 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale ε. We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter ε tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is non-degenerate at this point.

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Title: Geometry & Topology Seminar: A compactness theorem for Einstein metrics on the four-sphere

Speaker: Harish Seshadri (IISc Mathematics)

Date: Mon, 25 Feb 2019

Time: 4 pm

Venue: LH-1, Mathematics Department

It is open question if the standard round metric on $S^4$ is the unique positive Einstein metric, up to isometry and scaling. In this talk, I will discuss a compactness theorem which rules out nonstandard unit-volume Einstein metrics whose scalar curvatures lie within a certain range.

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Title: Algebra & Combinatorics Seminar: Cohomology of modules over H-categories and co-H-categories

Speaker: Samarpita Ray (IISc Mathematics)

Date: Fri, 22 Feb 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchel, an arbitrary small preadditive category may be understood as a “ring with several objects”. In this respect, for a Hopf algebra H, an H-category will denote an “H-module algebra with several objects” and a co-H-category will denote an “H-comodule algebra with several objects”. Modules over such Hopf categories were first considered by Cibils and Solotar. In this talk, we present a study of cohomology in such module categories.

In particular, we consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D,H)-Hopf modules over a Hopf comodule category D. These generalize relative (A,H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants.

Valiant (1979) showed that unless P = N P, there is no polynomial-time algorithm to compute the number of perfect matchings of a given graph – even if the input graph is bipartite. Earlier, the physicist Kasteleyn (1963) introduced the notion of a special type of orientation of a graph, and we refer to graphs that admit such an orientation as Pfaffian graphs. Kasteleyn showed that the number of perfect matchings is easy to compute if the input graph is Pfaffian, and he also proved that every planar graph is Pfaffian. The complete bipartite graph $K_{3,3}$ is the smallest graph that is not Pfaffian. In general, the problem of deciding whether a given graph is Pfaffian is not known to be in N P.

Special types of minors, known as conformal minors, play a key role in the theory of Pfaffian orientations. In particular, a graph is Pfaffian if and only if each of its conformal minors is Pfaffian. It was shown by Little (1975) that a bipartite graph $G$ is Pfaffian if and only if $G$ does not contain $K_{3,3}$ as a conformal minor (or, in other words, if and only if $G$ is $K_{3,3}$-free); this places the problem of deciding whether a bipartite graph is Pfaffian in co – N P. Several years later, a structural characterization of $K_{3,3}$-free bipartite graphs was obtained by Robertson, Seymour and Thomas (1999), and independently by McCuaig (2004), and this led to a polynomial-time algorithm for deciding whether a given bipartite graph is Pfaffian.

Norine and Thomas (2008) showed that, unlike the bipartite case, it is not possible to characterize all Pfaffian graphs by excluding a finite number of graphs as conformal minors. In light of everything that has been done so far, it would be interesting to even identify rich classes of Pfaffian graphs (that are nonplanar and nonbipartite).

Inspired by a theorem of Lovasz (1983), we took on the task of characterizing graphs that do not contain $K_4$ as a conformal minor – that is, $K_4$-free graphs. In a joint work with U. S. R. Murty (2016), we provided a structural characterization of planar $K_4$-free graphs. The problem of characterizing nonplanar $K_4$-free graphs is much harder, and we have evidence to believe that it is related to the problem of recognizing Pfaffian graphs. In particular, we conjecture that every graph that is $K_4$-free and $K_{3,3}$-free is also Pfaffian. The talk will be mostly self-contained. I will assume only basic knowledge of graph theory. For more details, see: https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.21882.

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Title: PhD Thesis defence: A study of some conformal metrics and conformal invariants on planar domains

Speaker: Amar Deep Sarkar (IISc Mathematics)

Date: Wed, 13 Feb 2019

Time: 11 am

Venue: LH-1, Mathematics Department

The main aim of this thesis is to explain the of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.

To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show the convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Caratheodory rigidity constant converges to 1 near smooth boundary points.

Next on the line is a conformal metric defined using holomorphic quadratic differentials. This was done by T. Sugawa and we will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasi-hyperbolic metric on a smoothly bounded domain.

We also study the Hurwitz metric that was introduced by D. Minda. Its construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner.

Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.

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Title: APRG Seminar: Universal models in ergodic theory

Speaker: Nishant Chandgotia (Hebrew University of Jerusalem, Israel)

Date: Tue, 12 Feb 2019

Time: 3:30 pm

Venue: LH-1, Mathematics Department

A topological dynamical system is a pair $(X,T)$ where $T$ is a homeomorphism of a compact space $X$. A measure preserving action is a triple $(Y, \mu, S)$ where $Y$ is a standard Borel space, $\mu$ is a probability measure on $X$ and $S$ is a measurable automorphism of $Y$ which preserves the measure $\mu$. We say that $(X,T)$ is universal if it can embed any measure preserving action (under some suitable restrictions).

Krieger’s generator theorem shows that if $X$ is $A^{\mathbb{Z}}$ (bi-infinite sequences in elements of $A$) and $T$ is the transformation on $X$ which shifts its elements one unit to the left then $(X,T)$ is universal. Along with Tom Meyerovitch, we establish very general conditions under which $\mathbb{Z}^d$ (where now we have $d$ commuting transformations on $X$)-dynamical systems are universal. These conditions are general enough to prove that the following models are universal:

A self-homeomorphism with non uniform specification on a compact metric space (answering a question by Quas and Soo and recovering recent results by David Burguet).
A generic (in the sense of dense $G_\delta$) self-homeomorphism of the 2-torus preserving Lebesgue measure (extending result by Lind and Thouvenot to infinite entropy).
Proper colourings of the $\mathbb{Z}^d$ lattice with more than two colours and the domino tilings of the $\mathbb{Z}^2$ lattice (answering a question by Şahin and Robinson).

Venue: LH-1, Mathematics Department

As a natural generalization of the classical uniformization theorem, one could ask if every conformal class of Riemannian metrics contains a metric with constant scalar curvature. In 1960, Yamabe attempted to solve the problem, and gave an incorrect “proof” of an affirmative answer to the above question. Yamabe’s idea was to to use techniques from calculus of variations and partial differential equations to minimize a certain functional, now named after him. Building on some work of Trudger, Aubin solved the Yamabe problem under an additional analytic condition, namely that the Yamabe invariant of the manifold is strictly smaller than the corresponding for the n-dimensional sphere. Aubin was able to verify this analytic condition for all manifolds of dimension strictly greater than 5 that are not locally conformally flat. Most remarkably, the problem was finally solved by Schoen in 1984 using the newly proven positive mass theorem of Schoen and Yau, which in turn has origins in general theory of relativity. A curious feature of Schoen’s proof is that it works precisely in the cases left out by Aubin’s result.

In the first part of the talk, I will provide some background in Riemannian geometry, and then describe the developments surrounding the Yamabe problem until Aubin’s work in 1976. A recurring theme will be the role played by the n-dimensional sphere and its conformal transformations, in all aspects of the problem. In the second part of the talk, I will describe the positive mass theorem, and outline Schoen’s proof of the Yamabe problem in low dimensions and for locally conformally flat manifolds. Most of the Riemannian geometric quantities (including scalar curvature, gradients, Laplacians etc.) will be be defined in the talk, and so the hope is that both the talks will be accessible to everyone.

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Title: Algebra & Combinatorics Seminar: Bisections and bicolorings of hypergraphs

Speaker: Niranjan Balachandran (IIT, Bombay)

Date: Wed, 30 Jan 2019

Time: 4 pm

Venue: LH-1, Mathematics Department

Given a hypergraph $\mathcal{H}$ with vertex set $[n]:={1,\ldots,n}$, a bisecting family is a family $\mathcal{A}\subseteq\mathcal{P}([n])$ such that for every $B\in\mathcal{H}$, there exists $A\in\mathcal{A}$ with the property $|A\cap B|-|A\cap\overline{B}|\in{-1,0,1}$. Similarly, for a family of bicolorings $\mathcal{B}\subseteq {-1,1}^{[n]}$ of $[n]$ a family $\mathcal{A}\subseteq\mathcal{P}([n])$ is called a System of Unbiased Representatives for $\mathcal{B}$ if for every $b\in\mathcal{B}$ there exists $A\in\mathcal{A}$ such that $\sum_{x\in A} b(x) =0$.

Venue: LH-1, Mathematics Department

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Title: APRG Seminar: Hermitian Vector Bundles over Jordan Manifolds

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Wed, 23 Jan 2019

Time: 4 pm

Venue: LH-1, Mathematics Department

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Title: APRG Seminar: Symmetric Spaces and Jordan Triples

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Tue, 22 Jan 2019

Time: 4 pm

Venue: LH-1, Mathematics Department

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Title: Geometry & Topology Seminar: Holomorphic differentials and stability in Fukaya categories

Speaker: Pranav Pandit (ICTS, Bangalore)

Date: Mon, 21 Jan 2019

Time: 4 pm

Venue: LH-1, Mathematics Department

Spectral networks are certain decorated graphs drawn on a Riemann surface. I will describe a conjectural picture in which spectral networks can be viewed as analogues of Hermitian-Einstein metrics on vector bundles, and in which holomorphic differentials on the Riemann surface arise as stability conditions on certain Fukaya-type categories. This talk is based on various joint projects with Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich, and Carlos Simpson.

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Title: Algebra & Combinatorics Seminar: Extensions of partial cyclic orders, boustrophedons and polytopes

Speaker: Sanjay Ramassamy (Ecole Normale Superiure, Paris, France)

Date: Fri, 18 Jan 2019

Time: 11 am

Venue: LH-1, Mathematics Department

While the enumeration of linear extensions of a given poset is a well-studied question, its cyclic counterpart (enumerating extensions to total cyclic orders of a given partial cyclic order) has been subject to very little investigation. In this talk I will introduce some classes of partial cyclic orders for which this enumeration problem is tractable. Some cases require the use of a multidimensional version of the classical boustrophedon construction (a.k.a. Seidel-Entringer-Arnold triangle). The integers arising from these enumerative questions also appear as the normalized volumes of certain polytopes.

This is partly joint work with Arvind Ayyer (Indian Institute of Science) and Matthieu Josuat-Verges (Laboratoire d’Informatique Gaspard Monge / CNRS).

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Title: Eigenfunctions Seminar: Some unpublished work of Harish-Chandra

Speaker: Ramesh Gangolli (University of Washington, Seattle, USA)

Date: Fri, 18 Jan 2019

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-1, Mathematics Department

When Harish-Chandra died in 1983, he left behind a voluminous pile of handwritten manuscripts on harmonic analysis on semisimple Lie groups over real/complex and p-adic fields. The manuscripts were turned over to the archives of the Institute for Advanced Study at Princeton, and are archived there.

Robert Langlands is the Trustee of the Harish-Chandra archive, and has always been interested in finding a way of salvaging whatever might be valuable in these manuscripts. Some years ago, at a conference in UCLA, he encouraged V. S. Varadarajan and me to look at these manuscripts.

The results of our efforts have resulted in the publication of the Volume 5 (Posthumous) of the Collected works of Harish-Chandra by Springer Verlag, which appeared in July this year. This volume covers only those manuscripts that deal with Real or Complex groups. The manuscripts dealing with p-adic groups remain in the archive.

My talk will be devoted to an outline of the results in this volume, without much detail. However, I will try to describe the main ingredient of the methods that cut across most of this work.

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Title: APRG Seminar: Totally nonnegative GCD matrices and kernels

Speaker: Dominique Guillot (University of Delaware, Newark, USA)

Date: Wed, 16 Jan 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

Let $X=(x_1, … ,x_n)$ be a vector of distinct positive integers. The $n \times n$ matrix with $(i,j)$ entry equal to gcd$(x_i,x_j)$, the greatest common divisor of $x_i$ and $x_j$, is called the GCD matrix on $X$. By a surprising result of Beslin and Ligh (1989), all GCD matrices are positive definite. In this talk, we will discuss new characterizations of the GCD matrices satisfying the stronger property of being totally nonnegative (i.e., all their minors are nonnegative).

Joint work with Lucas Wu (U. Delaware).

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Title: Algebra & Combinatorics Seminar: From word games to a Polymath project

Speaker: Apoorva Khare (IISc Mathematics)

Date: Fri, 11 Jan 2019

Time: 3 pm

Venue: LH-1, Mathematics Department

Consider the following three properties of a general group $G$:

Algebra: $G$ is abelian and torsion-free.
Analysis: $G$ is a metric space that admits a “norm”, namely, a translation-invariant metric $d(.,.)$ satisfying: $d(1,g^n) = |n| d(1,g)$ for all $g \in G$ and integers $n$.
Geometry: $G$ admits a length function with “saturated” subadditivity for equal arguments: $\ell(g^2) = 2 \; \ell(g)$ for all $g \in G$.

While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a “norm”.

We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and if time permits, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.

(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)

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Title: APRG Seminar: Dynamics of Schwarz reflections: mating rational maps with groups

Speaker: Sabyasachi Mukherjee (State University of New York, Stony Brook, USA)

Date: Mon, 07 Jan 2019

Time: 4:30 pm

Venue: LH-1, Mathematics Department (Skype talk)

A domain in the complex plane is called a quadrature domain if it admits a global Schwarz reflection map. Topology of quadrature domains has important applications to physics, and is intimately related to iteration of Schwarz reflection maps. As dynamical systems, Schwarz reflection maps produce various instances of “matings” of rational maps and groups.

We will introduce this new class of dynamical systems, and illustrate the above-mentioned mating phenomenon with a few concrete examples. We will then describe a specific one-parameter family of Schwarz reflection maps such that “typical” maps in this family arise as unique conformal matings of a quadratic anti-holomorphic polynomial and the ideal triangle group.

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Title: Algebra & Combinatorics Seminar: Tensor product decomposition for the general linear supergroup $GL(m|n)$

Speaker: Thorsten Heidersdorf (Max-Planck-Institut fur Mathematik, Bonn, Germany)

Date: Thu, 20 Dec 2018

Time: 11 am

Venue: LH-1, Mathematics Department

Let $\mathfrak{gl}(n)$ denote the Lie algebra of the general linear group $GL(n)$. Given two finite dimensional irreducible representations $L(\lambda), L(\mu)$ of $\mathfrak{gl}(n)$, its tensor product decomposition $L(\lambda) \otimes L(\mu)$ is given by the Littlewood-Richardson rule.

The situation becomes much more complicated when one replaces $\mathfrak{gl}(n)$ by the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. The analogous decomposition $L(\lambda) \otimes L(\mu)$ is largely unknown. Indeed many aspects of the representation theory of $\mathfrak{gl}(m|n)$ are more akin to the study of Lie algebras and their representations in prime characteristic or to the BGG category $\mathcal{O}$. I will give a survey talk about this problem and explain why some approaches don’t work and what can be done about it. This will give me the chance to speak about a) the character formula for an irreducible representation $L(\lambda)$, b) Deligne’s interpolating category $Rep(GL_t)$ for $t \in \mathbb{C}$ and c) the process of semisimplification of a tensor category.

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Title: APRG Seminar: Stopping cocyles of the CCR flow

Speaker: Alexander Belton (Lancaster University, Lancaster, UK)

Date: Thu, 20 Dec 2018

Time: 3 pm

Venue: LH-1, Mathematics Department

The non-commutative generalisation of the notion of stopping time was introduced by Hudson. Such quantum stopping times may be used to stop the CCR flow and its isometric cocycles, i.e., left operator Markovian cocycles on Boson Fock space. This stopping of the CCR flow yields a homomorphism from the semigroup of stopping times into the semigroup of unital endomorphisms of the algebra of bounded operators on the ambient Fock space, and this homomorphism is a generalised version of the shift semigroup. Moreover, stopped cocycles satisfy a non-deterministic version of the cocycle relation, which is a significant generalisation of a result of Applebaum. (Joint work with K. B. Sinha)

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Title: APRG Seminar: The projective spectrum: 1. Applications to group theory

Speaker: Rongwei Yang (State University of New York, Albany, USA)

Date: Wed, 19 Dec 2018

Time: 4 pm

Venue: LH-1, Mathematics Department

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Title: PhD Thesis defence: Decomposition of the tensor product of Hilbert modules via the jet construction and weakly homogeneous operators

Speaker: Soumitra Ghara (IISc Mathematics)

Date: Mon, 17 Dec 2018

Time: 11 am

Venue: LH-1, Mathematics Department

Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ be a sesqui-analytic function. We show that if $\alpha,\beta>0$ be such that the functions $K^{\alpha}$ and $K^{\beta}$, defined on $\Omega\times\Omega$, are non-negative definite kernels, then the $M_m(\mathbb{C})$ valued function $K^{(\alpha,\beta)} := K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^m$ is also a non-negative definite kernel on $\Omega\times\Omega$. Then we find a realization of the Hilbert space $(H,K^{(\alpha,\beta)})$ determined by the kernel $K^{(\alpha, \beta)}$ in terms of the tensor product $(H, K^{\alpha})\otimes (H, K^{\beta})$.

For two reproducing kernel Hilbert modules $(H,K_1)$ and $(H,K_2)$, let $A_n, n\geq 0$, be the submodules of the Hilbert module $(H, K_1)\otimes (H, K_2)$ consisting of functions vanishing to order $n$ on the diagonal set $\Delta:= \{ (z,z):z\in \Omega \}$. Setting $S_0=A_0^\perp, S_n=A_{n-1}\ominus A_{n}, n\geq 1$, leads to a natural decomposition of $(H, K_1)\otimes (H, K_2)$ into an infinite direct sum $\oplus_{n=0}^{\infty} S_n$. A theorem of Aronszajn shows that the module $S_0$ is isometrically isomorphic to the push-forward of the module $(H,K_1K_2)$ under the map $\iota:\Omega\to \Omega\times\Omega$, where $\iota(z)=(z,z), z\in \Omega$. We prove that if $K_1=K^{\alpha}$ and $K_2=K^{\beta}$, then the module $S_1$ is isometrically isomorphic to the push-forward of the module $(H,K^{(\alpha, \beta)})$ under the map $\iota$. We also show that if a scalar valued non-negative kernel $K$ is quasi-invariant, then $K^{(1,1)}$ is also a quasi-invariant kernel.

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Title: PhD Thesis colloquium: Some Results on Spectral Spaces and spectral sequences

Speaker: Samarpita Ray (IISc Mathematics)

Date: Thu, 13 Dec 2018

Time: 4 pm

Venue: LH-3, Mathematics Department

In the last twenty years, several notions of what is called the algebraic geometry over the “field with one element” has been developed. One of the simplest approaches to this is via the theory of monoid schemes. The concept of a monoid scheme itself goes back to Kato and was further developed by Deitmar and by Connes, Consani and Marcolli. The idea is to replace prime spectra of commutative rings, which are the building blocks of ordinary schemes, by prime spectra of commutative pointed monoids. In our work, we focus mostly on abstracting out the topological characteristics of the prime spectrum of a commutative pointed monoid. This helps to obtain several classes of topological spaces which are homeomorphic to the the prime spectrum of a monoid. Such spaces are widely studied and are called spectral spaces. They were introduced by M. Hochster. We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations like integral, saturation, Frobenius and tight closures on monoids. We prove that the collection of all continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.

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Title: APRG Seminar: The gamma function - an eclectic tour

Speaker: Gopal Srinivasan (IIT, Bombay)

Date: Thu, 13 Dec 2018

Time: 11 am

Venue: LH-1, Mathematics Department

In this talk we shall take an eclectic tour through three centuries since the inception of this function by Euler in 1729. The function has been the object of intense study by every great mathematician of the 19th century and continues to tantalize the modern mathematicians.

We shall dwell upon some of the more important identities, their modern ramifications and some new proofs of old results. We shall also discuss some recent developments that have occured in the last decade.

Lecture 1: Friday, Dec 7
Lecture 2: Monday, Dec 10
Lecture 3: Wednesday, Dec 12
Lecture 4: Friday, Dec 14
Lecture 5: Monday, Dec 17
Lecture 6: Wednesday, Dec 19

All lectures will take place in LH-1, Department of Mathematics.The first lecture will be at 3:00 pm. All subsequent lectures will be at 4:00 pm.

In this series, the speaker will discuss some basic material (connexions, curvature etc) and then cover, with full proofs, the Borel-Weil-Bott theorem and the Kodaira embedding theorem.

A second set of lectures will be announced subsequently.

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Title: On the structure and distances of repeated-root constacyclic codes

Speaker: Anuradha Sharma (IIIT, Delhi)

Date: Mon, 03 Dec 2018

Time: 3 pm

Venue: LH-1, Mathematics Department

The main aim of coding theory is to construct codes that are easier to encode and decode, can correct or at least detect many errors, and contain a sufficiently large number of codewords. To study error-detecting and error-correcting properties of a code with respect to various communication channels, several metrics (e.g. Hamming metric, Lee metric, Rosenbloom-Tsfasman (RT) metric, symbol-pair metric, etc.) have been introduced and studied in coding theory.

In this talk, we will establish algebraic structures of all repeated-root constacyclic codes of prime power lengths over finite commutative chain rings. Using their algebraic structures, we will determine Hamming distances, b-symbol distances, RT distances, and RT weight distributions of these codes. As an application of these results, we will identify MDS (maximum-distance separable) Hamming, MDS b-symbol and MDS RT codes within this particular class of constacyclic codes. We will also present an algorithm to decode these codes with respect to the Hamming, symbol-pair and RT metrics.

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Title: APRG Seminar: Polynomially convex embeddings of compact real manifolds

Speaker: Purvi Gupta (Rutgers University, USA)

Date: Mon, 19 Nov 2018

Time: 4 pm

Venue: LH-1, Mathematics Department

A compact subset of $\mathbb{C}^n$ is polynomially convex if it is defined by a (possibly infinite) family of polynomial inequalities. In this talk, we will discuss some questions and recent results regarding the minimum embedding (complex) dimension of abstract compact (real) manifolds subject to such convexity constraints. These embeddings arise naturally in connection with higher-dimensional analogues of the fact that the algebra of continuous functions on a circle can be generated by two smooth functions. We will expand on this connection, discuss the local and global challenges in constructing such embeddings, and discuss two distinct cases in which new bounds have been obtained. This is joint work with R. Shafikov.

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Title: Algebra & Combinatorics Seminar: Rings with several objects

Speaker: Abhishek Banerjee (IISc Mathematics)

Date: Fri, 16 Nov 2018

Time: 3 pm

Venue: LH-1, Mathematics Department

An ordinary ring may be seen as a preadditive category with just one object. This leads to the powerful analogy, first formulated explicitly by Mitchell in 1975, that a small preadditive category should be seen as a “ring with several objects”. We will trace the history and development of the category of modules over a preadditive category.

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Title: APRG Seminar: A $z^k$-invariant subspace without the wandering property

Speaker: Daniel Seco (ICMAT, Madrid, Spain)

Date: Wed, 14 Nov 2018

Time: 4 pm

Venue: LH-1, Mathematics Department

We study operators of multiplication by $z^k$ in Dirichlet-type spaces $D_\alpha$. We establish the existence of $k$ and $\alpha$ for which some $z^k$-invariant subspaces of $D_\alpha$ do not satisfy the wandering property. As a consequence of the proof, any Dirichlet-type space accepts an equivalent norm under which the wandering property fails for some space for the operator of multiplication by $z^k$ for any $k \geq 6$.

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Title: Eigenfunctions Seminar: A refinement of the Littlewood-Richardson rule

Speaker: K.N. Raghavan (IMSc, Chennai)

Date: Fri, 09 Nov 2018

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-1, Mathematics Department

Symmetric polynomials are interesting not just in their own right. They appear naturally in various contexts, e.g., as characters in representation theory and as cohomology classes of Grassmannians and other homogeneous spaces in geometry. There are many different bases for the space of symmetric polynomials, perhaps the most interesting of which is the one formed by Schur polynomials. The Littlewood-Richardson (LR) rule expresses the product of two given Schur polynomials as a linear combination of Schur polynomials. The first half of the talk will be a tour of these topics that should be accessible even to undergraduate students.

The second half will be an exposition (which hopefully will continue to be widely accessible!) of recent joint research work with Mrigendra Singh Kushwaha and Sankaran Viswanath, both of IMSc. The LR rule above has a natural interpretation as giving the decomposition as a direct sum of irreducibles of the tensor product of two irreducible representations of the unitary (or general linear) group. A generalised version of it (due to Littelmann, still called the LR rule) gives the analogous decomposition for any reductive group. On the tensor product of two irreducible representations, there is the natural “Kostant-Kumar” filtration indexed by the Weyl group. This consists of the cyclic submodules generated by the highest weight vector tensor an extremal weight vector. We obtain a refined LR rule that gives the decomposition as a direct sum of irreducibles of the Kostant-Kumar submodules (of the tensor product). As an application, we obtain alternative proofs of refinements of “PRV type” results proved by Kumar and others. (PRV = Parthasarathy, Ranga Rao, Varadarajan)

This is joint work with Roger Behrend.

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Title: On the formality of some contact manifolds

In this talk I will briefly describe the idea of a proof of the Mordell-Weil Theorem and introduce the n-Selmer group and Tate-Shafarevitch group associated to Elliptic Curves. I will define the L-function and some other arithmetic invariants attached to the Elliptic Curves and state the celebrated Birch-Swinnerton-Dyer Conjecture.

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Title: Introduction to Elliptic Curves

Speaker: Shubham Sinha (Graduate Student at UC San Diego, USA)

Date: Fri, 27 Jul 2018

Time: 2 pm

Venue: LH-2, Mathematics Department

Elliptic curves are important objects of study in various areas of research in modern mathematics. In this talk I will develop some algebraic and geometric tools to understand the group structures on elliptic curves and their Isogenies (certain kind of homomorphisms). I will specialise the general study of Elliptic Curves over finite fields, define zeta functions of associated Elliptic Curve and state the Weil Conjectures.

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Title: PhD Thesis colloquium: A Study of Conformal Metrics and Invariants on Planar Domains

Speaker: Amar Deep Sarkar (IISc Mathematics)

Date: Fri, 27 Jul 2018

Time: 3 pm

Venue: LH-3, Mathematics Department

The main aim of this thesis is to explain the behaviour of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated to the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.

To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Caratheodory rigidity constant converges to 1 near smooth boundary points.

We also study the Hurwitz metric that was introduced by D. Minda. It’s construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner.

Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.

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Title: On bases for local Weyl modules

Speaker: B. Ravinder (Chennai Mathematical Institute, Chennai)

Date: Thu, 26 Jul 2018

Time: 11 am

Venue: LH-3, Mathematics Department

Let G be a finite-dimensional complex simple Lie algebra and G[t] be its current algebra. The degree grading on the polynomial ring gives a natural grading on G[t] and makes it a graded Lie algebra. Local Weyl modules introduced by Chari and Pressley are interesting finite-dimensional graded G[t]-modules. Corresponding to a dominant integral weight x of G there is a local Weyl module denoted by W(x). The zeroth graded piece of W(x) is the irreducible G-module V(x). In this talk, we discuss how to obtain a basis for W(x) from the basis of V(x) given by Gelfand-Tsetlin patterns, when G is of type A and C.

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Title: Four Flavours of Combinatorics (Enumerative, Probabilistic, Extremal, and Geometric)

Speaker: Kunal Dutta (INRIA Sophia-Antipolis, France)

Date: Tue, 24 Jul 2018

Time: 4 pm

Venue: LH-3, Mathematics Department

In this talk we shall see three very different areas of applications of combinatorics in mathematics and computer science, illustrating four different flavours of combinatorial reasoning.

The first problem is on the decomposition, into irreducible representations, of the Weil representation of the full symplectic group associated to a finite module of odd order over a Dedekind domain. We shall discuss how a poset structure defined on the orbits of finite abelian p-groups under automorphisms can be used to show the decomposition of the Weil representation is multiplicity-free, as well as parametrize the irreducible subrepresentations, compute their dimensions in terms of p, etc. Joint works with Amritanshu Prasad (IMSc, Chennai).

Time: 11 am

Venue: LH-1, Mathematics Department

In this thesis, we have shown the following. Firstly, we have proven a conjecture about the density of 1’s in a system with arbitrary number of 0’s and 1’s. Secondly, we have made progress on a conjecture for the nonequilibrium partition function. In particular, we have given an alternate proof of the conjecture for the (1, n_1)-system and (n_0, 1)-system, using an enriched two-dimensional model.

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Title: An introduction to toric geometry

Speaker: Yifei Chen (Chinese Academy of Sciences, Beijing, China)

Date: Tue, 12 Jun 2018

Time: 3 - 5 pm

Venue: LH-3, Mathematics Department

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Title: MS Thesis defence: Duality for spaces of holomorphic functions into a locally convex topological vector space

Speaker: Kriti Sehgal (IISc Mathematics)

Date: Thu, 07 Jun 2018

Time: 2:30 pm

Venue: LH-1, Mathematics Department

In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function f: O → E, where O is an open subset of the complex plane and E a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.

For Ω_1 a subset of the Riemann sphere, we consider the space of locally holomorphic maps of Ω_1 into E vanishing at infinity if infinity belongs to Ω_1, denoted by P(Ω_1,E). For two complementary subsets Ω_1 and Ω_2 of the Riemann sphere we prove that given two locally convex topological vector spaces E and F in separating duality, under some general conditions, we can define a separating duality between P(Ω _1,E) and P(Ω_2,F).

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Title: An introduction to birational geometry in positive characteristic fields

Speaker: Yifei Chen (Chinese Academy of Sciences, Beijing, China)

Date: Tue, 05 Jun 2018

Time: 3 pm

Venue: LH-1, Mathematics Department

We will survey recent progress of birational geometry in positive characteristic fields. As well, we will introduce subadditiviy of Kodaira dimension and canonical bundle formula in positive characteristics. These are joint work with Caucher Birkar, Lei Zhang and Yi Gu.

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Title: Number of Solutions of Polynomial equations over Finite Fields

Speaker: Sudhir Ghorpade (IIT, Bombay)

Date: Mon, 04 Jun 2018

Time: 3 pm

Venue: LH-1, Mathematics Department

We will consider the following question:

Given a system of a fixed number of linearly independent homogeneous polynomial equations of a fixed degree with coefficients in a finite field F, what is the maximum number of common solutions they can have in the corresponding protective space over F?

Venue: LH-1, Mathematics Department

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is called \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, given any metric graph, its metric can be rescaled so that it can be essentially and isometrically embedded on a closed hyperbolic surface. The essential genus $g_e(G)$ of a metric graph $(G, d)$ is the lowest genus of a surface on which such an embedding of the graph is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ can be essentially and isometrically embedded (possibly after a rescaling the metric $d$) on a surface of genus $g$.

Next, we study minimal embeddings, where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_e^{\max}(G)$ a graph $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{\max}(G)$ are realized.

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Title: PhD Thesis defence: Computation of sparse representation of high dimensional time series data and experimental applications

Speaker: Sandeep K. Modi (IISc Mathematics)

Date: Thu, 26 Apr 2018

Time: 2:30 pm

Venue: LH-2, Mathematics Department

Obtaining a sparse representation of high dimensional data is often the first step towards its further analysis. Conventional Vector Autoregressive (VAR) modelling methods applied to such data results in noisy, non-sparse solutions with a too many spurious coefficients. Computing auxiliary quantities such as the Power Spectrum, Coherence and Granger Causality (GC) from such non-sparse models is slow and gives wrong results. Thresholding the distorted values of these quantities as per some criterion, statistical or otherwise, does not alleviate the problem.

We propose two sparse Vector Autoregressive (VAR) modelling methods that work well for high dimensional time series data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments using simulated data, our methods show consistently higher accuracy compared to other contemporary methods in recovering the true sparse model. The relative absence of spurious coefficients in our models permits more accurate, stable and efficient evaluation of auxiliary quantities. Our VAR modelling methods are capable of computing Conditional Granger Causality (CGC) in datasets consisting of tens of thousands of variables with a speed and accuracy that far exceeds the capabilities of existing methods.

Using the Conditional Granger Causality computed from our models as a proxy for the weight of the edges in a network, we use community detection algorithms to simultaneously obtain both local and global functional connectivity patterns and community structures in large networks.

We also use our VAR modelling methods to predict time delays in many-variable systems. Using simulated data from non-linear delay differential equations, we compare our methods with commonly used delay prediction techniques and show that our methods yield more accurate results.

We apply the above methods to the following real experimental data:

Analysis of data from the Human Connectome Project (HCP): fMRI data from the HCP database is used to compute sparse brain functional connectivity networks. The network and community structures obtained are consistent over independent recording sessions and show good spatial correspondence with known functional and anatomical regions of the brain.
Analysis of ADHD-200 data: fMRI data from children with ADHD (Attention Deficit Hyperactive Disorder) is used to compute sparse brain functional connectivity networks. Analysis of the network measures obtained provide new ways of differentiating between ADHD and typically developing children using global and node-level network measures. They also enable refinement of published results relating to the rFIC-ACC interaction in fMRI resting state data.
Time-delay prediction from LFP recordings:When applied to Local Field Potential (LFP) recordings from the rat and monkey, our methods predict consistent delays across a range of sampling frequencies.
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.

In addition to the above methods, we apply non-parametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed point-process and real time-series data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.

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Title: PhD Thesis colloquium: VAR Models, Granger Causality and Applications to Neuroscience

Speaker: Sandeep K. Modi (IISc Mathematics)

Date: Thu, 26 Apr 2018

Time: 2:30 pm

Venue: LH-2, Mathematics Department

We apply the above methods to the following real experimental data:

Analysis of data from the Human Connectome Project (HCP): fMRI data from the HCP database is used to compute sparse brain functional connectivity networks. The network and community structures obtained are consistent over independent recording sessions and show good spatial correspondence with known functional and anatomical regions of the brain.
Analysis of ADHD-200 data: fMRI data from children with ADHD (Attention Deficit Hyperactive Disorder) is used to compute sparse brain functional connectivity networks. Analysis of the network measures obtained provide new ways of differentiating between ADHD and typically developing children using global and node-level network measures. They also enable refinement of published results relating to the rFIC-ACC interaction in fMRI resting state data.
Time-delay prediction from LFP recordings:When applied to Local Field Potential (LFP) recordings from the rat and monkey, our methods predict consistent delays across a range of sampling frequencies.
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.

In addition to the above methods, we apply non-parametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed point-process and real time-series data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.

Speaker: Subhajit Goswami (IHES, Bures-sur-Yvette, France)

Date: Mon, 19 Mar 2018

Time: 2 pm

Venue: LH-1, Mathematics Department

In this talk I will give a brief introduction to Liouville first-passage percolation (LFPP) which is a model for random metric on a finite planar grid graph. It was studied primarily as a way to understand the random metric associated with Liouville quantum gravity (LQG), one of the major open problems in contemporary probability theory. In short the Liouville quantum gravity is a (conjectured) one parameter family of ``canonical’’ random metrics on a Riemann surface. I will discuss some recent results on this metric and the main focus will be on estimates of the typical distance between two points. I will highlight the apparent disagreement of these estimates with a prediction made in the physics literature about the LQG metric. I will also mention some (of many) future problems in this program. Based on a joint work with Jian Ding.

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Title: MS Thesis colloquium: Holomorphic functions into a locally convex topological vector space

Speaker: Kriti Sehgal (IISc Mathematics)

Date: Mon, 19 Mar 2018

Time: 11:15 am

Venue: LH-1, Mathematics Department

In the second half of this talk, a law of large numbers is proven for spaces with infinite “volume” (measure) as opposed to the above version for probability measures (“volume” =1).

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Title: APRG Seminar: Self-intersections and Poisson line processes on hyperbolic surfaces

Speaker: Jayadev Athreya (University of Washington, Seattle, USA)

Date: Wed, 17 Jan 2018

Time: 4 pm

Venue: LH-1, Mathematics Department

In joint work with Steve Lalley and Jenya Sapir, we study the tessellation of a compact, hyperbolic surface induced by a typical long geodesic segment. We show, that when properly scaled, the local behavior of a typical geodesic is that of a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.

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Title: APRG Seminar: Dynamics Of Determinantal Point Processes

Speaker: Alexander Bufetov (CNRS, Marseilles, France; and Steklov Mathematical Institute, IITP RAS, NRU Higher School of Economics, Russia)

Date: Wed, 17 Jan 2018

Time: 2:30 pm

Venue: LH-1, Mathematics Department

Determinantal point processes, which first appeared in Dyson’s work on random matrices, arise in diverse problems of asymptotic combinatorics, ergodic theory, representation theory. They have strong chaotic properties: for example, the sine-process has the Kolmogorov property and satisfies the Central Limit Theorem. A Functional Limit Theorem for the sine-process has been established in joint work with A. Dymov.

A delicate aspect of the behaviour of a determinantal point process is that particles interact at infinite radius. For instance, Ghosh and Peres showed that, under the sine-process, the number of particles in a bounded interval is determined by the configuration in the outside of the interval. For determinantal point processes with so-called integrable kernels, an explicit description is given of conditional measures of the process in a bounded interval with respect to the fixed exterior. These conditional measures are given by orthogonal polynomial ensembles with explicitly found weights. A key step in the argument is that projections inducing our processes satisfy a weaker analogue of the division axiom of de Branges: in fact, this weak division property, as shown in joint work with Roman Romanov, characterizes integrable kernels. Similar results for determinantal point processes governed by orthogonal projections onto Hilbert spaces of holomorphic functions are obtained in joint work with Y. Qiu. The talk is based on the preprints

https://arxiv.org/abs/1707.03463, (joint with R. Romanov), https://arxiv.org/abs/1701.00111 (joint with A. Dymov), and the paper

Alexander I. Bufetov, Yanqi Qiu, “Conditional measures of generalized Ginibre point processes”, J. Funct. Anal., 272:11 (2017), 4671–4708.

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Title: Eigenfunctions Seminar: Homogeneous length functions on Groups: A polymath adventure

Speaker: Siddhartha Gadgil and Apoorva Khare (IISc Mathematics)

Date: Mon, 15 Jan 2018

Time: 4 – 5 pm

Venue: LH-1, Mathematics Department

Terence Tao posted on his blog a question of Apoorva Khare, asking whether the free group on two generators has a length function $l: F_2 \to\mathbb{R}$ (i.e., satisfying the triangle inequality) which is homogeneous, i.e., such that $l(g^n) = nl(g)$. A week later, the problem was solved by an active collaboration of several mathematicians (with a little help from a computer) through Tao’s blog. In fact a more general result was obtained, namely that any homogeneous length function on a group $G$ factors through its abelianization $G/[G, G]$.

Venue: LH-5, Mathematics Department

Gorenstein rings are very common and significant in many areas of mathematics. The following are two important and widely open problems in commutative algebra and algebraic geometry:

characterization of possible Hilbert functions of Gorenstein K-algebras; and
classification of Gorenstein K-algebras with given Hilbert functions.

Recently, in a joint work with M.E. Rossi, we obtained partial results to these problems in some cases (K-algebras of socle degree 4). In this talk, we will discuss these new developments.

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Title: APRG Seminar: Compactness and Structural Stability of Nonlinear Flows

Speaker: Augusto Visintin (Universita' degli Studi di Trento, Italy)

Date: Thu, 11 Jan 2018

Time: 2:30 - 3:30 and 4 - 5 pm

Venue: LH-3, Mathematics Department

Given domains $D_1$, $D_2$ in complex Euclidean spaces, and a set ${(z_i,w_i): 1\leq i\leq N}\subset D_1\times D_2$, where $z_i$ are distinct and $N$ is a positive integer $\geq 2$, find necessary and sufficient conditions for the existence of a holomorphic map $F$ from $D_1$ into $D_2$ such that $F(z_i) = w_i$, $1\leq N$.

When such a map $F$ exists, we say that $F$ is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem – which we shall study in this thesis – have been of lasting interest:

INTERPOLATION FROM THE POLYDISC TO THE UNIT DISC: This is the case $D_1 = D^n$ and $D_2 = D$, where $D$ denotes the open unit disc in the complex plane and $n$ is a positive integer. The problem itself originates with Georg Pick’s well-known theorem (independently discovered by Nevanlinna) for the case $n=1$. Much later, Sarason gave another proof of Pick’s result using an operator-theoretic approach, which is very influential. Using this approach for $n\geq 2$, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur–Agler class. This is notable because when $n = 2$ the latter result completely solves the problem for the case $D_1 = D^2$, $D_2 = D$. However, Pick’s approach can also be effective for $n\geq 2$. In this thesis, we give an alternative characterization for the existence of a $3$-point interpolant based on Pick’s approach and involving the study of rational inner functions.

Cole, Lewis and Wermer lifted Sarason’s approach to uniform algebras – leading to a characterization for the existence of an interpolant in terms of the positivity of a large, rather abstractly-defined family of $(N\times N)$ matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a $D^n$-to-$D$ interpolant in terms of the positivity of a family of matrices parametrized by a class of polynomials.

INTERPOLATION FORM THE UNIT DISC TO THE SPECTRAL UNIT BALL: This is the case $D_1 = D$ and $D_2$ is the set of all $(n\times n)$ matrices with spectral radius less than $1$. The interest in this arises from problems in Control Theory. Bercovici, Fois and Tannenbaum adapted Sarason’s methods to give a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc – leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any $n$ and $N=2$. We shall present a necessary condition for the existence of a $3$-point interpolant. This we shall achieve by modifying Pick’s approach and applying the aforementioned result due to Bharali.

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Title: Derived categories and Ulrich bundles

Speaker: Kyoung-Seog Lee (Center for Geometry and Physics, Institute for Basic Science (IBS) Pohang, Republic of Korea)

Date: Mon, 04 Dec 2017

Time: 4 pm

Venue: LH-3, Mathematics Department

Derived category is an important tool in homological algebra invented by Grothendieck and Verdier. In these days derived categories play important roles in many areas of algebraic geometry. In this talk, I will discuss derived categories and their applications to study Ulrich bundles on some Fano manifolds.

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Title: Closed subroot systems of finite and affine root systems

Speaker: Krishanu Roy (IMSc, Chennai)

Date: Fri, 24 Nov 2017

Time: 10:30 am

Venue: LH-1, Mathematics Department

In this talk, I will tell you about the Borel-de-Sibenthal theorem which gives the classification of all maximal closed subroot systems of finite crystallographic root systems. I will start my talk by introducing the notion of finite root systems and it’s closed subroot systems.

Venue: LH-1, Mathematics Department

I will describe our work that establishes (akin to) von Neumann’s conjecture on condition number, the ratio of the largest and the smallest singular values, for sparse random matrices. Non-asymptotic bounds on the extreme singular values of large matrices have numerous uses in the geometric functional analysis, compressed sensing, and numerical linear algebra. The condition number often serves as a measure of stability for matrix algorithms. Based on simulations von Neumann and his collaborators conjectured that the condition number of a random square matrix of dimension $n$ is $O(n)$. During the last decade, this conjecture was proved for dense random matrices.

Sparse matrices are abundant in statistics, neural networks, financial modeling, electrical engineering, and wireless communications. Results for sparse random matrices have been unknown and requires completely new ideas due to the presence of a large number of zeros. We consider a sparse random matrix with entries of the form $\xi_{i,j} \delta_{i,j}, \, i,j=1,\ldots,n$, such that $\xi_{i,j}$ are i.i.d. with zero mean and unit variance and $\delta_{i,j}$ are i.i.d. Ber$(p_n)$, where $p_n \downarrow 0$ as $n \to \infty$. For $p_n < \frac{\log n}{n}$, this matrix becomes non-invertible, and hence its condition number equals infinity, with probability tending to one. In this talk, I will describe our work showing that the condition number of such sparse matrices (under certain assumptions on the moments of $\{\xi_{i,j}\}$) is $O(n^{1+o(1)})$ for all $p_n > \frac{\log n}{n}$, with probability tending to one, thereby establishing the optimal analogous version of the von Neumann’s conjecture on condition number for sparse random matrices.

Venue: LH-1, Mathematics Department

In this talk we discuss a formulation of Quantum Theory of Dark matter and discuss some operators on Hilbert spaces of singular measures.

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Title: For good measure ... and bad

Speaker: Malabika Pramanik (UBC, Canada)

Date: Wed, 02 Aug 2017

Time: 3:30 pm

Venue: LH-1, Mathematics Department

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Title: PhD Thesis colloquium: The Pick--Nevanlinna interpolation problem: complex-analytic methods in special domains

Speaker: Vikramjit Singh Chandel (IISc Mathematics)

Date: Mon, 17 Jul 2017

Time: 3 pm

Venue: LH-1, Mathematics Department

The Pick–Nevanlinna interpolation problem in its fullest generality is as follows:

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Title: The representation homology of spaces

Speaker: Ajay Ramadoss (Indiana University, USA)

Date: Wed, 12 Jul 2017

Time: 4 pm

Venue: LH-1, Mathematics Department

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Title: PhD Thesis colloquium: Analytic models, dilations, wandering subspaces and inner functions

Speaker: Monojit Bhattacharjee (IISc Mathematics)

Date: Mon, 03 Jul 2017

Time: 4 pm

Venue: LH-1, Mathematics Department

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Title: PhD Thesis colloquium: Homogeneous operators

Speaker: Somnath Hazra (IISc Mathematics)

Date: Mon, 19 Jun 2017

Time: 2 pm

Venue: LH-1, Mathematics Department

It is known that the characteristic function $\theta_T$ of a homogeneous contraction $T$ with an associated representation $\pi$ is of the form

$\theta_T(a) = \sigma_{L}(\phi_a)^{*} \theta(0) \sigma_{R}(\phi_a),$

where, $\sigma_{L}$ and $\sigma_{R}$ are projective representation of the Mobius group Mob with a common multiplier. We give another proof of the ``product formula’’. Also, we prove that the projective representations $\sigma_L$ and $\sigma_R$ for a class of multiplication operators, the two representations $\sigma_{R}$ and $\sigma_{L}$ are unitarily equivalent to certain known pair of representations $\sigma_{\lambda + 1}$and$\sigma_{\lambda - 1},$` respectively. These are described explicitly.

Let $G$ be either (i) the direct product of $n$-copies of the bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic automorphism group of the polydisc $\mathbb D^n.$ A commuting tuple of bounded operators $\mathsf{T} = (T_1, T_2,\ldots ,T_n)$ is said to be $G$-homogeneous if the joint spectrum of $\mathsf{T}$ lies in $\widebar{\mathbb{D}}^n$ and $\varphi(\mathsf{T}),$ defined using the usual functional calculus, is unitarily equivalent with $\mathsf{T}$ for all $\varphi \in G.$

We show that a commuting tuple $\mathsf{T}$ in the Cowen-Douglas class of rank $1$ is $G$ - homogeneous if and only if it is unitarily equivalent to the tuple of the multiplication operators on either the reproducing kernel Hilbert space with reproducing kernel $\prod_{i = 1}^{n} \frac{1}{(1 - z_{i}\overline{w}_{i})^{\lambda_i}}$ or $\prod_{i = 1}^{n} \frac{1}{(1 - z_{i}\overline{w}_{i})^{\lambda}},$ where $\lambda,$ $\lambda_i$ , $1 \leq i \leq n,$ are positive real numbers, according as $G$ is as in (i) or (ii).

Let $\mathsf T:=(T_1, \ldots ,T_{n-1})$ be a $G$-homogeneous $(n-1)$-tuple of rank $1$ Cowen-Douglas class, where $G$ is the the direct product of $n-1$-copies of the bi-holomorphic automorphism group of the disc. Let $\hat{T}$ be an irreducible homogeneous (with respect to the bi-holomorphic group of automorphisms of the disc) operator in the Cowen-Douglas class on the disc of rank $2$. We show that every irreducible $G$ - homogeneous operator, $G$ as in (i), of rank $2$ must be of the form $(T_1\otimes I_{\widehat{H}},\ldots , T_{n-1}\otimes I_{\widehat{H}}, I_H \otimes \hat{T}).$

We also show that if $G$ is chosen to be the group as in (ii), then there are no irreducible $G$- homogeneous operators of rank $2.$

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Speaker: Sanchayan Sen (McGill University, Canada)

Date: Thu, 08 Jun 2017

Time: 3 pm

Venue: LH-1, Mathematics Department

The aim of this talk is to give an overview of some recent results in two interconnected areas:

a) Random discrete structures: One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on $n$ vertices and degree exponent $\tau>3$, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}$. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.

More generally, recent research has provided strong evidence to believe that several objects, including (i) components under critical percolation, (ii) the vacant set left by a random walk, and (iii) the minimal spanning tree, constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the Gromov-Hausdorff sense, and these limiting objects are universal under some general assumptions. We will discuss recent developments in a larger program aimed at a complete resolution of these conjectures.

Venue: LH-1, Mathematics Department

The study of weighted inequalities in Classical Harmonic Analysis started in 70’s, when B. Muckenhoupt characterised in 1972 the weights $w$ for which the Hardy–Littlewood maximal function is bounded in $L^p(w)$. At that time the question about how the operator depended on the constant associated with $w$, which we denote by $[w]_{A_p}$, was not considered (i.e., quantitative estimates) were not investigated.

From the beginning of 2000’s, a great activity has been carried out in order to obtain the sharp dependence for singular integral operators, reaching the solution of the so-called $A_2$ conjecture by T. P. H\“ytonen.

In this talk we consider operators with homogeneous singular kernels, on which we assume smoothness conditions that are weaker than the standard ones (this is why they are called rough). The first qualitative weighted estimates are due to J. Duoandikoetxea and J. L. Rubio de Francia. For the norm of these operators in the space $L^2(w)$ we obtain a quantitative estimate which is quadratic in the constant $[w]_{A_2}$.

Venue: LH-1, Mathematics Department

The Hadamard product of two matrices is formed by multiplying corresponding entries, and the Schur product theorem states that this operation preserves positive semidefiniteness.

Speaker: Kartick Adhikari (IISc Mathematics)

Date: Tue, 31 Jan 2017

Time: 11 am

Venue: LH-1, Mathematics Department

Consider the infinite Ginibre ensemble (the distributional limit of the eigenvalues of nxn random matrices with i.i.d. standard complex Gaussian entries) in the complex plane. For a bounded set $U$, let $H_r(U)$ denote the probability (hole probability) that no points of infinite Ginibre ensemble fall in the region $rU$. We study the asymptotic behavior of $H_r(U)$ as $r \to \infty$. Under certain conditions on $U$ we show that $\log H_r(U)=C_U \cdot r^4 (1+o(1))$ as $r \to \infty$. Using potential theory, we give an explicit formula for $C_U$ in terms of the minimum logarithmic energy of the set with a quadratic external field. We calculate $C_U$ explicitly for some special sets such as the annulus, cardioid, ellipse, equilateral triangle and half disk.

Moreover, we generalize the above hole probability results for a class of determinantal point processes in the complex plane.

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Title: The Space of Metric Measure Spaces

Speaker: Sayantan Maitra (IISc Mathematics)

Date: Fri, 20 Jan 2017

Time: 11 am

Venue: LH-1, Mathematics Department

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Title: Linking numbers for algebraic cycles

Speaker: Souvik Goswami (ICMAT, Madrid, Spain)

Date: Wed, 18 Jan 2017

Time: 4 pm

Venue: LH-1, Mathematics Department

In algebraic geometry the concept of height pairing (a particular example of linking numbers) of algebraic cycles lies at the confluence of arithmetic, Hodge theory and topology. In a series of two talks, I will explain the notion of Beilinson’s height pairing for cycles homologous to zero. This will bring into picture the notion of Arakelov/arithmetic intersection theory. I will give sufficient background of this theory and provide examples. Finally, I will talk about my recent work with Dr. Jose Ignacio Burgos, about a generalization of Beilinson’s height pairing for higher algebraic cycles.

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Title: Approximate controllability of an impulsive sub-diffusion equation

Speaker: Lakshman Mahato (IIIT Dharwad)

Date: Fri, 13 Jan 2017

In this talk, I shall consider an abstract Cauchy problem for a class of impulsive sub-diffusion equation. Existence and regularity of solution of the problem shall be established via eigenfunction expansion. Further, I shall establish the approximate controllability of the problem by applying unique continuation property via internal control acts on a sub-domain.

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Title: Limiting measure for TASEP with a slow bond

Speaker: Sourav Sarkar (UC Berkeley, USA)

Date: Wed, 11 Jan 2017

Time: 3 pm

Venue: LH-1, Mathematics Department

It was shown by Basu, Sidoravicius and Sly that a TASEP starting with the step initial condition, i.e., with one particle each at every nonpositive site of $\mathbb{Z}$ and no particle at positive sites, with a slow bond at the origin where a particle jumping from the origin jumps at a smaller rate $r < 1$, has an asympototic current which is strictly less than 1/4. Here we study the limiting measure of the TASEP with a slow bond. The distribution of regular TASEP started with the step initial condition converges to the invariant product Bernoulli measure with density 1/2. The slowdown due to the slow bond implies that there is a long range effect near the origin where the region to the right of origin is sparser and there is a traffic jam to the left of the slow bond with particle density higher than a half. However, the distribution becomes close to a product Bernoulli measure as one moves far away from the origin, albeit with a different density ? < 1/2 to the right of the origin and ?’ > 1/2 to the left of the origin. This answers a question due to Liggett. The proof uses the correspondence between TASEP and directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times, and the geometric properties of the maximal paths there.

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Title: Upscaling of a System of Semilinear Diffusion Reaction Equations in a Heterogeneous Medium: MultiScale Modeling and Periodic Homogenization

Speaker: Hari Shankar Mahato (University of Georgia, USA)

Date: Tue, 10 Jan 2017

Time: 11 am

Venue: LH-1, Mathematics Department

A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multiscale medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed at the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods.

In this talk, diffusion and reaction of several mobile chemical species are considered in the pore space of a heterogeneous porous medium. The reactions amongst the species are modelled via mass action kinetics and the modelling leads to a system of multispecies diffusion; reaction equations (coupled semi-linear partial differential equations) at the micro scale where the highly nonlinear reaction rate terms are present at the right hand sides of the system of PDEs, cf. [2]. The existence of a unique positive global weak solution is shown with the help of a Lyapunov functional, Schaefer’s fixed point theorem and maximal Lp-regularity, cf. [2, 3]. Finally, with the help of periodic homogenization and two-scale convergence we upscale the model from the micro scale to the macro scale, e.g. [1, 3]. Some numerical simulations will also be shown in this talk, however for the purpose of illustration, we restrict ourselves to some relatively simple 2- dimensional situations.

As an extension to the previous model, we consider the mixture of two fluids. For such models, a system of Stokes-Cahn-Hilliard equations will be considered at the micro scale in a perforated porous medium. We first explain the periodic setting of the model and the existence results. At the end homogenization of the model will be shown using some extension theorems on Sobolev spaces, two-scale convergence and periodic unfolding.

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Title: The Perron-Frobenius theorem and its application to a class of dynamical systems

Speaker: Sandeep Krishna, NCBS.

Date: Fri, 28 Oct 2016

Venue: Simons Centre, Ground Floor.

The Perron-Frobenius theorem is a powerful and useful result about the eigenvalues and eigenvectors of a non-negative matrix. I will not be proving the theorem but will instead focus on its applications. In particular, I will discuss how it can be used to understand the behaviour of a certain class of dynamical systems, namely certain systems of first-order ordinary differential equations where the couplings between variables are specified by a graph. Some familiarity with graph theory will be useful, but I will try to recapitulate all the basic concepts needed for this talk.

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Title: Eigenfunctions Seminar: Algorithmic Applications of An Approximate Version of Caratheodory's Theorem

Title: Epimorphically Preserved Semigroup Identities

Speaker: Wajih Ashraf

Date: Fri, 23 Sep 2016

Time: 4 pm

Venue: LH-1

Title: Rank-level duality of Conformal Blocks

Speaker: Swarnava Mukhopadhyay University of Maryland

Date: Mon, 25 Jul 2016

Time: 3:30 - 4:30 p.m.

Venue: Lecture Hall I, Department of Mathematics

Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra.

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Title: Rank-level duality of Conformal Blocks

Speaker: Swarnava Mukhopadhyay (Univ of Maryland, College Park)

Date: Mon, 25 Jul 2016

Time: 3:30 pm

Venue: LH-1

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Title: Boundary behaviour of the Kobayashi distance and horosphere boundary in complete hyperbolic manifolds.

Speaker: Herve Gaussier Fourier Institute, Grenoble

Date: Wed, 13 Apr 2016

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Title: PhD Thesis colloquium: : Optimal Control Problems and Homogenization

Speaker: Mr. Bidhan Chandra Sardar Ph. D. Research Student

Date: Wed, 06 Apr 2016

Time: 11.00 a.m.

Venue: Lecture Hall I,Dept of Mathematics

We study asymptotic analysis (homogenization) of second-order partial differential equations(PDEs) posed on an oscillating domain. In general, the motivation for studying problems defined on oscillating domains, come from the need to understand flow in channels with rough boundary, heat transmission in winglets, jet engins and so on. There are various methods developed to study homogenization problems namely; multi-scale expansion, oscillating test function method, compensated compactness, two-scale convergence, block-wave method, method of unfolding etc.

In the 1980s, Goldman introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F. This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra.

In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures.

We use these results to identify the quotient of the Goldman Lie algebra of a non-closed surface by its center as a sub-algebra of the first Hochschild cohomology of the fundamental group.

Time: 2:15 – 3:15 pm

Venue: LH-1, Mathematics Department

Eigenvalues and eigenvectors appear in many physical and engineering problems, beginning with the noted study by Euler in 1751 of the kinematics of rigid bodies. From a mathematical point of view, for an operator of a suitable class, acting in a vector space, its point spectrum and associated subspaces refer to its “eigenvalues and eigenvectors”, while the subspaces associated with the continuous spectrum of the operator is said to consist of “eigenfunctions”. The basic ideas will be discussed mostly through examples, in some of which a natural connection with the representation of appropriate groups lurks behind.

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Title: PhD Thesis defence: On critical points of random polynomials and spectrum of certain products of random matrices

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Title: Iwasawa theory and the Birch and Swinnerton-Dyer conjecture:

Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)

Date: Mon, 31 Aug 2015

Time: 2 pm

Venue: LH 2

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Title: Contact Structures and Contact Invariants in Heegaard Floer Homology.

Speaker: Dheeraj Kulkarni, RKMVU, Kolkata

Date: Mon, 31 Aug 2015

Time: 3:30 p.m

Venue: L-1, Department of Mathematics, IISc

There is a dichotomy of contact structures– tight vs overtwisted. Classification of overtwisted contact structures up to isotopy, due to Eliashberg, is well understood. However, there have been few results towards classification of tight contact structures. In general, given a contact structure it is difficult to know whether it is tight or not.

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Title: Iwasawa theory and the Birch and Swinnerton-Dyer conjecture:

Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)

Date: Tue, 25 Aug 2015

Time: 2 pm

Venue: LH 2

Title: Koszul duality theory for algebras.

Speaker: Anita Naolekar (ISI, Bengaluru)

Date: Wed, 12 Aug 2015

Time: 11 am

Venue: LH4

Koszul duality theory is a homological method aiming at constructing an explicit quasi-free resolution for quadratic algebras. We introduce the concepts of bar (and cobar) construction for a quadratic algebra (and coalgebra) and provide a quasi-free resolution for quadratic algebras, under certain assumptions.

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Title: Dual pairs and intertwining distributions

Speaker: Angela Pasquale (Universite de Lorraine)

Date: Mon, 10 Aug 2015

Time: 4pm

Venue: LH-1

A reductive dual pair in the group Sp(W) of isometries of a symplectic space W, over a local

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Title: Reflected Backward SDEs with General Jumps

Speaker: Youssef Oukine (Universite Cadi Ayyad, Maroc)

Date: Fri, 07 Aug 2015

Time: 3-4 pm

Venue: LH-1

In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson random measure. The reflecting process is right continuous with left limits (RCLL for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the general result we use a fixed point argument in an appropriate space. The second part of the paper is related to BSDEs with two reflecting barriers. Once more we prove existence and uniqueness of the solution of the BSDE.

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Title: Operads: An Introduction.

Speaker: Anita Naolekar (ISI, Bengaluru)

Date: Wed, 05 Aug 2015

Time: 11 am

Venue: LH4

An operad is an algebraic device which encodes a type of algebra. The classical types of algebras, ie. associative, commutative and Lie algebras give the first examples of algebraic operads. Operadic point of view has several advantages. Firstly, many results known for classical algebras, when written out in operadic language, can be applied to other types of algebras. Secondly, operadic language simplifies the statements and proofs. Thirdly, even for classical cases, operad theory has provided new results. We start with several equivalent definitions, together with examples, and few basic properties.

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Title: PhD Thesis colloquium: On critical points of random polynomials and spectrum of certain products of Ginibre matrices.

Speaker: Tulasi Ram Reddy A

Date: Fri, 24 Jul 2015

Time: 11:00 am

Venue: Lecture Hall I, Department of Mathematics

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Title: PhD Thesis defence: Mechanisation of Knot Theory

Speaker: T. V. H. Prathamesh IISc

Date: Fri, 24 Jul 2015

Time: 11:00am - 12:00pm

Venue: Lecture Hall III, Department of Mathematics

Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many way of mechanising are: (1) generating results using Automated Theorem Provers, (2) Interactive theorem proving in a Proof Assistant which involves a combination of user intervention and automation.

In the first part of this thesis, we reformulate the question of equivalence of two Links in First Order Logic using Braid Groups. This is achieved by developing a set of Axioms whose canonical model is the Infinite Braid Group. This renders the problem of distinguishing Knots and Links, amenable to implementation in First Order Logic based Automated Theorem provers. We further state and prove results pertaining to Models of Braid Axioms.

The second part of the thesis deals with formalising Knot Theory in Higher Order Logic using the Isabelle Proof Assistant. We formulate equivalence of Links in Higher Order Logic. We obtain a construction of Kauffman Bracket in the Isabelle Proof Assistant. We further obtain a machine checked proof of invariance of Kauffman Bracket.

(i) We introduce the weight of a group which has a presentation with number of relations is at most the number of generators. We prove that the number of vertices of any crystallization of a connected closed 3-manifold $M$ is at least the weight of the fundamental group of $M$. This lower bound is sharp for the 3-manifolds $\mathbb{R P}^3$ , $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$ , $S^{\hspace{.2mm}2} \times S^1$ , $\TPSS$ and $S^{\hspace{.2mm}3}/Q_8$ , where $Q_8$ is the quaternion group. Moreover, there is a unique such vertex minimal crystallization in each of these seven cases. We also construct crystallizations of $L(kq-1,q)$ with $4(q+k-1)$ vertices for $q \geq 3$ , $k \geq 2$ and $L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent result of Swartz, our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.

(ii) We present an algorithm to find certain types of crystallizations of $3$-manifolds from a given presentation $\langle S \mid R \rangle$ with $\#S=\#R=2$ . We generalize this algorithm for presentations with three generators and certain class of relations. This gives us crystallizations of closed connected orientable 3-manifolds having fundamental groups $\langle x_1,x_2,x_3 \mid x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$ with $4(m+n+k-3)+ 2\delta_n^2 + 2 \delta_k^2$ vertices for $m\geq 3$ and $m \geq n \geq k \geq 2$ , where $\delta_i^j$ is the Kronecker delta. If $n=2$ or $k\geq 3$ and $m \geq 4$ then these crystallizations are vertex-minimal for all the known cases.

(iii) We found a minimal crystallization of the standard pl K3 surface. This minimal crystallization is a ‘simple crystallization’. Using this, we present minimal crystallizations of all simply connected pl $4$-manifolds of “standard” type, i.e., all the connected sums of $\mathbb{CP}^2$ , $S^2 \times S^2$ , and the K3 surface. In particular, we found minimal crystallizations of a pair of 4-manifolds which are homeomorphic but not pl-homeomorphic.

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Title: Fast algorithms for data analysis and elliptic partial differential equations

Speaker: Sivaram Ambikasaran (NYU)

Date: Wed, 24 Jun 2015

Time: 11-12

Venue: LH1

Large-dense matrices arise in numerous applications: boundary integral formulation for elliptic partial differential equations, covariance matrices in statistics, inverse problems, radial basis function interpolation, multi frontal solvers for sparse linear systems, etc. As the problem size increases, large memory requirements, scaling as O(N^2), and extensive computational time to perform matrix algebra, scaling as O(N^2) or O(N^3), make computations impractical. I will discuss some novel methods for handling these computationally intense problems. In the first half of the talk, I will discuss my contributions to some of the new developments in handling large dense covariance matrices in the context of computational statistics and Bayesian data assimilation. More specifically, I will be discussing how fast dense linear algebra (O(N) algorithms for inversion, determinant computation, symmetric factorisation, etc.) enables us to handle large scale Gaussian processes, thereby providing an attractive approach for big data applications. In the second half of the talk, I will focus on a new algorithm termed Inverse Fast Multipole Method, which permits solving singular integral equations arising out of elliptic PDE’s at a computational cost of O(N).

Speaker: Bidyut Sanki IISc

Date: Fri, 29 May 2015

Time: 10:00am - :11:00am

Venue: Lecture Hall III, Department of Mathematics

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.

A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.

Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

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Title: PhD Thesis defence: Shortest length geodesics on closed hyperbolic surfaces

Speaker: Bidyut Sanki IISc

Date: Fri, 29 May 2015

Time: 10:00am - :11:00am

Venue: Lecture Hall III, Department of Mathematics

$|\widehat{fd\sigma}(\xi)|\leq\ C\ (1+|\xi|)^{-\frac{n-1}{2}}.$

It follows that $\widehat{fd\sigma}\in L^p(\mathbb{R}^n)$ for all $p>2n/(n-1)$ . This result can be extended to compactly supported measure on $(n-1)$-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in $\mathbb{R}^n$ under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension $0<\alpha<n$ for $p\leq 2n/\alpha$ .

In 2004, Agranovsky and Narayanan proved that if $\mu$ is a measure supported in a $C^1$ -manifold of dimension $d<n$ , then $\widehat{fd\mu}\notin L^p(\mathbb{R}^n)$ for $1\leq p\leq \frac{2n}{d}$ . We prove that the Fourier transform of a measure $\mu_E$ supported in a set $E$ of fractal dimension $\alpha$ does not belong to $L^p(\mathbb{R}^n)$ for $p\leq 2n/\alpha$ . We also study $L^p$ -asymptotics of the Fourier transform of fractal measures $\mu_E$ under appropriate conditions on $E$ and give quantitative versions of the above statement by obtaining lower and upper bounds for the following:

$\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|\xi|\leq L}|\widehat{fd\mu_E}(\xi)|^pd\xi,$ $\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{L\leq |\xi|\leq 2L}|\widehat{fd\mu_E}(\xi)|^pd\xi.$

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Title: Eigenfunctions Seminar: Linearity in perceptual space

Title: Computational Aspects of Burnside Algebras I

Speaker: Prof. Dr. Martin Kreuzer University of Passau, Germany

Date: Wed, 25 Feb 2015

Time: 2:00 p.m. - 3:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

In the second half, I will define a variant of the monomer-dimer model on planar graphs and prove that the partition function of this model can be expressed as a determinant. I will use this result to calculate various quantities of interest to statistical physicists and end with some open questions.

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Title: What is Quantum Stochastic Calculus?

Speaker: Prof K. R. Parthasarthy ISI Delhi

Date: Fri, 13 Feb 2015

Time: 4:00 - 5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

TBA

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Title: Indeterminate moment problems and growth of associated entire functions

Speaker: Prof Christian Berg University of Copenhagen

Date: Fri, 06 Feb 2015

Time: 2:00 - 3:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

Speaker: Professor A. Raghuram Indian Institute of Science Education and Research (IISER) Pune, Maharashtra, India

Date: Thu, 16 Oct 2014

Time: 4:00 pm to 5:00 pm

Venue: Lecture Hall III, Ground Floor, Department of Mathematics, IISc, Bangalore

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Title: The BNS invariant and its application to twisted conjugacy

Speaker: Prof. Parameswaran Sankaran IMSc

Date: Tue, 14 Oct 2014

Time: 3:30 - 4:30 p.m.

Venue: Lecture Hall I, Department of Mathematics

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Title: PhD Thesis defence: Compactness theorem for the spaces of distance measure spaces and Riemann surface laminations.

Speaker: Mr. Divakaran D.

Date: Tue, 14 Oct 2014

Time: 11.00 a.m. - 12.00 noon

Venue: Lecture Hall I, Department of Mathematics

Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorff distance, is a landmark theorem with many applications. We give a generalisation of this result - more precisely, we prove a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple $(X, d, μ)$ , where $(X, d)$ forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and μ is a finite Borel measure.

Venue: LH-1, Mathematics Department

The goal of these talks is to discuss the history, the circumstances, and (the main aspects of) the proof of a recent result of ours, which says: a proper holomorphic map between two nonplanar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism.

Results of this character are motivated by a remarkable theorem of H. Alexander about proper holomorphic self-maps of the Euclidean ball. The first part of the talk will focus on the meanings of the terms in the above theorem. We will give an elementary proof that the automorphism group of a bounded symmetric domain acts transitively on it. But the existence of an involutive symmetry at each point is restrictive in many other ways. For instance, it yields a complete classification of the bounded symmetric domains. This follows from the work of Elie Cartan.

Venue: LH-1, Mathematics Department

Given n non-zero real numbers v_1,…,v_n, what is the maximum possible number of subsets {i_1,…,i_k} that can have the same subset sum v_{i_1}+…+v_{i_n}? Littlewood and Offord raised the question, showed that the answer is at most 2^nlog(n)/\sqrt{n} and conjectured that the log(n) can be removed. Erdos proved the conjecture and raised more questions that have continued to attract attention, primarily relating the arithmetic structure of v_1,…,v_n and the maximum number of subsets with a given subset-sum.

In the first lecture, we shall review and prove several of these results (due to Littlewood–Offord, Erdos, Moser, Stanley, Halasz, Tao–Vu, Rudelson–Vershynin…) and show an application to random matrices.

In the second part, we start with a question in random polynomials and see how it leads to a variant of the Littlewood–Offord problem that has not been considered before.

Title: Thurston-Teichmller theory and the dynamics of grafting

Speaker: Subhojoy Gupta Caltech

Date: Wed, 16 Jul 2014

Time: 3:30 - 4:30 p.m.

Venue: Lecture Hall III, Department of Mathematics

Complex projective structures on a surface S arise in the complex-analytic description of Teichmller space. The operation of grafting parametrizes the space P(S) of such structures, and yields paths in Teichmller space called grafting rays. I shall introduce these, and describe a result concerning their asymptotic behavior. This talk shall be mostly expository.

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Title: An introduction to preferential attachment schemes

Speaker: Sunder Sethuraman University of Arizona

Date: Tue, 15 Jul 2014

Time: (1) 10:00 - 11:00 am (2) 2:00 - 3:00 pm

Venue: Lecture Hall III, Department of Mathematics

Informally, a preferential attachment scheme is a dynamic (reinforcement) process where a network is grown by attaching new vertices to old ones selected with probability proportional to their weight, a function of their degrees. In these two talks, we give an introduction to these schemes and discuss a couple of approaches to the study of the large scale degree structure of these graphs. One fruitful approach is to view the scheme in terms of branching processes. Another is to understand it in terms of Markov decompositions and fluid limits.

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Title: On shift-like automorphisms of C^k

Speaker: Sayani Bera IISc

Date: Tue, 15 Jul 2014

Time: 4:00 - 5:00pm

Venue: Lecture Hall III, Department of Mathematics

We will use transcendental shift-like automorphisms of C^k ,k>2 to construct two examples of non-degenerate entire mappings with prescribed ranges. The first example generalizes a result of Dixon-Esterle in C^2, i.e., we construct an entire mapping of C^k, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. The second example shows the existence of a Fatou-Bieberbach domain in C^k, k>2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay-Rudin.

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Title: PhD Thesis defence: A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities

Speaker: Ms. Kamana Porwal IISc

Date: Mon, 14 Jul 2014

Time: 11:00 a.m. - 12:00 p.m.

Venue: Lecture Hall III, Department of Mathematics

Unlike the partial differential equations, the solutions of variational inequalities exhibit singularities even when the data is smooth due to the existence of free boundaries. Therefore the numerical procedure of these problems based on uniform refinement becomes inefficient due to the loss of the order of convergence. A popular remedy to enhance the efficiency of the numerical method is to use adaptive finite element methods based on computable a posteriori error bounds. Discontinuous Galerkin methods play a very important role in the local mesh adaptive refinement techniques.

The main focus in this thesis has been on the derivation of reliable and efficient error bounds for the discontinuous Galerkin methods applied to elliptic variational inequalities. The variational inequalities can be split into two kinds, namely, inequalities of the first kind and the second kind. We study an elliptic obstacle problem and a Signorini contact problem in the category of the first kind, while the frictional plate contact problem in the category of the fourth order variational inequalities of second kind. The mathematical analysis of error estimation in this class of problems crucially depends on a suitable nonlinear smoothing function that enriches the smoothness of the numerical solution. Another remarkable advantage of discontinuous Galerkin methods has been realized in the applications to higher order problems. Numerical experiments support the theoretical results and exhibit optimal convergence.

Title: PhD Thesis defence: Some Problems in Multivariable Operator Theory

Speaker: Mr. Santanu Sarkar (IISc)

Date: Mon, 23 Jun 2014

Time: 11.30 a.m.

Venue: Lecture Hall I, Department of Mathematics

This thesis investigates two different types of problems in multi-variable operator theory. The first one deals with the defect sequence for a contractive tuple and the second one deals with wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in (I) and (II) below.

Venue: Lecture Hall I, Department of Mathematics

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Title: Roots of Units, or A sum worthy of Gauss

Speaker: Chandan Singh Dalwat HRI

Date: Tue, 06 May 2014

Time: 3:30 - 4:30 p.m.

Venue: Lecture Hall I, Department of Mathematics

One looks at a certain sum G involving the p-th roots of unity (where p is a prime number), called the quadratic gaussian sum. It is easy to see that G^2=p, which means that G itself is either the positive or the negative square root of p. Which one ? It took Gauss many years to find the answer and to prove the result. Since then some other proofs of this result have been given, and it has become the central example of what is called the root number of an L-function. So the result is very important.

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Title: PhD Thesis colloquium: Contractivity and Complete contractivity

Speaker: Mr. Avijit Pal (IISc)

Date: Fri, 02 May 2014

Time: 2:00 - 3:00pm

Venue: Lecture Hall I, Department of Mathematics

We study homomorphisms $\rho_{V}$($\rho_{V}(f)=\left ( \begin{smallmatrix} f(w)I_n& \sum_{i=1}^{m} \partial_if(w)V_{i} \\ 0 & f(w)I_n \end{smallmatrix}\right ), f \in \mathcal O(\Omega_\mathbf A)$) defined on $\mathcal O(\Omega_\mathbf A)$ , where $\Omega_\mathbf A$ is a bounded domain of the form: $\begin{eqnarray*} \Omega_\mathbf A & := &\{(z_1 ,z_2, \ldots, z_m) :\|z_1 A_1 +\cdots + z_mA_m \|_{\rm op} < 1\} \end{eqnarray*}$ for some choice of a linearly independent set of $n\times n$ matrices $\{A_1, \ldots, A_m\}.$

Venue: Lecture Hall I, Department of Mathematics

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Title: PhD Thesis defence: On Walkup's class of manifolds and tight triangulations

Speaker: Mr. Nitin Singh

Date: Mon, 28 Apr 2014

Venue: Lecture Hall - I, Department of Mathematics, IISc

By a triangulation of a topological space $X$, we mean a simplicial complex $K$ whose geometric carrier is homeomorphic to $X$. The topological properties of the space can be expressed in terms of the combinatorics of its triangulation. Simplicial complexes have gained in prominence after the advent of powerful computers as they are especially suitable for computer processing. In this regard, it is desirable for a triangulation to be as efficient as possible. In this thesis we study different notions of efficiency of triangulations, namely, minimal triangulations, tight triangulations and tight neighborly triangulations.

Venue: Lecture Hall I, Department of Mathematics

The aim of this thesis is to give explicit descriptions of the set of proper holomorphic mappings between two complex manifolds with reasonable restrictions on the domain and target spaces. Without any restrictions, this problem is intractable even when posed for domains in $C^n$ . We present results for special classes of manifolds. We study, broadly, two types of structure results:

I. Descriptive: Our first result is a structure theorem for finite proper holomorphic mappings between products of connected, hyperbolic open subsets of compact Riemann surfaces. A special case of our result follows from the techniques used in a classical result of Remmert and Stein adapted to the above setting. However, the presence of factors that have no boundary, or boundaries that consist of a discrete set of points, necessitates the use of alternative techniques. Specifically: we make use of a finiteness theorem of Imayoshi.

II. Rigidity: A famous theorem of H. Alexander proves the non-existence of non-injective proper holomorphic self-maps of the unit ball in $C^n,\ n > 1$ . Several extensions of this result for various classes of domains have been established since the appearance of Alexander’s result. Our first rigidity result establishes the non-existence of non-injective proper holomorphic self-maps of bounded, balanced pseudoconvex domains of finite type (in the sense of D’Angelo) in $C^n,\ n > 1$ . This generalizes a result in $C^2$ due to Coupet, Pan and Sukhov to higher dimensions. In higher dimensions, several aspects of their argument do not work. Instead, we exploit the circular symmetry and a recent result in complex dynamics by Opshtein.

Our next rigidity result is for bounded symmetric domains. We prove that a proper holomorphic map between two non-planar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. Furthermore, our proof of this result does not rely on the fine structure (in the sense of Wolf et al.) of bounded symmetric domains. Thus, we are able to apply some of our techniques to more general classes of domains. We illustrate this through a rigidity result for certain convex balanced domains whose automorphism groups are only assumed to be non-compact. For the bounded symmetric domains, our key tool is that of Jordan triple systems.

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Title: A Characterisation of the Fourier transform on the Heisenberg group

Speaker: Dr. Lakshmi Lavanya IISc

Date: Fri, 11 Apr 2014

Time: 4:00 - 5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on R^n as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. Analogously, we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group. This is a joint work with Prof. S. Thangavelu.

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Title: A Characterisation of the Fourier transform on the Heisenberg group

Speaker: Dr. Lakshmi IISc

Date: Fri, 11 Apr 2014

Time: 4:00 - 5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

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Title: On open three-manifolds

Speaker: Gerard Besson Fourier Institut, Grenoble

Date: Fri, 07 Mar 2014

Time: 3:30 - 4:30 p.m.

Venue: Lecture Hall I, Department of Mathematics

We will describe quite carefully two classes of open three-manifolds. For one of them almost nothing is known about its Riemannian geometry and we will state open questions. For the second class a theorem is available as well as a complete classification.

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Title: On the Gromov hyperbolicity of convex domains

Speaker: Herve Gaussier Fourier Institut, Grenoble

Date: Wed, 05 Mar 2014

Time: 3:30 - 4:30 p.m.

Venue: Lecture Hall I, Department of Mathematics

Title: L-4 norms of cusp forms in the level aspect

Speaker: Prof Rizwanur Khan Texas A&M university at Qatar

Date: Fri, 03 Jan 2014

Time: 3:00 - 4:00 p.m.

Venue: Lecture Hall III, Department of Mathematics

Modular forms, particularly cusp forms, are ubiquitous objects in mathematics. A natural way to understand a cusp form is to study its L-p norms, for in principle the distribution of a function can be recovered from the knowledge of its moments. In this talk I will describe a new bound for the L-4 norm of a cusp form of prime level q, as q tends to infinity. This work is joint with Jack Buttcane.

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Title: On some generalizations and applications of Eisenstein-Dumas and Schonemann Criteria

Speaker: Prof Sudesh K. Khanduja IISER, Mohali

Date: Thu, 19 Dec 2013

Time: 11:30 - 12:30 p.m.

Venue: Lecture Hall I, Department of Mathematics

Speaker: D. Manjunath (IIT, Bombay)

Date: Mon, 11 Nov 2013

Time: 2:00pm

Venue: LH-II, Department of Mathematics, IISc

A multiqueue system serves a multiclass population. Classes differ in their valuation of time. Oblivious routing in which routing is not informed by current queue status or past decisions is assumed. First, we explore the structure of the routing fractions that maximise social welfare. We then analyse the case customer are strategic and the queues have an admission price. We then argue that admission prices can be set to achieve optimal routing.

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Title: On Abhyankar's inertia conjecture

Speaker: Prof Manish Kumar ISI Bangalore

Date: Fri, 08 Nov 2013

Time: 4:00 - 5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

The conjecture is about various subgroup that can occur as the inertia subgroup of an etale Galois cover of an affine line at a point above infinity over an algebraically closed field of positive characteristic. I will explain this conjecture and mention some positive results supporting this conjecture.

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Title: Beurling-Lax-Halmos Theorem and Operator Theory

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Title: Homotopy type and volume of locally symmetric spaces

Speaker: Prof C S Aravinda TIFR Bangalore

Date: Wed, 23 Oct 2013

Time: 3:30 - 4:30 p.m.

Venue: Lecture Hall I, Department of Mathematics

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Title: Ramanujan and hypergeometric series

Speaker: Dr.K. Srinivasa Rao, FNASc.,FTNASc., Senior Professor (Retd.), IMSc

Date: Thu, 12 Sep 2013

Time: 3:00 - 4:00 p.m.

Venue: Lecture Hall III, Department of Mathematics

Title: Generalized Radon transforms in image reconstruction problems

Speaker: Prof. Gaik Ambartsoumian University of Texas at Arlington

Date: Thu, 20 Jun 2013

Time: 4:00 - 5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

Title: A new class of high order finite volume schemes

Speaker: Prof. Philip Roe University of Michigan

Date: Mon, 13 May 2013

Time: 4:00 pm- 5:00pm

Venue: Department of Mathematics, LH-I

A robust, inexpensive, general-purpose CFD algorithm of better than second-order accuracy would have transformative impact on scientific computing. This talk will describes progress toward this objective.

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Title: Geometrical treatment of geometric shock waves

Speaker: Prof. Philip Roe University of Michigan

Date: Fri, 10 May 2013

Time: 4:00 pm- 5:00pm

Venue: Department of Mathematics, LH-I

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Title: Differential Modular forms on Shimura curves over totally real fields

Speaker: Prof. Debargha Benerjee Australian National University

Date: Thu, 02 May 2013

Time: 4:00 pm- 5:00pm

Venue: Department of Mathematics, LH-III

In this talk, we will introduce the theory of differential modular forms for compact Shimura curves over totally real fields. These are the modular forms obtained by applying the arithmetic jet space functor (adjoint to the Witt vector functor) to the ring of modular forms (global sections of certain line bundle). We will show that these differential modular forms help us to detect the ordinary quaternionic abelian schemes and schemes with Frobenius lifts (local analogue of CM abelian schemes). These are also useful to understand the categorical quotient of Shimura curves modulo Hecke correspondences/isogeny. We also construct the Serre-Tate expansions of such differential modular forms as a possible alternative to the Fourier expansion maps.

Title: Languages and Generalized Code Chains.

Speaker: Prof. R. D. Giri, Nagpur University.

Date: Wed, 20 Feb 2013

Time: 3:00 pm-4:00 pm

Venue: LH-III, Department of Mathematics

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Title: On condition numbers of a basis

Speaker: Professor B. V. Limaye (IIT Bombay)

Date: Tue, 19 Feb 2013

Time: 11:30 am-12:30 pm

Venue: LH-II, Department of Mathematics

Title: Switching in biological networks

Speaker: Prof. Badal Joshi University of Minnesota

Date: Mon, 11 Feb 2013

Time: 4:00 pm- 5:00pm

Venue: Department of Mathematics, LH-I

(Bio)chemical reaction networks are used in systems biology to model gene regulatory, protein, metabolic, and other cellular networks. Since existence of multiple steady states (MSS) provides the underpinnings for switching in chemical reaction networks, it is a fundamental problem to determine which network structures permit MSS. There exist several criteria which, when satisfied, establish that a network does not permit MSS regardless of the parameter values. On the other hand, results that establish that a network does permit MSS are rare. I will describe our recent work which provides a novel approach towards solving this problem. In the second part of the talk, I will describe stochastic switching which occurs in a network of neurons which is responsible for the distinct brain states of sleep and wake and for the transitions between the two states.

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Title: Derived Geometry and Topological String Theory

Speaker: Dr. Pranav Pandit, University of Vienna

Date: Thu, 07 Feb 2013

Time: 3:30 p.m.

Venue: LH-III, Department of Mathematics

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Title: Switching in biological networks

Speaker: Prof. Badal Joshi University of Minnesota

Date: Wed, 06 Feb 2013

Time: 4:00 pm- 5:00pm

Venue: Department of Mathematics, LH-I

Date: Mon, 29 Oct 2012

Time: 11:30 am- 12:30 pm

Venue: Department of Mathematics, Lecture Hall I

We characterize the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions.

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Title: Riesz transforms associated with Heisenberg groups and Grushin operator

Speaker: Mr. Sanjay P. K. IISc, Bangalore

Date: Mon, 29 Oct 2012

Time: 11:30 am- 12:30 pm

Venue: Department of Mathematics, Lecture Hall I

Title: Pricing Credit Derivatives in a Markov Modulated Market.

Title: Introduction to Harmonic analysis on NA groups

Speaker: Dr. Pratyoosh Kumar

Date: Wed, 08 Aug 2012

Time: 4:00 pm- 5:00 pm

Venue: Department of Mathematics, Lecture Hall 1

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Title: An introduction to GKM graphs

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Let D_1,…D_n be a system of commuting, formally self-adjoint, left invariant operators on a Lie group G. Under suitable hypotheses, we show that D_1,…D_n are essentially self-adjoint on L^2(G) and admit a joint spectral resolution, and we characterize their joint L^2 spectrum as the set of eigenvalues corresponding to a class of generalized joint eigenfunctions. Moreover, in the case G is a homogeneous group and D_1,…D_n are homogeneous, we obtain L^p-boundedness results for operators of the form m(D_1,…D_n), analogous to the Mihlin-Hormander and Marcinkiewicz multiplier theorems.

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Title: Joint spectral multipliers on nilpotent Lie groups

Speaker: Prof. Alessio Martini Universitaet zu Kiel

Date: Wed, 25 Jan 2012

Time: 4:00 p.m.

Venue: Department of Mathematics, Lecture Hall I

Title: Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture--Part I

Speaker: Prof. Jean-Pierre Demailly Institut Fourier, Universite de Grenoble France

Date: Wed, 23 Nov 2011

Time: 4:00-5:00 p.m.

Venue: Department of Mathematics, Lecture Hall I

Speaker: Prof. Sunil K. Chebolu Illinois State University, U.S.A.

Date: Fri, 20 May 2011

Time: 4:00-5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

Let G be the absolute Galois group of a field that contains a primitive p-th root of unity. This is a profinite group which is a central object of study in arithmetic algebraic geometry. In joint work with Ido Efrat and Jan Minac, we have shown that a remarkably small quotient of this big group determines the entire Galois cohomology of G. As application of this result, we give new examples of profinite groups that are not realisable as absolute Galois groups of fields. I will present an overview of this work. ALL ARE CORDIALLY INVITED

                           ALL ARE INVITED

Title: Forbidden configurations, extremal set systems, and Steiner designs

Speaker: Prof. Niranjan Balachandran Caltech U.S.A.

Date: Tue, 14 Dec 2010

Time: 4:00-5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

One of the fundamental problems in Extremal Combinatorics concerns maximal set systems with forbidden subconfigurations. One such open problem concerns the conjecture due to Anstee and Sali on the order of maximal configurations with certain forbidden subconfigurations. I shall talk about some well known results, talk about the Anstee-Sali conjecture, and finally talk about some of my recent work concerning Steiner designs occuring as maximal forbidden configurations for certain natural choices of subconfigrations. This generalizes a result of Anstee and Barekat.

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Title: Hilbert spaces based on complex Hermite polynomials, and related quantizations

Speaker: Prof. Jean-Pierre Gazeau Universit Paris-7 Denis Diderot France

Date: Mon, 13 Dec 2010

Time: 4:00-5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

I will report on the existence and properties of Hilbert spaces based on different families of complex, holomorphic or not, functions, like Hermite polynomials. The resulting coherent state quantizations of the complex plane will be presented. Some interesting issues will be examined, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules.

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Title: An iterated observer method for recovering the initial state for a class of PDE's

Speaker: Prof. Marius Tucsnak Institut Elie Cartan Universite Nancy-1 France

Date: Wed, 24 Nov 2010

Time: 4:00-5:00 p.m.

Venue: Lecture Hall I, Department of Mathematics

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Title: Distributed Processing and Temporal Codes in Neuronal Networks

Speaker: Prof. Wolf Singer Director, Max Planck Institute for Brain Research Frankfurt/Main Frankfurt Institute for Advanced Studies (FIAS)

Date: Thu, 18 Nov 2010

Time: lh 1, department of mathematics indian institute of science, bangalore

Venue: 11:00

Higher cognitive functions require the coordination of large assemblies of

spatially distributed neurons in ever changing constellations. It is

proposed that this coordination is achieved through temporal coordination

of oscillatory activity in specific frequency bands. Since there is no

supra-ordinate command centre in the brain, the respective patterns of

synchronous activity self-organize, which has important implications on

concepts of intentionality and top down causation. Evidence will be

provided that synchronisation supports response selection by attention,

feature binding, subsystem integration, short-term memory, flexible

routing of signals across cortical networks and access to the work-space

of consciousness. The precision of synchronisation is in the millisecond

range, suggesting the possibility that information is encoded not only in

the co-variation of discharge rates but also in the precise timing of

individual action potentials. This could account for the high speed with

which cortical circuits can encode and process information. Recent studies

in schizophrenic patients indicate that this disorder is associated with

abnormal synchronisation of oscillatory activity in the high frequency

range (beta and gamma). This suggests that some of the cognitive deficits

Venue: Lecture Hall III, Department of Mathematics

Speaker: K. R. Arun, Research student, Department of Mathematics, IISc

Date: Tue, 11 May 2010

We derive the conservation form of equations of evolution of a front propagating in three dimensions. We obtain a system of six conservation laws, known as 3-D kinematical conservation laws (KCL) in a ray coordinate system. The conservative variables of 3-D KCL are also constrained by a stationary vector constraint, known as geometric solenoidal constraint, which consists of three divergence-free type conditions. The 3-D KCL is an under-determined system, and therefore, additional closure relations are required to get a complete set of equations. We consider two closure relations for 3-D KCL: (1) energy transport equation of a weakly nonlinear ray theory (WNLRT) to study the propagation of a nonlinear wavefront, (2) transport equations of a shock ray theory (SRT) to study the propagation of a curved weak shock front. In both the cases we obtain a weakly hyperbolic system of balance laws. For the numerical simulation we use a high-resolution semi-discrete central scheme. The second order accuracy of the scheme is based on MUSCL type reconstructions and TVD Runge-Kutta time stepping procedures. A constrained transport technique is used to enforce the geometric solenoidal constraint and in all the test problems considered, the constraint is satisfied up to very high accuracy. We present the results of extensive numerical experiments, which confirm the efficiency and robustness of the method and also its ability to capture many physically realistic features of the fronts.

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Title: Some aspects of the Kobayashi and Caratheodary metrics on pseudoconvex domains.

Speaker: Dr. Harish Seshadri, IISc.

Date: Tue, 29 Dec 2009

Time: 10:00 a.m.

Venue: Lecture Hall - I, Dept. of Mathematics

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Title: String Topology

Speaker: Prof. Dennis Sullivan, Stony Brook.

Date: Mon, 21 Dec 2009

Time: 4:00 p.m.

Venue: Lecture Hall - I, Dept. of Mathematics

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Title: Can Computers do Mathematics? Ventures in Topology/Geometry

Speaker: Dr. Siddhartha Gadgil, IISc.

Date: Mon, 21 Dec 2009

Time: 4:00 p.m.

Venue: Lecture Hall - I, Dept. of Mathematics

Department of Mathematics

INDIAN INSTITUTE OF SCIENCE

Seminars

By Year

Title: Algebra & Combinatorics Seminar: Equivariant Cartan Homotopy formula

Speaker: Safdar Quddus (IISc Mathematics)

Date: Fri, 20 Mar 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

Title: Algebra & Combinatorics Seminar: Projective representations of Heisenberg groups

Speaker: Sumana Hatui (IISc Mathematics)

Date: Fri, 13 Mar 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

Title: Geometry & Topology Seminar: The two-systole of stretched positive scalar curvature metric on S^2xS^2

Speaker: Thomas Richard (Université Paris-Est Créteil, France)

Date: Wed, 11 Mar 2020

Time: 04:00 pm

Venue: LH-1, Mathematics Department

Title: MS Thesis defence: Determinantal processes and stochastic domination

Speaker: Raghavendra Tripathi (IISc Mathematics and University of Washington)

Date: Tue, 10 Mar 2020

Time: 11 am

Venue: LH-1, Mathematics Department

Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Tue, 10 Mar 2020

Time: 3 pm

Venue: LH-1, Department of Mathematics

Title: Geometry & Topology Seminar: Approximation of weak G-bundles in high dimensions

Speaker: Swarnendu Sil (ETH Zurich)

Date: Mon, 09 Mar 2020

Time: 4:00 pm

Venue: LH-1, Mathematics Department

Title: Algebra & Combinatorics Seminar: Formulas for the Gross-Stark units

Speaker: Matthew Honnor (King's College London, UK)

Date: Fri, 06 Mar 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

Title: Eigenfunctions Seminar: On the decimal expansion of log(2020/2019) and $e$

Speaker: Yann Bugeaud (IRMA, Université de Strasbourg, France)

Date: Fri, 28 Feb 2020

Time: 3 – 5 pm (with a 15 minute break at 3:45)

Venue: LH-5, Mathematics Department

Title: Algebra & Combinatorics Seminar: Joint extreme values of L functions (Joint with APRG Seminar)

Speaker: Akshaa Vatwani (IIT, Gandhinagar)

Date: Wed, 26 Feb 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

Title: Algebra & Combinatorics Seminar: Unit-root $L$-functions and $p$-adic differential equations

Speaker: Wansu Kim (KAIST, Daejeon, South Korea)

Date: Mon, 24 Feb 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

Title: Algebra & Combinatorics Seminar: Twisted conjugacy in linear groups over certain rings

Speaker: Parameswaran Sankaran (CMI, Chennai)

Date: Fri, 21 Feb 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

Title: Algebra & Combinatorics Seminar: Shintani descent and character sheaves on algebraic groups

Speaker: Tanmay Deshpande (TIFR, Mumbai)

Date: Fri, 07 Feb 2020

Time: 3 pm

Venue: LH-1, Mathematics Department

Title: APRG Seminar: Purely sequential estimation of a negative binomial mean with applications in ecology

Speaker: Sudeep Bapat (University of Calfornia at Santa Barbara, USA)

Date: Wed, 05 Feb 2020

Time: 2 pm

Venue: LH-1, Department of Mathematics

Title: APRG Seminar: Automorphic Forms in Analysis and Operator Theory

Speaker: Harald Upmeier (Universität Marburg, Germany; InfoSys Chair Professor, IISc)

Date: Wed, 05 Feb 2020

Time: 3 pm

Venue: LH-1, Department of Mathematics

Title: Geometry & Topology Seminar: Enumerative Geometry of the Quintic Threefold

Speaker: Ritwik Mukherjee (NISER, Bhubaneswar)

Date: Fri, 31 Jan 2020

Time: 10:00 am

Venue: LH-1, Mathematics Department

Title: Algebra & Combinatorics Seminar: An extremal problem related to weighted Davenport constant