28TH July

Title: Existence and asymptotic behaviour of solution of a heat equation


Abstract: We shall discuss solution of a Hardy-Henon parabolic equation

(which is a heat equation with inverse square potential and

inhomogeneous nonlinearity) in the Lebesgue space L^q. First we shall

talk about a local existence result in super critical regime. For this,

a linear estimate for the associated heat semigroup and inhomogeneous

nonlinearity will be proved. Later, we shall also briefly discuss some

global existence and asymptotic behaviour of global solution when the

initial data satisfies certain smallness.

(ISI, Bangalore)

August 11

Title: Commutant lifting theorem on the polydisc 


Abstract: Sarason's one variable commutant lifting theorem is a key result in the theory of linear operators, complex analysis, and Hilbert function space theory, which has a stellar reputation in its application to classical results like Nevanlinna-Pick interpolation, Caratheodory-Fejer interpolation problem, Nehari interpolation problem, von Neumann inequality, isometric dilations, just to name a few. The expanded list easily includes control theory and electrical engineering. However, Sarason's lifting theorem does not hold in the setting of polydisc in general. Comprehending the subtleties of the lifting theorem on the polydisc is considered to be one of the challenging problems.

In the first half of this talk, we will provide a quick historical overview (within the span of little more than a century), present an introduction to the commutant lifting theorem, and explore how it interacts with the Nevanlinna-Pick interpolation. The second half of the talk will go over some recent advances in the commutant lifting theorem on the polydisc and its applications to interpolation and perturbation problems.




Title of the talk: Constructing Bipartite Graphs for certain Triangle Groups.


Abstract: Given three positive integers p, q, r, all >1, a triangle group Δ(p, q, r) is a group of isometries, of the Euclidean plane, or 2-sphere, 

or the hyperbolic plane, generated by the reflections in the sides of a triangle with angles π/p, π/q and π/r.

In this talk we will focus on the triangle group of isometries of the hyperbolic plane. 


After setting up heavy notations, we will describe two methods of constructing bipartite graphs 

(famously known as dessin d’enfant aka child's drawings) for finite index subgroups of the Hecke triangle group Δ(2, q, oo).

Time permitting we will end with some remarks, based on some ongoing works, about how the study triangle groups 

and associated dessin d’enfant plays a role in the modern arithmetic geometry 

via ambitious programs envisaged by Alexandre Grothendieck and Henri Darmon.



Title: Verlinde's formula and its twisted analogs


Abstract: In 1988, E. Verlinde gave a formula that predicts the dimension of the "state spaces" of a Chern-Simon theory or dimensions of conformal blocks associated with a conformal field theory. Verlinde's formula has since gotten a lot of attention from mathematicians especially in algebraic geometry as these "state spaces" or "conformal blocks" can be realized as the non-abelian generalizations of the space of classical theta functions. Verlinde's formula was proved in full generality by the combined works of Tschiya-Ueno-Yamada, Faltings, and Teleman. 

In the first part of the talk, we will define conformal blocks for a simply connected group G and discuss various aspects of the Verlinde formula and its role in studying moduli of bundles on curves, braided tensor categories, invariants of tensor product representations of a Lie group. In the second part of the talk, we will consider twisted analogs of conformal blocks and the twisted Verlinde formula. We will also discuss the appropriate connection to associated areas like twisted theta functions and the crossed modular categories. This is a joint work with Tanmay Deshpande. 



5.Anilatmaja Aryasomayajula (27 th October). 

Title: Estimates of Mumford forms


Abstract: Estimates of sections of holomorphic line bundles defined over complex manifolds is a topic of great interest in complex analytic geometry. Optimal estimates of sections of holomorphic line bundles defined over compact complex manifolds are well understood, from the works of Tian, Zeldtich, Demailly et al. However, uniform estimates in the setting of noncompact complex manifolds are difficult to obtain, and are known only in the setting of automorphic forms. 


Mumford forms are global sections of the determinant bundle of the Hodge bundle defined over the universal curve of the moduli space of compact Riemann surfaces of genus g>1. Via the Torelli map, one can think of Mumford forms as analogues of automorphic forms. In this talk we discuss estimates of Mumford forms, and bring out the comparison with automorphic forms. This is joint work with my PhD student Debasish Sadhukhan, and part of his PhD thesis. 



6. 31st Oct (Tuesday) Soumen Sarkar (IIT Chennai).  

Title: Generalized equivariant cohomology theory of weighted Grassmanns.


Abstract: Weighted Grassmanns were discussed by Corti and Reid (2020) as a generalization of Grassmanns.

They are projective varieties with orbifold singularities. In 2014 Abe and Matsumura defined them explicitly and studied their equivariant cohomology with rational coefficients. In this talk I'll give an equivalent definition of weighted Grassmanns and compute their generalized equivariant cohomology with integer coefficients. This is a joint work with Koushik Brahma.



7. T. R, Ramadas  (3 rd November). 

Title: Integrals over the $SU(2)$ character variety and lattice gauge

theory.


Abstract: Given a genus-g Riemann surface Σ, the moduli space of rank

two vector bundles with trivial determinant is, by the

Narasimhan-Seshadri Theorem, in bijection with the space of (equivalence

classes of) representations in SU(2) of the fundamental group of the

surface .


In the latter avatar, this space has a symplectic structure and a

corresponding finite measure, the Liouville measure. Normalising to

total mass one gives a probability measure. There is a natural class of

real-valued functions  parameterised by isotopy classes of loops on the

surface. These are called Wilson loop functions by physicists and

Goldman functions by mathematicians. I present a simple scheme to

compute joint distributions of these functions for families of loops.

This is possible because of the miracle of symplectic geometry called

the Duistermaat-Heckman formalism (whose applicability in this context

is due to L. Jeffrey and J. Weitsman) and a continuous analogue of the

Verlinde algebra.


The large $g$ asymptotics can be easily read off.


This leads to the second (speculative) part of the talk, which suggests

an approach to rigorous analysis of (lattice) gauge theories, as a

preliminary step to quantum-field-theoretic constructions.

8. Jacques Tilouine (10th November)

Title: Congruences and period relations for transfers, a new case.


Abstract: In a joint work in progress with K. Prasanna, we study

congruences and period relations for the transfer from GSp_4 to GL_4

of a cohomological cuspidal representation. An unexpected period occurs in the

period relations. It might be interpretable as a Beilinson regulator, 

although it is not yet done.


9. Rajaram Bhat (17th November)

Title: Peripheral Poisson boundary

Abstract

It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a

von Neumann algebra has a C*-algebra structure. This extends the notion of non-commutative Poisson boundary by including the point

spectrum of the map contained in the unit circle. The main ingredient is dilation theory. This theory provides a simple formula

for the new product. The notion has implications to our understanding of quantum dynamics. For instance, it is shown that

the peripheral Poisson boundary remains invariant in discrete quantum dynamics. This talk is based on a joint work with Samir Kar and Bharat Talwar.


10. Sneha Choubey (24th November).

Title: Distribution of spacings of real-valued sequences


Abstract: The topic on the distribution of sequences saw its light with the seminal paper of Weyl. While the classical notion of equidistribution modulo one addresses the “global” behaviour of the fractional parts of a sequence, quantities such as k-point correlations and nearest neighbour gap distributions are useful in investigating the sequence on finer scales. 

In this talk, we discuss these fine-scale statistics for real-valued arithmetic sequences, and show that the limiting distribution of the nearest neighbour gaps of real-valued lacunary sequences is Poissonian. We also prove the Poissonian behavior of the 2-point correlation function for certain classes of real-valued vector sequences. This is achieved by extrapolating conditions on the number of solutions of Diophantine inequalities using twisted moments of the Riemann zeta function. 


11. Ved Datar,  IISc Bangalore (6th December)



Title: Diameter rigidity for Kahler manifolds with positive bi-sectional curvature

Abstract: A classical result in Riemannian geometry (the so-called Myers theorem) says that the diameter of an n-dimensional Riemannian manifold with Ricci curvature bounded below by one, is bounded above by the diameter of the round sphere with (constant) Ricci curvature one. In the 1970's, Cheng proved that equality holds if and only if the Riemannian manifold is isometric to the round sphere. In the Kahler setting, the analogue of Myers' theorem, with Ricci curvature replaced by bi-sectional curvature, was proved recently by Wang-Li. In 2021, in collaboration with Harish Seshadri, we were able to establish a version of Cheng's result in the Kahler setting. Our proof relies on complex analytic techniques and was a departure from previously used methods.  In my talk, I will survey some of these recent developments. 



12. Dipendra Prasad, IIT, Bombay (8th December)



Title: Degenerate Whittaker models.

Abstract: Whittaker models have played a large role in the representation theory of

reductive groups over finite, local and global fields. They are not available for all

representations. There are the degenerate Whittaker models which are available

more generally, usually studied when there are no Whittaker models, however,

they have interest even when there is a Whittaker model. The lecture will be an

exposition of some results, some due to me and others due to others.


August 2023 semester