Talks given by students during the year 2024 are listed below.

Fall 2024

EP2403: Representation of Compact Groups and the Peter-Weyl Theorem

Speaker: Ritabrata Bhattacharyya (M. Math, 2026)
Abstract: In this talk, we’ll give a short overview of Representation of Finite Groups and Haar Measure on Locally Compact Groups. Then we’ll define unitary representations and develop the language of Representation theory for Compact Groups. For a compact group $G$, we’ll give $L^2(G)$, the space of square integrable functions on $G$ a continuous action of $G$ and see how we can decompose it into simpler looking finite dimensional representations of $G$. We’ll study certain continuous functions on $G$, called “Matrix Coefficients”, which arise from various representations of $G$. Then we’ll prove the Peter Weyl Theorem which states how $L^2(G)$ decomposes and the space of matrix coefficients is dense in $L^2(G)$. As a consequence, we’ll show that for any compact hausdorff group $G$ and any neighborhood $U$ of the identity, we can find a closed normal subgroup $H$ inside $U$ so that $G/H$ can be embedded inside some general linear group $GL_n(\mathbb{C}).$
Pre-requisites: Measure Theory, Point-Set Topology. Some knowledge on Representation of Finite Groups would be helpful, but not necessary.
Video: [TBU]
Notes: [TBU]
Date and Time: Saturday, 26th October 2024, 4:00 PM - 6:30 PM (IST)
Venue: Online (Google Meet)

EP2402: Extreme Singular Values of Random Matrices

Speaker: Saraswata Sensarma (M. Math, 2026)
Abstract: Random Matrix Theory (RMT) originated in the late 1940s with Wigner and Eisenbud’s efforts to model strong interactions in heavy nuclei. A pivotal advancement occurred in 1973 when Dyson and Montgomery uncovered a connection between the zeros of the zeta function and the eigenvalues of random Hermitian matrices. Since then, RMT has developed significantly and is now connected with numerous mathematical disciplines. Traditionally, RMT examines the limiting distributions of spectral statistics as matrix size increases. However, many practical problems require precise bounds for matrices of a fixed finite size, leading to the development of non-asymptotic random matrix theory, which employs techniques from geometric functional analysis and has applications across theoretical computer science, statistics, and signal processing.
In this talk, we will explore bounds on the extremal singular values of square random matrices with independent and identically distributed (iid) entries. We derive bounds for the largest singular value using concentration inequalities and a covering argument. However, this approach is insufficient for the smallest singular value. For this, we employ geometric methods developed by Rudelson and Vershynin, which involve analyzing weighted sums of iid random variables. This will involve a brief detour into Littlewood-Offord Theory, and its recent advancements by Tao and Vu. We will also see some generalizations of these ideas to other setups. The emphasis throughout will be on the proof techniques, given their broad applicability.
Pre-requisites: Probability, Linear algebra
Video: Available here
Notes: Available here
Date and Time: Saturday, 28th September 2024, 6:00 PM - 7:30 PM (IST)
Venue: Online (Google Meet)

EP2401: On Dirichlet’s Problem on the Circle

Speaker: Nilaksh Pundir (M. Math, 2025)
Abstract: Given an $L^1$ function on a circle, it can be shown that its Poisson integral defines a harmonic function on the unit disc. This then relates to the Dirichlet problem on the circle, which asks whether a function on the circle can be “extended” to a harmonic function on the unit disc. It is known that for continuous functions, this gives a complete solution to the Dirichlet problem. In the case of $L^1$ functions, we will see that convergence of the Poisson integral depends on the path you take to reach the boundary. We will also define the Hardy-Littlewood maximal function on the circle and study the method of maximal functions.
Pre-requisites: Basic measure theory
Video: (Not Recorded)
Date and Time: Thursday, 26th September 2024, 4:00 PM - 5:30 PM (IST)
Venue: 508, 5th floor, SN Bose Bhavan