Statistics and Probability seminar
Speaker: Dr. Debapratim Banerjee (Ashoka University)
Host: Koushik Saha
Title: Some results on random matrices with dependent entries
Time, day and date: 11:00:00 AM, Tuesday, March 11
Venue: Ramanujan Hall
Abstract: In the last few years, we have seen remarkable progress on the theory of random matrices with independent entries. For example, one might consider the successful resolution of the Dyson Mehta conjectures. After the independent case being solved, recently some amount of interest has been shown in matrices with dependent entries. In this talk, we shall discuss our result on the spectral norm of Wigner matrices with dependent entries. When the entries of the matrix are correlated centered Gaussians, we show under some assumptions, the largest eigenvalue converges to the support of the limiting spectral distribution. Our result is in some sense optimal as we show counter examples (i.e. the largest eigenvalue goes beyond the support) under minor violations of the assumptions. We also have weak results in the non-centered case. Towards the end of the talk, I shall mention briefly the work I did during my PhD and my most recent research on operator limits of Wigner matrices. Due course of the talk, I shall also discuss some ongoing projects with my collaborators.

Partial Differential Equations seminar
Speaker: Anup Biswas (IISER Pune)
Host: Mayukh Mukherjee
Title: Pointwise convergence of the solutions to the initial data for the abstract heat
equation
Time, day and date: 2:30:00 PM, Tuesday, March 11
Venue: Ramanujan Hall
Abstract: In a recent study, Hartzstein, Torrea, and Viviani characterized all the weights $v$ for which the solution to the classical heat equation with initial data $f$, where $f\in L^p_v(\mathbb{R}^n)$, converges to $f$ as $t\to 0$, almost everywhere and for every $f\in L^p_v(\mathbb{R}^n)$. This work is, of course, in the spirit of Carleson’s program, where similar investigations have been conducted for the Schrödinger operators. In this talk, we will extend the results of Hartzstein et al. to a broader class of operators on metric measure spaces with a volume doubling condition, including $\phi$-nonlocal operators, mixed local-nonlocal operators, the Laplacian with a Hardy potential, the Laplacia-Beltrami operators, Laplacian on fractals and many others.

This talk is based on a recent joint work with Bhimani and Dalai.

Speaker: Dr. Devendra R, Mathematics Department, IIT Bombay

Time: 3.30 pm

Title: Gaussian channels

Abstract: In this talk, we shall discuss the three well-known definitions
of finite mode quantum Gaussian channels found in the literature. We will
rigorously establish the equivalence of all these definitions, even though
this equivalence is generally acknowledged.  As an application, we answer
some of the questions asked by Parthasarathy in Indian J Pure Appl Math
46, 419–439 (2015).

Speaker: Debsoumya Chakraborti (University of Warwick)
Host: Niranjan Balachandran
Title: Results in Extremal Combinatorics
Time, day and date: 4:30:00 PM, Tuesday, March 11
Venue: Online talk only (https://tel.meet/hqy-hswu-jho?hs=5)
Abstract: A key objective of extremal combinatorics is to investigate various conditions on combinatorial structures (such as graphs, set systems, and simplicial complexes) that guarantee the existence of specific substructures. In this talk, I will concentrate on two central topics within this theme of extremal combinatorics:
1. Tur\'an problems and
2. Embedding spanning subgraphs.
I will begin with a gentle introduction to the first topic, highlighting a few fundamental questions in the field. In this context, I will introduce the Erd\H{o}s--Sauer problem that asks for the maximum possible number of edges that an $n$-vertex graph can have without containing an $r$-regular subgraph. The problem
had seen no progress since Pyber's work in 1985 until recently when Janzer and Sudakov resolved this problem up to a multiplicative constant depending on $r$. We resolve the Erd\H{o}s--Sauer problem up to an absolute constant factor (not depending on $r$) as follows. There exists an absolute constant $C$ such that for
each positive integer $r$, every $n$-vertex graph with at least $Cr^2n\log \log n$ edges contain an $r$-regular subgraph. Moreover, we show this to be tight up to the value of $C$ for all $r\geq 3$ and $n\geq n(r)$.

Next, I will transition to the second topic, starting with two classical results on embedding the Hamilton cycle (a cycle that visits every vertex exactly once):
(i) Dirac's theorem, which establishes a sharp minimum degree condition on a graph to ensure the existence of a Hamilton cycle, and
(ii) Theorems on various orientations of Hamilton cycles in tournaments.
In the last decade, extending subgraph embedding problems to the setting of transversals over a collection of graphs has sparked significant interest in the literature. I will introduce this concept and then discuss the transversal generalizations of (i) and (ii). Some of these include results from my own work in various papers.