Introduction to Root Systems

Speaker: Anantha Krishna B. (Ph.D. Student)

Date: 16th September, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: Root systems are ‘highly symmetrical sets’ of vectors in Euclidean space that arise in the study of semisimple Lie algebras. In this talk, we will introduce reduced root systems and study their basic properties. We will also discuss the classification of irreducible root systems and Dynkin diagrams. If time permits, we will also discuss affine root systems.

Prerequisites: Basic linear algebra.

The Coupon Collector Problem: variants and converse

Speaker: Sourish Parag Maniyar (Integrated Ph.D. Student)

Date: 02nd September, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: No course in discrete probability is complete without considering the coupon collector problem. It combines elementary ideas to solve a non-trivial waiting time problem. The setup is simple: given n types of coupons drawn uniformly and independently, what is the expected number of trials required to collect at least one of each? 
The talk will begin with a derivation of the solution in the classic setting. We will then explore two slightly more involved variants of this problem, such as the case of non-uniform probabilities. In the last part of the talk, we will discuss the ‘Unseen Species Problem,’ a fascinating converse in which the number of coupon types is unknown. In this setting, we will investigate how to estimate the total number of “species” from a large sample of observations, an estimation problem with implications in fields like ecology and cryptography.

Polynomial Approximation in Complex Variable

Speaker: Biplab Das (Ph.D. Student)

Date: 9th September, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: Polynomial approximation is one of the most classical and beautiful themes in complex analysis. I will begin with the classical Weierstrass theorem, which shows that continuous functions on a real interval can be approximated by polynomials, and then move to Runge’s and Mergelyan’s theorems, which extend this approximation idea to functions in one complex variable. After setting up these familiar results, I will venture into the several complex variables setting, where new phenomena appear. In particular, I will introduce CR manifolds and CR functions—objects that naturally generalize holomorphic functions to higher dimensions. The main focus of the talk will be the Baouendi–Trèves approximation theorem, a powerful result about approximating CR functions by holomorphic polynomials.

Since the proof of this theorem is quite technical, I will instead illustrate it with an example. In particular, I will show a situation where CR singularities are present, yet polynomial approximation still works.

Prerequisites: No prior knowledge beyond basic complex analysis will be assumed, and all new notions will be introduced gently along the way. My goal is to give a broad overview of this approximation theme, conveying the main ideas without heavy technicalities.

Uncertainty Principles: Old and New.

Speaker: Sujit Sakharam Damase (Ph.D. Student)

Date: 19th August, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: The phrase uncertainty principle refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be highly localized. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools from analysis, from the elementary to the very deep.

This talk will explore the concept of uncertainty principles, beginning with a review of classical results before presenting a modern, unifying framework. Drawing on recent work by Wigderson and Wigderson (2021), we will demonstrate that a broad range of uncertainty principles can be derived from a single, surprisingly simple result.

The proof of this key result relies on a fundamental property of the Fourier transform: that it and its inverse are bounded operators from the space . This elegant approach not only provides a unified perspective on many established principles but also offers a powerful tool for developing new variants. We will show how this framework can be used to prove new variants of uncertainty principles that apply to different measures of localization and to operators beyond the Fourier transform. If time permits, I will talk about uncertainty on non-commutative groups.

Prerequisites: Although I will try to keep it self-contained, familiarity with Fourier analysis would be assumed.

Fermat’s Last Theorem over F_p.

Speaker: Vyshnav V (PhD Student)

Date: 12th August, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: In 1637, Fermat conjectured that the equation has no positive integer solutions for . A common strategy in the study of Diophantine equations is to examine them locally, for example, modulo a prime to gain insight into their global behaviour. However, in 1916, Issai Schur demonstrated in his paper “Über Kongruenzen modulo p” published in the Jahresbericht der Deutschen Mathematiker-Vereinigung, that this approach is futile for Fermat’s Last Theorem. Specifically, he proved that for any sufficiently large prime , there exist nontrivial solutions to Fermat’s equation in the finite field .

This result can be established using either Fourier analysis or Ramsey theory. In this talk, we will follow Schur’s original work and establish his result through the lens of Ramsey theory, highlighting its elegant combinatorial nature.

Prerequisites: The talk will assume no background. 

On Erdős matrices and related problems.

Speaker: Raghavendra Tripathi (NYU, Abu Dhabi)

Date: 05th August, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: Marcus and Ree in 1959 showed that any doubly stochastic matrix satisfies

Erdős asked to characterize doubly stochastic matrices that saturate this inequality. We refer to such matrices as Erdős matrices. Marcus and Ree provided some partial answers, but this problem received little attention thereafter. There is a renewed interest in this problem, but so far, the complete characterization of Erdős matrices is known only in dimensions $3$ and $4$ (and, possibly $5$ by a private communication). It is known that for each $n$, there are only finitely many Erdős matrices, and these matrices have only rational entries. But beyond this, very little is known about Erdős matrices.

In this talk, I will survey the few known results on this problem. The problem of characterizing Erdős matrices has naturally led to several interesting questions in combinatorics, random matrix theory, and representation theory that are of independent interest. I will describe several of these problems — most of which are largely unexplored.

Prerequisites: The talk will assume no background beyond linear algebra and a first course in probability theory. 

Hyperbolic LEGO to Teichmüller space.

Speaker: Akash Sudhanshu (UG)

Date: 22nd April, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: Like LEGO bricks, one can construct any closed Riemann surface (except a torus) by gluing together hyperbolic trousers as the picture shows.

However, varying the lengths of trousers’ cuffs and twisting while gluing them together, changes both the geometry and the marking (a topological “labelling”) of the surface. This produces distinct (marked hyperbolic) Riemann surfaces. Remarkably, just these lengths and twists parameters completely parameterize the space of all possible marked hyperbolic structures on a closed orientable surface.

         We will start with basic hyperbolic geometry and build an example of a marked genus 2 surface step-by-step. Then, we will go on to study the parameters and define the above parametrization. This will be an introductory talk on Teichmüller theory.

Prerequisites: Familiarity with Complex analysis, fundamental group, and definition of a Riemann surface.  No prior knowledge of hyperbolic geometry is assumed.

A Tour of Surfaces

Speaker: M V Ajay Kumar Nair (PhD)

Date: 08th April, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract:  Surfaces are an important object in geometric topology. They are very simple to visualise as they are two dimensional objects- but beneath the surface (pun intended) lies a rich theory. In this talk, we will take a guided tour through the world of surfaces. We will have a proof-free survey-style talk and go through various aspects like hyperbolic structures, complex structures etc. 

Prerequisites: Some Topology and Complex Analysis

Analytic functionals and the Fantappiè transform.

Speaker: Agniva Chatterjee

Date: 01st April, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract:  Analytic functionals are elements of the topological dual space of the space of holomorphic functions on open or compact subsets of . This naturally leads to the question of whether the space of analytic functionals can be identified with certain well-known function spaces.

In the first half of the talk, after reviewing definitions and examples of various analytic functionals, we introduce the Fantappiè transform. When the underlying set is equipped with a suitable notion of convexity, the Fantappiè transform enables us to identify the space of analytic functionals with the space of holomorphic functions on a closely related set, known as the dual of the original set. In the latter half, we explore some applications of the Fantappiè transform, particularly its role in representing certain holomorphic functions as a sum of partial fractions.

Prerequisites: Familiarity with Complex analysis.

Inverse Galois Problem

Speaker: Irish Debbarma

Date: 25th March, 2025

Time: 9:15 PM-10:15 PM

Venue: LH-1

Abstract: We will discuss an age-old problem of deciding whether every finite group can be realised as the Galois Group of some field extension over the rational numbers. Initially, we will tackle finite abelian groups and handle difficult groups afterwards. We will explore variants/refinements of the problem and if time permits, try to understand why the problem is hard. 

Prerequisites: Finite Galois Theory, Algebraic Topology (Deck Transformations and Seifert-van Kampen theorem), Complex Analysis. Most of the talk will be self-contained.