Class groups and Selmer groups

Abhishek Shukla

Abstract: Class groups and Selmer groups are two important objects that appear in number theory. The class group measures how far a ring of integers of a number field is from having unique factorisation, while the Selmer group encodes information about rational points on abelian varieties, especially on elliptic curves.

In this talk, we will introduce these two groups and explain how they fit together in a common cohomology group. If time permits, we will also discuss how this connection helps in understanding the arithmetic of elliptic curves.

An overview of the anisotropic Calderón problem 

Susovan Pramanik

Abstract: In this talk, we introduce the formulation of the Calderón problem, inspired by real world inverse boundary value problems. We explain how this problem fundamentally seeks to determine the internal properties of  a medium from boundary measurements. Emphasizing the anisotropic case (which is geometric in nature), we discuss foundational results and some of our recent findings in this context. Click here for the notes.

Product of conjugacy classes in finite groups

Harish Kishnani

Abstract: Let G be a finite group and C1, C2, · · · , Ck be non-trivial conjugacy classes of G. The product of these classes defined by    C1 · C2 · · · Ck := {x1 · x2 · · · xk | xi ∈ Ci , 1 ≤ i ≤ k} is invariant under conjugation and thus a union of conjugacy classes of G. In particular, for an integer k ≥ 2, the k-th power of a conjugacy class C of G, denoted by Ck, is simply the set C · C · · · C (k-times). The main question is, for a given collection of non-trivial conjugacy classes of a finite group G, how large is the set C1 · C2 · · · Ck ? In particular, when does this set cover the entire group G? It is a well-established theme in finite group theory which is very active at the current moment. The works of Bertram, Herzog, Lev, Kaplan, Guralnick, Malle, etc. are good sources of motivation for this problem.

In this talk, I will first discuss some already known results in this direction. Then, a joint work with Dr. Rijubrata Kundu and Dr. Sumit C. Mishra will be discussed. In the symmetric group, we take the power of a conjugacy class of cycles of a fixed length and determine conditions under which it will cover the alternating group. We will provide a complete answer to both the conjectures of Herzog, Lev and Kaplan [Herzog, Marcel; Kaplan, Gil; Lev, Arieh, Covering the alternating groups by products of cycle classes. J. Combin. Theory Ser. A 115 (2008), no. 7, 1235–1245]. Click here for the notes.

On a problem of Pólya

Sudipa Das

Abstract: If α is an algebraic integer, we know that Trℚ(α)/ℚ (αi) is an integer for all i ≥ 0. It is therefore natural to ask whether the converse is true. In other words, if α is an algebraic number such that Trℚ(α)/ℚ (αi) is an integer for all i ≥ 0, must α necessarily be an algebraic integer? In 1915, G. Pólya gave an affirmative answer to this question. Later, in 1993, B. de Smit established a finite version of this result, which depends on the degree of α. In this talk, we will discuss group ring and polynomial analogues of B. de Smit’s result. If time permits, we will also touch upon a linear recurrence analogue of the problem. This is a joint work with Dr. A. Bharadwaj and Prof. R. Thangadurai. Click here for the notes.

Structure of twisted Jacquet modules for cuspidal representations of GL(n, Fq)

Himanshi Khurana

Abstract: Let G = GL(n, Fq) be the general linear group over a finite field. According to the philosophy of cusp forms, cuspidal representations are the basic building blocks from which every irreducible representation is constructed via parabolic induction. The Jacquet module, a functor adjoint to parabolic induction, is an important tool in understanding the structure of representations of G. For a cuspidal representation of G, the Jacquet module is trivial. This motivates the study of its twisted versions, referred to as twisted Jacquet modules. In this talk, we will discuss some recent results about the structure of twisted Jacquet modules. This is based on joint work with Kumar Balasubramanian and Krishna Kaipa. 

A linear independence criterion for infinite products

Oais Ahmad Bhat

Abstract: Let l ≥ 2 be an integer,  {an} be a sequence of positive integers and 𝛼1, 𝛼2, · · · , 𝛼l be real algebraic numbers greater than 1. We study the linear independence of infinite products of the form 

along with 1 over the field of algebraic numbers under some hypothesis on {an} and 𝛼1, 𝛼2, · · · , 𝛼l . This result generalizes earlier work of Hančl and Corvaja. One of the key ingredients used in the proof is the Subspace Theorem. Click here for the notes.

On the existence, nonexistence, and uniqueness of weak solutions to the nonlinear fragmentation equation

Ram Gopal Jaiswal

Abstract: In this talk, we will discuss the existence, nonexistence, and uniqueness of mass-conserving weak solutions to the continuous nonlinear fragmentation equation. This equation describes the evolution of a large system of particles undergoing binary collisions that lead to breakage into a finite or infinite number of smaller fragments. Click here for the notes.

p-adic L-functions and Perrin-Riou big logarithm

Bhargab Das

Abstract: Suppose V be a (p-adic) Galois representation. One can associate an p-adic avatar of Artin L-function to V . Perrin-Riou’s theory says that p-adic L-function can beobtained from a cohomology class in the associated Selmer group of V , via some “Big” logarithm map. We will discuss some known example of this phenomenon. Click here for the notes.

Multi-zeta values and their ℚ-Algebra structure

Guru Sharan N

Abstract: This talk will be expository in nature. We first define the multi-zeta values (MZVs) and their ℚ-Algebra structure. We will then discuss the Shuffle and Stuffle relations satisfied by the MZVs. We will discuss a few conjectures on the ℚ-Algebra and its connections with the mixed Tate motives. The content of the talk is mainly based on the works of Zagier, Gangl-Kaneko-Zagier, and Francis Brown. This talk will be built on basic concepts in Analytic number theory and Algebra. 

Riemannian maps and their Geometric Significance 

Arkadeepta Roy

Abstract: The notion of a Riemannian map, introduced by Fischer in 1992, provides a unified framework that generalizes both Riemannian immersions and Riemannian submersions, two classical constructions in differential geometry. A smooth map between Riemannian manifolds is called a Riemannian map if its differential preserves the lengths of vectors orthogonal to its kernel. This concept not only bridges the gap between immersions—where the differential is injective—and submersions—where the differential is surjective—but also extends their geometric and analytical features to a broader setting. In this talk, we will explore the geometric significance of such maps. Click here for the notes.

On a conjecture of P. Erdös 

Adarsh Mishra 

Abstract: In this talk, for any integer k ≥ 9, we prove that

v3(C(2k+1, 2k)) = |Ak| ≥ |Bk| − 1 

where v3(n) denotes the maximum power of 3 dividing the integer n, C(n, r) denotes the binomial coefficient,

Ak = { j ∈ ℕ : { 2k/ 3j} ≥ 0.5},    Bk = {0 ≤ j ≤ l : 2k = al3l + · · · + a13 + a0, ai ∈ {0, 1, 2} and aj = 2}, 

and {x} denotes the fractional part of the real number x. This is related to a conjecture of P. Erdös. We note that the lower bound

|Bk| − 1 can be computed easier than the actual divisibility by the powers of 3 of C(2k+1, 2k) . Click here for the notes.

On the Classification of Integrable Modules for τ^ (d, q)

Suman Rani

Abstract: We classify irreducible integrable modules with finite-dimensional weight spaces for toroidal Lie algebras coordinated with rational quantum tori under trivial central action. This completes the work of S. Eswara Rao and K. Zhao, who treated the nontrivial central case, by describing all such modules for the extended algebra τ^ (d, q). Click here for the notes.

Sign changes in the Fourier coefficients of automorphic forms

Amrinder Kaur

Abstract: The study of sign changes in the Fourier coefficients of automorphic forms has attracted the interest of number theorists, as it provides an understanding of their distribution and oscillatory behaviour. In this talk, we will examine recent results concerning the sign changes of Fourier coefficients of modular forms and Maass forms. We will also see how the results vary when we consider the sign changes of Fourier coefficients on sparse subsequences like the sum of squares. Click here for the notes.

Motivating Motives: Examples and Applications 

Arnab Kundu

Abstract: Grothendieck introduced the idea of a "motif" (now called a motive) in a 1964 letter to Serre as a framework to capture the common structure behind cohomology theories of algebraic varieties. In this talk with some motivation and problems, we will present basic examples and intuition, and explain how motives provide a unifying language for cohomological realizations and if time permits some arithmetic applications.