Abstracts

Saikat Panja

Abstract:  Starting with a brief introduction to Hopf-Galois module structures and Skew braces, we will see how they are related via a simple group theoretic criterion. Although this correspondence is not bijective, it is enough to conclude about non-realizability (this will be defined in the talk). The talk will primarily serve as an exposé on this topic. Click here for the notes.

Bhargab Das

Abstract: Euler systems are certain norm-compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. These families are connected with p-adic L-functions and have consequences towards Iwasawa main conjectures. In this talk, we will discuss the properties of Euler systems with some examples. 

Sohan Ghosh

Abstract: In this talk, we will explore the topic of finite flat group schemes. We will begin by emphasizing the significance of studying these schemes, followed by some illustrative examples. If time permits, we will also look at some examples of connected étale sequences.  Click here for the notes.

Pradeep Bisht

Abstract: I will talk about toroidal Lie algebras, which are universal central extensions of the underlying loop algebras, and hence are n-variable generalization of the Kac- Moody algebras. In the later half I will recall the notion of integrable representation and discuss the classification of integrable representation with some restrictions at a glance. 

Dilip Kumar Sahoo

Abstract: In this talk we study the absolute convergence of multiple Dirichlet series using Abel summation formula. 

Muskan Bansal

Abstract:  Determining the algebraic rank of an elliptic curve has always been a difficult task. But under some conditions we can use Heegner points to supply the rational points. In this talk, we define the Heegner points and see how they are related to the first derivative of the L-functions of elliptic curves. We use this to give an application to the Birch and Swinnerton-Dyer conjecture. Click here for the notes.

Subhash Chandra

Abstract: In this talk, we discuss a novel method to construct fractal functions with graphical illustrations. Fractal measures play an important role in the theory of fractal dimensions. We also show the existence of fractal measures supported by the attractor of the iterated function system satisfying the strong separation condition. Click here for the slides.

Arpan Das

Abstract:  In this talk, I will try to give an exposition of modular representations by constructing mod p representations of SL2 over a finite field of characteristic p (a prime). This is a classical work of Brauer and Nesbitt from 1940s, but here we will use an elegant idea of Robert Steinberg (from around late 60s), motivated by highest weight theory which can be generalized to rational representations of a much broader class of groups (called Chevalley groups), mod p representations of SL2 being a concrete example. Click here  for the slides

Amit Patra

Abstract:  In this talk, we will discuss some basic definitions, examples, theorems etc. of derivation modules. Click here for slides. 

Aparna Tripathi

Abstract: The real numbers whose base b-expansion contains every string of finite sequence in {0,1,...,b-1} with the frequency 1/ b^(length of the sequence) are called normal numbers. Emile Borel proved in 1909 that 'almost all' real numbers are normal numbers; and conjectured, "algebraic irrational numbers are normal numbers". In the talk we shall briefly discuss some results in the direction of the above conjecture, majorly about the work by Mahler and its extensions. In 1973, Mahler proved that for a given algebraic irrational $alpha$ in it's base b-expansion, there exists an integer X ('with some bound') such that any given finite sequence in {0,1,...,b-1} appears infinitely often in the fractional part of X $alpha$ . There are results on improvements of bounds of integer X with respect to length of given finite sequence. We shall discuss the quantitative extension of Mahler's Theorem i.e in terms of frequency, done in 2017, by N. K. Meher, K. Senthil Kumar, R. Thangadurai; and a recent development in the result. Click here for slides.

Swapnil Raghunath Sanap

Abstract:  Since the dawn of time, human beings have asked some fundamental questions: who are we? why are we here? is there life after death ? unable to talk about any of these, in this talk I will discuss about existence of fundamental solution of differential operator. Click here  for the slides.

Oais Ahmad Bhat

Abstract : Let b ≥ 2 be an integer. The b-ary expansion of every rational number is eventually periodic, but what can be said about the b-ary expansion of an irrational algebraic number? This question was addressed for the first time by ´Emile Borel, who made the conjecture that such an expansion should satisfy some of the same laws as do almost all real number. Let b ≥ 2 be an integer. We prove that the b-ary expansion of every irrational algebraic number cannot have low complexity. Our main tool is a new, combinatorial transcendence criterion. Click here for slides

Pallabi Manna

Abstract : In this talk I will discuss about the Super graphs based on the different equivalence relations on a group G. I will focus on the various graph theoretic properties for this graph. Click here for slides.

Jitendra Rathore

Abstract : In this talk, we will discuss a certain homology theory for associative algebras known as Hochschild homology. We will begin with the definition and certain examples. We will then explore the relationship of Hochschild homology groups with other important invariants attached to associative algebras. Later, we will also discuss its topological counterpart, known as topological Hochschild homology. Click here for the notes.