Glimpses of Waring-type problems
Saikat PanjaAbstract: Taking motivations from Waring's problem in number theory, there has been growing interest in Waring-type problems in several contexts, e.g. finite groups of Lie type, Lie groups, matrix algebras, polynomial algebras, Lie algebras, etc. We will discuss some recent developments in these areas. If time permits, I will mention some of my collaborative works in this direction.Universal quadratic forms and Northcott property for infinite extensions of rationals
Srijonee Shabnam ChaudhuryAbstract: In this talk, I will discuss on Northcott Property for infinite Galois extension of ℚ and I will show that how this property determines the existence of universal quadratic forms over those fields. In particular, I will talk about the following existing results:- No universal quadratic form exists over the compositum of all totally real Galois fields of a fixed prime degree over ℚ.
- Under some conditions, no classical universal quadratic form exists over the compositum of all such fields of degree 3d (For any fixed odd integer d).
I will also try to discuss some open problems on this topic.An Introduction to Knot Theory: Fundamental Problems and Knot Invariants
Amrendra SinghAbstract: This talk will be a beginner's introduction to Knot Theory. We will discuss the mathematical theory of knots and try to understand the fundamental problems of the subject. Knot invariants are central in the study of knots, and a major part of the talk will be devoted to explaining a few examples of knot invariants coming from different directions. If time allows, we can possibly discuss the connection between knots and braid groups.INTRODUCTION TO INVERSE GALOIS PROBLEM
Rohit PokhrelAbstract: In this talk, we shall give a short introduction to Emmy Noether’s Inverse Galois Theory. We will introduce the concept of Hilbertian Field and state the Hilbert irreducibility theorem. Then we shall see its relationship with the inverse Galois problem.The Hilbert-Speiser theorem
Sudipa DasAbstract: Let L/K be a finite Galois extension of number fields with its Galois group G. The well-known normal basis theorem states that L is a free module of rank 1 over the group algebra K[G]. An interesting and difficult problem is that of determining whether the ring of integers 𝒪L is free over an appropriate subring of K[G]. If 𝒪L is free of rank 1 over 𝒪K[G] then we say that 𝒪L has a normal integral basis. In this talk, we would like to present the first major result concerning the existence of normal integral basis, namely, the Hilbert-Speiser theorem. We shall also discuss some generalizations of this theorem.Geometry of Warped Product
Mohammad AqibAbstract: Warped products, a profound extension of Cartesian products, beckon us into a realm of boundless possibilities. The concept of 'warped products' was first unveiled by Bishop and O'Neill back in 1964. They introduced this elegant mathematical construct as a gateway to construct a large class of complete manifolds of negative curvature. In this talk, I will talk about relationship between geodesics and curvature within these warped products, where the interplay of base and fibers creates a tapestry of geometric intrigue.Algebraic points on graphs of functions
Gorekh Prasad SenaAbstract: The problem of counting the number of algebraic points on the graphs of functions has been intensely studied for the last few decades. General bounds for analytic functions have been obtained by Bombieri and Pila. But, as pointed out by Masser, much better bounds are valid for special functions. In fact, he proved such a bound for the Riemann zeta function. In this talk, after briefly surveying this problem, we shall discuss our results for the Weierstrass sigma function; this is a joint work with Dr. K. Senthil Kumar. Saikatul HaqueAbstract: In this talk, I planned to present ill-posedness in Fourier amalgam spaces for a nonlinear Schrödinger equation (NLS) with convolution type nonlinearity. I shall describe meanings of several types of ill-posedness. This includes mild ill-posedness, strong ill-posedness, norm inflation (NI) and NI with infinite loss of regularity. At the end, I shall present a recently published work dealing with ill-posedness in Fourier amalgam spaces. Sieve Method and it's application
Bina JhaAbstract: We estimated the number of twin primes up to a large integer N using Sieve method. We discussed about sieve method and estimated it for a smaller set. We studied a better estimation for the number of twin primes first by using Brun's inequality, which gives an upper bound including the loglog(x) term. After that, we studied how is Brun Hooley sieve method is useful for replacing the loglog(x) term by log(x).Finite set-theoretic solutions of Yang-Baxter equation and related algebraic structures
Arpan KanrarAbstract: The presentation is organized as follows. Motivation for studying solutions of the Yang-Baxter equation. Some algebraic structures related to this will be discussed such as cycle sets, brace, skew-brace, and regular subgroups of Holomorph. Finally, some interesting results will be presented.Automorphic representations of GL(2) coming from GL(1)
Sudipa MondalAbstract: Let F be a number field and Π be an automorphic representation of GL(2, 𝔸F). If Π is obtained from aHecke character of a quadratic extension E of F, then we say Π is automorphically induced from E. In this talk, we will discuss about these representations.Correspondence between 1-stably elementary matrices and stably free projective modules
Kuntal ChakrabortyAbstract: In 1992, Rao and van der Kallen proved that if R is a non-singular algebra over a perfect C1 field of dimension d ≥ 2, then a stably elementary matrix of size d+1 is an elementary matrix. In this talk we will prove some correspondence between 1-stably elementary matrices of size n over the ring Rst and stably free R-projective modules of rank n and type 1. Using this result we will explain why some condition is needed on the base field.Introduction to the character theory of finite groups
Hassain MAbstract: In this talk, we define characters of finite groups and discuss some basic examples and interesting results