List of the Talks
Talk: 12 (27.04.2023)
Title:-Primes in Short Intervals
Speaker: Aditi Dineshbhai Savalia
Abstract:- It is well-known that the number of primes up to $x$ is approximately $x/ log x$. If primes are "well-distributed" over positive integers, one would expect that in an interval $(x,x+h(x))$, there are $h(x)/ log x$ many primes. In other words, we may ask that for which functions $h$ we should expect this to hold. Well, it is not easy to answer this question, but what we can do is understand for which functions this does not hold. We will answer this using Maier's matrix method and discuss some consequences of this elegant method.
Talk: 11 (27.04.2023)
Title:-Good Reduction Criterion for Elliptic Curves
Speaker: Sudip Pandit
Abstract:- Elliptic curves are central objects in arithmetic and geometry. Let E be an elliptic curve over rational numbers. Given a prime p, it is crucial to know whether this curve remains smooth after reduction modulo p or not. If it remains smooth, we say E has a good reduction at p. In this talk, we will discuss the good reduction criterion of E in terms of the Galois representation associated with E. This is a beautiful instance where E's arithmetic and geometric properties encode each other.
Talk: 10 (19.04.2023)
Title:- PDE Problems and Diffusions
Speaker:- Priyank Oza
Abstract:- It is fascinating to observe interactions between probability theory and partial differential equations(PDEs). We will discuss how stochastic processes contribute to a better understanding of certain aspects of PDEs. We will begin by familiarizing the audience with the essential elements of stochastic processes and thus provide a relationship with PDEs.
Talk: 9 (05.04.2023)
Title:- Orthogonal Decompositions and Twisted Isometries
Speaker:- Shreema Bhatt
Abstract:- We introduce the notions of Twisted Isometries and Doubly Twisted Isometries on Hilbert Spaces. We show the existence of Orthogonal Decomposition for the above classes of isometries. We also identify tuples of isometries that admit a Wold-Von Neumann Decomposition.
Talk:8 (29.03.2023)
Title:- Surreal Numbers and Their Construction
Speaker:- Guru Sharan N
Abstract:- As Mathematicians, we at some point learn about how number systems were constructed. We will try and understand a new unconventional way of constructing a wider (much much) set of numbers. All is fun & games until we start getting confused, don’t worry, there is a definition for us to get confused as well. It all starts with a simple game, Hackenbush, which we try and win, which will lead us to a newer type of numbers. Surreal numbers they are called, which almost seem to be real. This talk will include Analysis, Combinatorics, and some puzzles. Sharpen your axes before the talk!
Talk:7 (22.03.2023)
Title:- Distribution Theory and Applications to PDE
Speaker:- Somnath Gandal
Abstract:-Distributions are an important tool in modern analysis, especially in the field of partial differential equations. Distributions are generalized functions that allow for operations, such as differentiation and convolution, on objects that fail to be functions. The talk aims to introduce the basic theory of distributions.
Talk:6 (15.03.2023)
Title:- Polynomial Ideals associated with Finite Graphs
Speaker:- Dr. Kamalesh Saha
Abstract:-We will discuss how finite graphs are associated with the ideals of polynomial rings and the motivation behind studying those ideals. We will see how the combinatorics of graphs play a crucial role in determining the primary decompositions of those ideals.
Talk:4-5 (15.02.2023, 22.02.2023)
Title:- Smooth Representation
Speaker:- Rajat Kumar Mishra
Abstract:- Smooth representations are representations defined over locally profinite groups with some extra properties. We will start with the definition of representations on any group and will try to see some of its basic properties and examples over finite groups first to get used to the concept. Then, we will define what a locally profinite groups is and try to see some examples. Next, we will define smooth representations and prove the Schur’s lemma. We will also see some examples of various types of smooth representations and try to extend them from their counter-parts in the case of finite groups. Finally, if time permits, we will discuss about the Local Langlands Correspondence (LLC), which gives us a bijection between a certain class of smooth representations of the Weil group of a local field F and smooth irreducible representations of GLn(F). For most part of the lecture (except the part about LLC), pre-requisites will be basic topology and linear algebra.
Talk:3 (01.02.2023)
Title:- The Frobenius Problem: An Algebraic Approach
Speaker:- Om Prakash
Abstract:- This problem was discussed by Georg Ferdinand Frobenius (1849-1917) in the late 1800s. The problem is finding the largest integer, which can not be written as a non-negative integral combination of the given natural numbers with one as their greatest common divisor. This integer is called the Frobenius number. This problem seems fundamental in arithmetic but is known as NP-hard. In this lecture, we will discuss the proof of the fact that the Frobenius number of a numerical semigroup is the degree of the Hilbert series (as a rational function) of the associated semigroup algebra.
Talk:2(18.01.2023)
Title:- Polyominoes and polyomino algebras
Speaker:- Dharm Veer
Abstract:- Combinatorial commutative algebra is the intersection of combinatorics and commutative algebra, and utilizes the tools of one to study the fundamental structures of the other. In this talk, we study the $K$-algbera $K[\mathcal{P}]$ (over a field $K$) associated to a polyomino $\mathcal{P}$, which is a combinatorial object. We see that some algebraic properties of $K[\mathcal{P}]$ can be related to the combinatorial properties of $\mathcal{P}$.
Talk:1(11.01.2023)
Title:-Introduction to Game Theory
Speaker:- Saraswati Girish Nanoti
Abstract:- Game theory is the study of mathematical models of strategic interactions among rational agents. It has applications in all fields of social science, as well as in logic, systems science and computer science. In this talk, we will formally define what are two player and n-player games and demonstrate some examples such as Prisoner’s dilemma. We will cover the concepts of payoff matrices, zero sum games, pure strategy and mixed strategy. We will define the concept of Pure Strategy Nash Equilibrium (PSNE) and Mixed Strategy Nash Equilibrium (MSNE) and we will cover more topics if time permits. This talk will not assume any prior knowledge of algorithms or complexity theory from the audience.
Talk:3(22.11.2022)
Title:-d-holomorphic vector bundles
Speaker:- Ayush Jaiswal
Abstract:- In 1957, Michael Atiyah had developed the theory of holomorphic connection on principle bundles over compact complex manifolds, and gave a criterion (now known as Atiyah-Weil criterion) for the existence of holomorphic connection on holomorphic vector bundles over comapct Riemann surfaces. In 1882, Felix Klein had introduced Klein surface and further analytic theory on Klein surfaces was studied in more depth by Schiffer and Spencer in 1954. Norman Alling and Newcomb Greenleaf studied correspondence between Klein surfaces and real algebraic function fields. In this talk, we will discuss the theory of d holomorphic vector bundle and if time permits we will talk about d-holomorphic connections.
Talk:1,2 (08.11.2022 & 15.11.2022)
Title:- Euler Gamma Function and its various generalizations.
Speaker:- Shivajee Gupta
Abstract:- We will define Euler’s gamma function and its various generalized and related special functions. Also, introduce its important properties like a functional equation, difference equation, reflection formula, etc. We will also look for a result on the generalized Mittag-Leffler functions and technique used in its proof.