Introduction to Morse function and complex

Speaker: Arun Maiti

Date: 19/02/2019

Time and Venue: 9:15-10:15PM, LH-I

Abstract: A Morse function on a Riemannian manifold can be used to study topology of the manifold. In this talk we will see construction of a cellular chain complex known as the Morse complex associated to a Morse function. If time permits we will briefly talk about how geodesics can be studied using Morse theory. 

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Modular forms and associated L- functions

Speaker: Abhash Kumar Jha

Date: 12th February, 2019

Time: 9:15pm-10:15pm

 

Abstract: Modular forms are complex-valued holomorphic function defined on complex upper half plane satisfying certain transformation law with respect to the action of the SL2(Z). A modular form is uniquely determined by its Fourier coefficients. One can associate an L-function to a modular form. In this talk, we shall discuss how special values of certain L-functions appear as a Fourier coefficients of modular forms.

 

Prerequisites:

1) Basic familiarity with analysis and linear algebra

Introduction to Finite Element Method(FEM)

Speaker: Rahul Biswas

Date: 5th, February, 2019

Time: 9:15pm-10:15pm

 

Abstract:  Finite Element analysis is a numerical method to solve PDEs. FEM turns a PDE into a algebraic system which can be solved to get an approximation to the original solution of the PDE.

This will be a very basic introduction to Finite Element Method. I will discuss some abstract results which can can applied to study the existence and uniqueness of solution of elliptic PDEs. I will also discuss how FEM works with an example ( Poisson equation with zero boundary). If time permits I can also run some MATLAB codes to find discrete solution to the Poisson equation.

 

Prerequisites:

1) Basic Functional Analysis

Determinantal processes and their properties

Speaker: Raghavendra Tripathi

Date: 29th January, 2019

Time: 9:15pm-10:15pm

 

Abstract: Determinantal processes were introduced by Macchi in 1957, although the examples of such processes were known before. In this talk, we will define determinantal processes—which are point processes whose correlation functions are given by determinants— and we will prove some nice properties such processes enjoy. In order to motivate the definition we shall also present some examples where such processes naturally arise.

 

Prerequisites:

1) Introductory probability theory/measure theory

2) A little bit of Functional Analysis

The field of p-adic numbers – An introduction

Speaker: Hassain Maliyekkal

Date: 22nd January, 2019

Time: 9:15pm  – 10:15pm

Abstract: In this talk I will discuss about the construction of “p-adic numbers” from rational numbers. Note that “real numbers” are also “constructed” from rational numbers. Also i will discuss the “interesting” differences between real and p-adic numbers.

 

Prerequisites: None

Finding the solutions to diophantine equation a/(b+c) + b/(a+c) + c/(a+b) = 4

Speaker: K. Hariram

Date: 20th November, 2018

Time: 9:15pm-10:15pm

 

Abstract:

We will go through a tour of trying to solve for positive integer solutions for the above equation. First we will see how far we can go with elementary methods. Then I will introduce elliptic curves and use them to finally get a solution.

Reference: http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf

On von Neumann’s inequality for contractions on a Hilbert space

Speaker: Soumitra Ghara

Date: 13th November, 2018

Time: 9:15pm-10:15pm

Abstract:

In 1951, von Neumann proved that if T is a contraction on a Hilbert

space, i.e. the operator norm of T is less than or equal to 1, then for any polynomial p,

the operator norm of p(T) is less than or equal to the supremum of |p(z)| over
|z|<1. In this talk, I will start with a proof of this theorem when the Hilbert space

is finite dimensional, and then use a limiting argument to obtain a proof forthe

infinite dimensional case. Then we will see another proof of this theorem using

Sz.-Nazy’s dilation theorem (which will only be stated). If time permits, I will

also discuss some multivariate generalizations of this theorem.

Prerequisites: Linear algebra and elementary functional analysis.