A proof of beurling’s theorem

Speaker: Abhay Jindal (Int. PhD)

Date: 26 October, 2021

Time: 9:15pm-10:15pm

Abstract:  Does every operator on an infinite-dimensional Hilbert space have a non-trivial invariant subspace? The question is still unanswered. A possible approach is to classify all invariant subspaces of all known operators in the hope of getting an insight that will lead to proof or a counterexample. Beurling’s theorem is a step in that direction. In this talk, we aim to prove Beurling’s theorem which gives a complete characterization of the invariant subspaces of the Hardy shift. 

Prerequisite: Familiarity with Complex Analysis and Hilbert Spaces.

Frucht’s theorem and automorphism groups of field extensions over Q

Speaker: Jayakumar.R (PhD student, IMSc Chennai)

Date: 2nd April, 2019

Time: 9:15pm-10:15pm

 

 

Abstract: In this talk, We will see the Cayley’s colour graphs and we will prove Frucht’s theorem which says “Given any finite group G there exists a connected simple graph X whose graph automorphism group is isomorphic  to G”. Then if time permits I will try to give an idea of proof of the interesting problem “Given any finite group there exists a finite extension over rational numbers whose automorphism group is isomorphic to the given group” using the Frucht’s theorem to find such extensions.

 

Prerequisite: Basic Graph Theory.

Introduction to random graph models and some percolation results

Speaker: Sanjay Kumar Jhawar

Date: 19th March, 2019

Time: 9:15pm-10:15pm

 

Abstract: This discussion consists of introduction to some random graph models. The models are bond & site percolation model on $\mathbb{Z}^d$, Erdos- Reyni model, nearest neighbor model, Boolean model, random connection model (RCM), enhanced random connection model (eRCM), inhomogeneous eRCM (IeRCM) on Poisson point process of intensity $\lambda$. We define some events on these models and that will helps us to study the percolation behaviors. In the end we state some results on non-trivial phase transition and non-percolation at criticality for the model eRCM. The Russo-Seymour-Welsh (RSW) lemma is a key ingredient for the proof of the results.

 

Prerequisite: Basic Probability Theory.

Introduction to Calculus of Variations

Speaker: Abu Sufian

Date: 12th March, 2019

Time: 9:15pm-10:15pm

 

Abstract: In this talk, we will start by studying a brief introduction to Calculus Variations or variational Calculus via some examples.  Then we will derive one of the most important equations “Euler- Lagrange ”  equation, from that we will deduce the  Beltrami identity. Using this equation and  identity, we will solve the following problems:  the geodesic problem on a plane and on  a sphere, the brachistochrone problem.  If time permits, we will see the “catenary problem” also.

 

Pre-requisites: None

Gelfand’s proof of Weiner’s theorem

Speaker: Abhay Jindal

Date: 26th February, 2019

Time: 9:15pm-10:15pm

 

Abstract: Weiner’s theorem is a simple statement which states that if the Fourier coefficients of a continuous functions f are absolutely summable, then Fourier coefficients of 1/f also is also absolutely summable provided f is non-vanishing on circle. There are many known proofs for this elagant result. One particularly interesting proof is due to Gelfand which we will describe in this talk.

 

Pre-requisites: We will keep the talk self contained. Anyone with the understanding of basic undergraduate mathematics is welcome.

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. 

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