## 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, February2019

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