**Speaker:**Anwoy Maitra

**Date:**21st August 2018

**Time:**9:15pm – 10:15pm

**Venue**: Lecture Hall I, Department of Mathematics

**Abstract:**

**Prerequisites:**Basics of complex analysis.

**Area:**Complex Analysis

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# Author: studentseminarsiiscmath

## The Weierstrass Factorisation Theorem

**Speaker:** Anwoy Maitra
**Date:** 21st August 2018
**Time:** 9:15pm – 10:15pm
**Abstract:**
**Prerequisites:** Basics of complex analysis.
**Area: **Complex Analysis
## Conformal Densities on Limit Sets

**Date**: 1st May 2018
**Time**: 09:15 pm-10:15 pm
**Venue**: Lecture Hall I, Department of Mathematics
**Area**: Hyperbolic geometry, Ergodic theory
**References: **
## What are modular forms?

**Date**: 24th April 2018
**Time**: 09:15 pm-10:15 pm
**Venue**: Lecture Hall I, Department of Mathematics
**Area**: Number Theory
## Discrete Groups and Limit Sets

**Date**: 17th April 2018 (Tuesday)
**Time**: 09:15 pm-10:15 pm
**Venue**: Lecture Hall I, Department of Mathematics
**Area**: Hyperbolic Geometry, Ergodic Theory
**References: **
## Two Summation Formulas for Arithmetical Functions and a Few Applications

**Date**: 10th April 2018 (Tuesday)
**Time**: 09:15 pm-10:15 pm
**Venue**: Lecture Hall I, Department of Mathematics
**Area**: Number Theory
## Basics of discrete time Markov chains with finite state space

**Date**: 3rd April 2018 (Tuesday)
**Time**: 09:15 pm-10:15 pm
**Venue**: Lecture Hall I, Department of Mathematics
**Area**: Probability
## Alternating Sign Matrices (ASMs)

**Date**: 27th March 2018 (Tuesday)
**Time**: 09:15 pm-10:15 pm
**Venue**: Lecture Hall I, Department of Mathematics
**Area**: Combinatorics

**Venue**: Lecture Hall I, Department of Mathematics

In this talk, I will outline a proof of the Weierstrass Factorisation Theorem. This theorem, which deals with the factorisation of an entire function into a product of possibly infinitely many factors, is a cornerstone of one-variable complex analysis. Closely associated to this theorem is another fundamental theorem that assures us of the existence of holomorphic functions on arbitrary regions having zeros at prescribed points with prescribed multiplicities. These theorems have interesting applications to analytic function theory and, in particular, can be used to derive Wallis’s product formula for pi.

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NOTE: THIS TALK WAS CANCELLED.

**Speaker**: Chaitanya Tappu

**Abstract**:

We will quickly recap the last talk (hyperbolic space, discrete group of hyperbolic motions, limit sets). We will show the construction of measures on the boundary which possess nice ergodicity properties with respect to the action of the group on the boundary. Specifically, if the group is non-elementary or has bounded fundamental domain, then the action is ergodic on the conical limit set.

**Prerequisites:**

I will assume familiarity with linear algebra and metric space topology. Attendance in the previous talk on 17 April would help.

- Nicholls, P.J. (1989). The Ergodic Theory of Discrete Groups, volume 143 of the London Mathematical Society Lecture Note Series. Cambridge University Press.

**Speaker**: Pramath Anamby

**Abstract**:

Modular forms are interesting number theoretic objects and are used widely to study arithmetic functions. In this talk I will briefly introduce modular forms, mainly through examples.

**Prerequisites:**

Basics of number theory, linear algebra and complex analysis.

**Speaker**: Chaitanya Tappu

**Abstract**:

We will define the n-dimensional hyperbolic space. An object of study in hyperbolic geometry is the group of isometries, i.e. distance preserving transformations of the hyperbolic space, and its action on the hyperbolic space. These correspond to Mobius transformations, and their action extends onto the boundary of the hyperbolic space. The action of discrete subgroups of this group has orbits that accumulate on the limit set, which lies on the boundary. We will show the construction of measures on the limit set which possess nice ergodicity properties with respect to this action.

**Prerequisites: ** Familiarity with linear algebra and metric space topology will be assumed.

- Nicholls, P.J. (1989). The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press

**Speaker**: Ritwik Pal

**Abstract**:

Various summation formulas are very important tools in analytic number theory. We here wish to discuss two elementary (they don’t need complex analysis) summation formulas, namely, Abel’s summation formula and Dirichlet’s hyperbola method. We will also show some applications of them to estimate sums. I hope that the talk will be accessible to students who have had a course in real analysis in one variable.

**Prerequisites: **Basic real analysis

**Speaker**: Subhajit Ghosh

**Abstract**:

In this talk, I shall discuss discrete time Markov chains with finite state space briefly. Mainly, I’ll prove a version of the Perron-Frobenius theorem, which ensures the existence and uniqueness of the stationary distribution for irreducible chains; moreover, if the chain is aperiodic, then the distribution after the *n*th transition will converge to the stationary distribution.

**Prerequisites: **Basic Probability and Linear Algebra

**Speaker**: Nimisha Pahuja

**Abstract**:

ASMs are a class of square matrices in combinatorics. The formula enumerating all nxn ASMs was conjectured in 1983 and it took nearly two decades to find a bijective proof for the identity. In this talk, I will briefly go over the proof and will discuss a few of the many objects like plane partitions, fully packed loops etc. which are in bijection with ASMs. Further, I will give some of their applications.

**Prerequisites: **Basic Linear Algebra and elementary ideas about bijections and recursions.