Discrete Groups and Limit Sets

Speaker: Chaitanya Tappu

Date: 17th April 2018 (Tuesday)
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics

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.

 

Area: Hyperbolic Geometry, Ergodic Theory
References: 
  1. Nicholls, P.J. (1989). The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press
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Two Summation Formulas for Arithmetical Functions and a Few Applications

Speaker: Ritwik Pal

Date: 10th April 2018 (Tuesday)
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics

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

 

Area: Number Theory

Basics of discrete time Markov chains with finite state space

Speaker: Subhajit Ghosh

Date: 3rd April 2018 (Tuesday)
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics

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 nth transition will converge to the stationary distribution.

Prerequisites: Basic Probability and Linear Algebra

 

Area: Probability

Alternating Sign Matrices (ASMs)

Speaker: Nimisha Pahuja

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

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.

Area: Combinatorics

Identities vs. Bijections

Speaker: Abhay Jindal

Date: 20th March 2018 (Tuesday)
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics

Abstract:

This talk is based on the chaper “Identities vs. Bijection” from the celebrated book “Proofs from the Book”.
As is well known that infinite products are replete with unexpected identities, in many cases the coefficients of the monomials can be given a combinatorial meaning. And, in this talk, we would discover many such fascinating interpretations.
Area: Combinatorial Identities in Number Theory
References:
  1. Proofs from THE BOOK, by Martin Aigner and Gunter M. Ziegler.

An Introduction to the Theory of Homogenisation

Speaker: Abu Sufian

Date: 6th March 2018 (Tuesday)
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics
Abstract:

The theory of homogenisation helps us to understand multi-scale phenomena of heterogeneous materials like composite materials, porous media, thin structures etc. The main goal is to understand a complex, rapidly-varying medium by a homogeneous medium in which the micro-scale structure is averaged out in an appropriate way.

In this talk, I will first give a brief introduction to homogenisation, and then we will see some examples. This talk will be at a very basic level and I will try my best to keep the presentation self-contained.

Prerequisites: Basic knowledge of Functional Analysis
Area: Partial Differential Equations

Finding Integer Values of the Riemann Zeta Function Using Doubly Periodic Complex Functions

Speaker: K. Hariram

Date: 27th February 2018 (Tuesday)
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics
Abstract: The sin function is an example of a periodic complex function. In this talk, we will explore the space of doubly periodic functions of a complex variable, i.e., the space of all f:ℂ→ ℂ such that f(z) = f(z+ω1) = f(z+ω2) ∀z∈ℂ, for two “real”ly independent periods ω1, ω2 ∈ℂ. Finally, by letting one of these periods go to infinity, we get a method to extract values of the Riemann ζ function.
Prerequisites: Complex Analysis
Area: Functions of One Complex Variable
References:
  1. Introduction to Elliptic Curves and Modular Forms, by Neal Koblitz.