## 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.

## Lie Algebras and Dynkin Diagrams

Speaker: Teja G.

Dates: 13th & 20th February 2018 (Tuesday)
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics
Abstract: I will start with an introduction to Lie algebras and present some examples like $sl(2,F)$, $sl(3,F)$, $O(5,F)$ etc and general classical Lie algebras of type $A_l$, $B_l$, $C_l$, $D_l$. Then I will discuss about root systems, bases for root systems, irreducible root systems and briefly mention some results on these. Finally, I will discuss about Dynkin diagrams and construction of root systems/bases for each type, with brief comments on the semisimplicity and simplicity of the classical Lie algebras.
Prerequisites: Elementary linear algebra
Area: Lie Algebras
References:
1. Introduction to Lie Algebras and Representation Theory, by James E. Humphreys.

## Packing Polynomials and the Fueter-Polya Theorem

Speaker: Chaitanya Tappu

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

Abstract: A polynomial of two (real) variables that maps $\mathbb N \times \mathbb N$ bijectively onto $\mathbb N$ is called a packing polynomial. Cantor’s diagonal enumeration provides us with one packing polynomial, and is used to prove that rationals are countably infinite. Packing polynomials find other uses too—to store two-(multi-)dimensional arrays in linear memory. The Fueter-Polya theorem (1923) asserts that there are just two quadratic packing polynomials—Cantor’s polynomial and its reflection. In this talk I’ll present an elementary proof of this theorem due to M.A. Vsemirnov (2001).
Pre-requisites: School algebra. Knowledge of congruences/modular arithmetic would help, but can be covered in the talk if necessary.
Area: Computer Science, Number Theory
References:
1. Lew, J. S. (1981). Polynomials in two variables taking distinct integer values at lattice-points. The American Mathematical Monthly, 88(5):344–346.
2. Nathanson, M. B. (2016). Cantor polynomials and the Fueter-Polya theorem. The American Mathematical Monthly, 123(10):1001–1012. https://arxiv.org/abs/1512.08261.

## A Finite Division Ring is a Field

Speaker: Poornendu Singh

Date: 30th January 2018 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

In this talk I will present a proof of the celebrated result: “finite division ring is a field”. The talk will be mostly self-contained but basic knowledge of algebra will be assumed. If time permits, I shall also talk about where this theorem is useful and applied.

Area: Algebra

## Holomorphic Foliation on complex manifold

Speaker:  Sahil Gehlawat

Date: 23rd January 2018 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

In this talk, I’ll recall the definition of complex manifold and then define holomorphic foliation on a complex manifold. We will see some methods of finding holomorphic foliations using some basic theorems of differential geometry and workout some examples. In the end, I will give a brief introduction to singular holomorphic foliation.

Prerequisites: Basic definitions and facts from Differential Geometry and Complex Analysis.

Area: Complex Analysis, Differential Geometry

## Proofs and Applications of the CSB (Cauchy-Schwarz-Bunyakowsky) and AM-GM (arithmetic mean-geometric mean) Inequalities

Speaker:  Raghavendra Tripathi

Date: 16th January 2018 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

In this talk I will revisit two very basic gems of modern analysis, namely the CSB inequality and the AM-GM inequality. I will present an elegant proof of the AM-GM inequality which uses induction.
As a consequence of these two simple inequalities I shall prove three seemingly unrelated results. (Two from analysis (the first result concerns the positions of roots of monic polynomial with real roots; the second concerns the bounds on tangential triangle and rectangle of monic polynomials which are positive in (-1,1)) and one from Graph Theory, namely a special case of Turan’s theorem).

Area: Analysis