## 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

## Quantifying Distinctions / The Random Walk

Speaker:  Yash Mehta

Date: 14th November 2017 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

The talk will basically consist of two different solutions of the random walk problem, and one of the key techniques used in one of the proofs would be used to provide a method to estimate the number of “types” of an entity in a region. Basic proofs will be outlined along the way (some might be skipped depending on the availability of time). Actual examples of its utilisation will be discussed at the end.

Prerequisites: Basic arithmetic, elementary ideas of calculus and limits.

Area: Probability Theory

## Gromov Hausdorff convergence, ultralimits, and some applications

Speaker:  Sayantan Khan

Date: 7th November 2017 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

This talk will outline a very useful generalization of Hausdorff convergence (which will also be defined in the talk), and will involve proofs of some essential properties of this form of convergence (some proofs will be sketched, and some easy ones will be skipped). We’ll also look at ultralimits, which are a nice way of abstracting out the diagonal argument used in many proofs in analysis. Finally, we’ll have a look at some applications of the theory of Gromov Hausdorff convergence, if time permits.

Prerequisites: Basic knowledge of analysis, familiarity with compactness, and basic topology, i.e. definition of metric spaces.

Area: Topology