## Determinantal processes and their properties

Speaker: Raghavendra Tripathi

Date: 29th January2019

Time: 9:15pm-10:15pm

Abstract: Determinantal processes were introduced by Macchi in 1957, although the examples of such processes were known before. In this talk, we will define determinantal processes—which are point processes whose correlation functions are given by determinants— and we will prove some nice properties such processes enjoy. In order to motivate the definition we shall also present some examples where such processes naturally arise.

Prerequisites:

1) Introductory probability theory/measure theory

2) A little bit of Functional Analysis

## The field of p-adic numbers – An introduction

Speaker: Hassain Maliyekkal
Date: 22nd January, 2019
Time: 9:15pm  – 10:15pm

Abstract: In this talk I will discuss about the construction of “p-adic numbers” from rational numbers. Note that “real numbers” are also “constructed” from rational numbers. Also i will discuss the “interesting” differences between real and p-adic numbers.

Prerequisites: None

## Finding the solutions to diophantine equation a/(b+c) + b/(a+c) + c/(a+b) = 4

Speaker: K. Hariram

Date: 20th November, 2018

Time: 9:15pm-10:15pm

Abstract:
We will go through a tour of trying to solve for positive integer solutions for the above equation. First we will see how far we can go with elementary methods. Then I will introduce elliptic curves and use them to finally get a solution.

## On von Neumann’s inequality for contractions on a Hilbert space

SpeakerSoumitra Ghara

Date: 13th November, 2018

Time: 9:15pm-10:15pm

Abstract:
In 1951, von Neumann proved that if T is a contraction on a Hilbert
space, i.e. the operator norm of T is less than or equal to 1, then for any polynomial p,
the operator norm of p(T) is less than or equal to the supremum of |p(z)| over
|z|<1. In this talk, I will start with a proof of this theorem when the Hilbert space

is finite dimensional, and then use a limiting argument to obtain a proof forthe
infinite dimensional case. Then we will see another proof of this theorem using
Sz.-Nazy’s dilation theorem (which will only be stated). If time permits, I will
also discuss some multivariate generalizations of this theorem.
Prerequisites: Linear algebra and elementary functional analysis.

## The great Picard theorem

Speaker: Amardeep Sarkar

Date: 6th November, 2018

Time: 9:15pm-10:15pm

Abstract:
In complex analysis, one of the well-known theorems is great Picard theorem. Roughly speaking, it says that a holomorphic function near an essential singularity takes each complex number infinitely many times except at most one. There are many ways to prove this result, and I am going to present a proof of this using method of conformal metrics.
Prerequisites: Basic calculus and definition of a holomorphic function in one variable.

## Algebra in Practice

Speaker: Sruthi Sekar
Date: 23rd October and 30th October, 2018.
Time: 9:15pm-10:15pm
Abstract:
Over the years, techniques from Algebra and Computational Number Theory have played an important role in fields of theoretical computer science like Cryptography, Coding Theory and Computational Complexity Theory. The blend of theory and practice is important and extremely motivated.
In this talk, I would begin by giving a quick glimpse of some common algebraic/number theoretic techniques that are used in building computer algebra systems. Then very specifically, I would move on to an application, which lies in the field of my research, Cryptography.
I will start by introducing the study of Lattices and a specific problem in Lattice that is NP-hard. Lattice based Cryptography is an extremely vibrant field, specifically as they help in building what are known as “post-quantum schemes” (i.e., Crypto-systems that would be safe even if quantum computers come into use!). I aim to give an introduction to Lattices and then describe the RSA encryption scheme, as is, followed by giving an attack to the scheme, that is based on this problem in lattice called the shortest vector problem (SVP).
**This would be a two part talk and is meant to be accessible to a general audience.
Pre-requisites: Basic linear algebra, elementary probability theory.

## Characterization of Normed Linear Spaces

Speaker: Babhrubahan Bose

Date: 16th October, 2018

Time: 9:15pm-10:15pm

Abstract:

In functional analysis, we often encounter characterization of Hilbert spaces. Two nice examples are: 1) when parallelogram law holds in a Banach space and 2) when given any two vectors of the same norm in the Banach space, there exists an isometry of the space that takes one of the points to the other. Here we are asking the next natural question, i.e., when does a vector space, equipped with a metric, becomes normed linear. It is elementary to note that if the metric is translation invariant and scaling equivariant, the metric comes from a norm. In the paper that I shall present, Peter Semrl proves that if the metric is translation invariant and every one-dimensional subspace of the vector space has and isometry with the reals, then the vector space is normed linear if the dimension is 2 or more.

Prerequisites: Familiarity with vector spaces and a little bit of metric spaces (though not necessary). The talk will be self-contained.