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

Speaker: Soumitra 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.

Advertisements

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.

The Theory of Optimal Control Problems Governed by PDE

Speaker: Ramesh Chandra Sau

Date: 9th October, 2018

Time: 9:15pm – 10:15pm

 Abstract: In this talk, I will give a brief introduction to the optimal control problem through real life examples. These optimal control problems play a very important role in the modern scientific world for example in Aerospace Engineering and Medical Sciences. Firstly, I would introduce weak formulation of an elliptic PDE. Then I will proceed to prove the existence and uniqueness of the solution of the optimal control problem for the same. A necessary and sufficient optimality condition will be derived. 

Prerequisites: Basics of  PDE and Functional  Analysis.

Area: Optimal Control Theory

Commutator of operators

Speaker: Paramita Pramanick

Date: 2nd october,2018

Time: 9:15pm-10:15pm

 

Abstract:

The commutator of two operators A and B is defined as the operator AB-BA. In this talk we will start by proving a basic theorem about commutator of matrices : a complex matrix has trace zero if and only if it is a commutator. Then we will prove Kleinecke-Shirokov theorem.If time permits we may discuss limit of commutators and distance from a commutator to the identity.

 

Prerequisites: Basic Linear Algebra and Introduction to Functional Analysis.

Fixed point theory and some of its applications

Speaker: Nidhi Rathi

Date: 18th September 2018

Time: 09:15 pm-10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract: 

We will start by looking at the classical theorem in fixed point theory — Brouwer’s Fixed point theorem. Furthermore, I will talk about its combinatorial equivalent — Sperner’s lemma and its set-theoretic equivalent — KKM lemma. We will see proofs of these equivalences. Time permitting, I would love to introduce the celebrated area of ‘Fair Division’ in Game theory and see how wonderfully it is connected to the world of topology.

Prerequisites:

Basic Analysis and Introduction to Topology (though not necessary). The talk will be self-contained and thus I highly encourage UG students to attend it.