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.
Advertisements

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.

 

Closest Unitary Matrix to a given matrix

Speaker: Poornendu Kumar

Date: 11th-September- 2018

Time: 9:15pm-10:15pm

Abstract: 

In various physics and chemistry experiments we get the measurements as vectors. And, theoretical considerations often suggest that these vectors should be orthogonal/orthonormal. One way to achieve this is to apply Gram orthogonalisation to obtain a new set of orthogonal vectors. But, often the orthogonal set of vectors obtained via Gram process is not close to the original set of vectors. This led to the problem of ‘orthonormalising’ a set of vectors without changing the vectors much. Put more precisely, we can see the set of vectors as a matrix (square) A and we may ask for the unitary matrix U which is closest to A (in operator norm sense).

Interestingly, the solution of this problem goes through the polar decomposition of the matrix A, which gives an answer but conceals much of the geometry. In this talk, I will present a proof of the fact that the closest unitary matrix to given matrix A is the one which is obtained in it’s polar decomposition. There are natural generalizations of this problem which I shall discuss if time permits.

 

Prerequisites: 

Basic Linear Algebra

(Knowing the statement of polar decomposition would be enough.)

 

Fractional power of Laplacian

Speaker: Rakesh Balhara
Date: 4th September 2018
Time: 9:15pm-10:15pm
Venue: Lecture Hall 1, Department of Mathematics
 
Abstract:
We will consider an extension problem and relate its solution to the fractional powers of the Grushin operator. The same stuff has been done for Laplacian on R^n, so we will start with Laplacian case in order to understand this technique of defining fractional power of an operator via extension problem. Further we will derive trace-Hardy inequality and subsequently Hardy’s inequality for the fractional power of Grushin operator using the extension problem technique.
Prerequisites:  Basic Functional Analysis

Introduction to Mass Transport

Speaker: Raghavendra Tripathi
Date: 28th August 2018
Place: Lecture Hall 1, Department of Mathematics
Abstract:
Theory of Mass Transport deals with optimal transportation and allocation of resources. Theory of mass transport has been a hot topic in mathematics and economics alike. Shaw prize (2018) to Caffareli and Fields Medal (2018) to A. Figalli was announced for their contribution in this area. It’s obvious that it has gained huge popularity in mathematics community.
In this talk I would introduce the idea behind the mass transport and sketch a proof of Kantorovich duality for optimal transport. This talk is intended to present a brief and simple introduction to a rather complicated and huge subject. I will not get into rigorous details of a proof, but rather focus on highlighting the connections of mass transport with other subjects and I will try to highlight the tools and tricks frequently used in mass transportation.
Prerequisites:
1. Undergraduate level mathematics
2. A little bit of measure theory would be helpful