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.

 

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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

The Weierstrass Factorisation Theorem

Speaker: Anwoy Maitra
Date: 21st August 2018
Time: 9:15pm  – 10:15pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:
In this talk, I will outline a proof of the Weierstrass Factorisation Theorem. This theorem, which deals with the factorisation of an entire function into a product of possibly infinitely many factors, is a cornerstone of one-variable complex analysis. Closely associated to this theorem is another fundamental theorem that assures us of the existence of holomorphic functions on arbitrary regions having zeros at prescribed points with prescribed multiplicities. These theorems have interesting applications to analytic function theory and, in particular, can be used to derive Wallis’s product formula for pi.
Prerequisites: Basics of complex analysis.
Area: Complex Analysis

Conformal Densities on Limit Sets

NOTE: THIS TALK WAS CANCELLED.

Speaker: Chaitanya Tappu

Date: 1st May 2018
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics

Abstract:

We will quickly recap the last talk (hyperbolic space, discrete group of hyperbolic motions, limit sets). We will show the construction of measures on the boundary which possess nice ergodicity properties with respect to the action of the group on the boundary. Specifically, if the group is non-elementary or has bounded fundamental domain, then the action is ergodic on the conical limit set.

Prerequisites:

I will assume familiarity with linear algebra and metric space topology. Attendance in the previous talk on 17 April would help.

 

Area: Hyperbolic geometry, Ergodic theory
References: 
  1. Nicholls, P.J. (1989). The Ergodic Theory of Discrete Groups, volume 143 of the London Mathematical Society Lecture Note Series. Cambridge University Press.

What are modular forms?

Speaker: Pramath Anamby

Date: 24th April 2018
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics

Abstract:

Modular forms are interesting number theoretic objects and are used widely to study arithmetic functions. In this talk I will briefly introduce modular forms, mainly through examples.

Prerequisites:

Basics of number theory, linear algebra and complex analysis.

 

Area: Number Theory