Speaker: Babhrubahan Bose
Date: 16th October, 2018
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.
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
Speaker: Nidhi Rathi
Date: 18th September 2018
Time: 09:15 pm-10:15 pm
Venue: Lecture Hall I, Department of Mathematics
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.
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.
Speaker: Rakesh Balhara
Date: 4th September 2018
Venue: Lecture Hall 1, Department of Mathematics
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
Speaker: Raghavendra Tripathi
Date: 28th August 2018
Place: Lecture Hall 1, Department of Mathematics
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.
1. Undergraduate level mathematics
2. A little bit of measure theory would be helpful