Many moduli spaces in algebraic geometry are constructed as quotients of algebraic varieties by a reductive group action using geometric invariant theory. In this talk we explain two such examples: moduli of coherent sheaves on a projective variety and moduli of quiver representations. In both cases, we introduce and compare two stratifications: a Harder-Narasimhan stratification associated to the notion of stability for the moduli problem and a stratification coming from the geometric invariant theory construction. In nice cases, these stratifications can be used to give recursive formulas for the Betti numbers of the moduli spaces.

When can the rows and columns of a non-negative square matrix be scaled so that it becomes doubly stochastic? In 1964, Sinkhorn proposed and analyzed a natural iterative procedure that produces such a scaling when possible. In this talk, we will see this procedure and see some algorithmic and (if time permits) combinatorial Applications.

I will outline a proof of the h-cobordism theorem in these two lectures

I will outline a proof of the h-cobordism theorem in these two lectures

We will give a combinatorial proof of a product formula for the number of spanning trees of the n-dimensional hypercube. The proof we will present is a simplified version of the proof given by Bernardi.

We will prove that the second stable homotopy group of the Eilenberg Maclane space is completely determined by the Schur multiplier. Then we will discuss about the Schur multipliers of Noetherian groups. Time permitting, we will also discuss Noether's Rationality problem. All of the above will be shown as an application of a group theoretical construction.

In recent years multiple branch of mathematics have been 'quantised', in other words - made noncommutative. In this talk we will show how this noncommutation procedure is applied to the field of probability, more precisely - to stochastic calculus. We will mainly focus on the ideas leading us to quantising the notion of Wiener chaos via multiple Wiener integrals. This is joint work with Prof. J. Martin Lindsay.

http://www.math.iitb.ac.in/~seminar/algebra/watanabe-02-nov-16.pdf