Given a Noetherian Local Ring, with a maximal ideal $m$, we will present the definition and existence of Mixed Multiplicities of filtrations of $m-$ primary ideals. We will show that this mixed multiplicity for general filtrations satisfies the standard theorems and inequalities for mixed multiplicities for $m-$ primary ideals including the Minkowski inequalities. (This is joint work with Dale Cutkosky and Parangama Sarkar )

Given a Noetherian Local Ring, with a maximal ideal $m$, we will present the definition and existence of Mixed Multiplicities of filtrations of $m-$ primary ideals. We will show that this mixed multiplicity for general filtrations satisfies the standard theorems and inequalities for mixed multiplicities for $m-$ primary ideals including the Minkowski inequalities. (This is joint work with Dale Cutkosky and Parangama Sarkar )

Stochastic Approximation (SA) refers to iterative algorithms that can be used to find optimal points or zeros of a function, given only its noisy estimates. In this talk, I will review our recent advances in techniques for analysing SA methods. This talk has four major parts. In the first part, we will see a motivating application of SA to network tomography and, alongside, discuss the convergence of a novel stochastic Kaczmarz method. In the second part, we shall see a novel analysis approach for non-linear SA methods in the neighbourhood of an isolated solution. The main tools here include the Alekseev formula, which helps exactly compare the solutions of a non-linear ODE to that of its perturbation, and a novel concentration inequality for a sum of martingale differences. In the third part, we will extend the previous tool to the two timescale but linear SA setting. Here, I will also present our ongoing work to obtain tight convergence rates in this setup. In parallel, we will also see how these results can be applied to gradient Temporal Difference (TD) methods such as GTD(0), GTD2, and TDC that are used in reinforcement learning. For the analyses in the second and third parts to hold, the initial step size must be chosen sufficiently small, depending on unknown problem-dependent parameters; or, alternatively, one must use projections. In the fourth part, we shall discuss a trick to obviate this in context of the one timescale, linear TD(0) method. We strongly believe that this trick is generalizable. We also provide here a novel expectation bound. We shall end with some future directions.

I. Shafarevich has raised the following very general question. 'Is the universal covering space of every smooth connected projective variety holomorphically convex ?' This is a generalization of the famous Uniformization Theorem for Riemann Surfaces. We will discuss some applications of a positive solution of the Sharafevich question, viz. A conjecture of Madhav Nori is true, and the second homotopy group of a connected smooth projective surface is a free abelian group. We will also mention positive solutions for the Shafarevich question in several interesting cases.

In this talk, we will continue the discussion on the control of wave equation from where we stopped at the previous lecture. The observability inequality for wave equation will be proved by multiplier method.

We show how multiplicities of (not necessarily Noetherian) filtrations on a Noetherian ring can be computed from volumes of appropriate Newton Okounkov bodies. We discuss applications and examples.

We show how multiplicities of (not necessarily Noetherian) filtrations on a Noetherian ring can be computed from volumes of appropriate Newton Okounkov bodies. We discuss applications and examples.

The generators of symbolic powers of an ideal, in general, are hard to determine. A natural question is the relation between symbolic powers and ordinary powers. In this context, Bocci and Harbourne gave an asymptotic quantity called resurgence. Though this is hard to determine, in some cases it is known. In this talk, we focus on certain monomial curves. We discuss the regularity for symbolic powers and the resurgence.

The Grothendieck-Serre formula was proved by J.P.Serre in 1955. The formula expresses the difference of the Hilbert function and Hilbert polynomial of a finite graded module over a standard graded Noetherian ring, in terms of length of certain local cohomology modules. In this talk, we will look at the proof of the formula.

In this talk, we will continue the discussion on the control of wave equation from where we stopped at the previous lecture. The observability inequality for wave equation will be proved by multiplier method.