Abstract: Testing whether a set F of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). In this work we put the problem in AM & coAM. In particular, dependence testing is unlikely to be NP-hard. Our proof method is algebro-geometric-- estimating the size of the image/preimage of the polynomial map F over the finite field.Next, we introduce a new problem called *approximate* polynomials satisfiability (APS). We show that APS is NP-hard and, using projective algebraic-geometry ideas, we put APS in PSPACE (prior best was EXPSPACE via Grobner bases). This has many unexpected applications to approximative complexity theory.

Abstract: The BSD Conjecture is a million-dollar question that relates the algebraic structure of an elliptic curve to its analytic behaviour. As a motivation we will start with a classical problem that is closely related to the conjecture. Then we will introduce the basic ingredients before stating the conjecture. We will conclude by briefly mentioning the progress made towards the conjecture.

Abstract: Weierstrass approximation theorem is a very important theorem in analysis with deep connections to topology and probability theory. One of the popular proofs of this theorem uses Bernstein polynomials; see, for example, Introduction to Topology and Modern Analysis written by George F. Simmons. However, this proof is generally presented in a rather hard-analytic fashion without giving any motivation behind it. In particular, it remains a mystery why this proof even works.

Abstract: Let A be a 2n x 2n real strictly positive matrix. Then there exists a symplectic matrix L and a positive diagonal nxn matrix D such that A=L T (D D) L. This is known as Williamson’s normal form and the diagonal entries of D are known as symplectic eigenvalues of A. We present the infinite dimensional version of this theorem. This is a joint work with Tiju Cherian John.Quantum Gaussian states on Bosonic Fock spaces are quantum versions of Gaussian distributions. They are type I quasi-free states. Here infinite mode quantum Gaussian states have been explored and to do this we need Williamson’s normal form in infinite dimensions. We extend many of the results of K R Parthasarathy from finite mode to the infinite mode setting. This include various characterizations, convexity and symmetry properties. This is a joint work with Tiju Cherian John and R. Srinivasan.

Abstract: Using the Jordan normal form, the conjugacy classes of nilpotent n x n matrices can be parametrized by partitions of n. On the other hand, these partitions also parametrize irreducible representations of the permutation group S_n. Is this merely a curious coincidence? The Springer correspondence provides a deeper geometric understanding of the above correspondence.

Abstract: This talk is about computing approximately) the “best" map between two polygons taking vertices to vertices. It arises out of a real-life problem, namely, surface registration.
Our notion of “best" is extremal quasiconformality (least angle-distortion). I will try to keep the talk as self-contained as possible. It is based on a joint-work with M. Goswami, G. Telang, and X. Gu.

Abstract: Sharp bound for Calder\'{o}n-Zygmund operators on weighted L^p-spaces, known as A_2 conjecture, was open for almost 30 years. Many individual operators met with success and finally in 2012 conjecture was proved in full generality with an intimidating proof. In 2016 "Sparse operator technique", evolved from previous attempts, provided an alternate easily accessible proof of A_2 conjecture. At present this technique is having an overwhelming impact on research activities in Fourier analysis. In this talk we will see the growth of this technique and some applications.

Abstract: In this talk, I will the describe the structure of entropy solutions of scalar conservation laws with source term.

Abstract: Komlos (1967) studied the question of non singularity of an $n \timesn$ matrix whose entries are randomly chosen to be 0 or 1. In the last decade this question has attracted a lot of attention subsequent to the work of Tao and Vu (2006). We present an overview of this topic.

Abstract: A log-gas in potential V is a system of n points on the real line with joint density \exp{-\beta H(x_1,..,x_n)} where H(x_1,...,x_n)=sum_i V(x_i) + \sum_{i0 is a fixed number. The question of interest is the behaviour of the random Variable \max{x_1,...,x_n}. When \beta=0, this reduces to the classical extreme value theory. But the story is completely different when \beta>0. For \beta=2 and V(x)=x^2, it was first solved by Tracy and Widom in a landmark paper and then generalized to \beta=2 and general V by methods of integrable systems and Riemann-Hilbert problems (Pastur-Scherbina, Deift, many others). These methods do not work for other values of \beta. In this lecture, we describe one approach to the study of the maximum by constructing tridiagonal random matrices whose eigenvalues are distributed according to the log-gas. Building upon earlier works of idea of Trotter, Dumitriu, Edelman, Sutton, Virag, Rider, Ramirez, Valko, we show that there is universality in V for every \beta. This is joint work with Brian Rider and Balint Virag. No prior knowledge of random matrix theory is necessary.

Abstract: In this talk, I will attempt to describe some of the most fundamental contributions of R. G. Douglas to Operator Theory and Operator Algebras. Apparent in much of his work was unexpected Connections between several distinct areas of mathematics. This began with his very early work connecting function theory and Toeplitz operators. Then along with Brown and Fillmore, he solved one of the problems from the list of "ten problems in Hilbert spaces" of Halmos using fundamental ideas from Algebraic topology. Over the years these ideas acquired an identity of their own and evolved into "non-commutative geometry". This was followed by finding connections between operator theory and complex geometry. This remains an active area of research today and would probably remain so for years to come.
He also initiated the use of modules in study of questions involving Hilbert spaces operators systematically.

Abstract: In this talk we plan to describe how to prove sharp Hardy inequality for fractional powers of Laplacians on Euclidean spaces and sublaplacians on Heisenberg groups using solutions of the associated extension problems.

Abstract: I'll present a new formulation of the notion of Levi cibita connection for noncommutative Riemannian manifolds and give a sufficiently general existence-uniqueness theorem. A brief sketch of the proof of the main result will be given if time permits.

Abstract: Finding solutions in integers to systems of polynomial equations in several variables and studying their properties forms the heart of number theory.
A famous example is the Pythagoras equation: the sum of two squares is again a square. One of the main methods is to study solutions over congruences modulo various integers (which will be called local), and try to recover the solutions over integers (called global).
I will begin by finding the solutions of the Pythagoras equation. I will then outline the geometric method of rational parametrization, which yields solutions of conic sections given the existence of a single solution. I will then outline a proof of a theorem of Legendre, about the existence of a non-trivial global solution for a diagonal conic provided the local solutions exist.
Finally, I will give an introduction to one of the Clay problems, which is the form of an expected solution to the local-global principle for elliptic curves.
The talk will be accessible to 1st year undergraduate students.

Abstract: After introducing the concept of controllability first for ODEs and then PDEs, I will talk about the linear viscoelastic models:
Jeffreys and Maxwell systems. The controllability of these linear models will then be discussed.

Abstract: In his first memoir, Galois gave a criterion for an irreducible equation of prime degree to be solvable by radicals. In the second memoir, he defined primitive equations and showed that if a primitive equation is solvable by radicals, then its degree is the power of a prime. His results can be reformulated in terms of extensions of fields. We will show how to extend this reformulation and parametrise all primitive solvable extensions of an arbitrary field. (An extension is called primitive if there are no intermediate extensions, and it is called solvable if the Galois group of its Galois closure is a solvable group).
All these concepts will be recalled and illustrated through examples. If time permits, we will discuss an arithmetic application.

Abstract: Defining holomorphic Cartan geometries, some of their properties as well as some recent results on them will be described.