Let F_q be the field with q elements. We will consider the task of testing if a m-variate function f over F_q is close to being a polynomial of degree at most d. This task was originally considered in the setting where d was small compared to q and the resulting analyses found many applications in probabilistic checking of proofs. In this talk I will describe more recent results where we consider the setting where d > q. The natural test would be to pick a random subspace of dimension roughly d/q and see if the f restricted to this subspace is a polynomial of degree d. Earlier analyses had shown that this natural test rejects functions that are, say 1%, far from being a degree d polynomial, with probability at least q^{-d/q}. In this talk I will describe our improved analyses which show that this same test rejects such functions with constant probability for constant q. Time permitting I might also mention some applications where the setting of d > q is useful. Based on joint works with Arnab Bhattacharyya, Elad Haramaty, Swastik Kopparty, Noga Ron-Zewi, Grant Schoenebeck, Amir Shpilka and David Zuckerman.

This will be for the most part a survey talk. I will define Siegel modular forms, their number theoretic significance, and explain the link between Siegel modular forms of degree 2 and automorphic representations of GSp(4). I will talk about the significance of their Fourier coefficients, and describe several known results and still unproven conjectures. I will also talk about certain lifts to and from spaces of Siegel modular forms that are special cases of Langlands' general conjectures.

Quadrature domains in the plane are those on which a given test class of functions (say, the class of holomorphic functions) satisfies a generalized mean value property. The purpose of the talk will be to see how the Schwarz reflection principle leads to an understanding of these domains. The talk will be elementary.

We shall review some recent results concerning the local stabilization, around unstable steady states, of fluid flows and of fluid structure systems. In all these problems the fluid flows will be described by the incompressible Navier-Stokes equations. The control is either a control acting at the boundary of the fluid domain, or a control acting in the structure equation. We consider models in which the structure is located at the boundary of the fluid domain and described by either a damped beam equation in 2D or a plate equation in 3D. Another fluid structure model, that we consider, consists in coupling the incompressible Navier-Stokes equations with the Lame system of linear Elasticity.

In this expository talk, we give a gentle introduction to the theory of Ginzburg Landau vortices. We will mostly be talking about the two dimensional theory, because in this context complex variables methods are quite useful. This development by Bethuel Brezis and Helein paved way to the theory of weak Jacobians which proved crucial for the problem in higher dimensions. Time permitting, we will briefly describe this and some related time dependent problems.

The compact torus S^1�S^1 has a structure of Riemann surface and therefore is a complex projective manifold. On product of odd dimensional spheres S^{2p+1}�S^{2q+1 with p > 0 or q > 0, complex structures were obtained by H. Hopf (1948) and Calabi-Eckmann (1953). These complex manifolds are one of the first examples of non-K ?ahler, and hence non-projective, compact complex manifolds. The aim of this talk is to describe construction of a new class of non-Kahler compact complex manifolds. Let K be an even dimensional compact Lie group and G be its universal complexification. We will show that if a smooth K -principal bundle EK---->M over complex manifold M, is obtained after reduction of the structure group of a holomorphic G-principal bundle EG---->M, then the total space EK admits a complex structure. If K is non-abelian then the complex manifold EK will be non-Kahler. In certain special cases we will discuss Picard group, deformation and algebraic dimension of EK.This talk is based on an ongoing joint work with Mainak Poddar.

The epsilon-pseudospectrum ?(a) of an element a of an arbitrary Banach algebra A is studied. Its relationships with the spectrum and numerical range of a are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of a. Suppose for some_x000f_ epsilon > 0 and a,b \in A, ?(ax) = ?(bx) for all x \in A. It is shown that a = b if: (I) a is invertible. (ii) a is Hermitian idempotent. (iii) a is the product of a Hermitian idempotent and an invertible element. (iv) A is semisimple and a is the product of an idempotent and an invertible element. (v) A = B (X) for a Banach space X. (vi) A is a C*-algebra. (vii) A is a commutative semisimple Banach algebra.

A famous theorem of Norbert Wiener states that for a function f in L^1(R), span of translates f(x?a) with complex coefficients is dense in L^1(R) if and only if the Fourier transform of f is nonvanishing on R. That is the ideal generated by f in L^1(R) is dense in L^1(R) if and only if the Fourier transform of f is nonvanishing on R. This theorem is well known as the Wiener Tauberian theorem. This theorem has been extended to abelian groups. The hypothesis (in the abelian case) is on a Haar integrable function which has nonvanishing Fourier transform on all unitary characters. However, back in 1955, Ehrenpreis and Mautner observed that Wiener Tauberian theorem fails even for the commutative Banach algebra of integrable radial functions on SL(2,R). In this talk we shall discuss about a genuine analogue of the theorem for real rank one, connected noncompact semisimple Lie groups with finite centre.

This will be a description of precise conjectures (due mostly to K. Prasanna) about the p-adic valuation of quadratic twisted L-functions at the centre of the critical strip, and an approach we propose to prove them, based on work of Wei Zhang.

Let f be a non-zero Bernstein function, i.e., f(s) = a+bs+\int_0^{\infty}(1?e?xs)d?(x), a,b \geq 0,? \geq 0.There exists a uniquely determined product convolution semigroup (\rho_c)_c>0 on (0,\infty) such that (1) \int_0^{\infty} x^n d\rho_c(x) = (f(1)�...f(n))_c, c>0 , n= 0,1,...,... The Stieltjes moment sequence in (1) is always determinate when c \leq 2as an easy consequence of the Carleman criterion. However, for c >2, it can be determinate or indeterminate depending on f. In fact, in the case f(s) =s,where the moment sequence is (n!)c, it was proved that the moment sequence is indeterminate. In this case \rho_c=e_c(t)dm(t), where m is Lebesgue measure on the half-line and (2) ec(t) = 12\pi \int_{-\infty}^{\infty} t^{ix?1} \Gamma(1?ix)^c dx, t >0. The proof of the indeterminacy was quite delicate based on asymptotic formulas for stable distributions due to Skorokhod. In recent work with Jose Lopez, Spain, we have found the asymptotic behaviour of e_c at infinity. Then, it is easy to derive the indeterminacy of e_c from a criterion of Krein. In the lecture I will give the necessary background for the above.