The Riemann hypothesis asks about the location of zeros of the Riemann zeta function. More generally, one may consider the analogue of this hypothesis for L-functions. It is also of interest to study the distribution of non-zero values. We will discuss some old and recent results on this problem.

A theorem of Bayer and Stillman asserts that if I is an ideal in a polynomial ring S over a field (in finitely many variables), then the projective dimension and regularlity of S/I are equal to those of S/Gin(I), where Gin(I) is the generic initial ideal of I in the reverse lexicographic order. In this series of talks, we will discuss the necessary background material, and prove the above theorem.

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.

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 Lam ?e system of linear Elasticity

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.

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.

This lecture is based on the paper "Generalized Hamming Weights for Linear Codes" by V.K. Wei. In this lecture I will define generalized Hamming weights and then discuss monotonicity of Hamming weights and duality theorem. Determination of complete weight hierarchy of binary Reed-Muller code will also be outlined.

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.

We will define principal bundles, connections and curvature. With the basics defined, we will construct the Chern-Weil homomorphism. Let E be a principal G-bundle on M with a connection D. Let F be the curvature of D, and g=Lie(G). The Chern-Weil homomorphism associates to each Ad-invariant polynomial on g, a well defined cohomology class in the de Rham cohomology H_{dR}^*(M). Let P be an Ad-invariant homogeneous polynomial of degree k on g. The Chern-Weil image of P is given by the closed 2k-form P(F^{2k}). Its class in H^{2k}_{dR}(M) does not depend on the choice of the connection. This class is functorial. We will conclude with a few examples.

This lecture is based on the paper "Generalized Hamming Weights for Linear Codes" by V.K. Wei. In this lecture I will define generalized Hamming weights and then discuss monotonicity of Hamming weights and duality theorem. Determination of complete weight hierarchy of binary Reed-Muller code will also be outlined.