CR functions are certain generalizations of holomorphic functions and CR manifolds are those that support CR functions. For instance, a pseudoconvex hypersurface in $\mathbb{C}^N$ is a CR manifold and CR functions are locally boundary values of holomorphic functions. We will begin by describing this holomorphic extension result before proceeding to discuss the codimension two case. Codimension two submanifolds of $\mathbb{C}^N$ generically have isolated CR singularities and we are interested in studying the behaviour of the extension of CR functions near CR singularities. We prove that under certain nondegeneracy conditions on the CR singularity this extension is smooth up to the CR singularity. This is joint work with Jiri Lebl and Alan Noell.

This is the first in a series of lectures on class field theory. We begin with Chapter 1 in Cassels and Frohlich.

Let P(z) = \sum_{i=0}^m z^i A_i, where A_i are complex n cross n matrices for i= 0, . . . , m, be a regular matrix polynomial. The matrix polynomial P(z) is said to be structured if the matrices A_i, for i= 0, . . . , m belong to a special subset S of (C^{nxn})^{ m+1}. Structured matrix polynomials have occurred in many engineering applications and have been studied widely for the last two decades. Structured eigenvalue-eigenpair backward error analysis of structured matrix polynomials is important in order to know the backward stability of algorithms that compute them without losing the structure of the polynomial. In this talk, I will derive formulas for the structured eigenvalue backward errors of matrix polynomials that have Hermitian and related structures, like skew-Hermitian, ?-even, ?-odd. This involves a reformulation of the original problem of computing eigenvalue backward error into an equivalent problem of minimizing the maximum eigenvalue of a parameterized Hermitian matrix. Numerical experiments show that there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations

A famous conjecture of Sato and Tate (now a celebrated theorem of Taylor et al) predicts that the normalised p-th Fourier coefficients of a non-CM Hecke eigenform follow the semicircle distribution as we vary the primes p. In 1997, Serre obtained a distribution law for the vertical analogue of the Sato-Tate family, where one fixes a prime p and considers the family of p-th coefficients of Hecke eigenforms. In this talk, we address a situation in which we vary the primes as well as families of Hecke eigenforms. In 2006, Nagoshi obtained distribution measures for Fourier coefficients of Hecke eigenforms in these families. We consider another quantity, namely the number of primes p for which the p-th Fourier coefficient of a Hecke eigenform lies in a fixed interval I. On averaging over families of Hecke eigenforms, we obtain a conditional central limit theorem for this quantity. This is joint work with Kaneenika Sinha.

Given a compact group G and a real representation pi of G, there is a sequence of interesting invariants of pi which lie in the group cohomology of G, called the Stiefel-Whitney classes. The first class gives the determinant of pi, and the second class is related to the spinoriality of pi, that is whether it lifts to the spin group. We survey work on this problem when G is a connected Lie group, and also when G is the symmetric group. This is joint work with my Ph.D. students Rohit Joshi and Jyotirmoy Ganguly.

The Steiner diversity is a type of multi-way metric measuring the size of a Steiner tree between vertices of a graph and it generalizes the geodetic distance. The Steiner Wiener index is the sum of all Steiner diversities in a graph and it generalizes the Wiener index. Recently the Steiner Wiener index has found an interesting application in chemical graph theory as a molecular structure descriptor composed of increments representing interactions between sets of atoms, based on the concept of the Steiner diversity. Amon other results a formula based on a vertex contributions of the Steiner Wiener index by a newly introduced Steiner betweenness centrality, which measures the number of Steiner trees that include a particular vertex as a non-terminal vertex, will be presented. This generalizes Krekovski and Gutman's Vertex version of the Wiener Theorem and a result of Gago on the average betweenness centrality and the average distance in general graphs.

In a finite projective 3-space considered as an incidence geometry, an ovoid is the analogue of the sphere in Euclidean 3-space, introduced independently by Segre and Tits. Elliptic quadrics are generic examples of ovoids. A projectively nonequivalent family of ovoids were constructed by Tits which is closely related to the Suzuki groups and Moufang sets. These are the only known families of ovoids. Apart from being objects of intrinsic interest, they are fundamentally related to some important combinatorial structures like inversive planes, generalised quadrangles, permutation polynomials, group divisible designs, etc. Their classification and understanding their distribution in the projective 3-space are the fundamental problems regarding them. However, several first questions about them are yet to be settled. To name a few: the structure of the intersection of any two of them, packing the projective 3- space by ovoids, the number of such objects, up to projective equivalence. As an introduction to this topic, we discuss some interesting results and mention some open problems. I will make an effort to keep the talk elementary.

We will describe our work with I. Biswas and A. Dey on the classification of torus equivariant principal bundles over toric varieties. We will relate this to equivariant analogues of the Serre problem, Grothendieck's theorem on bundles over the projective line, and Hartshorne's conjecture on bundles of small rank over projective space.

This work considers goodness-of-fit for the life test data with hybrid censoring. An alternative representation of the Kolmogorov-Smirnov (KS) statistics is provided under Type-I censoring. The alternative representation leads us to approximate the limiting distributions of the KS statistic as a functional of the Brownian bridge for Type-II, Type-I hybrid and Type-II hybrid censored data. The approximated distributions are used to obtain the critical values of the tests in this context.We found that the proposed KS test procedure for Type-II censoring has more power than the available one(s) in literature.

A famous result of H. Alexander asserts that any proper holomorphic self-map of the unit (Euclidean) ball in higher dimensions is an automorphism. Alexander's result has been extended to various classes of domains including strictly pseudoconvex domains (by Pinchuk) and weakly pseudoconvex domains with real-analytic boundary (by Bedford and Bell). It is conjectured that any proper holomorphic self-map of a smoothly bounded pseudoconvex domain in higher dimensions must be an automorphism. In this talk, I shall first briefly survey some of the prominent Alexander-type results. I shall then talk about an extension of Alexander's Theorem to a certain class of balanced, finite type domains. I shall also highlight how the use of dynamics in the proof offers some insight on the aforementioned conjecture.