Given two semistable elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{\'e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $B\to C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $\tilde{A}_{n-1}$ to that of $\tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.

Conformal blocks are refined invariants of tensor product of representations of a Lie algebra that give a special class of vector bundles on the moduli space of curves. In this talk, I will introduce conformal blocks and explore connections to questions in algebraic geometry and representation theory. I will also focus on some ``strange" dualities in representation theory and how they give equalities of divisor classes on the moduli space of curves.

We shall describe a construction of non-arithmetic lattices in SO(n,1) following Agol.

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.

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.

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.

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.

I will talk about formal group laws and their relation to complex oriented cohomology theories. In the end I will state Landweber exact functor theorem.

We shall continue the discussion on the results of Marina Ratner on unipotent flows.