We study the extremes of branching random walks under the assumption that underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. In the lighter-tailed case, however, the behaviour is much more subtle, and the scaling of the position of the rightmost particle in the n-th generation depends on the family of stepsize distribution, not just its parameter(s). In all of these cases, we discuss the convergence in probability of the scaled maxima sequence. Our results and methodology are applied to study the almost sure convergence in the context of cloud speed for branching random walks with infinite progeny mean. The exact cloud speed constants are calculated for regularly varying displacements and also for stepsize distributions having a nice exponential decay. This talk is based on a joint work with Souvik Ray (Stanford University), Rajat Subhra Hazra (ISI Kolkata) and Philippe Soulier (Univ of Paris Nanterre). We will first review the literature (mainly, the PhD thesis work of Ayan Bhattacharya) and then talk about the current work. Special care will be taken so that a significant portion of the talk remains accessible to everyone.

The symmetric product is an interesting and important construction that is studied in Algebraic Geometry, Complex Geometry, Topology and Theoretical Physics. The symmetric product of a complex manifold is, in general, only a complex space. However, in the case of a one-dimensional complex manifold (i.e., a Riemann surface), it turns out that the symmetric product is always a complex manifold. The study of the symmetric product of planar domains and Riemann surfaces has recently become very important and popular. In this talk, we present two of our recent contributions to this study. The first work (joint with Divakaran, Bharali and Biswas) gives a precise description of the space of proper holomorphic mappings from a product of Riemann surfaces into the symmetric product of a bordered Riemann surface. Our work extends the classical results of Remmert and Stein. Our second result gives a Schwarz lemma for mappings from the unit disk into the symmetric product of a Riemann surface. Our result holds for all Riemann surfaces and yet our proof is simpler and more geometric than earlier proved special cases where the underlying Riemann surface was the unit disk or, more generally, a bounded planar domain. This simplification was achieved by using the pluricomplex Green's function. We will also highlight how the use of this function can simplify several well-know and classical results.

We shall introduce smooth and formally smooth morphisms and study their basic properties. We shall complete the proof of CST (Cohen’s structure theorem for complete local rings).

We will indicate proofs (based on L. Avramov's paper) of some of the descent properties of rings of invariants of a finite pseudo-reflection group acting on a local ring.

The critical group of a graph is an interesting isomorphic invariant. It is a finite abelian group whose order is equal to the number of spanning forests in the graph. The Smith normal form of the graph's Laplacian determines the structure of its critical group. In this presentation, we will consider a family of strongly regular graphs. We will apply representation theory of groups of automorphisms to determine the critical groups of graphs in this family

Let M be a finitely generated seminormal submonoid of the free monoid \mathbb Z_+^n and let k be a field. Then Anderson conjectured that all finitely generated projective modules over the monoid algebra k[M] is free. He proved this in case n=2. Gubeladze proved this for all n using the geometry of polytopes. In a series of 3 lectures, we will outline a proof of this theorem.

In rank one Riemannian symmetric spaces of noncompact type, we shall characterize the eigenfunctions of the Laplace--Beltrami operator with arbitrary eigenvalues through an asymptotic version of the ball mean value property. This is joint work with Muna Naik and Swagato K Ray.

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, partitions of n also parametrize irreducible representations of the permutation group S_n. In this talk, I will describe the Springer correspondence which provides a deeper geometric understanding of the above coincidence. Towards the end, I will sketch the ideas involved in the proof of the Springer correspondence and their relationship with the theory of character sheaves on reductive groups.

We will present Chevalley's proof of this important result. As applications, we will state several results from the paper of L. Avramov. Proofs of some of these will be indicated.

We give a gentle introduction to the geometry of numbers. We start with the classical theory and then treat some of the modern aspects of this subject. This talk will be accessible to the general audience.