The classical version of Glivenko Cantelli thm asserts uniform convergence of the empirical cdf to the true cdf for iid real valued random variables. In this talk we extend that result to regenerative sequences, exchangeable sequences and stationary sequences all with possible delays. We discuss the extension to the vector case. This is based on joint work with Vivek Roy.

We shall discuss the homological invariants like depth and regularity of the edge ideals and their powers of bipartite graphs. We shall mostly concentrate to the cases where the edge ideal is unmixed or Cohen-Macaulay. Some recent progress and direction for future reseach shall be discussed.

We study some properties of Fatou and Julia sets for a family of holomorphic endomorphisms of C^k, k >1. In particular, the family we consider is a semigroup generated by various classes of holomorphic endomorphisms of C^k. Here we generalize a result from the dynamics of the iterates single holomorphic function in C^k, which was proved by Fornaess-Sibony(1998). Also we define recurrent Fatou components in this setup and give a classification result.

The drift of a real-valued random sequence at a particular time is equal to the conditional expected change in the sequence over the next time step, given the information known about the sequence up to the given time. If the drift is zero the sequence is known as a martingale. The actual change in the sequence is equal to the drift plus a conditional mean zero deviation. After each time step, a new drift can be calculated, and the random deviations from the drift add up over time. It is thus important to bound the cumulative effect of the deviations, to quantify whether the values of the sequence over a long period of time evolve according to the drift. This talk identifies an incomplete list of bounds implied by drift that have been used in many applications, including to analyze the performance of randomized algorithms for non-convex global optimization problems.

In this work, we investigate the extremal behaviour of left-stationary symmetric \alpha- stable random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls (in the Cayley graph) of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying quasi-invariant action of the free group but is different from what happens in the case of alpha stable random fields indexed by Zd. The presence of this new dichotomy is confirmed by the study of stable random fields generated by the canonical action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Sullivan. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a new class on point processes that we have termed as randomly thinned cluster Poisson processes. This limit too is very different from that in the case of a lattice. This talk is based on a joint work with Sourav Sarkar, who carried out a significant portion of the work in his master’s dissertation at Indian Statistical Institute.

Let (A,m) be a Noetherian local ring with depth(A) > 1, I an m-primary ideal, M a finitely generated A-module of dimension r, and G_n, the associated graded module of M with respect to I^n. We will discuss a necessary and sufficient condition for depth (G_n) > 1 for all sufficiently large. This talk is based on a paper by Tony Joseph Puthenpurakal (Ratliff-Rush filtration, regularity and depth of higher associated graded modules: Part I )

In this talk algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. We will show that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined in. Tensor, dual, pullback and pushforward operations are also defined for parabolic Bundles.

Brauer-Thrall conjectures for representation theory of Artin algebra's was proved many years ago (in 1968). However the techniques invented by Auslander to prove this conjecture has found more applications than just proving the original conjectures. These techniques have been extended in commutative algebra to study Maximal Cohen-Macaulay modules over Cohen-Macaulay isolated singularities. I will also discuss a result of mine in this direction.