Let f be a step function on the circle with zero mean and rational discontinuities while alpha is a quadratic irrational. The point-wise ergodic theorem tells us that the ergodic sums, f(x)+f(x+alpha)+...+f(x+(n-1)alpha) is o(n) for almost every x but says nothing about its deviations from zero, that is, its discrepancy; the study of these deviations naturally draws us to the study of ergodic transformations on infinite measure spaces, viz., skew products over irrational rotations. In this talk, after a brief introduction to these terms, we will learn how the temporal statistics of the ergodic sums for x=0 can be studied via random affine transformations (RATs) leading to a central limit theorem and other fine properties like the visit times to a neighbourhood of 0 vis-a-vis bounded rational ergodicity (all of course time permitting). This is reporting on joint work with Jon Aaronson and Michael Bromberg.

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

A total weighting of a graph G is a mapping f which assigns to each element z a real number f(z) as its weight. The vertex sum of v with respect to f is the sum of weight of v and weights of edges adjacent to v. A total weighting is proper if vertex sums of adjacent vertices are distinct. A (k, k')-list assignment is a mapping L which assigns to each vertex v a set L(v) of k permissible weights, and assigns to each edge e a set L(e) of k’ permissible weights. We say G is (k, k')-choosable if for any (k,k')-list assignment L, there is a proper total weighting f of G with f(z) L(z) for each z. It was conjectured by Wong and Zhu that every graph is (2,2)-choosable and every graph with no isolated edge is (1, 3)-choosable. We will see a proof of the statement in the title, due to Wong and Zhu.

Open de Rham spaces refer to certain moduli spaces of meromorphic connections on the projective line. I will discuss certain aspects of these, namely their relation to quiver varieties, the existence of hyperkaehler metrics, and the computation of their E-polynomials.

One knows that any compact riemann surface of genus > 2 carries a riemanniann metric of constant curvature. In higher dimension even the existence of compact manifolds of constant negative curvature is by no means that abundant. In this lecture we will show how arithmetic enables us to construct such manifolds in every dimension greater than equal to 2.

We shall present a simple proof due to Vijay Kodiyalam. This proof makes use of the fact that transform of a complete ideal in a two-dimensional regular local ring R in a quadratic transform of R is again complete. It also uses a structure theorem, due to Abhyankar, of two-dimensional regular local rings birationally dominating R.

Suppose we want to label the edges of the complete bipartite graph K_{n,n} with elements of F_2^d in such a way that the sum of labels over any simple cycle is nonzero. What is the smallest possible value of d be for such a labeling to exist? It was proved by Gopalan et. al. that log^2(n) \leq d \leq nlog(n). Kane, Lovett and Rao recently proved that d is in fact linear in n. In particular we have n/2-2 \leq d < 6n. Upper bound is established by explicit construction while lower bound is obtained by bounding the size of independent sets in certain Cayley graphs of S_n.

We shall continue the discussion on the results of Marina Ratner on unipotent flows, and the techniques involved.

Consider a sample of size N from a linear process of dimension p where n, p are tending to infinity, p/n is tending to a finite non-negative number. We prove, in a unified way, that the limiting spectral distribution (LSD) of any symmetric polynomial in these matrices exist. Our approach is through the intuitive algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. We suggest statistical uses of these LSD and related results in problems such as order determination and white noise testing. The ideas extend to several independent processes and is useful for statistical tests in to sample problems.

Let n be any positive integer, n]:={1,2,...,n}, and suppose $D\subset\{-n,-n+1,..,-1,0,1,...,n}$. Let F be a family of subsets of [n]. A family F' of subsets of [n] is said to be D-secting for F if for every A in the family F, there exists a subset A'in F' such that $|A\cap A'|-|A\cap ([n]\setminus A')| = i$, for some $i\in D$. A D-secting family F' of F, where D = {-1,0,1}, is a bisecting family ensuring the existence of a subset $A'\in F'$ such that $|A\cap A'|\in{\lfloor |A|/2\rfloor, \lceil |A|/2\rceil\}$ for each $A\in F$. We consider the problem of determining minimal D-secting families F' for certain families F and some related questions. This is based on joint work with Rogers Mathew, Tapas Mishra, and Sudebkumar Prashant Pal.