I will begin by introducing the class of symplectic manifolds and explain their connections to other well known geometric objects. In the second part of the talk, I will introduce symplectic reduction whereby one constructs symplectic quotients of symplectic manifolds under suitable actions of lie groups.

The classical version of Glivenko Cantelli theorem says: let X sub I , i = 1, 2, 3, ..be real valued r.v. with cdf F(x) then the empirical cdf based on X sub i , i = 1,2,3,....n converges wp1 as n goes to infinity to F uniformly over R. In this talk this theorm is generalised to cover Markov chains, exchangeable sequences and regenerative sequences. Some open problems will also be posed.

Graph complexes are simplicial complexes arising from graphs. In this talk we mainly focus on two types of complexes: Neighborhood complexes and Hom complexes. The topology of these complexes are closely related to the chromatic number of the underlying graphs. We give a brief survey of the research have been done with respect to them in recent years. We also discuss some open problems related to them.

In this talk we will assume that certain cohomology theories exist, and demonstrate how to prove rationality and functional equation of the zeta function. A large part of the talk will be using tools from linear algebra and algebraic topology. We will mostly be following the nice set of notes available here: http://www.math.lsa.umich.edu/~mmustata/lecture5.pdf

Consider a rational map from P^{n-1}—>P^n parametrized by homogeneous polynomials f_0,\dots,f_n of degree d. We study the equations defining the graph of the map whose coordinate ring is the Rees algebra of the ideal generated by f_0, .., f_n. We provide new methods to construct these equations using work of Buchsbaum and Eisenbud. Furthermore, for certain classes of ideals, we show that our construction is general. These classes of examples are interesting, in that, there are no known methods to compute the defining ideal of the Rees algebra of such ideals. These new methods also give rise to effective criteria to check that φ is birational onto its image.

Consider a rational map from P^{n-1}—>P^n parametrized by homogeneous polynomials f_0,\dots,f_n of degree d. We study the equations defining the graph of the map whose coordinate ring is the Rees algebra of the ideal generated by f_0, .., f_n. We provide new methods to construct these equations using work of Buchsbaum and Eisenbud. Furthermore, for certain classes of ideals, we show that our construction is general. These classes of examples are interesting, in that, there are no known methods to compute the defining ideal of the Rees algebra of such ideals. These new methods also give rise to effective criteria to check that φ is birational onto its image.

In this presentation, I will provide some personal perspectives on the so-called Big Data movement, its impact, potential benefits and challenges. As part of this, I will discuss developments in, and applications of, machine learning with a focus on supervised learning. I will also spend a few minutes on the new field of Data Science. This will be a non-technical talk.

In the modern big data era, data sets with a large number of features is a common phenomenon. In this talk, I will present an overview of high dimensional variable selection methods along with some recent advances both from the frequentist and Bayesian viewpoints. Both computational and theoretical challenges and ways to address them will be discussed.

In this talk we will assume that certain cohomology theories exist, and demonstrate how to prove rationality and functional equation of the zeta function. A large part of the talk will be using tools from linear algebra and algebraic topology. We will mostly be following the nice set of notes available here: http://www.math.lsa.umich.edu/~mmustata/lecture5.pdf Time: 4-6 PM. Venue: Room 215

In this talk we shall see three very different areas of applications of combinatorics in mathematics and computer science, illustrating four different flavours of combinatorial reasoning. First, we shall look at Haussler's Packing Lemma from Computational Geometry and Machine Learning, for set systems of bounded VC dimension. We shall go through its generalization to the Shallow Packing Lemma for systems of shallow cell complexity, and see how it can be used to prove the existence of small representations of set systems, such as epsilon nets, M-nets, etc. Joint works with Arijit Ghosh (IMSc, Chennai), Nabil Mustafa (ESIEE Paris), Bruno Jartoux (ESIEE Paris) and Esther Ezra (Georgia Inst. Tech., Atlanta). Next, we consider lower bounds on the maximum size of an independent set, as well as the number of independent sets, in k-uniform hypergraphs, together with an extension to the maximum size of a subgraph of bounded degeneracy in a hypergraph. Joint works with C. R. Subramanian (IMSc, Chennai), Dhruv Mubayi (UIC, Chicago) and Jeff Cooper (UIC, Chicago) and Arijit Ghosh. The last problem is on the decomposition, into irreducible representations, of the Weil representation of the full symplectic group associated to a finite module of odd order over a Dedekind domain. We shall discuss how a poset structure defined on the orbits of finite abelian p-groups under automorphisms can be used to show the decomposition of the Weil representation is multiplicity-free, as well as parametrize the irreducible subrepresentations, compute their dimensions in terms of p, etc. Joint works with Amritanshu Prasad (IMSc, Chennai).