Monday 09-Feb-2015: Lecture Hall 4 15:00 Rajeeva Karandikar 16:00 Sushmita Venugopalan Tuesday 10-Feb-2015: Seminar Hall 15:30 Senthamarai Kannan (unusual time) Wednesday 11-Feb-2015: Lecture Hall 4 15:00 T. Geetha 16:00 Sourav Chakraborty Thursday 12-Feb-2015: Lecture Hall 2 14:00 Vijay Ravikumar (unusual time) Friday 13-Feb-2015: Lecture Hall 2 15:00 Upendra Kulkarni 16:00 Clare D'Cruz ************************************ Titles and Abstracts Sourav Chakraborty, Wednesday 11-Feb-2015, 16:00 Title: Understanding the complexity of Boolean functions through various combinatorial measures Abstract: Given a Boolean function f:{0,1}^n \to {0,1} a natural question is to ask how complex or complicated is this function. There is no unique or canonical way of defining the complexity of the function. A number of natural complexity measures are studied. Understanding of these measures and the relationships between different measures is important in understanding the inherent hardness of the functions and that in turn helps us in other applications. At the same time these problems give rise to many interesting problems in combinatorics, approximation theory and Fourier analysis on finite groups. In the talk we will look a few such measures of Boolean functions and some long standing open problems and my recent attempts to solve them. ************************************ Clare D'cruz, Friday 13-Feb-2015, 16:00 Title: Cohen-Macaulayness of certain Symbolic Rees Algebras. Abstract: Symbolic Rees Algebra has been of interest after a result of Cowsik where he showed that they are related to the problem of Set Theoretic Complete Intersection. In 1994, Goto showed produced a class of monomial curves in an affine n-space whose symbolic Rees Algebras were Noetherian. He also showed that for these curves the Symbolic Rees Algebra was Cohen-Macaulay for n=3,4. This problem was open for n atleast 5. We have a method to handle the general case. This is joint work with Shreedevi Masuti. ************************************ T. Geetha, Wednesday 11-Feb-2015, 15:00 Title: Schur-Weyl dualities and the partition algebras Abstract: Schur-Weyl duality is a cornerstone of representation theory which connects the representation theory of general linear groups and the symmetric groups. In this talk we will see in detail about partition algebras which arises out of this duality by replacing the general linear group by its subgroup symmetric group. These algebras were introduced in the early 90's by Martin in the context of statistical mechanics and independently by Jones in his work related to C*-algebras. ************************************ Senthamarai Kannan, Tuesday 10-Feb-2015, 16:00 Title: On wonderful compactifications Abstract: Let $G$ be a semisimple algebraic group of adjoint type over $\mathbb{C},$ and $H$ be the subgroup of fixed points of an automorphism group of $G$ of order two. In 1983, De Concini and Procesi constructed a $G$-equivariant smooth projective embedding $X$ of the symmetric space $G/H$ with some nice properties, which are now =known as wonderful compactifications. Minimal rank symmetric varieties $G/H$ were introduced by Brion. He and Joshua studied the the geometry of the $B$-orbit closures in $X$ whenever $G/H$ is of minimal rank ($B$- a Borel subgroup of $G.$) In this talk, we present a result saying that an equivariant vector bundle on $X$ is nef (respectively, ample) if and only if its restriction to a $B$-orbit closure is nef (respectively, ample). We also present a construction of a smooth compactification of $PGL(n, C)/T$ with some nice properties ($T$- amximal torus of $G$). We obtain this compactification as a Geometric Invariant Theoretic quotient of the wonderful compactification of $PGL(n)$ modulo $T$ for a suitable polarisation. This is a joint work with Indranil Biswas and D. S. Nagaraj. ************************************ Rajeeva Karandikar, Monday 09-Feb-2015, 15:00 Title: Martingales: An overview Abstract: Martingales is a powerful tool in study of various aspects of probability theory and it's applications. Indeed, the notion is at the centerstage of connections of probability theory with mathematical finance- with the term appearing in what is known as the fundamental theorem of asset pricing. In this talk I will describe various aspects of martingales and the role it plays in the theory of stochastic integration and also touch upon application to finance. ************************************ Upendra Kulkarni, Friday 13-Feb-2015, 15:00 Title: Polynomial functors and symmetric group representations Abstract: Classical Schur-Weyl duality concerns commuting actions of GL(V) and of the symmetric group S_r on r-fold tensor power of a vector space V. This leads, among other things, to certain equivalences of categories of representations. I will give an exposition of some topics of current interest taking this story further. One can simultaneously consider all V and all r. The resulting equivalence (between categories of polynomial functors and of vector species) allows one to transport structures from one side to the other. I will show an interpretation of the tensor product on the symmetric group side via this bridge. Depending on time I will discuss additional structures/constructions related to these categories: (1) The notion of twisted commutative algebras (2) FI-modules (3) Categorical sl_2 and Heisenberg algebra actions on each side. ************************************ Vijay Ravikumar, Thursday 12-Feb-2015, 14:00 Title: The Cohomology Ring of the Complex Grassmannian Abstract: The Grassmannian is a simple example of a moduli space, and its geometry is very well understood. We give an introduction to the geometry of the complex Grassmannian X by studying its cohomology ring H*(X). First we describe the Schubert varieties of X and show how the corresponding Schubert classes form an additive basis for H*(X). By relating the intersections of Schubert varieties to the products of Schubert classes, we determine the multiplicative structure on H*(X) with respect to the Schubert basis. In particular, we describe Pieri's rule for multiplying arbitrary Schubert classes with certain special Schubert classes, and show how it can be used to solve classical problems in enumerative geometry. ************************************ Sushmita Venugopalan, Monday 09-Feb-2015, 16:00 Title: Moment map and the vortex equation Abstract: A moment map is a function associated to the Hamiltonian action of a Lie group on a symplectic manifold. It is a function from the symplectic manifold to the dual of the Lie algebra. The symplectic quotient is the zero level set of this map quotiented out by the group action. By the Kempf-Ness theorem, when a complex reductive group acts on a non-singular variety, the GIT quotient coincides with the symplectic quotient. In gauge theory, there are infinite dimensional analogs of this phenomenon. I present one such case involving the space of holomorphic fiber bundles on curves with some additional data. In this case the moment map corresponds to the vortex equation.