Schedule of Talks Monday 3rd February 2014: Seminar Hall 14:00 M Sundari, The Uncertainty Principle (This talk is earlier than usual.) 16:00 S Sundar, Semigroup actions and groupoids. Tuesday 4th February 2014: Seminar Hall 15:00 Narasimha Chary, Automorphism group of a BSDH variety. 16:00 Priyavrat Deshpande, Reflections on manifolds & generalized Artin groups. Wednesday 5th February 2014: Lecture Hall 3 15:00 T R Ramadas, Rational and global forms of certain Chiral Conformal Field Theories; Vertex Algebras I 16:00 K V Subrahmanyam, Commuting $U_q(2) \otimes U_q(m)$ action on $V_{q,\lambda}(2m)$. Thursday 6th February 2014: Seminar Hall 15:00 Sourish Das, Bayesian Solution to Some Ill-Posed Problems 16:00 Sukhendu Mehrotra, Hilbert schemes of points on K3 surfaces and deformations II Monday 10th February 2014: Seminar Hall 15:30 B V Rao, Brownian Motion Titles/Abstracts Narasimha Chary, Tue 04-02-2014 15:00 Title: Automorphism group of a BSDH variety. Abstract: Let G be a simple, simply connected algebraic group over the field of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G. Let X(w) be the Schubert variety in G/B corresponding to an element w of the Weyl group W. In general Schubert varieties are not smooth. Demazure constructed for each reduced expression of w, a desingularization Z(w) of the Schubert variety X(w), which is now known as Bott-Samelson-Demazure-Hansen variety (for short BSDH variety). Unlike Schubert varieties it is not clear from the construction of BSDH varieties whether they are independent of the reduced expression of w. In this talk, we will discuss the automorphim groups of BSDH varieties. More precisely, we will show that the connected component of the identity element of the automorphism group of Z(w) is a parabolic subgroup of G either if G is simply laced and the action of B on X(w) is faithful or X(w)=G/B. We will also describe explicity this parabolic subgroup. In particular, we will see examples of BSDH varieties corresponding to different reduced expressions for the same Weyl group element having different non-isomorphic automorphism groups and hence the BSDH varieties do depend on the reduced expressions under consideration. (joint work with S.S Kannan and A.J Parameswaran). Sourish Das, Thu 06-02-2014 15:00 Title: Bayesian Solution to Some Ill-Posed Problems Abstract: When the problem Xb=y, is not well-posed (means either non-existence or non-uniqueness) of b, ordinary least square method does not yield required solution. Such problem is typically known as ill-posed problems or ill-posed systems. In statistics, this problem is known as either "Large-P, small N" or "Multicollinearity" problem. In order to prefer particular solution with desirable properties, a Tikhonov’s regularization term can be included in the objective function. It turns out that Tikhonov’s regularization yields a Bayesian solution popularly known as Ridge Regression. In recent times, some variant Bayesian solution like ‘LASSO’ and ‘Elastic Net’ is being proposed. On the same ill-posed systems, multiple testing problems can be considered and nice Bayes solution is available. Similarly, in functional data analysis, Bayes solution is achievable with Gaussian process prior. Finally if time permits, we will present some empirical results on the application of multiple testing in Finance. Priyavrat Deshpande, Tue 04-02-2014, 16:00 Title : Reflections on manifolds and generalized Artin groups. Abstract: The Artin groups form an interesting class of groups as they appear in the study of various areas like mapping class groups, configuration spaces, moment angle complexes, reflection groups and hyperplane arrangements. Moreover, these groups have intriguing properties; most of them are torsion-free, have trivial center, have solvable word problem and in general are bi-automatic. The study of Artin groups display a rich interplay between algebra, combinatorics and topology. In this talk I will begin with an introduction to these groups, describe some of their interesting properties and discuss several instances where they appear naturally. Finally I will propose a generalization of these groups and ongoing work to understand the aspects of so-called K(\pi, 1) problem. Sukhendu Mehrotra, Thu 06-02-2014 16:00 Title: Hilbert schemes of points on K3 surfaces and deformations II Abstract: The Hilbert scheme of points of a K3 surface X admits a 21-dimensional space of deformations, while the moduli space of K3 surfaces is 20-dimensional. The goal of this talk is to provide an interpretation of this extra modulus of the deformation space the Hilbert scheme X[n] in terms of deformations of the derived category D(Coh(X)). We present the concluding results of a joint project with Eyal Markman (UMass); the initial work done under this project was described at this seminar last year. T R Ramadas, Wed 05-02-2014 15:00 Title: Rational and global forms of certain Chiral Conformal Field Theories; Vertex Algebras I Abstract: By rational forms we mean constructions on the complex projective line that replace power series, formal or otherwise, by rational functions. By global forms we understand extensions to smooth projective curves of arbitrary genus. I will show how definitions and basic results around vertex algebras are made simple when power seris are replaced by rational expressions. B V Rao, Mon 10-02-2014 15:30 Title: Brownian Motion Abstract: I shall explain ways of arriving at the Brownian Motion and then explain Ito integrals. This part is introductory. If time permits, I shall explain semi martingales and associated quadratic variation Process. K V Subrahmanyam, Wed 05-02-2014 16:00 Title: Commuting $U_q(2) \otimes U_q(m)$ action on $V_{q,\lambda}(2m)$. Abstract: We describe the steps and the proofs in the construction of quantum deformations of irreducible $GL(2m)$ modules with a commuting $U_q(m)$ and $U_q(2)$ action. This is joint work with Bharat Adsul and Milind Sohoni. Some open questions and possible directions to pursue, in order to get a better understanding of this construction will also be discussed. S Sundar, Mon 03-02-2014 16:00 Title: Semigroup actions and groupoids. Abstract: We will discuss the construction of a groupoid coming out from a semigroup action. Let P \subset G be a semigroup. The Wiener-Hopf algebra is the C*-algebra generated by the compression of the group C*-algebra to L^{2}(P). We will discuss how this can be realised as groupoid C*-algebras. M Sundari, Mon 03-02-2014 14:00 (This talk is earlier than usual.) Title: The Uncertainty Principle Abstract: It is a well known fact in Fourier Analysis that a function and its Fourier transform cannot both be `concentrated' unless f =0. Depending on the definition of `concentration' we get a host of uncertainty principles. We shall look at a few of these in this talk.