Schedule of Talks Monday 8th September 2014: Seminar Hall 15:00 Suratno Basu 16:00 Manoj Kummini Tuesday 9th September 2014: Seminar Hall 15:00 Santanu Sarkar 16:00 R Srinivasan Wednesday 10th September 2014: Lecture Hall 2 15:00 Priyavrat Deshpande 16:00 V Balaji Thursday 11th September 2014: Lecture Hall 2 15:00 Ronno Das 16:00 Ananya Lahiri Friday 12th September 2014: Lecture Hall 1 15:00 Prem Prakash Pandey 16:00 K G Arun Titles/Abstracts Suratno Basu, Monday, 8th September, 15:00 Title: Analogue of a theorem of Mumford-Newstead for singular curves. Abstract: For a smooth projective curve $X$ of genus $\geq 2$, the theorem of Mumford-Newstead shows that the Weil-Griffiths intermediate jacobian of the moduli space $M$ of vector bundles of rank $2$ and fixed determinant of degree $1$ is canonically isomorphic to the Jacobian of the curve $X$. As a consequence they derive a Torrelli type theorem for these moduli spaces. These questions have been generalized on many fronts over smooth curves, but we focus our attention on a generalization of this to singular curves. We prove an analogous result for ceratin moduli spaces of bundles on reducible curves, these moduli spaces (constructed by Nagaraj-Seshadri) occur as degenerations of $M$, and provide some insight into the variations of the classical intermediate jacobians and also answer a new Torelli type theorem for singular curves having two smooth components meeting tranversally at a point. Manoj Kummini, Monday, 8th September, 16:00 Title: Free Resolutions of some determinantal-like varieties. Abstract: We describe minimal free resolutions of some varieties that are like the determinantal varities using Schubert geometry and the Kempf-Lascoux-Weyman geometric technique. This is joint work with V. Lakshmibai, Pramath Sastry and C. S. Seshadri. Santanu Sarkar, Tuesday, 9th September, 15:00 Title: Cryptanalysis of RSA Variants. Abstract: The famous RSA public key cryptosystem is possibly the most studied topic in cryptology. Apart from the basic RSA proposal there are several variants of it for efficiency and security purposes. In this talk we will discuss two RSA variants: Common Prime RSA and Prime Power RSA. An important tool to crypanalyse RSA and its variants is Coppersmith root finding technique using LLL algorithm. We first consider the Common Prime RSA as proposed by Wiener (IEEE-IT, 1990). We present a new range of parameters in Common Prime RSA for which it is not secure. Next we consider an RSA variant with Modulus $N=p^r q$, known as Prime Power RSA. We present technique which improve the attack region when $r \leq 5$. R Srinivasan, Tuesday, 9th September, 16:00 Title: Cohomology for super-product systems. Abstract: Super-product system is a generalization of product system of Hilbert spaces introduced by Bill Arveson. They arise naturally in the theory of $E_0-$semigroups on factors. We propose a cohomology theory for spatial super-product systems, and describe the 2-cocycles for some of the basic examples. This consequently classifies a family of $E_0-$semigroups on type III factors associated with canonical anti-commutation relations. This is a joint work with Oliver T. Margetts. Priyavrat Deshpande, Wednesday, 10th September, 15:00 Title: Cutting, counting and the ubiquitous Mobius inversion. Abstract: In combinatorial geometry the so-called partitioning problems deal with determining the number of pieces into which a certain geometric set is divided by given subsets. The study of partitioning problems for Euclidean and projective spaces go way back to 1830's. Interestingly enough there are some questions which are still unanswered; I plan to mention some of those during the talk. In recent years there has been an increase in interest to study these problems in the context of manifolds. The talk will begin with an introduction to arrangements of hyperplanes and some of their enumerative aspects. Then I will show how to use measure theory for posets in order to derive some face counting formulas. A portion of this talk is based on the joint work with Karthik Chandrashekhar regarding arrangements of toric lines. V Balaji, Wednesday, 10th September, 16:00 Title: Degeneration of moduli of Higgs bundles. Abstract: We construct and study a degeneration of the moduli of Higgs bundles. As an application we obtain some new compactifications of the classical Picard varieties of stable curves. This is based on joint work with Barik-Nagaraj. Ronno Das, Thursday, 11th September, 15:00 Title: Artin groups related to reflections on manifolds. Abstract: The natural action of a reflection group on euclidean space fixes a union of hyperplanes. However the group acts fixed point freely on the complexified complement of this union of hyperplanes. The fundamental group of this space is the pure Artin group and that of the quotient space is the Artin group. These groups are of independent interest to geometric group theorists. Many of their properties have been established by analyzing the topology of the complement. In this talk we shall look at reflection groups that appear as groups of diffeomorphisms. Drawing parallels with the classical theory we will define reflection arrangements on manifolds. The main focus will be on the idea of tangent bundle complement as the generalization of the complexified complement. We will demonstrate how the combinatorial data of the reflection group can be used to build a regular cell complex which has the same homotopy type as that of the tangent bundle complement. Time permitting, we will also discuss some applications. This talk is based on joint work with Priyavrat Deshpande. Ananya Lahiri, Thursday, 11th September, 16:00 Title: Spectrum of heavy Tailed Random Matrices. Abstract: For random symmetric matrix with independent and identically distributed entries, the empirical distribution of the eigenvalues normalized by square root of size of matrix, converges almost sure and in expectation to the so called semicircular law, as size of the matrix goes to infinity. We will discuss same question where the constituting random variables follow alpha stable law. Prem Prakash Pandey, Friday, 12th September, 15:00 Title: A structure theorem in additive number theory. K G Arun, Friday, 12th September, 16:00 Title: Gravitational Wave Astronomy. Abstract: Gravitational Waves (GWs) are ripples in the curvature of space time which carry energy. I will give an overview of the generation of gravitational waves and astrophysical sources of GWs. I will discuss the ongoing experimental efforts to directly detect GWs and the proposed LIGO-India project. The talk will be aimed at a general audience.