Title: Operads and Infinite loop space theory

Time, Day and Date: 11:30 a.m,  Wednesday 28th August.

Venue: Ramanujan hall

Host: Rekha Santhanam

Speaker: Sahin Mandal

Abstract: In this series of talks we will define Operads and discuss their
properties. We will then define Segal's "gamma" space to prove
Barratt-Priddy-Quillen theorem and discuss uniqueness theorem for infinite
loop spaces machines as proven by May and Thomason.
 

Description: Spectral theory of automorphic forms
Date, time, venue: 28/08, 11:30am -- 1pm, Room 215
Hosts: Keshav Aggarwal and Mallesham Kummari
Speaker: Suraj Panigrahy and Aditi Savalia, IIT Bombay
Abstract: We'll discuss sections 1.3 -- 1.4 from Iwaniec's Topics in
classical automorphic forms. Suraj will continue with his presentation and
discuss section 1.3. Aditi will discuss section 1.4 (and 1.5 if time
allows).

Commutative Algebra Seminar:
Speaker: Tony J. Puthenpurakal (IIT Bombay)
Title: K-theoretic methods in local algebra
Time, Day and Date: 12:30 p.m., Tuesday, August 27
Venue: Room 215, Department of Mathematics
Abstract: We continue over studies in k-theory of complexes

Colloquium:

Speaker: Najmuddin Fakhruddin (School of Mathematics, TIFR, Mumbai)

Title: The Hodge theory of the KZ equations and enriched representation rings

Time, Day and Date: 4:00 p.m. on Wednesday, August 28

Venue: Ramanujan Hall

Abstract: The representation ring of a simple Lie algebra over the field of complex numbers is the free abelian group on the isomorphism classes of irreducible representations with the product structure given by decomposing the tensor product of two irreducible representations as a direct sum of irreducible representations (with multiplicities). In recent joint work with Prakash Belkale and Swarnava Mukhopadhyay we have defined certain families of "enriched" representation rings: these are free modules over the integral polynomial ring in one variable on the set of irreducible representations, with a product which specializes to the usual product when the variable of the polynomial ring is set to 1. Furthermore, the enriched "multiplicities" are polynomials with non-negative coefficients. These rings arose in our work on the Hodge theory of the Knizhnik-Zamolodchikov (KZ) equations: these are certain linear partial differential equations associated to a simple Lie algebra, a finite set of irreducible representations and an auxiliary complex number kappa, and were defined in the context of conformal field theory in the 1980s, but various aspects of these equations, in particular their monodromy, have been studied by mathematicians such as Kohno, Drinfeld and many others. Our work builds on the work of Schechtman-Varchenko and Looijenga which led to the proof that these equations for kappa rational are of Gauss-Manin type, i.e., arise from the cohomology of families of algebraic varieties. In my talk I will explain the construction of the enriched representation rings and how they are relevant to computing the ranks of the Hodge filtration on KZ local systems. The general constructions will be made explicit throughout the talk in the concrete case of the Lie algebra sl_2.