Lecture series on algebraic stacks

Monday 23 October, 11.30 am

=========================

Venue: Ramanujan Hall
Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (Retd)

Title: The classifying stack BG for an algebraic group

Abstract: To any Lie group, there is classically associated a  topological space BG with the requisite universal property in the homotopy category of paracompact topological spaces. For example, for G = GL(n) the space BG is the infinite Grassmannian. However, when we go to the algebraic category (say schemes or algebraic spaces and their morphisms), such a space BG does not exist. This is a paradigmatic example where algebraic stacks rescue the situation. In this lecture, we will explain the construction of an algebraic stack BG which has the requisite universal property of classifying principal G-bundles, where G is an algebraic group. The algebraic cohomology of this stack gives the algebraic cohomological version of the characteristic classes of principal G-bundles.

Number Theory Seminar 

Monday, 23 October 2023, 14:30

======================

Venue: Ramanujan hall

Host: U. K. Anandavardhanan 

Speaker: Anand Chitrao 

Affiliation: TIFR Mumbai 

Title: Reductions mod $p$ of semi-stable representations.

Abstract: We compute the reductions mod $p$ of irreducible two-dimensional semi-stable representations of the absolute Galois group $\GQp$ of $\Qp$. We use the compatibility with respect to reduction mod $p$ between the $p$-adic Local Langlands Correspondence and an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence. By estimating certain logarithmic functions on $\Qp$ by polynomials on open subsets of $\Zp$, we compute the reductions mod $p$ completely for weights at most $p + 1$. We also state how this method can be used, in theory, to compute the reductions mod $p$ of semi-stable representations of arbitrarily large weights. In passing, we extend some results on Iwahori induction to the case of non-commutative Hecke algebras.