


Lecture series on algebraic stacks
Monday 23 October, 11.30 am
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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 Gbundles, where G is an algebraic group. The algebraic cohomology of this stack gives the algebraic cohomological version of the characteristic classes of principal Gbundles.
Number Theory Seminar
Monday, 23 October 2023, 14:30
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Venue: Ramanujan hall
Host: U. K. Anandavardhanan
Speaker: Anand Chitrao
Affiliation: TIFR Mumbai
Title: Reductions mod $p$ of semistable representations.
Abstract: We compute the reductions mod $p$ of irreducible twodimensional semistable 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 semistable representations of arbitrarily large weights. In passing, we extend some results on Iwahori induction to the case of noncommutative Hecke algebras.