


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 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
======================
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.
Lecture series on Hodge Theory
Wednesday, 25 October, 11.301.00
=======================
Venue: Ramanujan Hall
Host: Sudarshan Gurjar
Speaker: V. Srinivas, IIT Bombay
Title: Kahler condition and its consequences
Abstract: In the next couple of talks, I will give a sketch of the proof of the Hodge theorem, that any smooth form can be expressed as a sum of a harmonic form and a form in the image of the Hodge Laplacian.
Topology and Related Topics Seminar
Thursday, 26 Oct. 2:30 pm  3:45 pm
==========================
Venue: Room 215
Host: Rekha Santhanam
Speaker: Omkar Javadekar
Affiliation: IIT Bombay
Title: A review of derived category and related topics2
Abstract: We will continue the discussion on derived categories and the HopkinsNeeman theorem for small Rcomplexes.
Online Number Theory Seminar
Thursday, 26 October, 3.15 pm
======================
Venue: https://meet.google.com/zrcowygukt
Host: Ravi Raghunathan
Speaker: Aditi Savalia
Affiliation: IIT Gandhinagar
Title: Limitations to equidistribution in arithmetic progressions
Abstract: It is well known that prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the BombieriVinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions “on average” for moduli $q$ in the range $q ≤ x^{1/2−\epsilon}$ for any $\epsilon > 0$. Building on the idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli $q$ is extended to $q ≤ x/(log x)^B$ for any $B > 1$. We discuss variants of this result and give some applications. This is joint work with Akshaa Vatwani.
Commutative Algebra seminar
Thursday, 26 Oct. 4.005.30 pm
=======================
Venue: Ramanujan Hall
Host: Tony Puthenpurakal
Speaker: Jugal Verma, IIT Bombay
Title: The GrothendieckSerre formula for Hilbert functions
Abstract: We will prove the GrothendieckSerre formula for Hilbert functions of finite graded modules over graded Noetherian rings.
Algebraic Groups Seminar
Friday, 27 October 2023, 4 pm
===================
Venue: Ramanujan Hall
Host: Shripad Garge
Speaker: Chayan Karmakar
Affiliation: IIT Bombay
Title: Lie algebrasI
Abstract: We introduce the notion of Lie algebra of a linear algebraic group.