Speaker: Prof. R. Parimala

Title: Cohomological invariants for algebraic groups

Abstract: We shall discuss the degree three cohomological invariant for torsors under simply connected absolutely simple linear algebraic groups over function fields of curves over local fields and number fields.

Day and Date: Wednesday 4 Dec 2024

Time: 16.00 hrs

Venue: Ramanujan Hall

Analysis Seminar

Date & Time: 10:00 am, Thursday, December 5, 2024

Speaker: Ravi Jaiswal, TIFR-CAM

Meeting link: https://meet.google.com/twn-hsiu-mze

*Title: *Boundary Behaviour of Biholomorphic Invariants on Infinite Type
Domains
*Abstract: *On domains in $\mathbb{C}^n$, $n > 1$, there is a deep
interplay between the boundary geometry of the domain and function theory
on the domain. The interplay is often captured in the boundary behaviour of
various canonical objects associated to the domain, many of which are also
biholomorphic invariants. Examining the boundary behaviour of these objects
provides insights into the behaviour of holomorphic mappings and the
classification of domains up to biholomorphic equivalence.

Motivated by the above facts, we will prove optimal lower and upper bounds
of the Bergman and Szeg\H{o} kernels near the boundary of bounded smooth
generalized decoupled pseudoconvex domains in $\mathbb{C}^{n}$. Generalized
decoupled domains may have complex tangential directions that are not
necessarily decoupled individually, and their boundary points may possess
both finite and infinite type directions.

We will then proceed to study exponentially flat infinite type domains. On
this class of domains, we will prove nontangential asymptotic limits of the
following at exponentially flat infinite type boundary points of smooth
bounded pseudoconvex domains in $\mathbb{C}^{n}$: Bergman kernel and
metric, Kobayashi and Kobayashi--Fuks metrics, holomorphic sectional, Ricci
and scalar curvatures of the Bergman metric, and Bergman canonical
invariant.

Finally, I will discuss some future research plans in the directions
mentioned above.

Number theory seminar
Speaker: Manish Mishra (IISER Pune)
Title: Types and Hecke Algebras
Time, day and date: 4:00:00 PM - 5:00:00 PM, Friday, December 6
Venue: Ramanujan Hall

Abstract:
Let R(G) denote the category of smooth complex representation of G(F), where G is a connected reductive group defined over a non-archimedean local field F. Bernstein decomposition expresses R(G) as a product of indecomposable subcategories called Bernstein blocks. Each Bernstein block is equivalent to the module category of the "Hecke algebra" associated with that "type". I will go over the basic theory mentioned above. To each Bernstein block, the theory of Moy and Prasad associates a number called depth. I will describe a result, jointly done with Jeff Adler, Jessica Fintzen and Kazuma Ohara, which states that each Bernstein block is equivalent to a depth-zero Bernstein block of a certain subgroup of G, when the residue characteristic is not too small.