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.

PhD Presynopsis seminar

Speaker : Deep Makadiya

Day/date/Time : Monday, 9 December, 11:00 am

Venue : Ramanujan hall

Title : Some Properties of Twisted Chevalley Groups

Abstract : We will discuss some results from my PhD research, which
investigates certain structural properties of twisted Chevalley groups
over commutative rings.

Speaker: Devdatta Hegde

Day/Date/Time : Tuesday 10th December, 11:30

Venue : Ramanujan Hall

Title : Poles of non-cuspidal Eisenstein series

Abstract : We will show how to determine the poles of non-cuspidal
Eisenstein series by a straightforward global argument in the simplest
example of unramified degenerate Eisenstein series. The location of the
poles is determined by the highest weights occurring in the decomposition
of a natural SL_2 representation.

Speaker: Basudev Pattanayak, Univ of Hong Kong

Date & Time: 4 pm, Wednesday, 11 December 2024

Title: Algorithms for Computing Parabolic Inductions and Jacquet Modules for the General Linear Group

Abstract: Let G be a general linear group over a non-Archimedean local field. Parabolic inductions and Jacquet modules are basic tools in the representation theory of G. In this talk, we will discuss algorithms for computing specific Jacquet modules in the form of 'derivatives' and certain parabolic inductions in the form of 'integrals' of irreducible smooth representations of G. These derivatives and integrals are essential for understanding quotient branching laws, particularly in identifying generalized GGP relevant pairs in G. This talk is based on ongoing work with Kei Yuen Chan.

Speaker: R. Venkatesh, Department of Maths, IISc
Date and time: 12th Dec 2024, 11:00-12:00 hrs
Venue: Ramanujan hall, our department.

Title: Partially commutative algebraic structures and graph invariants

Abstract:  Let G be a finite simple graph. Various algebraic structures
can be associated with G. In this lecture, we will focus on some of the
partially commutative algebraic structures related to G, and explore the
connections between these structures and some key graph invariants of G.

Program Title:  Advances in High Dimensional Statistical Learning Conference
Date: 15-16 Dec 2024
Venue: Ramanujan Hall, Department of Mathematics
Program Brochure: Attached
Program Webpage: Under Construction

Program Title:  Advances in High Dimensional Statistical Learning Conference
Date: 15-16 Dec 2024
Venue: Ramanujan Hall, Department of Mathematics
Program Brochure: Attached
Program Webpage: Under Construction

Thesis Defence
Speaker: PMS Sai Krishna (IIT Bombay)
Title: Exponential maps and their applications
Time, day and date: 11:30:00 AM - 12:39:00 PM, Monday, December 16
Venue: Room 216

Abstract
Exponential maps of k-domains generalize locally nilpotent derivations, and they coincide when k is of zero characteristic. They have proven to be a useful tool in approaching problems in Affine Algebraic Geometry. In the first part of the talk, we will look at some results related to locally nilpotent derivations. Next, we introduce exponential maps and define some related invariants, using which we give an algebraic characterization of the affine plane and affine three-space. The second part of the talk will be about some results related to the rigidity and triangularity of exponential maps. The last part of the talk will be about the rigidity problem, which is about the existence of a non-trivial exponential map of a k-domain. We will see some results related to the rigidity of the ring of invariants of an exponential map, and we provide a sufficient condition under which the ring of invariants of an exponential map of k^[3] is k^[2].

Title:  Oddtowns, partial ovoids, and the rank-Ramsey problem

Speaker: Prof. Anurag Bishnoi, TU Delft, Netherlands

[https://anuragbishnoi.wordpress.com/]

Abstract :

What is the largest size of a family of subsets of {1, ..., m} such that
every set in the family has odd cardinality and among any three distinct
sets there is at least one pair that intersects in an even number of
elements?  We'll show that this generalisation of the classic Oddtown
problem is equivalent to finding the largest size of a partial 2-ovoid in a
binary symplectic space and that of finding the largest set of nearly
orthogonal vectors in F_2^m. Moreover, we show that it's intimately linked
to the following recently introduced rank-Ramsey problem: for a given m,
find the largest n for which there exists a triangle-free graph with n
vertices whose adjacency matrix A  satisfies rank(A+I) <= m.  If we
consider the rank over the binary field, then the problem is equivalent to
generalized oddtowns, while over other fields it has a different behaviour.

We give new constructions of these objects, improving the state of art,
using a triangle-free Cayley graph associated with BCH codes.  Moreover, by
using binary projective caps, that is, sum-free sets in F_2^n,  we improve
the best construction for this rank-Ramsey problem over the reals.

Joint work with John Bamberg and Ferdinand Ihringer.

Dr. Aditi Savalia will give a number theory seminar on Thursday, December 26 at 10:00 a.m. in Ramanujan Hall.

Title: Approximate functional equation for $\zeta(s)\zeta(s-\alpha)$
Abstract:
The Riemann zeta function is defined by
$\zeta(s)=\sum_{n}1/n^s=\prod_{p-prime}(1-1/p^s)^{-1}$, for $Re(s)>1$.
Further, it can be analytically continued to the whole complex plane
except for a simple pole at $s=1$. The behavior of this function is
fascinating in the critical strip $0<Re(s)<1$. One way to understand this
is by truncating the series or product to finite terms and studying its
relation with the zeta function, known as the approximate functional
equation. In this talk, we will discuss identities related to approximate
functional equation for $\zeta(s)\zeta(s-\alpha)$, for some complex number
$\alpha$.