Mathematics Colloquium

Time and Date: 4-5 pm Wednesday, 21 September 2022

Speaker: Prof. Saikat Mazumdar, IIT Bombay

Title: Searching conformally for metrics with constant curvature.

Abstract: I will start by surveying the Yamabe problem, which asks to find a (conformal) metric with constant scalar curvature on a compact Riemannian manifold. This amounts to solving a nonlinear PDE involving the Laplacian. The solution to the Yamabe problem highlighted the role played by local and global geometry of the manifold and the unexpected connection to the positive mass theorem of general relativity.

I will first discuss the case of compact surfaces, and introduce some tools and techniques from Calculus of Variations, Nonlinear Analysis on Manifolds, and PDEs. In the remaining time, I will discuss the higher-order version of the Yamabe problem: “Given a compact Riemannian manifold, does there exists a conformal metric with constant Q-curvature”?

Note: The speakers make an effort to make Mathematics Colloquia accessible to the general audience.

We will have *Basudev Pattanayak* speaking in our RTAG seminar from 2pm to
3:30pm on Thursday.


Here are the necessary details for his talk:

Time: Thursday, 22 September, 2:00 – 3:30 pm.

Venue : Ramanujan Hall, Department of mathematics.

Title: A Visit to the Local Langlands Conjecture - 2

Abstract: In this series of talks, we first recall some important results
of class field theory. Then we will discuss the representation theory of
p-adic groups. Here we will discuss the Hecke algebra attached to
Bushnell-Kutzko types. With little basic setup, later we will state the
local Langlands Conjecture and its enhancement. For some special cases, we
will discuss their proofs.



We have now a dedicated website where one can find the notes and resources
from the past meets and announcements of the upcoming meetings:

https://sites.google.com/view/rtag/

Please join us!

Speaker: P Amrutha, IISER Thiruvananthapuram

Date & Day: September 22, 2022, Thursday Time: 4.00--5.00 pm

Venue: Room 215

Title: On the partitions and multipartitions not divisible by powers of 2

Abstract:

Given a finite group G and a natural number p, an interesting question one can ask is to count

the number of inequivalent irreducible representations of G whose degree is not divisible by p. This
question originated in a paper by I. G. Macdonald for the case of prime numbers. MacDonald’s
paper was a motivation for the McKay conjecture. The announcement of McKay conjecture in 1971
is the origin of a different kind of counting conjectures of finite groups. Extending Macdonald’s

results to all integers is a much harder problem to study. Motivated by a question from chiral rep-
resentations of the wreath products, we will see a generalization of the above question to composite

numbers of the form 2k and a recursive formula for the groups Sn, An, and (Z/rZ)≀ Sn. Regardless
of the description of the count, even for the smaller integers, a complete characterization of the
irreducibles with a degree not divisible by a given prime number is still missing in the literature.
We will see such characterization for some special cases at the end and further open problems in
this direction. This is joint work with T. Geetha.