Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

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Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology. 

Commutative algebra seminar

3.30 pm on Tuesday, 18 October 2022

Venue: Ramanujan Hall

Speaker: H. Ananthnarayan

Title: Boij-Soderberg Theory and multiplicity conjecture-III

Abstract: 
In an article published in 2008, Boij and Soderberg introduced the notion of a cone related to the graded Betti numbers of a graded
module over the polynomial ring over a field, and stated a couple of conjectures related to the extremal rays of this cone. They also showed
that a positive answer to these conjectures resolves the Multiplicity conjecture. Eisenbud-Schreyer (2009) show that the Boij-Soderberg
conjectures are true.

In these talks, we will introduce the multiplicity conjectures, indicate their connection to the Boij-Soderberg conjectures, and give an idea of 
how Eisenbud-Schreyer resolve the latter conjectures. We explore similar results over other standard graded rings.

Department Colloquium: Prof. Chandrashekhar Khare:  University of California Los Angeles

Title: Modularity of elliptic curves over number fields

Abstract: I will give an account of developments arising from Wiles’s  work on modularity of elliptic curves over the rational numbers  and Fermat’s Last Theorem.  I will focus on recently announced results of Ana Cariani and James Newton which prove modularity of all elliptic curves over Gaussian numbers. Their result uses as a key step a result of Patrick Allen, Jack Thorne and myself which proves the modularity of mod 3 representations arising from such elliptic curves. This provides a starting point for Cariani and Newton in the same way as a result of Langlands-Tunnel was a starting point for Wiles.

This lecture will be a guided tour of Wiles’s breakthrough in 1994 and the numerous developments since in this very active area of number theory.

 Prof. Nitin Nitsure will deliver his next talk on Thursday, 20th Oct at
5:00 pm in Ramanujan Hall. Meanwhile he has shared the following link to
the recent notes by Jarod Alper on 'Moduli and Stacks', which has a vast
overlap with what Prof. Nitsure plans to cover. Jarod Alper is one of the
leading experts in this area.

https://sites.math.washington.edu/~jarod/moduli.pdf

We will have *Basudev Pattanayak* speaking in the RTAG seminar from *11AM
to 12:30PM* on *Friday(tomorrow)*.


Here are the necessary details for his talk:

Time: Friday, 21 October, 11:00AM – 12:30 PM.

Venue : Room 215, Department of mathematics.

Title: A Visit to the Local Langlands Conjecture - 4

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/