Lecture series on Lie groups 

Date and Time: Six Mondays at 4 pm
Tea: 3.50 pm
Venue: A1-A2, CDEEP, Mathematics Department
Host: Dipendra Prasad

Speaker: M. S. Raghunathan, CEBS, Mumbai

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, π_1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is  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.
 

Algebraic Groups Seminar
Date and time: Tuesday, 17 Jan. 2023, 4-5.30 pm

Venue: Room 215
Host: Shripad Garge

Speaker: Arghya Pramanik
Affiliation: Dept of Mathematics, IIT Bombay

Title: Some algebraic geometry – II
Abstract:  This is the second lecture on this topic. We are following the first chapter of Tony A. Springer's book. In this lecture, we will discuss locally ringed spaces and then give the definition of an affine algebraic variety.

Seminar on stochastic processes
Date and Time: Tuesday, 17 January, 2023, 4 pm

Venue: Ramanujan Hall

Host: Ayan Bhattacharya

Speaker: Siva Athreya, ICTS, Bengaluru

Title: Graphon-valued Stochastic Processes

Abstract: We will present our attempts thus far to develop a theory of 
graphon-valued stochastic processes. We will present a brief review of 
the theory of Graphons and dynamics constructed on the space of 
graphons. We shall construct and analyze a natural class of such 
processes arising from population genetics. In conclusion, we shall 
present the challenges in our ongoing work on constructing dynamics 
where the edges and vertices interact with each other. This is joint 
work with Frank den Hollander and Adrian Roellin.

Seminar on number theory

Date and time: Wednesday, 18 January 2023 at 11:30 AM

Venue: Ramanujan Hall

Host: Dipendra Prasad

Speaker:  Rahul Dalal

Affiliation: Johns Hopkins University

Title: Explicit Trace Formulas and Statistics of Families of "Nice" Automorphic Representations

Abstract: The lecture will discuss a bit of the famous Arthur-Selberg trace formula with a view to applying it to families of automorphic representations. 

Mathematics Colloquium
Date and Time: Wednesday, 18 January, 2023, 4 pm
Host: Sudhir Ghorpade

Speaker: Sudesh Kaur Khanduja
Affiliation: Panjab University, Chandigarh

Title: When is Z[θ] the ring of integers?
Abstract: Let K = Q(θ) be an algebraic number field with θ an algebraic integer having minimal polynomial f(x) over Q. Let AK denote the ring of algebraic integers of K. In this talk, we shall discuss some necessary and sufficient conditions to be satisfied by f(x) so that AK = Z[θ]. In particular when f(x) is an irreducible trinomial x^n+ax^m +b ∈ Z[x], then we shall describe a set of necessary and sufficient conditions in terms of prime powers dividing a, b, m and n, for any prime p to divide the group index [A_K : Z[θ]]. Using the well known Dedekind Criterion, we shall also discuss a generalisation of this result for a simple ring extension R[η] of a valuation ring R to be integrally closed when η is a root of an irreducible trinomial x^n+ax^m +b belonging to R[x]. The latter result yields interesting number theoretic applications. This is partly based on joint works with A. Jakhar, B. Jhorar, Sumandeep Kaur, M. Kumar, and N. Sangwan.
 

Commutative Algebra Seminar
Date and time: Thursday 19 January, 2023, 4 pm

Venue: Ramanujan Hall
Host: Tony Puthenpurakal

Speaker: Tony Puthenpurakal
Affiliation: Mathematics Department, IIT Bombay

Title: An analogue of Rees Theorem for filtrations
Abstract: Let A be an analytically unramified Cohen-Macaulay local ring. Let {I_n} be a filtration of m-primary ideals. Let I be an m-primary ideal contained in I_1. It is easily seen that the multiplicity of {I_n} is at least multiplicity of I. We show that if equality holds then the Rees algebra of the filtration is a finite module over Rees algebra of I.