Ph. D. Defence seminar

Date and time: Monday, January 9, 2023 Time: 11 AM-12 PM 
Venue: Ramanujan Hall
Host: Sanjay Pusti
Google meet link: https://meet.google.com/mjb-ghwp-tgk

Speaker: Mr. Tapendu Rana

Title: Wiener Tauberian theorems on Lie groups and Pseudo-differential operators on symmetric spaces and homogeneous trees 
Abstract: In this seminar, first, we will discuss the L^p-boundedness property of the pseudo-differential operators associated with a symbol on the rank one Riemannian symmetric spaces of noncompact type, where the symbol satisfies Hörmander-type conditions near infinity. We will also investigate the same problem in the setting of homogeneous trees, which are considered to be the discrete version of the rank one noncompact symmetric spaces.
We will talk about the Wiener Tauberian theorem on Lie groups in the second part of our seminar. We will discuss a genuine analogue of Wiener Tauberian theorem for L^{p,1}(SL(2, R)) (1 ≤ p < 2). Finally, we will prove Wiener Tauberian theorem type results for various Banach algebras and Lorentz spaces of radial functions on real rank one semisimple Lie group G, which is noncompact, connected, and has a finite center. This is a natural generalization of the Wiener Tauberian theorem for the commutative Banach algebra of the radial integrable functions on G.
 

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.