Lecture series on Lie groups

Monday, 27 Feb at 4 pm

Tea: 3.50 pm

==============================

Venue: A1-A2, CDEEP, Mathematics Department

Host: Dipendra Prasad

Speaker: M. S. Raghunathan

Affiliation: 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 center 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 the basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups, and also with some basic results from algebraic topology.

Presynopsis seminar

Monday, 28 Feb. 2023, 4.15 pm

Venue: Conference room

Host: Siuli Mukhopadhyay

Speaker: Savita Pareek

Affiliation: IIT Bombay

Title: On Some Problems in Mixed Effect Models

Algebraic Geometry Seminar

Wednesday, 1 March 2023,  4.00pm
===========================

Venue: Room No 215

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Local Criterion of Flatness-II

Commutative algebra seminar

Thursday, 2 March 2023, 4 pm

=========================

Venue: Room 215

Host: Tony Puthenpurakal

Speaker: Sudeshna Roy

Affiliation: TIFR, Mumbai

Title: Asymptotic behavior of the Castelnuovo-Mumford regularity of (saturations of) products of powers of homogeneous ideals.

Abstract: In this talk, we will give a brief survey on results regarding the asymptotic linear bounds of the Castelnuovo-Mumford regularity of (saturations of) products of powers of ideals. We also aim to discuss a finer description of the asymptotic behavior of this invariant under some extra assumptions.

Virtual Commutative Algebra Seminar

Friday,  3 March 2023, 6:30pm

==============================

Venue: meet.google.com/mjq-ahwy-oxo

Speaker: Vaibhav Pandey

Affiliation: Purdue University, West Lafayette, IN, USA

Title: Linkage and F-regularity of generic determinantal rings

Abstract: We prove that the generic link of a generic determinantal ring of maximal minors is strongly F-regular, hence it has rational singularities. In the process, we strengthen the result of Chardin and Ulrich. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that they are, in fact, strongly F-regular. 

In the mid-1990s, Hochster and Huneke showed that generic determinantal rings are strongly F-regular; however, their proof is quite involved. The techniques that we discuss will allow us to give a new and simple proof of the strong F-regularity of generic determinantal rings defined by maximal minors. Time permitting, we will also share new proof of the strong F-regularity of determinantal rings defined by minors of any size. This is joint work with Yevgeniya Tarasova.