Lecture series on Lie groups.

Monday, 20 March at 4 pm

Tea: 3.50 pm

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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 a 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 abelianization is trivial is compact.

Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch 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.  

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PDE Seminar

Tuesday, 21 March at 4 pm

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Venue: 216, Mathematics Department

Host: Saikat Mazumdar

Speaker: Dr. Souptik Chakraborty

Affiliation: IIT Bombay

Title: Quantitative stability results of some functional inequalities

Abstract: In 1992, Bianchi-Egnell proved that the "Sobolev deficit" is bounded below by the distance of the function to the manifold of optimizers which gives a quantitative version of Lions result. In recent years, Figalli and other mathematicians obtained quantitative stability results for the critical exponent problem which is the Euler-Lagrange equation associated to Sobolev inequality. In this talk, I'm going to present recent developments in this direction. In particular, I will talk about some progress in the quantitative stability aspects of Hardy-Sobolev inequality. This is an ongoing joint project with Saikat Mazumdar.

Algebraic groups Seminar

Tuesday, 21 March at 4 pm

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Venue: 114, Mathematics Department

Host: Shripad Garge

Speaker: Shripad M. Garge

Affiliation: IIT Bombay

Title: Linear algebraic groups, basic themes. V

Abstract: We discuss the last section in Springer's second chapter. It is on determining a group by its representations.

Commutative Algebra Seminar

Thursday, 23 March 2023, 4 pm

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Venue: Ramanujan Hall

Host: Tony Puthenpurakal

 Speaker: Tony Puthenpurakal

Affiliation: IIT Bombay

Title: Asymptotic primes and analytic spread

Abstract: We discuss two results of Brodmann. The first result shows that over a Noetherian ring, Ass M/I^nM stabilizes for n large. The second result shows that depth M/I^nM is constant for n large.

Algebraic Geometry Seminar

Friday 24 March 2023,  11.00am
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Venue: Ramanujan Hall, Department of Mathematics

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Local Criterion of Flatness-II

GGT Seminar 

Date and time: Friday 24 March, 2023 at 12:35 pm

Venue: Room No 114, Department of Mathematics

Host: Rekha Santhanam

Speaker:  Ankit Rai 

Affiliation: IIT Bombay

Title: The Ping-Pong Lemma.

Abstract : The talk will be based on the chapter titled The Ping-Pong Lemma from the book Office hours with a geometric group theorist. The  chapter consists of ping-pong lemma and it's application(s), a construction of Farey tree and a gentle introduction to hyperbolic upper half plane.

CACAAG seminar

4 PM Friday, 24 March 2023

Please note the deviation in the time from the previous seminars.

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Venue: Ramanujan Hall

Host: Madhusudan Manjunath

Speaker: Madhusudan Manjunath

Affiliation: IIT Bombay

Title: Unimodality and Log concavity in Algebra, Geometry, and Combinatorics: h-vectors of Polytopes and Toric Geometry.

Abstract:  We discuss unimodality questions on h-vectors of polytopes and sketch an approach via toric geometry.

Virtual Commutative Algbera Seminar

Friday, 24 March 2023, 6.30 p.m.

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Venue: meet.google.com/upz-rfxh-uob

Host: J. K. Verma

Speaker: Takumi Maruyama

Affiliation: Purdue University, USA

Title: Uniform bounds on symbolic powers in regular rings via closure theory

Abstract: The containment problem asks: For a fixed ideal I, which symbolic powers of I are contained in an ordinary power of I? We present a closure-theoretic proof of the theorem which says that for ideals I in regular rings R, there is a uniform containment of symbolic powers of I in ordinary powers of I.