Lecture series on Lie groups

Monday, 20 Feb 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 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 Geometry Seminar

Tuesday, 21 February 2023, 11.30 am
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Venue: Ramanujan hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Local Criterion of Flatness-II

Algebraic groups seminar

Tuesday, 21 February 2023, 4 pm

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Venue: Room 113

Host: Shripad M. Garge

Speaker: Deepkumar Makadiya

Affiliation: IIT Bombay, Mumbai

 Title: Linear algebraic groups, basic themes- III

Abstract: We begin the study of Jordan decomposition in linear algebraic groups.

Bombay Seminar Series on Probability and Statistics (BSSPS)

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Venue: Ramanujan Hall

Host: Ayan Bhattacharya

Dates: Thursday, 23 February 2023, 11.30 am and 4 pm

            Friday, 24 February 2023, 10 am

Speaker: Prof. Arup Bose

Affiliation: ISI, Kolkata

Title: Non-commutative probability and its connections to random matrices

Abstract: In this talk, the connection of the non-commutative probability to the large-dimensional random matrices will be discussed. This talk will be accessible to a wide audience.

Statistics Seminar

Thursday, 23 February 2023, 2.30 pm

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Venue: Ramanujan Hall

Host: Ayan Bhattacharya

Speaker: Prof. Rishideep Roy

Affiliation: Indian Institute of Management Bangalore

Title: Some results in interacting particle systems

Abstract: We will introduce two disease spread models, the frog model and the rumour model. In the frog model we show co-existence of multiple rumours first for finite types of rumours, followed by infinite co-existence, on Z^d. In rumour models we show the speed of spread of rumours on the integer line.

Bombay Seminar Series on Probability and Statistics (BSSPS)

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Venue: Ramanujan Hall

Host: Ayan Bhattacharya

Dates: Thursday, 23 February 2023, 11.30 am and 4 pm

            Friday, 24 February 2023, 10 am

Speaker: Prof. Arup Bose

Affiliation: ISI, Kolkata

Title: Non-commutative probability and its connections to random matrices

Abstract: In this talk, the connection of the non-commutative probability to the large-dimensional random matrices will be discussed. This talk will be accessible to a wide audience.

PhD defence seminar:

Title: Totally positive field extensions and pythagorean closures of
formally real fields.
Day-Date: Friday, 24.02.2023
Time: 10 AM - 11 AM
Venue: Room no 216, Department of Mathematics.

Bombay Seminar Series on Probability and Statistics (BSSPS)

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Venue: Ramanujan Hall

Host: Ayan Bhattacharya

Dates: Thursday, 23 February 2023, 11.30 am and 4 pm

            Friday, 24 February 2023, 10 am

Speaker: Prof. Arup Bose

Affiliation: ISI, Kolkata

Title: Non-commutative probability and its connections to random matrices

Abstract: In this talk, the connection of the non-commutative probability to the large-dimensional random matrices will be discussed. This talk will be accessible to a wide audience.

Virtual Commutative algebra seminar

Friday, 24 February 2023, 5:30pm IST

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 Host: J. K. Verma

Venue: Gmeet link: meet.google.com/pur-voek-uvo

Speaker: Luis Duarte

Affiliation: University of Genoa, Genoa, Italy 

Title: Perturbations of ideals in local rings

Abstract: Let I be an ideal of a Noetherian local ring R. We study how properties of the ideal change under small perturbations, that is, when I is replaced by an ideal J which is the same as I modulo a large power of the maximal ideal. In particular, assuming that R/J has the same Hilbert function as R/I, we show that the Betti numbers of R/J coincide with those of R/I. We also compare the local cohomology modules of R/J with those of R/I.

For more information and links to previous seminars, visit the website of VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar