


Lecture series on Lie groups
Date and Time: Six Mondays at 4 pm
Tea: 3.50 pm
Venue: A1A2, 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 subalgebras of the Lie group and Lie subgroups of the Lie groups, also with some basic results from algebraic topology.
Ph. D. Defence
Tuesday, 14 February 2023, 11.30 am
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Host: Shripad Garge
Venue: Ramanujan Hall
Speaker: Arghya Pramanik
Title: A Study in Topology and Geometry
Department of Mathematics
Geometric Group Theory Learning Seminar
Tuesday, 14th Feb 2023, 12:30 PM
Host: Rekha Santhanam
Venue: Room 216
Speaker: Radhika Gupta (TIFR)
Affiliation: TIFR, Mumbai
Title: Introduction to the seminar series
Abstract: I will give a (biased) introduction to geometric group theory,
mentioning some questions and some sample theorems.
Algebraic groups seminar
Tuesday, 14 February 2023, 4 pm
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Venue: Ramanujan Hall
Host: Shripad M. Garge
Speaker: Shripad M. Garge
Affiliation: IIT Bombay, Mumbai
Title: Some examples I
Abstract: We will see some examples to understand some of the basic concepts in the theory of linear algebraic groups.
Statistics seminar
Wednesday, 15 February, 8.00 am
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Venue: Zoom
Host: Ayan Bhattacharya
Speaker: Ayan Bhattacharya
Affiliation: IIT Bombay
Title: Online lecture series on extreme value theory
Abstract: In Part I of this lecture series, I will try first to provide an overview of the modelling aspects, avoiding mathematical details as much as possible. A basic understanding of probability is a must. Basic real analysis is also a must (convergence of a sequence and sequence of functions are essential). It will involve some knowledge of the convergence concepts in probability. I will try to explain this whenever needed and feel free to ask any questions in case of any confusion. However, a good understanding of the convergence concepts in probability and functions will make the lectures more accessible and appealing.
Algebraic Geometry and Commutative Algebra seminar
Tuesday, 15 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 for Flatness
Commutative Algebra Seminar
Thursday, 16 February, 4 pm
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Venue: Ramanujan Hall
Host: Tony Puthenpurakal
Speaker: Prof. R. V. Gurjar
Affiliation: IIT Bombay
Title: Positively Graded Domains
Abstract: I will continue my lectures on this topic. Following results will be discussed. 1. Demazure's construction of normal affine positively graded domains. Some applications of this will be discussed. 2. Flenner and Keiichi Watanabe's rationality of singularities criterion for positively graded affine domains. 3. A very general result I conjectured around 1990 and proved by O.Mathieu In 2002 will be discussed. It has some new consequences for rings of invariants of reductive algebraic group action on an affine space. 4. Divisor Class Groups of positively graded domains. Works of Brieskon Flenner, Samuel, SchejaStorch, Anurag Singh etc, will be mentioned. The connection with the topology of these results will be discussed.
Ph. D. Defence
Friday, 17 February 2023, 11.00 am
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Host: Jugal Verma
Venue: Ramanujan Hall
Speaker: Saipriya Dubey
Title: Tight Hilbert polynomials and Frational local rings