


Lecture series on Lie groups
Monday, 27 Feb at 4 pm
Tea: 3.50 pm
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Venue: A1A2, 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 subalgebras 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
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Venue: Room No 215
Host: Sudarshan Gurjar
Speaker: Nitin Nitsure
Affiliation: TIFR, Mumbai (retd)
Title: Local Criterion of FlatnessII
Commutative algebra seminar
Thursday, 2 March 2023, 4 pm
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Venue: Room 215
Host: Tony Puthenpurakal
Speaker: Sudeshna Roy
Affiliation: TIFR, Mumbai
Title: Asymptotic behavior of the CastelnuovoMumford 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 CastelnuovoMumford 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
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Venue: meet.google.com/mjqahwyoxo
Speaker: Vaibhav Pandey
Affiliation: Purdue University, West Lafayette, IN, USA
Title: Linkage and Fregularity of generic determinantal rings
Abstract: We prove that the generic link of a generic determinantal ring of maximal minors is strongly Fregular, 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 Fregular.
In the mid1990s, Hochster and Huneke showed that generic determinantal rings are strongly Fregular; however, their proof is quite involved. The techniques that we discuss will allow us to give a new and simple proof of the strong Fregularity of generic determinantal rings defined by maximal minors. Time permitting, we will also share new proof of the strong Fregularity of determinantal rings defined by minors of any size. This is joint work with Yevgeniya Tarasova.