


Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students
on a topic of basic importance to all of mathematics. Title and abstract are given below.
The course will begin on Monday 17th October at 4pm in the room A1A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run
roughly through the middle of December, so about 8 lectures. Since Monday 24th October is Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.
It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:
https://us06web.zoom.us/j/86048537440?pwd=c2FUTXYzOU84dkpPc1NQTGpDSUYvQT09
Meeting ID: 860 4853 7440
Passcode: 156831
Hope to see you there!
Best wishes, Dipendra

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, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a 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.
This is a reminder for the upcoming talk on Thursday 27 October. https://sites.google.com/math.iitb.ac.in/geometricanalysis/home Speaker: Panchugopal Bikram (NISERBhubaneswar) Time: October 27, Thursday, 5 pm (Indian Standard Time) Title: On the noncommutative Neveu decomposition and stochastic ergodic theorems Abstract: In this talk we discuss the noncommutative analogue of Neveu decomposition for actions of locally compact amenable groups on finite von Neumann algebras. In addition, we assume $G = \Z_+$ or $G$ is a locally compact group of polynomial growth with a symmetric compact generating set $V$, then for a state preserving action $\alpha$ of $G$ on a finite von Neumann algebra $M$, discuss the convergence in bilateral almost uniformly of the ergodic averages associated with the predual action on $M_{*}$ corresponding to the F\o lner sequence $\{K_n\}_{n \in \N}$ (where $K_n = \{ 0, 1, \ldots n1 \}$ for $G= \Z_+$ and $K_n = V^n$ otherwise) . At the end, using these results, we establish the stochastic ergodic theorem. Google Meet joining info: Video call link: https://meet.google.com/nirpyzjhxf Or dial: (US) +1 9786159747 PIN: 439 474 149#
Virtual commutative algebra seminar Speaker: Xianglong Ni, University of California, Berkeley, CA, USA Date/Time: 28 October 2022, 6:30pm IST/ 1:00pm GMT /9:00am ET (joining time 6:20 pm IST) Gmeet link: meet.google.com/ngrtekrvfb [1] Title: Linkage in codimension three Abstract: All perfect ideals of codimension two are in the linkage class of a complete intersection (licci), but in codimension three and beyond this is no longer the case. I will share some ongoing work, joint with Lorenzo Guerrieri and Jerzy Weyman, which illustrates how the theory of "higher structure maps" originating from Weyman's generic ring may be used to distinguish licci ideals within the broader class of perfect ideals of codimension three. For more information and links to previous seminars, visit the website of VCAS: https://sites.google.com/view/virtualcommalgebraseminar [2]