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 A1-A2 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:

Hope to see you there!

Best wishes, Dipendra

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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 sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology. 

Commutative algebra and algebraic Geometry seminar
Speaker: R. V. Gurjar
Dates: Tuesday, 8 and 15 November 2022
Time: 3.30-5.00 pm
Venue: Ramanujan Hall

Title  : Positively Graded Domains

Abstract : We will discuss positively graded affine domains over complex
field from algebraic, geometric, and topological viewpoints.
Important results by M. Demazure, I. Dolgachev, H. Flenner, S. Goto, A.
Grothendieck, S. Mori, W. Neumann, P. Orlik- P. Wagreich,
H. Pinkham, Keiichi Watanabe will be mentioned. A very general result
"conjectured" by me in 1990 and proved by O. Mathieu
around 2003 will be discussed. It has important consequences for rings of
invariants of reductive algebraic groups.
Many naturally occurring examples of positively graded domains will be
discussed.

If time permits, I will mention closely related results proved recently by
A. Pramanik-S. Thandar-R.V. Gurjar
about affine surfaces with finite fundamental group at infinity.

Date 16 November at 4 pm.

Speaker: Michel Waldschmidt, University of Sorbonne, Paris

Title
On the degree of hypersurfaces with given singularities

Abstract
Let $n$, $t$ be positive integers and $S$ be a finite set of points in
$\C^n$. We denote by $\omega_t(S)$ the least degree of a nonzero polynomial
vanishing with multiplicity at least $t$ at each point of $S$. The sequence
$(\omega_t(S)/t)_{t\ge 0}$ has a limite $\Omega(S)$ as $t$ tends to
infinity. This invariant was introduced in 1975 for the proof of a Schwarz
Lemma in several variables which occurs in the solution by Bombieri in 1970
of a conjecture of Nagata dealing with a generalization of a transcendence
result of Schneider and Lang. The same invariant occurs in connection with
another conjecture that Nagata introduced in 1959 in his work on Hilbert's
14th problem. It is closely related with Seshadri's constant.

Speaker: Mitul Islam (Heidelberg University)

Time: November 17, Thursday, 5 pm (Indian Standard Time)

Title: Understanding linear groups via real convex projective structures

Abstract: In recent years, real convex projective geometric structures
(which are a generalization of hyperbolic structures) have played an
important role in understanding discrete subgroups of projective general
linear groups. This has connections with several other areas like (higher)
Teichmüller theory and Anosov representations. In this talk, I will
discuss the notion of real convex projective structures and convex
co-compact groups and then study them from the perspective of geometric
group theory. In particular, I will discuss results (joint work with A.
Zimmer) that provide a complete geometric characterization of relatively
hyperbolic convex co-compact groups (with respect to any peripheral
subgroups).


Google Meet joining info
Video call link: https://meet.google.com/jmv-jpox-fcr
Or dial: ‪(US) +1 208-715-5833‬ PIN: ‪835 386 658‬#

Seminar on Linear Algebra

Friday,18 November 2022 at 3.30 pm

Venue: Ramanujan Hall

Speaker: Rajesh Sharma, Himachal Pradesh University, Shimla

Title: On some inequalities related to the Cauchy-Schwarz inequality in 
matrix algebra

Abstract: We focus on the non-commutative versions of some inequalities 
related to the Cauchy-Schwarz inequality in matrix algebra. We discuss 
some inequalities involving positive unital linear maps on matrix 
algebra and demonstrate how positive linear maps can be used to obtain 
bounds for the spreads of matrices. The particular cases provide 
inequalities of statistical interest involving moments of discrete and 
continuous random variables.

Date: Friday, 18th November 2022 @4:35 pm

Venue: Ramanujan Hall

Speaker: T. N. Venkataramana, TIFR Mumbai

Title: Unipotent Generators for Higher Rank arithmetic Groups.

Abstract: Old results of Tits, Vaserstein, Raghunathan and myself say that the  subgroup generated by elementary matrices, in  any arithmetic higher rank group - namely the G(Z) of integer points of a simple algebraic group G defined over Q, is also arithmetic. The proofs rely on constructing a suitable completion of the group G(Q) of rational points and showing that this completion is a central extension of the (finite) adelic completion of G(Q). The other main ingredient of the proof relies on "Moore's uniqueness of reciprocity laws", which  is used to deduce that
this extension is finite.

In this talk I describe a modification of the proof which shows that only the centrality is enough; the technically complicated second step may be avoided.

Virtual Commutative algebra seminar

Date and Time: Friday, 18 November 2022, 5:30pm

Gmeet link: meet.google.com/nrf-fugo-xzp [1]

Speaker: Mina Bigdeli, IPM, Tehran, Iran

Title: Quadratic monomial ideals with almost linear free resolutions

For more information and links to previous seminars, visit the website 
of VCAS:

  https://sites.google.com/view/virtual-comm-algebra-seminar [2]