- Time:
- 4:00pm-4:45pm
- Location:
- Room No. A1A2, CDEEP, Department of Mathematics
- Description:
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.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Title : Group Testing Designs : A Combinatorial Marvel Speaker: Bikas K Sinha Retired Professor of Statistics Indian Statistical Institute, Kolkata Abstract : Group Testing is a technique to test a collection of units in several groups, rather than in isolation (i.e., one-at-a-time), in order to ascertain the 'status' of each individual unit in the collection in respect of a well-deﬁned 'feature'. The problem is to plan the testing procedure so as to be able to do so without any ambiguity and with a minimum number of such tests [called Group Tests (GTs)]. The response to be extracted from each unit is on the same 'feature' and it is 'binary' in nature. It is tacitly assumed that 'possession' of the feature by at least one unit within a group [so formed] would render the group 'identiﬁable' as 'possessed'. When this happens, we need to 'open up' the group and go for further exploration of the status of individual units of the group, possibly by sub-group(s) testing or by other means. Other possibility is that the group would be declared as 'passed', and consequently, it would mean that all constituent units within the group would be declared as 'passed' and 'at one go'! This interpretation is accepted for Group Testing schemes to work. When this latter phenomenon happens, the merit of Group Testing prevails over testing individual units in terms of reduction in the required number of tests. For a given collection of units, we may adopt one-at-a-time testing or Group Testing with formation of suitable groups, or even a combination of the two strategies. As is mentioned above, the sole purpose is to minimize the number of GTs in such situations for a given collection of test items. The above formulation looks deceptively simple! Hidden are probabilistic and combinatorial challenges. In this talk, we will discuss some issues related to combinatorial challenges only. Key Words.... Group tests; Hypergeometric group tests, Sequential group tests, t-completeness, Detecting power of order t, Group Divisible Designs, Petersen graphs.

- Time:
- 11:00am-12:30pm
- Location:
- Room No 215, Department of Mathematics
- Description:
We will have *Basudev Pattanayak* speaking in the RTAG seminar from *11:00 to 12:30 AM* on *Thursday(tomorrow)*. Here are the necessary details for his talk: Time: Thursday, 10 November, 11:00 – 12:30 AM. Venue: Room 215, Department of Mathematics. Title: A Visit to the Local Langlands Conjecture.

- Time:
- 12:00pm-1:00pm
- Location:
- Video call link: https://meet.google.com/rps-gagj-wtb
- Description:
Speaker: Prajakta Sahasrabuddhe Title: Minimal dilations for commuting contractions and Q-commutant lifting Date and time: Thursday, November 10· 12:00 pm – 1:00pm Google Meet joining info Video call link:

- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Commutative algebra seminars Please note the unusual time for the seminars this week. Thursday, 10 Nov. @ 2.30 pm Speaker: R. V. Gurjar 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.

- Time:
- 10:30am-11:30am
- Location:
- Room 216, Mathematics Department
- Description:
Title: Totally positive field extensions and pythagorean closures of formally real fields. Day-Date: 11th November 2022, Friday Time: 10:30 am - 11:30 am Venue: Room no 216, Department of Mathematics

Tiltle and Abstract: https://www.math.iitb.ac.in/~seminar/Priyabrata_Mandal.pdf

- Time:
- 12:30pm-1:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Speaker: Ramlal Debnath

Title: Dilations of W-hypercontractions

Abstract: Attached with this email

Date and Time: Friday, November 11· 12:30 pm – 1:30 pm

Venue: Ramanujan Hall

- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
There is a change in timing of Prof. Nitsure's talk tomorrow (11th). He will lecture at 2:30 pm in Ramanujan hall.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Mathematics Colloquium Date: 11 Nov 2022, 4 pm. Speaker: Haruzo Hida University of California, Los Angeles, CA, USA

Title: Background of modular p-adic deformation theory and a brief outline Abstract: The deformation theory of modular forms is increasingly attracting many researchers in arithmetic geometry as it has been an important step in the proof of Fermat's last theorem by Wiles (and Taylor) and supplied an effective tool for the study of the p-adic Birch and Swinnerton Dyer conjecture in the proof by Skinner-Urban of divisibility of the characteristic power series of the Selmer group of a rational elliptic curve by its p-adic L-function under appropriate assumptions. I try to give my background motivation of creating the theory and describe an outline of the theory.

- Time:
- 5:30pm
- Location:
- Gmeet link: meet.google.com/ezs-fiec-gxd
- Description:
**Virtual Commutative Algebra seminars****Speaker:**Ramakrishna Nanduri, IIT Kharagpur**Date/Time:**Friday 11 November 2022, 5:30pmGmeet link: meet.google.com/ezs-fiec-gxd

**Title:**On the regularity of (symbolic) Rees algebra and (symbolic) powers of edge & vertex cover ideals of graphs**Abstract:**In this talk, we discuss the Castelnuovo-Mumford regularity (or regularity) of Rees algebras and symbolic Rees algebras of certain ideals associated with finite simple graphs and we give various combinatorial upper bounds. Also, we study the upper bounds for symbolic and ordinary powers of edge and vertex cover ideals of simple graphs.For more information and links to previous seminars, visit the website of VCAS:

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