Representation theory in Arithmetic and Physics
30th April 2024 - 30th August 2024
In this trimester, we plan to organize two virtual seminars on Tuesday and Thursday every week. The tentative timing of talks would be between 16:00-21:00 Hrs IST.
Urgent changes and extra talks may be possible once in a while.
The Brief Plan of the Third Trimester
1. Introduction to Langlands Program.
(With special emphasize on the theory of Heegner points, Gross-Zagier's work and generalizations.)
Rahul Dalal, John Hopkins University.
Solomon Friedberg, Boston College.
Spencer Leslie, Boston College.
M. Ram Murty, Queen's University, Canada.
C. S. Rajan, Ashoka University.
Sujatha Ramdorai, UBC Vancouver.
Freddy Saia, University of Illinois Chicago.
Vaibhav Vaish, IISER Mohali.
2. Cohomology of Arithmetic Groups.
(The goal on this topic is to discuss recent work of Prasanna-Venkatesh and Venkatesh's conjectures. )Mathilde Gerbilli-Gauthier, McGill University.
Arvind Nair, TIFR Mumbai.
Kartik Prasanna, University of Michigan, Ann Arbor.
Ankit Rai, CMI Chennai.
3. Representation Theory and Theoretical Physics.
(The goal on this topic is to foster interactions between physicists and mathematicians. )
Abhijit Gadde, TIFR Mumbai.
Minhyong Kim, ICMS Edinburgh.
Sunil Mukhi, IISER Pune.
Daniel Persson, Chalmers University.
Sreerup Raichaudhuri, TIFR Mumbai.
More announcements will follow soon...
Trimester III
From 30th April 2024 to 30th August 2024
Week 1
7:00 PM, 30th April 2024, Tuesday
Title: A non-expert’s introduction to Langlands functoriality. (video is here)
Speaker: Minhyong Kim, ICMS Edinburgh.
Chair for the Talk: Dipendra Prasad, IIT Bombay.
Abstract: This is a talk for non-experts by a non-expert who has tried fairly hard to understand the Langlands programme
over many years and thereby sympathises with many difficulties facing beginners. After a brief review of the classification theory
of various kinds of representations, I will try to motivate Langlands functoriality and mention some ‘applications’
to number theory, namely, the study of motives.
Seminars in May 2024
7:00 PM (IST), 2nd May 2024, Thursday
Title: From Harish-Chandra to the Langlands program: an introduction. ( video is here)
Speaker: Solomon Friedberg, Boston College.
Chair for the Talk: Manish Mishra, IISER Pune.
Abstract: Harish-Chandra's 1950 PNAS paper "Automorphic forms on a semisimple Lie group" and his 1968 book "Automorphic Forms on Semisimple Lie Groups" reflect his deep interest in this subject, which builds in essential ways on his study of harmonic analysis on real semisimple Lie groups. The Langlands program connects automorphic forms and harmonic analysis a la Harish-Chandra to constructions in algebra/number theory and algebraic geometry, and is a cornerstone of modern number theory.
In this talk I will give an introduction to this circle of ideas and also provide a brief description of some recent progress.
Week 2
3:00 PM (IST), 7th May 2024, Tuesday
Title: Cohomology of Arithmetic Groups. (Slides are here and video is here)
Speaker: Ankit Rai, CMI, Chennai.
Chair for the Talk: Santosh Nadimpalli, IIT Kanpur.
Abstract: On SL_2, we have three points of view: modular forms, cohomology classes, and automorphic forms.
and we establish the dictionary between these three and we convey the idea that the cohomology of arithmetic groups is a generalization of modular forms.
3:00 PM (IST), 9th May 2024, Thursday
Title: Cohomology of Arithmetic Groups -II. (Slides are here and video is here)
Speaker: Ankit Rai, CMI, Chennai.
Chair for the Talk: Santosh Nadimpalli, IIT Kanpur.
Abstract: We introduce the notion of (g, K)-cohomology and relate the cohomology of upper half plane and Lie algebra cohomology. Introduce the notion of cocompact quotients and Matsushima formula. Finally, specializing to the case of SL(2, R),
we will try to see that modular forms and certain infinite dimensional representation of SL(2,R) are intimately related.
These infinite dimensional representations are called as discrete series.
Week 3
7:00 PM (IST), 14th May 2024, Tuesday
Title: Cohomological Representations -I. ( video is here)
Speaker: Mathilde Gerbelli-Gauthier, McGill University.
Chair for the Talk: Rahul Dalal, John Hopkins University.
Abstract: In this talk, we will discuss cohomological representations, i.e. the representations that contribute nontrivially to the cohomology of arithmetic groups via Matsushima's formula. By looking at their Harish-Chandra modules, will see that they are very well-understood and explicitly classified: for example, given a reductive group, there are finitely many representations that can contribute to cohomology with trivial coefficients.
7:00 PM (IST), 16th May 2024, Thursday
Title: Cohomological Representations -II. ( video is here)
Speaker: Mathilde Gerbelli-Gauthier, McGill University.
Chair for the Talk: Tanusree Khandai, IISER Mohali.
Abstract: In this talk, we will continue our study of cohomological representations with a focus on the group Sp_4.
We will explicitly describe all possible cohomological representations with trivial coefficients, situate them in the Langlands classification, and use them as pretext to say a few words about functoriality in the Langlands program.
Week 4 No talks this week.
Week 5 No talks this week.
Seminars in June 2022
Week 1
3:00 PM (IST), 4th June 2024, Tuesday
Title: Lie Groups and Gauge theories in Particle Physics - I.
Speaker: Sreerup Raychaudhuri, TIFR Mumbai.
Chair for the Talk:
Abstract: U(1) and SU(2) gauge theory.
11:00 AM (IST), 6th June 2024, Thursday
Title: What is an L-function?
Speaker: M. Ram Murty, Queen's University, Canada.
Chair for the Talk:
Abstract: We will give a survey of L-functions, from Euler to Langlands, and indicate how L-functions encode many arithmetic properties of prime numbers.
3:00 PM (IST), 6th June 2024, Thursday
Title: The Sato-Tate conjecture.
Speaker: M. Ram Murty, Queen's University, Canada.
Chair for the Talk:
Abstract: The Sato-Tate conjecture formulated independently by Michio Sato and John Tate is about the distribution of "angles of Frobenius" attached to an elliptic curve. This is now a theorem thanks to the work of many people in number theory and the theory of automorphic representations. I will survey these developments and explain in simple terms the depth and delight contained in the
breakthroughs.
3:00 PM (IST), 7th June 2024, Tuesday
Title: Lie Groups and Gauge theories in Particle Physics - II.
Speaker: Sreerup Raychaudhuri, TIFR Mumbai.
Chair for the Talk:
Abstract: Standard Model and symmetry-breaking.
Week 2
3:00 PM (IST), 11th June 2024, Tuesday
Title: TBA
Speaker: Kartik Prasanna, University of Michigan, Ann Arbor.
Chair for the Talk: Manish Mishra, IISER Pune.
Abstract:
3:00 PM (IST), 13th June 2024, Thursday (90 minute talk)
Title: Conformal and Superconformal Representation Theory -I.
Speaker: Abhijit Gadde, TIFR Mumbai.
Chair for the Talk:
Abstract:
Week 3
3:00 PM (IST), 18th June 2024, Tuesday (90 minute talk)
Title: Conformal and Superconformal Representation Theory -II.
Speaker: Abhijit Gadde, TIFR Mumbai.
Chair for the Talk:
Abstract:
3:00 PM (IST), 20th June 2024, Thursday (90 minute talk)
Title: Conformal and Superconformal Representation Theory -III.
Speaker: Abhijit Gadde, TIFR Mumbai.
Chair for the Talk:
Abstract:
3:00 PM (IST), 21st June 2024, Friday (90 minute talk)
Title: Conformal and Superconformal Representation Theory -IV.
Speaker: Abhijit Gadde, TIFR Mumbai.
Chair for the Talk:
Abstract:
Week 4
3:00 PM (IST), 25th June 2024, Tuesday
Title: Trace formula and Automorphic Reps-I.
Speaker: Rahul Dalal, John Hopkins University.
Chair for the Talk:
Abstract: We'll first heuristically describe Arthur's trace formula and many of its applications, with a particular focus on the most conceptually direct ones involving computing explicit information about families of automorphic representations. We'll then discuss practical considerations towards realizing these applications, emphasizing examples of simple trace formulas and various computational difficulties that arise.
3:00 PM (IST), 27th June 2024, Thursday
Title: Trace formula and Automorphic Reps-II.
Speaker: Rahul Dalal, John Hopkins University.
Chair for the Talk:
Abstract: After summarizing both topics, we'll illustrate some of the power of the stabilization of the trace formula and the endoscopic classification through discussing how it helps us compute explicit statistical information about automorphic representations (as in the framework of the first talk). We'll discuss further practical difficulties created, particularly in computing information about various "transfers" of functions.
3:00 PM (IST), 2nd July 2024, Tuesday
Title: TBA.
Speaker: C. S. Rajan, Ashoka University.
Chair for the Talk:
Abstract:
3:00 PM (IST), 4th July 2024, Thursday
Title: TBA.
Speaker: C. S. Rajan, Ashoka University.
Chair for the Talk:
Abstract:
3:00 PM (IST), 9th July 2024, Tuesday
Title: CM points on modular curves.
Speaker: Freddy Saia, University of Illinois Chicago.
Chair for the Talk: TBA.
Abstract: This talk is aimed to be a gentle introduction to CM points, and moreover Heegner points, on modular curves.
This will be with a slight view towards the applications of interest to this seminar, namely the work of Gross—Zagier and generalizations thereof. We will also spend some time discussing new work on CM points over number fields,
which provides a computationally useful vantage.
.
3:00 PM (IST), 11th July 2024, Thursday
Title: TBA.
Speaker: TBA.
Chair for the Talk: TBA.
Abstract: TBA.
3:00 PM (IST), 16th July 2024, Tuesday
Title: TBA.
Speaker: Arvind Nair, TIFR Mumbai.
Chair for the Talk:
Abstract:
3:00 PM (IST), 18th July 2024, Thursday
Title: TBA.
Speaker: Arvind Nair, TIFR Mumbai.
Chair for the Talk:
Abstract:
3:00 PM (IST), 23rd July 2024, Tuesday
Title: TBA.
Speaker: TBA.
Chair for the Talk: TBA.
Abstract: TBA.
3:00 PM (IST), 25th July 2024, Thursday
Title: Heegner Points and Rep theory - I.
Speaker: Vaibhav Vaish, IISER Mohali.
Chair for the Talk:
Abstract:
3:00 PM (IST), 30th July 2024, Thursday
Title: Heegner Points and Rep theory - II.
Speaker: Vaibhav Vaish, IISER Mohali.
Chair for the Talk:
Abstract:
3:00 PM (IST), 1st August 2024, Thursday
Title: TBA.
Speaker: Sujatha Ramdorai, UBC Vancouver.
Chair for the Talk:
Abstract:
3:00 PM (IST), 6th August 2024, Thursday
Title: TBA.
Speaker: Sujatha Ramdorai, UBC Vancouver.
Chair for the Talk:
Abstract:
3:00 PM (IST), 8th August 2024, Thursday
Title: Periods and L-functions.
Speaker: Spencer Leslie, Boston College.
Chair for the Talk:
Abstract: We introduce the notions of periods of automorphic forms and their relations to special values of L-functions.
We will look at several classical examples, with the goal of motivating the `Relative Langlands program.'
3:00 PM (IST), 13st August 2024, Tuesday
Title:
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Chair for the Talk:
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3:00 PM (IST), 14th August 2024, Wednesday
Title: Arithmetic QFT.
Speaker: Minhyong Kim, ICMS Edinburgh.
Chair for the Talk:
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3:00 PM (IST), 16th August 2024, Wednesday
Title: Lie Groups and Gauge theories in Particle Physics - III.
Speaker: Sreerup Raychaudhuri, TIFR Mumbai.
Chair for the Talk:
Abstract: SU(5) grand unification.
3:00 PM (IST), 20th August 2024, Tuesday
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3:00 PM (IST), 22nd August 2024, Thursday
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3:00 PM (IST), 27th August 2024, Tuesday
Title: Scattering Amplitudes and Automorphic Representations.
Speaker: Daniel Persson, Chalmers University.
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3:00 PM (IST), 28th August 2024, Wednesday
Title: Scattering Amplitudes and Automorphic Representations.
Speaker: Daniel Persson, Chalmers University.
Chair for the Talk:
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3:00 PM (IST), 30th August 2024, Friday.
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More updates will follow soon.