Trimester III

From 30th April 2024  to 30th  August 2024

Week 1 

7:00 PM, 30th April 2024, Tuesday, at Zoom and YT Livestream.

Title: A non-expert’s introduction to Langlands functoriality.  (Slides are here and 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, at Zoom and YT Livestream.

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, at Zoom and YT Livestream.

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, at Zoom and YT Livestream.

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, at Zoom and YT Livestream. 

Title:  Cohomological Representations -I.  (Slides are here and  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, at Zoom and YT Livestream. 

Title: Cohomological Representations -II.  (Slides are here and 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
     

11:00 AM (IST), 6th June 2024, Thursday, at Madhava Hall, Zoom and YT Livestream.

Title: What is an L-function?   (Slides are here and video is here)

Speaker: M. Ram Murty, Queen's University, Canada.

Chair for the Talk: S. A. Katre, BP Pune.

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, at Madhava Hall, Zoom and YT Livestream.

Title: The Sato-Tate conjecture.   (video is here)

Speaker: M. Ram Murty, Queen's University, Canada.

Chair for the Talk: Manish Mishra, IISER Pune.

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, Friday, at Zoom and YT Livestream.

      Title: Lie Groups and Gauge theories in Particle Physics - I.  (video is here)

Speaker: Sreerup Raychaudhuri, TIFR Mumbai.

Chair for the Talk: Suresh Govindarajan, IIT Madras.

Abstract: In these two lectures, the idea of a gauge theory, based on a Lie group symmetry, will be elaborated, taking U(1) and SU(N) as examples. A brief introduction will be made to the concept of hidden symmetry and the Higgs mechanism, leading to the Standard Model of electroweak interactions. In a subsequent lecture, the extension of Lie group symmetries to get a Grand Unified Theory (GUT) will be described. 

Week 2 

3:00 PM (IST), 10th June 2024, Monday, at Madhava Hall, Zoom and YT Livestream.

Title:  An introduction to the motivic action conjectures.  (video is here)

Speaker: Kartik Prasanna, University of Michigan, Ann Arbor.

Chair for the Talk: Manish Mishra, IISER Pune.

Abstract: This talk will give an introduction to the web of conjectures relating the cohomology of locally symmetric spaces (and the coherent cohomology of Shimura varieties) to motivic cohomology. We will focus on a number of classical examples (Hilbert modular forms, Bianchi modular forms and Siegel modular forms), where we will explicate the conjectures and indicate the relation with the conjectures of Stark and Beilinson on special values of L-functions. 

3:00 PM (IST), 13th June 2024, Thursday, at Zoom and YT Livestream.

Title: Lie Groups and Gauge theories in Particle Physics - II. (video is here)

Speaker: Sreerup Raychaudhuri, TIFR Mumbai.

Chair for the Talk: Poonam Mehta, JNU New Delhi.

Abstract: Same as the previous talk by the speaker on 7th June.

              Week 3 

7:00 PM (IST), 18th June 2024, Tuesday, at Zoom and YT Livestream.

Title: Trace formula and Automorphic Reps-I.   (Slides are here and video is here)

Speaker: Rahul Dalal, John Hopkins University.

Chair for the Talk: Suryateja Gavva, CUNY New York.

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.  

           Week 4 

7:00 PM (IST), 25th June 2024, Tuesday, at Zoom and YT Livestream. 

Title: CM points on modular curves.   (Slides are here and video is here)

Speaker: Freddy Saia,  University of Illinois Chicago.

Chair for the Talk: Debargha Banerjee, IISER Pune.

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.

7:00 PM (IST), 27th June 2024, Thursday, at Zoom and YT Livestream. 

Title: Trace formula and Automorphic Reps-II.  (Slides are here and video is here)

Speaker: Rahul Dalal, John Hopkins University.

Chair for the Talk: Aprameyo Pal, HRI Prayagraj.

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.

7:00 PM (IST), 28th June 2024, Friday, at Zoom and YT Livestream. 

Title: Quantum Unique Ergodicity (QUE) and Arithmetic. (Video is here and Slides are here)

Speaker: Suryateja Gavva, CUNY New York.

Chair for the Talk: Kummari Mallesham , IIT Bombay.

Abstract: The Quantum Unique Ergodicity (QUE) problem concerns the quantization of a classically chaotic system, specifically focusing on the equidistribution of eigenfunctions in the high-energy limit. This talk will survey the QUE problem for arithmetic surfaces, highlighting its connections to automorphic forms and the subconvexity problem for L-functions.

3:00 PM (IST), 1st July 2024, Monday, at Madhava Hall, Zoom and YT Livestream. 

Title: Introduction to Langlands program. (Video is here)

Speaker:  C. S. Rajan, Ashoka University.

Chair for the Talk:  Chandrasheel Bhagwat, IISER Pune.

Abstract: We will give an impressionistic, fuzzy picture (no details, certainly no proofs) of the Langlands program and the developments leading up to the notion of endoscopy, etc. We hope that the talk will be accessible to graduate students. 

3:00 PM (IST), 3rd July 2024, Wednesday,  at Madhava Hall, Zoom and YT Livestream. 

Title:  Introduction to Langlands program. (Video is here)

Speaker: C. S. Rajan, Ashoka University.

Chair for the Talk:  Manish Mishra, IISER Pune. 

Abstract: Same as the previous talk by the speaker on 1st July.

7:00 PM (IST), 11th July 2024, Thursday, at Zoom and YT Livestream. 

Title: Automorphic vector bundles on Sp_4.  (Video is here and Slides are here)

Speaker: Mathilde Gerbelli-Gauthier, McGill University.

Chair for the Talk: Srilakshmi K, IISER Trivandrum.

Abstract:  In preparation for the upcoming talks on the motivic action conjectures, we will introduce automorphic vector bundles on Shimura varieties, with a focus on Sp_4. We will then describe circumstances under which the same Hecke eigensystem can appear in different degrees in the associated coherent cohomology groups. 

11:00 AM (IST), 12th July 2024, Friday (90 minute talk), at Madhava Hall, Zoom and YT Livestream. 

Title: Conformal and Superconformal Representation Theory -I.

Speaker: Abhijit Gadde, TIFR Mumbai.   (Video is here)

Chair for the Talk:  Sunil Mukhi, IISER Pune.

Abstract: I'll give a quick introduction to path integral formulation of quantum field theory starting with finite dimensional integrals. I'll focus on the symmetries of path integral and Noether theorem. I'll then discuss different types of symmetries encountered in physics and the focus on conformal symmetry and its representations.

3:00 PM (IST), 12th July 2024, Friday (90 minute talk), at Madhava Hall, Zoom and YT Livestream. 

Title: Conformal and Superconformal Representation Theory -II.

Speaker: Abhijit Gadde, TIFR Mumbai.  (Video is here)

Chair for the Talk: Sunil Mukhi, IISER Pune.

Abstract: In this lecture, I'll construct unitary representations of the conformal group. The conformal dimension for such representations need to satisfy the so-called unitarity bounds. When these bounds are saturated we get null states in the representations. I'll demonstrate this for free fields and conserved currents. I'll also introduce elements of conformal bootstrap.

3:00 PM (IST), 15th July 2024, Monday (90 minute talk), at Madhava Hall, Zoom and YT Livestream. 

Title: Conformal and Superconformal Representation Theory -III.

Speaker: Abhijit Gadde, TIFR Mumbai.  (Video is here)

Chair for the Talk: Manish Mishra, IISER Pune.

Abstract: I'll start with a review of supersymmetry and show how to marry it with the conformal symmetry to get superconformal symmetry. I'll discuss classification of superconformal symmetry that is relevant for physics and then focus on representation theory for 4d N=1 superconformal symmetry. If possible, I'll briefly discuss the superconformal index.

3:00 PM (IST), 23rd July 2024, Tuesday, at Zoom and YT Livestream.

Title: Heegner Points and the Gross-Zagier Formula.

Speaker:  Vaibhav Vaish, IISER Mohali.   (Video is here and Slides are here)

Chair for the Talk:  Manish Mishra, IISER Pune.

Abstract:  We will begin with a survey of the objects that play a role in the Gross Zagier Formula.
Following Gross, we will also discuss the related representation theoretic picture which has eventually led
to vast generalizations and further conjectures.

7:00 PM (IST), 8th August 2024, Thursday 

Title:  Periods and L-functions.

Speaker:  Spencer Leslie, Boston College.   (Video is here and Slides are here)

Chair for the Talk: Manish Mishra, IISER Pune.

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.'

7:00 PM (IST), 13st August 2024, Tuesday 

Title: Motivic action for Siegel modular Forms.

Speaker: Kartik Prasanna, University of Michigan, Ann Arbor.  (Video is here)

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: This talk will focus on joint work with Horawa in which we study an action of motivic cohomology on the coherent cohomology of Siegel threefolds. We will also discuss the relation between this case and the motivic action conjecture for Bianchi modular forms. 

7:00 PM (IST), 22nd August 2024, Thursday 

Title: The AI Mathematician.     (Video is here and Slides are here)

Speaker: Yang Hui-He, London Institute of Mathematical Sciences, London.  

Chair for the Talk: Sunil Mukhi, IISER Pune.

Abstract: We summarize how AI can approach mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and string theory, we present a number of recent experiments on how various standard machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to combinatorics, and to number theory. At the heart of the programme is the question how does AI help with mathematical discovery.

5:00 PM (IST), 26th August 2024, Monday 

Title: Scattering Amplitudes and Automorphic Representations.   (Video is here and Slides are here)

Speaker: Daniel Persson, Chalmers University.

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: Automorphic forms attached to certain “minimal” representations of Lie groups play a central role in many different contexts, both in mathematics and in physics. In particular, they appear in theta correspondences as well as in string amplitudes.
Such minimal automorphic forms are characterized by having very few non-vanishing Fourier coefficients.
In these lectures I will present an overview of these mathematical structures, with comments on their role in string theory.

More specifically, I will begin by discussing certain aspects of automorphic forms, with emphasis on their Fourier expansions. 

Of particular interest are Fourier-Whittaker coefficients attached to so called degenerate Whittaker models. 

I will then focus “small” representations and discuss recent results on minimal and next-to-minimal representations of simple Lie groups. 

5:00 PM (IST), 27th August 2024, Tuesday 

Title: Scattering Amplitudes and Automorphic Representations.  (Video is here and Slides are here)

 bSpeaker: Daniel Persson, Chalmers University.

Chair for the Talk: Sunil Mukhi, IISER Pune.

Abstract: In the the second lecture talk I will present extensions of these results that suggest the existence of minimal automorphic representations on Kac-Moody groups (infinite-dimensional generalizations of Lie groups).
Along the way I will provide the necessary background on automorphic forms and Kac-Moody groups.
I will also mention conjectural applications of these results in string theory. 

More updates will follow soon.