Lie Groups and Their Representations
1st September 2023 - 31st December 2023
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In this trimester, we plan to organize two seminars on Tuesday and Thursday. 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.
1) Euclidean Harmonic Analysis.
2) Representation theory of Compact Lie groups.
3) A crash course on Lie Groups and Lie Algebra.
Speakers
1) Jotsaroop Kaur, IISER Mohali.
2) Parasar Mohanty, IIT Kanpur.
3) Malabika Pramanik, UBC Vancouver.
4) Surya Teja Gavva, Rutgers University, NJ.
5) Varadharajan Muruganandam, IIT Palakkad.
6) Muna Naik, IISc. Bangalore.
7) Aprameyan P, IIT Madras.
8) Amber Habib, Shiv Nadar University.
9) Samir Hazra, HRI Prayagraj.
10) Sreerup Raychaudhuri, TIFR Mumbai.
11) Tanusree Khandai, IISER Mohali.
12) Sunil Mukhi, IISER Pune and ICTS Bengaluru.
Trimester I
From 1st September 2023 to 31st December 2023
Seminars in September 2023
3:00 PM (IST), 1st September 2023, Friday at LHC-203, Zoom and YT Livestream.
Title: Fourier transform on Locally Compact Abelian Groups -I. (Slides are here and video is here)
Speaker: Parasar Mohanty, IIT Kanpur.
Chair for the Talk: Dipendra Prasad, IIT Bombay.
Abstract: In these two talks, we will cover the Fourier series and transform, Fourier Analysis on abelian groups,
Pontryagin Duality, Schwartz space, convolutions, and tempered distributions.
3:00 PM (IST), 5th September 2023, Tuesday, at LHC-108, Zoom and YT Livestream.
Title: Fourier transform on Locally Compact Abelian Groups -II. (Slides are here and video is here)
Speaker: Parasar Mohanty, IIT Kanpur.
Chair for the Talk: Manish Mishra, IISER Pune.
Abstract: Same as the previous talk by speaker on 1st September.
8:00 PM (IST), 7th September 2023, Thursday, at Zoom and YT Livestream.
Title: Plancherel's Theorem -I. (Slides are here and video is here)
Speaker: Malabika Pramanik, UBC Vancouver.
Chair for the Talk: Gadadhar Misra, IIT GN.
Abstract: Plancherel's theorem is a fundamental result in harmonic analysis with myriad applications and connections to other fields. The two talks will be devoted to the derivation of Plancherel's theorem in the Euclidean setting, and to some of its applications.
3:00 PM (IST), 12th September 2023, Tuesday, at Zoom and YT Livestream.
Title: Fourier transform as Spectral Decomposition and Paley Wiener Theorem -I. (Slides are here and video is here)
Speaker: Jotsaroop Kaur, IISER Mohali.
Chair for the Talk: Divyang Bhimani, IISER Pune.
Abstract: We will introduce the spectral theory of differential operators on a suitable Hilbert space. We will define the spectrum of Laplacian in \R^n. We will also view the Plancherel theorem as a spectral resolution of the identity of the Laplacian. We will also define the Fourier transform of generalized functions. We will prove the Paley Wiener theorem for distributions.
3:00 PM (IST), 14th September 2023, Thursday, at Zoom and YT Livestream.
Title: Fourier transform as Spectral Decomposition and Paley Wiener Theorem -II.(Slides are here and video is here)
Speaker: Jotsaroop Kaur, IISER Mohali.
Chair for the Talk: Anisa Chorwadwala, IISER Pune.
Abstract: Same as the previous talk by speaker on 12th September.
8:00 PM (IST), 19th September 2023, Tuesday, at Zoom and YT Livestream.
Title: Applications of Harmonic Analysis. (Slides are here and video is here)
Speaker: Surya Teja Gavva, Queens College, New York.
Chair for the Talk: Amit Priyadarshi, IIT Delhi.
Abstract: Harmonic analysis focuses on deriving quantitative estimates for functions and operators on various interesting spaces such as Euclidean spaces, integer lattices, and Lie groups and as a result interacts with different areas of mathematics including group representation theory, number theory, complex analysis, geometry, spectral and scattering theory, additive combinatorics, ergodic theory, PDE and functional analysis. These ideas can be effectively applied to yield significant results in analysis, differential geometry, analytic number theory, combinatorics, computational complexity, mathematical physics, and also find practical applications in areas such as signal processing, and approximations/simulations of PDEs.
In this talk, we provide a survey of applications specifically related to the Abelian case (Euclidean spaces, Toruses, Boolean cube, etc.) and motivate the importance of studying similar questions within a group setting.
8:00 PM (IST), 21st September 2023, Thursday, at Zoom and YT Livestream.
Title: Plancherel's theorem -II. (Slides are here and video is here)
Speaker: Malabika Pramanik, UBC Vancouver.
Chair for the Talk: Shashi Mohan Srivastava, IACS, Kolkata.
Abstract: Same as the previous talk by speaker on 7th September.
3:00 PM (IST), 27th September 2023, Wednesday, at Zoom and YT Livestream.
Title: Peter Weyl Theorem -I. (Slides are here and video is here)
Speaker: Muna Naik, HRI Prayagraj.
Chair for the Talk: Ramesh Manna, NISER Bhubaneswar.
Abstract: In these two lectures we will prove Peter-Weyl theorem for compact groups which states that for a compact group $G$ the matrix coefficients of the finite-dimensional irreducible unitary representations, suitably normalized, form an orthonormal basis of $L^2(G)$.
3:00 PM (IST), 28th September 2023, Thursday, at Zoom and YT Livestream.
Title: Peter Weyl Theorem - II. (Slides are here and video is here)
Speaker: Muna Naik, HRI Prayagraj.
Chair for the Talk: V. Muruganandam, IIT Palakkad.
Abstract: Same as the previous talk by speaker on 27th September.
Seminars in October 2023
3:00 PM (IST), 17th October 2023, Tuesday, at Madhava Hall, Zoom and YT Livestream.
Title: Harmonic Analysis of Unitary Groups I. (Slides are here and video is here)
Speaker: V. Muruganandam, IIT Palakkad.
Chair for the Talk: Divyang Bhimani, IISER Pune.
Abstract: Haar measure on SU(2) - Unitary dual of SU(2) and SO(3) - Structure Theory of SU(n), SO(n) - Complexification.
3:00 PM (IST), 19th October 2023, Thursday, at Madhava Hall, Zoom and YT Livestream.
Title: Harmonic Analysis of Unitary Groups - II. (Slides are here and video is here)
Speaker: V. Muruganandam IIT Palakkad.
Chair for the Talk: V. M. Sholapurkar, BP, Pune.
Abstract: Irreducible representation of U(n)- Highest weight theorem-Weyl’s integration formula- Character Formula- Parameterizing \hat{SU(n)} by its dominant weights.
3:00 PM (IST), 24th October 2023, Tuesday, at Madhava Hall, Zoom and YT Livestream.
Title: Harmonic Analysis of Unitary Groups - III.
Speaker: V. Muruganandam IIT Palakkad. (Slides are here and video is here)
Chair for the Talk: S. A. Katre, BP Pune.
Abstract: Holomorphic extension of irreducible representations of U(n) into GL(n, C)-Gelfand pair- Spherical harmonics on SO(n)/SO((n −1)).
3:00 PM (IST), 26th October 2023, Thursday, at LHC-108, Zoom and YT Livestream.
Title: Geometric Realisations of Representations for Compact Connected Lie Groups -I.
Speaker: Aprameyan P, IIT Chennai. (Slides are here and video is here)
Chair for the Talk: Debargha Banerjee, IISER Pune.
Abstract: In this lecture, we will look at the Borel-Weil theorem for U(n) in detail and make remarks on the general situation. We’ll mention in passing about generalisations in the non-compact setting including Harish Chandra’s own ideas.
3:00 PM (IST), 30th October 2023, Monday, at Madhava Hall, Zoom and YT Livestream.
Title: Geometric Realizations of Representations for Compact Connected Lie Groups - II.
Speaker : Aprameyan P, IIT Chennai. (Slides are here and video is here)
Chair for the Talk: Chandrasheel Bhagwat, IISER Pune.
Abstract: We will discuss how the geometry of coadjoint orbits of U(n) (symplectic objects - the classical side) correspond to its unitary irreducible representations (the quantum side) via the quantum-classical correspondence, and how integrality conditions occur naturally in this context. We’ll briefly touch upon induced representations in the course of this lecture.
Seminars in November 2023
3:00 PM (IST), 2nd November 2023, Thursday, at Zoom and YT Livestream.
Title: An Introduction to Lie Groups and Algebras - I.
Speaker : Amber Habib, SNU Noida. (Slides are here and video is here)
Chair for the Talk: Amiya Kumar Mondal, IISER Berhampur.
Abstract: Linear Lie groups, exponential map, Lie algebra associated to a linear Lie group, relationship between representations of Lie groups and Lie algebras, examples of maximal tori and their use (e.g. showing certain groups are simple), complexification of real Lie algebras.
3:00 PM (IST), 7th November 2023, Tuesday, at Zoom and YT Livestream.
Title: An Introduction to Lie Groups and Algebras - II.
Speaker : Amber Habib, SNU Noida. (Slides are here and video is here)
Chair for the Talk: Shashi Mohan Srivastava, IACS, Kolkata.
Abstract: Same as the previous talk by speaker on 2nd November.
3:00 PM (IST), 9th November 2023, Thursday, at Zoom and YT Livestream.
Title: An Introduction to Lie Groups and Algebras - III.
Speaker : Amber Habib, SNU Noida. (Slides are here and video is here)
Chair for the Talk: Amiya Kumar Mondal, IISER Berhampur.
Abstract: Same as the previous talk by speaker on 2nd November.
3:00 PM (IST), 21st November 2023, Tuesday, at Zoom and YT Livestream.
Title: An Introduction to Lie Groups and Algebras -IV.
Speaker : Samir Hazra, GITAM University, Bangalore. (Slides are here and video is here)
Chair for the Talk: Muna Naik, IISc Bangalore.
Abstract: After introducing the simple and semisimple Lie algebras, we shall state Weyl's theorem on complete reducibility. Then we introduce the notion of Cartan subalgebra, roots, and the root space decomposition. We shall also need to introduce some notions from the general theory of root systems and the Weyl group. After this much preparation, we shall move to universal enveloping algebra and understand the PBW theorem. Finally, we shall focus on Harish-Chandra's isomorphism and understand Harish-Chandra's theorem on central characters.
3:00 PM (IST), 23rd November 2023, Thursday, at Zoom and YT Livestream.
Title: An Introduction to Lie Groups and Algebras -V.
Speaker: Samir Hazra, GITAM University, Bangalore. (Slides are here and video is here)
Chair for the Talk: Devendra Tiwari, BP Pune.
Abstract: Same as the previous talk by speaker on 21st November.
3:00 PM (IST), 28th November 2023, Tuesday at Madhava Hall, Zoom and YT Livestream.
Title: Lie Groups and the Eightfold Way - I.
Speaker: Sreerup Raychaudhuri, TIFR Mumbai. (Slides are here and video is here)
Chair for the Talk: Sunil Mukhi, IISER Pune.
Abstract: These three lectures will introduce the use of Lie Groups and Lie Algebras to describe systems having symmetries in quantum mechanics and quantum field theory, and then go on to the early theories of strongly-interacting particles, or hadrons.
It will be shown how a large number of such particles discovered in the 1950s could be systematically understood in terms of representations of small unimodular unitary groups. It will then be shown how these ideas led to the discovery of quarks and the sub-structure of hadrons, paving the way for a more fundamental theory of strong interactions.
3:00 PM (IST), 30th November 2023, Thursday at Madhava Hall, Zoom and YT Livestream.
Title: Lie Groups and the Eightfold Way - II.
Speaker: Sreerup Raychaudhuri, TIFR Mumbai. (video is here)
Chair for the Talk: Sachin Jain, IISER Pune.
Abstract: Same as the previous talk by speaker on 28th November.
Seminars in December 2023
3 PM (IST), 1st December 2023, Friday at Madhava Hall, Zoom and YT Livestream.
Title: Lie Groups and the Eightfold Way - III.
Speaker: Sreerup Raychaudhuri, TIFR Mumbai. (video is here)
Chair for the Talk: Manish Mishra, IISER Pune.
Abstract: Same as the previous talk by speaker on 28th November.
3 PM (IST), 5th December 2023, Tuesday at Zoom and YT Livestream.
Title: Highest Weight Representations of Semisimple Lie algebras-I.
Speaker: Tanusree Khandai, IISER Mohali. (video is here)
Chair for the Talk: Manish Mishra, IISER Pune.
Abstract: The finite-dimensional representations of a complex semisimple Lie algebra $\mathfrak g$ are completely reducible. A Cartan subalgebra acts completely reducibly on any finite-dimensional irreducible \mathfrak g-module and decomposes it into simultaneous eigenspaces called weight spaces. With respect to a positive system of roots of \mathfrak g and a fixed Cartan subalgebra of \mathfrak g, one can talk about the "highest weight spaces" of a finite-dimensional representation.
The Theorem of the Highest weight says that the irreducible finite-dimensional representations of a semisimple Lie algebra are characterized by their highest weights and the highest weights can be any "dominant algebraically integral ,.linear functional on the fixed Cartan subalgebra". The Verma Modules are the universal highest weight modules of a semisimple Lie algebra and these play an important role in the theory of the highest weight modules.
In the lectures, we will discuss the Theorem of Highest weights and introduce the notion of Verma modules. The basic results required to understand the statements and proofs of the results will be recalled.
3 PM (IST), 7th December 2023, Thursday at Zoom and YT Livestream.
Title: Highest Weight Representations of Semisimple Lie algebras-II.
Speaker: Tanusree Khandai, IISER Mohali. (video is here)
Chair for the Talk: Anisa Chorwadwala, IISER Pune.
Abstract: Same as the previous talk by speaker on 5th December.
2 PM (IST), 8th December 2023, Friday at Zoom and YT Livestream.
Title: Highest Weight Representations of Semisimple Lie algebras - III.
Speaker: Tanusree Khandai, IISER Mohali. (video is here)
Chair for the Talk: Srilakshmi K, IISER Trivandrum.
Abstract: Same as the previous talk by speaker on 5th December.
3 PM (IST), 26th December 2023, Tuesday at Madhava Hall, Zoom and YT Livestream.
Title: Representations of Lorentz Group with Historical perspective (Work of Dirac and Harish-Chandra).
Speaker: Sunil Mukhi, IISER Pune and ICTS Bengaluru. ( Slides are here and video is here)
Chair for the Talk: Mainak Poddar, IISER Pune.
Abstract: In this lecture I will provide a brief review of the Lorentz groups, and discuss why its unitary representations (and more generally, those of non-compact Lie groups) are necessarily infinite-dimensional. Finally I will review Dirac’s 1945 paper on unitary representations of the Lorentz group, which by the above result are necessarily infinite-dimensional. Dirac referred to his representations as “expansors”, an infinite-dimensional version of tensors.
3 PM (IST), 28th December 2023, Thursday at Madhava Hall, Zoom and YT Livestream.
Title: Representations of Lorentz Group with Historical Perspective (Work of Dirac and Harish-Chandra).
Speaker: Sunil Mukhi, IISER Pune and ICTS Bengaluru. (Slides are here and video is here)
Chair for the Talk: Sachin Jain, IISER Pune.
Abstract: Following his own 1945 work, Dirac suggested to Harish-Chandra to investigate general representations of the Lorentz group. Harish-Chandra’s results were published in 1947. Here he achieved a general characterization of irreducible representations of the Lorentz group, which are infinite-dimensional under some conditions and unitary under additional conditions. In particular, he found classes of representations that he called “expinors”, and proposed a way in which they might describe particles of half-integer spin in quantum theory.
Here concludes the first trimester of the Harish-Chandra lecture series.
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