Reading seminar in Representation Theory and Algebraic Geometry

This is an informal reading seminar that is intended to explore a wide range of topics in representation theory and algebraic geometry.

This platform will be used to publish previous and upcoming details about the lectures in this seminar series. The meetings will mostly happen on Tuesdays' from 02:00 PM to 03:30 PM. The meetings will be held online. Any change regarding the venue or the timings will be mentioned in this webpage.

If you wish to participate in this seminar please email the organisers at mathstudentsemiitb@gmail.com.

21st October, 2022, A Visit to the Local Langlands Conjecture - 4, Basudev Pattanayak.

Abstract: In this series of talks, we first recall some important results of class field theory. Then we will discuss the representation theory of p-adic groups. Here we will discuss the Hecke algebra attached to Bushnell-Kutzko types. With little basic setup, later we will state the local Langlands Conjecture and its enhancement. For some special cases, we will discuss their proofs.

27th September, 2022, A Visit to the Local Langlands Conjecture - 3, Basudev Pattanayak.

Abstract: In this series of talks, we first recall some important results of class field theory. Then we will discuss the representation theory of p-adic groups. Here we will discuss the Hecke algebra attached to Bushnell-Kutzko types. With little basic setup, later we will state the local Langlands Conjecture and its enhancement. For some special cases, we will discuss their proofs.

22nd September, 2022, A Visit to the Local Langlands Conjecture - 2, Basudev Pattanayak.

Abstract: In this series of talks, we first recall some important results of class field theory. Then we will discuss the representation theory of p-adic groups. Here we will discuss the Hecke algebra attached to Bushnell-Kutzko types. With little basic setup, later we will state the local Langlands Conjecture and its enhancement. For some special cases, we will discuss their proofs.

15th September, 2022, A Visit to the Local Langlands Conjecture - 1, Basudev Pattanayak.

Abstract: In this series of talks, we first recall some important results of class field theory. Then we will discuss the representation theory of p-adic groups. Here we will discuss the Hecke algebra attached to Bushnell-Kutzko types. With little basic setup, later we will state the local Langlands Conjecture and its enhancement. For some special cases, we will discuss their proofs.

26th August, 2022, Okounkov-Vershik approach to the representation theory of symmetric groups -4, Sridhar Narayan.

Abstract: In this series of talks we will bootstrap the representation theory of symmetric groups inductively, following the 2005 revision of Vershik's and Okounkov's seminal paper on the topic.

16th August, 2022, Okounkov-Vershik approach to the representation theory of symmetric groups -3, Sridhar Narayan.

Abstract: In this series of talks we will bootstrap the representation theory of symmetric groups inductively, following the 2005 revision of Vershik's and Okounkov's seminal paper on the topic.

2nd August, 2022, Okounkov-Vershik approach to the representation theory of symmetric groups -2, Sridhar Narayan.

Abstract: In this series of talks we will bootstrap the representation theory of symmetric groups inductively, following the 2005 revision of Vershik's and Okounkov's seminal paper on the topic.

26th July, 2022, Okounkov-Vershik approach to the representation theory of symmetric groups -1, Sridhar Narayan, notes.

Abstract: In this series of talks we will bootstrap the representation theory of symmetric groups inductively, following the 2005 revision of Vershik's and Okounkov's seminal paper on the topic.

17th May, 2022, Bezout's theorem - 2, Suraj Panigrahy.

Abstract: Named after Étienne Bézout, The Bezout's Theorem states that in the Complex Projective Plane two polynomials curves(without any common factors) intetsect at atmost mn points(counting multiplicity), where m, n are the degrees of the two polynomials. To prove this result, we will see basics of Algebraic Geometry and/or Projective Geometry.

26th April, 2022, Bezout's theorem - 1, Suraj Panigrahy.

Abstract: Named after Étienne Bézout, The Bezout's Theorem states that in the Complex Projective Plane two polynomials curves(without any common factors) intetsect at atmost mn points(counting multiplicity), where m, n are the degrees of the two polynomials. To prove this result, we will see basics of Algebraic Geometry and/or Projective Geometry.

19th April, 2022, Tannakian categories and some applications - 3, Ankit Rai, Notes.

Abstract: There is a classical result of Tannaka which provides a way to reconstruct compact Lie groups from the category of its finite dimensional representations. Later, Grothendieck and Saavedra developed the so-called Tannakian formalism which allows for an extension of the above result in the context of algebraic groups. In the first two talks I gave a sketch of the proof of the classical theorem of Tannaka and its reformulation in algebraic groups. In the upcoming talks I shall give a short introduction to neutral tannakian categories and discuss a few applications.

7th April, 2022, Tannakian categories and some applications - 2, Ankit Rai, Notes.

Abstract: There is a classical result of Tannaka which provides a way to reconstruct compact Lie groups from the category of its finite dimensional representations. Later, Grothendieck and Saavedra developed the so-called Tannakian formalism which allows for an extension of the above result in the context of algebraic groups. In the first talk I gave a sketch of the proof of the classical theorem of Tannaka. In the upcoming talks I shall give a short introduction to Tannakian formalism and discuss a few applications.

31th March, 2022, Tannakian categories and some applications - 1, Ankit Rai, Notes.

Abstract: There is a classical result of Tannaka which provides a way to reconstruct compact Lie groups from the category of its finite dimensional representations. Later, Grothendieck and Saavedra developed the so-called Tannakian formalism which allows for an extension of the above result in the context of algebraic groups. In the two talks I shall give a short introduction to Tannakian formalism and discuss a few applications.

24th March, 2022, Introduction to Lie group representations-3, Sudarshan Gurjar, slides.

Abstract: I will introduce representations of Lie groups and prove a few basic properties of them. This will prepare ground for the Peter Weyl theorem to be discussed in the second lecture.

Note ! This talk is an offline talk and is going to take place at the Ramanujan hall in the Math building, IIT Bombay.

17th March, 2022, Introduction to Lie group representations-2, Sudarshan Gurjar, slides.

Abstract: I will introduce representations of Lie groups and prove a few basic properties of them. This will prepare ground for the Peter Weyl theorem to be discussed in the second lecture.

Note ! This talk is an offline talk and is going to take place at the Ramanujan hall in the Math building, IIT Bombay.

10th March, 2022, Introduction to Lie group representations-1, Sudarshan Gurjar, slides.

Abstract: In the first of the two talks that I will give, I will introduce representations of Lie groups and prove a few basic properties of them. This will prepare ground for the Peter Weyl theorem to be discussed in the second lecture.

Note ! This talk is an offline talk and is going to take place at the Ramanujan hall in the Math building, IIT Bombay.

3rd March, 2022, Modular representation theory- 5, Sarjick Bakshi, slides.

Abstact: We have studied the Borel-Weil-Bott theorem and Kempf vanishing theorem. Using Serre duality and Borel-Weil-Bott we will first discuss the irreducibility of the Weyl modules over a field of characteristics 0. Then moving to the prime characteristic field and we will briefly discuss the Steinberg Tensor product theorem. Finally we will wrap up the talk by browsing through some later developments and recent trends. So far the main references have been Jantzen's book ``Representation theory of Algebraic Groups'' and a note by Andersen ``Modular representation of Algebraic groups and Relations to Quantum groups''. For this talk besides the previous references I would also like to point out the following two articles which discuss a few recent big developments in representation theory of reductive groups in prime characteristics and geometric representation theory.

1. Lectures on Geometry and modular representation theory of algebraic groups by Joshua Ciappara and Geordie Williamson https://arxiv.org/pdf/2004.14791.pdf.

2. Modular representations and reflection subgroups by Geordie Williamson https://arxiv.org/pdf/2001.04569.pdf.

24th February, 2022, Modular representation theory- 4, Sarjick Bakshi, slides.

Abstract : We will discuss a few important and classical theorems in the representation theory of reductive algebraic groups like the Borel-Weil-Bott theorem, Kempf's vanishing theorem. The main reference would be Jantzen's book ``Representation theory of Algebraic Groups'' and a note by Andersen ``Modular representation of Algebraic groups and Relations to Quantum groups''.

17th February, 2022, Modular representation theory- 3, Sarjick Bakshi, slides.

Abstract : We will discuss a few important and classical theorems in the representation theory of reductive algebraic groups like the Borel-Weil-Bott theorem, Kempf's vanishing theorem. The main reference would be Jantzen's book ``Representation theory of Algebraic Groups'' and a note by Andersen ``Modular representation of Algebraic groups and Relations to Quantum groups''.

10th February, 2022, Modular representation theory- 2, Sarjick Bakshi, slides.

Abstract : We will discuss a few important and classical theorems in the representation theory of reductive algebraic groups like the Borel-Weil-Bott theorem, Kempf's vanishing theorem. The main reference would be Jantzen's book ``Representation theory of Algebraic Groups'' and a note by Andersen ``Modular representation of Algebraic groups and Relations to Quantum groups''.

3rd February, 2022, Modular representation theory -I , Sarjick Bakshi, slides.

Abstract : We will discuss a few important and classical theorems in the representation theory of reductive algebraic groups like the Borel-Weil-Bott theorem, Kempf's vanishing theorem. The main reference would be Jantzen's book ``Representation theory of Algebraic Groups'' and a note by Andersen ``Modular representation of Algebraic groups and Relations to Quantum groups''.