# Student seminar

### Representation Theory Student Seminar 2021 Semester 2 and 2022 Semester 1&2

Organisers: Ting Xue, Yaping YangGufang Zhao

Student organiser (2022S2):  Davood Nejaty

## Overview

This is a learning seminar on representation theory and related topics, aimed at 3rd year undergraduate students, MSc students, as well as PhD students in mathematics or related fields.

In the learning seminar, participants are expected to learn a topic based on the reference given, and present their work to the other participants. The audience are expected to ask questions and make comments during the presentation. Discussions tangential to the topics are welcome. The idea is for all participants to actively learn and discuss together.

This semester the topic will be Lie groups and their representations. We will follow the chapter on Lie groups by Graeme Segal in the reference below.

Reference: Carter, Roger; Segal, Graeme; Macdonald, Ian, Lectures on Lie groups and Lie algebras. With a foreword by Martin Taylor. London Mathematical Society Student Texts, 32. Cambridge University Press, Cambridge, 1995.

Time: Wednesdays 11-12

# Exercises

## Schedule (2022S2)

Time: Wednesdays 11-12

Aug 17 Haris Rao The Peter-Weyl theorem

## Schedule (2022S1)

Time: Mondays 11-12

Student organisers (2022S1):  Weiying Guo and Davood Nejaty

May 30 Beaudon Anasson Maximal compact subgroups

May 23 Zhongtian Chen  Compact groups and integration, continued

May 16 Zhongtian Chen  Compact groups and integration

May 9 Kevin Fergusson Fourier series and Representation theory

May 2 Davood Nejaty Review, continued

Grace Yuan Bruhat decomposition Notes

Apr 11 Organisational meeting and review of previous topics

## Schedule (2021S2)

Time: Wednesdays 3:15-4:15pm

Dec 1 Linfeng Wei Lie’s theorems, continued. Notes

Nov 24 Linfeng Wei Lie’s theorems

Oct 27 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map, continued. Notes

Oct 20 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map, continued.

Oct 6 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map

Sep 29 Abraham Zhang Diagonalisation and maximal tori, continued. Notes

Sep 22 Abraham Zhang Diagonalisation and maximal tori

Sep 8 Grace Yuan Polar decomposition, Graham-Schmidt

Sep 1 Yifan Guo Homogeneous spaces, continued. Notes

Aug 25 Yifan Guo Homogeneous spaces Notes

Aug 18 Eskander Salloum $SU_2,SO_3,SL_2\mathbb{R}$, continued. Notes

Aug 11 Eskander Salloum $SU_2,SO_3,SL_2\mathbb{R}$

Aug 4 Benjamin Gerraty Examples

 Date Topics Reference Speaker Aug 4, 2021 Examples [S] 1 Benjamin Gerraty Aug 11/18, 2021 $SU_2,SO_3,SL_2\mathbb{R}$ [S] 2 Eskander Salloum Aug 25/Sep 1, 2021 Homogeneous spaces [S] 3 Yifan Guo Sep 8, 2021 Polar decomposition, Graham-Schmidt, Bruhat decomposition [S] 4 Grace Yuan Sep 22/29, 2021 Diagonalisation and maximal tori [S] 4 Abraham Zhang Oct 6/20/27, 2021 Smooth manifolds, tangent space, one parameter subgroups and the exponential map [S] 5 Adam Monteleone Nov 24/Dec1, 2021 Lie’s theorems [S] 5 Linfeng Wei May 9/2022 Fourier series and Representation theory [S] 6 Kevin Fergusson May 16, 23/2022 Compact groups and integration [S] 7 Zhongtian Chen May 30/2022 Maximal compact subgroups [S] 8 Beaudon Anasson Aug 17, 2022 The Peter-Weyl theorem [S] 9 Haris Rao Functions on $\mathbb{R}^n$ and $S^{n-1}$ [S] 10 Yuhan Gai Induced representations [S] 11 Amit Ben Harim The complexification of a compact group [S] 12 Ali Khalili The unitary groups and the symmetric groups [S] 13 Weiying Guo The Borel-Weil theorem [S] 14 Ali Khalili References [S] Graeme Segal, Lie groups, in Carter, Roger; Segal, Graeme; Macdonald, Ian, Lectures on Lie groups and Lie algebras. With a foreword by Martin Taylor. London Mathematical Society Student Texts, 32. Cambridge University Press, Cambridge, 1995.

Past seminar

Special lectures by Arun Ram summer 2021/22

Student seminar 2020S2 and 2021S1