# Student seminar

### Representation Theory Student Seminar 2021 Semester 2

Organisers: Ting Xue, Yaping YangGufang Zhao

## 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 3:15-4:15pm

# Exercises

## Schedule

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

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 Examples [S] 1 Benjamin Gerraty Aug 11/18 $SU_2,SO_3,SL_2\mathbb{R}$ [S] 2 Eskander Salloum Aug 25/Sep 1 Homogeneous spaces [S] 3 Yifan Guo Sep 8 Polar decomposition, Graham-Schmidt, Bruhat decomposition [S] 4 Grace Yuan Sep 22/29 Diagonalisation and maximal tori [S] 4 Abraham Zhang Oct 6/20/27 Smooth manifolds, tangent space, one parameter subgroups and the exponential map [S] 5 Adam Monteleone Nov 24 Lie’s theorems [S] 5 Linfeng Wei Fourier series and Representation theory [S] 6 Joshua Culbert Compact groups and integration [S] 7 Justin Tan Maximal compact subgroups [S] 8 Beaudon Anasson The Peter-Weyl theorem I [S] 9 Haris Rao The Peter-Weyl theorem II [S] 9 Haris Rao Functions on $\mathbb{R}^n$ and $S^{n-1}$ [S] 10 Induced representations [S] 11 The complexification of a compact group [S] 12 The unitary groups and the symmetric groups [S] 13 Weiying Guo The Borel-Weil theorem [S] 14 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

Student seminar 2020S2 and 2021S1