Introduction

Lie theory

0.00 This course is an introduction to the theory of Lie groups and Lie algebras, which was initiated by the Norwegian mathematician Sophus Lie in the 19th century, and then developed by Cartan, Killing, Weyl and others throughout the 19th, 20th and into the 21st century, and continues to be a source of ideas and inspiration for mathematicians and physicists today.

0.30 The first thing you need to know about Lie theory is that it's pronounced "Lee", not "Lye" (it's named after a guy called Lie). The second thing you need to know is that it's about the interplay between analysis and algebra; it uses tools from analysis and calculus to prove things about groups and their representations. So if you're more of an "algebra person" or more of an "analysis person", the likelihood is that you'll be discomfited by something during this course. Don't worry about this: it's perfectly normal. You should think of it as an opportunity to revisit your background in whichever of the subjects you don't like as much, and appreciate it being used in action in Lie theory, where it might suddenly seem more relevant or intuitive than before.

1.30 The third thing you need to know is that Lie theory is much too big a subject for me to cover in a 10 week course like this one. The way I've structured the course will hopefully allow you to focus on those parts you find most interesting.

Course structure

Each week

Pre-class videos and exercises

2 live sessions

Post-class problems

Each fortnight

Submit post-class problems

Live workshop session

Over the whole term

In-depth project (approx 10 pages written plus a presentation)

due after Christmas

Grade

No exam

Fortnightly problems: 40%

Presentation during workshop: 10%

In-depth project 30% written, 20% presentation

2.00 Each week there will be 2 "live sessions": these are the contact time we have with one another. As preparation for these sessions there will be some pre-class videos you need to watch, and some pre-class exercises for you to think about. In the live sessions:

• we will talk a bit about the exercises, and try to solve them together;

• we'll go through some worked examples as a group;

• I might ask you some questions to see how much you've understood;

• you can ask me questions if there is stuff you're confused about.

They are discussion sessions for us to communicate about the material; it's not an opportunity for me to lecture.

3.10 After the live sessions there will be post-class problems which you can now solve, armed with the ideas and techniques from the videos and live sessions. These will be the problems you submit for your assessed coursework. I will set more problems than you need to do and you will need to submit 4-5 of these every fortnight: you can pick which ones you want to submit. This is the first aspect of the structure which will let you focus on what you're interested in. Each fortnight we will have an additional live workshop session for you to present your solutions. This coursework component counts for 40% of the grade.

EDIT! Since recording the video, I found that the assessment model is different from what I thought. What is written here is now correct. During one of the fortnightly workshops you will give a presentation explaining your solution to one of the problems; this presentation will count for 10% of your grade.

4.15 The course doesn't have an exam, so the remaining 50% will come from one in-depth project which will give you the chance to focus in on the aspect of the course that interests you most: do some further reading beyond the videos, write about it and tell us about it. This will have a written component (around 10 pages, worth 30% of the grade) and a presentation component (worth the remaining 20%).

5.00 Let me reiterate: there will be more problems than you need to do. Of course it'll benefit you to try more than 4-5 per fortnight, because you'll learn a lot by doing them. There will also be additional videos which are completely optional for those who want to dig a bit deeper into aspects of the course which are somewhat off the beaten track, which may be useful for you while you're doing your projects.

6.00 The written project and presentation will be due after Christmas because you may want to focus on aspects of the course which are covered later in term.

EDIT! This year, although you only need to do 4-5 questions, the more and harder questions you do, the better your grade will be. See this page to learn more about how your work will be assessed.