Math 110.619: Introduction to Lie Groups and Lie Algebras, Spring 2020

Course description:
There are many different angles to take in introducing this material as Lie groups and Lie algebras are present in a diverse range of topics, including physics (Gauge theory, Yang-Mills equations) and number theory (Langlands program). The tact we will take, especially early on in the course, will be on the use of symmetry groups in solving certain ODE and PDE - this was in fact a motivation for Sophius Lie to study Lie groups. This is a good place to start because it is not very abstract but clearly indicates the usefulness of symmetry/presence of Lie group and provides historical context.

The first part of the class will have the added bonus of not using the language of differential geometry/manifolds to give students less familiar with these topics time to brush up. After getting some experience and motivation with Lie groups in a concrete setting we will transition to a more abstract/traditional approach. If there is time and depending on the interests of the class we will say something about symmetric spaces from a different source to finish.

Who, where, when:
Instructor: Alex Mramor, amramor1 'at'
Office hours 3-5 Thursdays at Krieger hall 311

The course is scheduled to meet Tuesdays and Thursdays from 9 to 10:15 in Kreiger 204. Since the class size is small depending on everyones schedule we may try to meet at a different time and have lecture in my office.

The text for the start of the class will be Symmetry Methods for Differential Equations, by Peter Hydon, ISBN: 9780511623967.

After spending a few weeks or so with Hydon, we will transition to An Introduction to Lie Groups and Lie Algebras, by Alexander Krylov Jr., ISBN: 0521889693.

Depending on the needs of the students we might also spend a short amount of time between Hydon and Krylov on some preliminaries in manifolds. There are many good books on the subject but if you need to get the definitions quickly searching online for lecture notes might be the better option - feel free to ask me if you are unsure.

Grading: Per department policy, there must be some graded work in 600 level graduate courses so there will be some homework due every couple weeks or so. The homework will be kept at unintrusive levels but will hopefully be interesting.

Academic integrity:
The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of students and/or the chairman of the Ethics Board beforehand. Read the “Statement on Ethics” at the Ethics Board website for more information.

Disability accommodations policy:
Students with documented disabilities or other special needs who require accommodation must register with Student Disability Services. After that, remind the instructor of the specific needs at two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.

Course schedule and assignments (updated often!):

Week 1:
This week we will cover chapter 1 of Hydon (and possibly more). No homework will be assigned.

Week 2:
This week we wrapped up chapter 2 of Hydon on Tuesday and will cover chapter 3 on Thursday.

This week there is homework (assigned 2/4, due 2/11): fill out the details (as in, give a complete exposition) of examples 2.10, 2.11, 2.15, 2.17 and do problems 2.2, 2.5 out of Chapter 2 of Hydon.

Week 3, 4
These weeks we covered more or less chapters 3-5 of Hydon. On thursday 2/20 we will start discussing some differential geometry for our more advanced foray into Lie groups and algebras. Our goal is to discuss the correspondence between Lie groups and Lie algebras - this will take 2 or 3 lectures.

Homework 2 (assigned 2/19, due 3/3): do problems 5.1, 5.5, 5.7, 6.1