I just noticed that Greg Moore has been teaching a wonderful course in recent years with the misleadingly bland title of Applied Group Theory. His choice of the topics he wants to cover given here is an excellent one and a good outline for anyone trying to get themselves a serious education in the modern overlap of math and physics.
The problem with this outline is that it’s far too ambitious to cover in a one-semester course, starting just from basics. Moore notes that in 2008 and 2009 versions of the course he only got through roughly half the topics, with students still complaining about the fast pace of the course. In 2013 he only made it through two out 21 topics, but in doing so generated two book-length documents of notes:
- Chapter 1: Abstract Group Theory
- Chapter 2: Linear Algebra User’s Manual
These each contain a wealth of valuable material. I do hope he someday writes up the other 19 topics, but if he does it the way he has been going, the length might turn out to be around 4000 pages, so that might take a while. In the meantime, an account of some of them is available here.
In addition, there’s also a list of suggested topics for term papers, nearly a hundred of them, each with a description of an interesting issue that has been a topic of significant research, with references for where to start learning about the topic.
A gold mine!
It’s far less “sexy” than group cohomology (these days), but y’all can find a very applied “users manual” to Lie Algebra representation theory as 9 course module pdfs on my webpage,
http://mf23.web.rice.edu/
(scroll down to “Lecture notes”)
These are basically a more fleshed-out version of Cahn’s excellent 1984 book _Semi-simple Lie Algebras and Their Representations_. My idea was to try to make representation theory accessible to (upper division) undergrads, which means developing the ideas without bringing in applications to quantum field theory. I haven’t figured out the best way to distribute these yet.
Since the reading list for Physics 618 mentions books related to history and culture associated with group theory, two additional books are suggested below, both available from Dover Publications. (1) The Theory of Groups & Quantum Mechanics by Weyl. This is a classic and since last published in 1930, provides an insight into a time when a lot was happening in quantum physics. Interestingly, when discussing the Dirac equation, and trying to reconcile theory with observed facts, Weyl writes, “indeed, according to it the mass of a proton should be the same as the mass of an electron”. At that point, one recollects that the positron had not been discovered at the time. It was discovered two years later. (2) The Genesis of the Abstract Group Concept by Hans Wussing. This book discusses the history of Group Theory from a purely mathematical perspective and attributes the development of the subject to the theory of algebraic equations (Galois Theory and the solution of the quintic equation), number theory , and geometry.
Thanks for pointing that out, looks great.
-drl
On the topic of being “far too ambitious to cover in a one-semester course” and “complaining about the fast pace of the course”, my PhD supervisor once delivered a Nuclear Theory course with so much content that one his students wrote on the backboard before he came in to the lecture theater “Question: when can a lecturer go faster than light? Answer: when he conveys no information!”
He laughed and took all the students to the pub but it is not clear whether he slowed down.
That said, both “Chapters” are fascinating and inspiring.
I have heard the same joke about the pace of growth of the Physical Review.
There is certainly a wealth of material in these notes, and hopefully Moore will find time to cover the other topics as well. I’m curious about chapter 15 “Kac-Moody and affine Lie algebras, and beyond”. In particular what he has to say about the “beyond” part.