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:
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.