I'll be doing Introductory General Relativity and Graduate Quantum Mechanics II next semester. I still need to choose 2 (or maybe 3, but I don't want to overload too much) from the following:
Algebraic Topology I: Sounds useful in both mathematical physics, and things like string theory, so leaning towards taking this one.
Number Theory: Sounds interesting, not sure if very useful though. Things like the Riemann-Zeta function will be discussed, and other topics may include elliptic curves etc.
Graduate Complex Analysis: My fear is that it may be too focused on proofs, so not sure how useful it'll be as far as use in physics goes. I'm already familiar with the usual complex analysis methods that physicists use, but sharpening my understanding of it can't hurt. How useful would this be to learning the sort of complex geometry stuff that seems indispensible these days (reimann surfaces and complex manifolds etc)?
Lie Algebras and Representation Theory: Same concern as the one with complex analysis, i.e offered by the math department and designed for math grad students, so not sure how useful it will end up being.
Graduate Statistical Mechanics: Haven't had an undergrad class yet so it may be a bit tough going. Also, its at the same time as the Lie Algebras class.
I would really appreciate it if someone with experience in fields such as string theory, quantum gravity, mathematical physics etc can comment on the relative usefulness of these courses and give their recommendations on which ones to choose. I know that all these topics are potentially useful and I would have to learn them all eventually if I am serious about these fields but which ones are the most immediately useful ones in terms of progressing to the next stage.