Self-learning math for a physics major

tut_einstein

I'm a physics major and I'm pretty sure that I want to go into theoretical physics research. Due to scheduling difficulties and such, I haven't been able to systematically study math. I'm am very comfortable with all the basic math that any physics major should know - multivariable calc, PDEs, linear algebra, complex analysis (Riemann sheets, Cauchy integration techniques etc..), tensors, a little bit of group theory...I have also worked enough with differential geometry and topology since I do gravitational physics research.

I want to fill the gaps in my math education by reading books (rigorous, challenging ones) and teaching myself things that I probably missed. I have been following the MIT open courseware site, but since I do have a time constraint, I was wondering which topics I must focus on and which books/lecture notes are good for them (considering that I would like these topics to give me greater insight into physics.) I don't know if this matters, but I am deciding between theoretical condensed matter physics and high energy theory.

Thanks!

homeomorphic

Since I'm feeling lazy, I'll just refer you to Baez's book list:

http://math.ucr.edu/home/baez/books.html

I'll add a couple things. Thurston's 3-dimensional Geometry and Topology has a good discussion of how hyperbolic space ties in with Minkowski space-time. This is a key to understanding why SL(2, C) is the double cover of the identity component of the Lorentz transformations.

This page also has a lot of interesting stuff in it:

http://math.ucr.edu/home/baez/QG.html

twofish-quant

The two traditional gaps that physicists have are numerical mathematics (i.e. most non-trivial PDE's involve some sort of computer simulation) and probability and statistics (extremely important in interpreting observations in cosmology and HEP.)

One problem is that it's an infinite pool, and you'll never learn everything even if you spend your entire life at it.

One thing that I've found useful is to start with the papers that are relevant to the physics that I'm interested and then focus on the mathematics that is used for that physics. One reason this is useful for me is that I'm dreadful at "pure mathematics" but if you can point at that such and such math is used in such and such situation, then I can use my physical intuition to make up for weakness in mathematics.