What Are the Best Physics Courses to Apply Advanced Mathematics?

[D_Miller]

I have almost finished my bachelor's degree in mathematics, with a minor in physics. A while ago I decided not to continue with physics on my master's degree, but to focus on pure mathematics while learning a bit of computer science as well.

However, after planning out the two semesters of next year, I still have two 7-week periods open, and I would like to spend them learning a bit of physics. In the last one of them, I have decided to take a course on cosmology, a subject I've always had a soft spot for. I'm still undecided with respect to the first seven weeks though. I think I'll take a directed individual study, and this is where I'd love some advice:

I think it's a shame that I haven't had more chances to apply what I've learned in mathematics to my physics studies. This is why I would like some suggestions on physics subjects that require skills in more exotic kinds of mathematics than just calculus and linear algebra. For example, Hilbert space theory, complex analysis and measure-based probability theory all seem like subjects one could apply to physics. The same probably goes for group theory. My university is already running a semester class on geometry in physics, though, so I assume they would be reluctant to let me conjure up such a course. Furthermore, would prefer to have a course on something which isn't just a different approach to something I've already seen. Here I am thinking of the Von Neumann formulation of quantum mechanics, and the Caratheodory formulation of thermodynamics. Remember, I have to be able to cover it in seven weeks, so it can't be too wide.

Sorry if I've ranted a bit too much. Any suggestions would be greatly appreciated.

 

[mistergrinch]

If you're going to study cosmology, shouldn't you study General Relativity first? You need some fairly sophisticated differential geometry to understand GR (which is useful in many other areas of physics), so the math is interesting. 7 weeks isn't much time, but if you're very motivated and smart you should be able to get the basics. If you want to do some self-study of GR, I highly recommend these lectures: http://www.physics.mcgill.ca/~maloney/514/

 

[Nabeshin]

If you're going to study cosmology, shouldn't you study General Relativity first? You need some fairly sophisticated differential geometry to understand GR (which is useful in many other areas of physics), so the math is interesting.

Of course, from a top-of-the-mountain perspective, this is true, but realistically one doesn't need previous exposure to GR in order to study cosmology. Similarly, one does not need previous exposure to differential geometry to understand general relativity -- you learn it as you go, and only the results which are directly relevant to the theory. Realistically speaking you must do this when studying physics, because to attempt to make the theory axiomatic in the way mathematics is is quite problematic.

 

[D_Miller]

The only non-obvious prerequisite for the cosmology course (which is actually about more than just cosmology) is a course on Riemann geometry and Einstein metrics - a course I'm taking as we speak. This is another reason why I'd prefer that the other course has something to do with a branch of mathematics that isn't geometry. Am I completely wrong in assuming that a discipline like complex analysis could be very useful in physics?