Which Pathway is Best for Theoretical Physics Grad School?

[cesaruelas]

I want to go to grad. school in theoretical physics after I'm done with undergrad (aiming quite high: Perimeter Institue, Cambridge's DAMTP, MIT, etc. I know it's hard but I want to keep optimistic about it). Which pathway would be more appropiate/realistic?

route 1: I self study math courses online from mit ocw or something like that with, possibly, help from a professor who would encourage me and help me out every now and then or on my own and asking questions online etc. (we don't have a math undergrad program so, even though there is a math department, the courses stop at linear algebra/partial diff. eqs. and the rest is primarily external industry research) I have my mind on analysis 1 (both the one with proof introduction and then the other one), analysis 2, algebra 1, differential geometry, Fourier analysis and introduction to functional analysis (from the mit ocw page, that is: http://ocw.mit.edu/courses/mathematics/).

route 2: I self study one or several mathematical physics book (I can get my hands on hassani, byron & fuller, morse & feshbach, courant & hilbert, that sort of books). I'm guessing a professor from the physics department would be more willing to help me out on this one, probably.

route 3: I stop worrying about it and stick to my curriculum (we cover practically all of arfken and of boas over the course of a year except for probability and other chapters that we should already know).

I just don't want to go to grad school and have to take "catching-up" courses or simply have to study a relatively HUGE amount of time because I don't know my maths.

Some background: I'm an engineering physics major, sophomore year, taking math methods 1 this semester (covering Fourier analysis, tensor analysis, series solutions of ode's, pde's and calculus of variations (not necessarily in that order )). Will cover functions of a complex variable, special functions, basic group theory, and basic operator theory next semester. I've already taken calculus 1-3, ode's, and linear algebra.

There is also the concern that I'll burn out after a while or, even worse, just as I'm off to grad school. I also work 20 hours a week, I'm involved in a research project, help out rural high school kids through an online program and do amateur triathlon training (I could cut the triathlon and stop working but I'd rather not if possible).

Opinions? Experiences? Advice?

Thank you.

 

[Nabeshin]

I just don't want to go to grad school and have to take "catching-up" courses or simply have to study a relatively HUGE amount of time because I don't know my maths.

This is probably not going to happen. If you're smart enough to get into a top tier physics grad program for theory, you can probably pick up any maths you need along the way quite quickly. But realistically, what you need is not far beyond the standard undergrad math curriculum. The only real example where you might want to study an extra book would be something like Nakahara's text "Geometry, Topology, And Physics" if you were planning on taking a serious crack at string theory. Otherwise, the advanced topics like group theory, Fourier analysis, algebra, and even differential geometry are covered along with physics material with a strong emphasis on what is relevant to the physicist.