I'm a physics major and I'm pretty sure that I'm interested in theoretical physics research. However, I haven't been able to take math courses in a systematic way (as a math major would) due to scheduling difficulties. Therefore, I would like to fill the gaps in my math education by reading books on my own. Are there any suggestions on what topics I should concentrate on and what books are the best, considering that my main objective is to improve my physics insight. (I am very familiar with rigorous proofs and stuff.)
Thanks!
A short (but might be helpful) answer is to consult the MIT opencourse website. See what you found interesting and you may borrow the book from the library to read it. A good thing is one can learn math entirely on one own with a computer. For physics you need to learn how to do experiments, etc.
A topic you might be interested is differential geometry and geometrical topology if your interest is " theoretical physics research". Similarly you may read functional analysis, PDE, etc....
I also recommend this page which I know since high school and never finished learning from it...