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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!

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    It seems to me now that any future math you learn will be heavily dependent on what area of research you are going into. Is this not the case? $T$hat is, other than recommending bigger and better analysis/PDE/complex analysis texts, I don't know what I would recommend2012-04-22

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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...

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    Why don't you try to study differential analysis? I know nothing about that field, but it should be indispensable if you want to understand any substantial PDE, harmonic analysis, etc.2012-04-23