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Surprisingly, the university (a major tech school) I attended does not offer a mathematical physics class. Consequently, I often get asked by my physics friends what are some good math classes to take to help them in their studies. The problem is two fold. One, I haven't taken more than the minimum physics classes required for a degree...so I can't say I have been exposed to much. And secondly, a math class often covers significantly more than is required; consequently, I am hesitant to suggest taking a full course over self study.

So rather than telling me a generic course physics students should take (e.g. complex analysis), can you tell me what specific topics (e.g. of complex analysis) have utility in physics (of any level). Moreover, can you explain the utility of these topics in the context of physics? The more specific, the better. It would be cool if this could become a repository of information I could point said friends to.

It may be the case that one has to take a whole math class to understand the topics you suggest. However, I believe that a specific list of topics and corresponding uses in physics provides significant motivation to take such classes. Apologies if this is a repeat question.

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Group analysis and symplectic geometry are indispensable to physics. –  Alexei Averchenko Jul 17 '11 at 9:27
    
Out of curiosity, why'd you choose to ask here and not at the Physics StackExchange? :) –  J. M. Jul 17 '11 at 10:18
    
I know that (at least at my university) a large portion of students who have a math background also have a strong physics background, the converse not being as true. Consequently, I presumed I would have a higher chance of obtaining answers here. I think it is clearly an appropriate question for this forum. A better question is why did I not post it on both sites :P –  user13255 Jul 17 '11 at 10:34
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The problem with mathematical physics is that it is very broad. You could do a course of differential geometry with specific accents towards GR and string theory. Or you could do a course on functional analysis, C*-algebras, Von Neumann algebras with accents towards quantum field theory or solid state physics. Or you could do a course on chaos theory and consider applications towards fields like fluid dynamics or other systems. There's such a large spectrum of possibilities. –  Raskolnikov Jul 17 '11 at 12:27
    
Please allow me to offer a plug for two different math texts that approach their subject with physical applications in mind but cover vastly different topics mathematically: one is Lieb & Loss's "Analysis" and the other is Frankel's "Geometry of Physics". –  John M Jul 18 '11 at 3:02

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