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Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I should take to get to the modern frontiers of mathematical physics research. My mathematical knowledge covers basic graduate analysis, algebra and topology with an emphasis in algebraic geometry, and as for physics, I know up to basic quantum field theory at the level of P&S and basic general relativity at the level of Wald.

However, unlike those fields that I have studied so far, I am not sure what to learn in order to learn the basics to get to the research level, i.e., able to fully comprehend and dissect research papers in the mathematical physics journals. Most mathematical physics books that I have seen so far are only mathematical methods used in physics. To re-emphasize, I don't simply want references of mathematics used in physics, I wish to know the fundamentals that mathematical physics have to master and concrete examples of modern topics in the field. However, if it is essential, I would like to know what the main relevant mathematical topics are.

I suppose one of the main things I'm confused about is, before one does any actual research, what exactly is the difference between the training/preparation for a mathematical physicist and a pure mathematician? It seems that mathematical physicists basically just learns mathematics, except it's not focused in a particular field and has some physical applications.Do mathematical physicists often get insights behind the way a physicist thinks about problems as well?

Thus, for my main question: What specific books/papers should I start reading to understand the fundamentals of mathematical physics at this point and in what order should I read/study them?

As for side questions: I do not really understand the basic knowledge that a mathematical physicist should have. Do they specialize in a particular area of mathematics or is it mostly topology and geometry or must they know other applicable areas such as functional analysis as well and to what depth? Would me continuing to self-study algebraic geometry be compatible with learning mathematical physics at the same time? What main fields are there now and what advanced books/papers could I read regarding them after learning the fundamentals as addressed in my previous question? What are the most relevant mathematical topics? Off the top of my head, I can think of mostly functional analysis and topology and geometry.

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marked as duplicate by Qmechanic Oct 10 '14 at 0:28

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    $\begingroup$ I don't think it is a duplicate because I'm not looking for books for a background to read more advanced physics texts. I'm asking for a guide to the subject of mathematical physics specifically. Moreover, I believe I have sufficient background in algebraic topology and differential topology and lie algebras already, so that is not what I am asking for $\endgroup$ – FireBlade Oct 10 '14 at 0:39
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    $\begingroup$ Who is the best mathematical physicist you know at your university or anywhere else in the world? Storm trough the office door of that person (if needed tackling the entire departmental staff that tries to stop you), fall on your knees and don't stop begging until he/she takes you as an apprentice. This, by the way, is the one and only proven success recipe for a science career in ANY field. $\endgroup$ – CuriousOne Oct 10 '14 at 0:46
  • $\begingroup$ Lol, I suppose I could ask to do research with them next term. But for now, what would be the things I could do in accordance with my questions above? $\endgroup$ – FireBlade Oct 10 '14 at 0:50
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    $\begingroup$ Seriously, if I have learned anything about early indicators for career success/failure by watching others, it's that the time a top notch mentor is willing to invest in a student is one of the best measures. You can get a thousand book titles from us, but if nobody who is right up there with the best thinks that you are a good use of their time... then you may want to pick a different field. I watched a highly gifted freshman throw a potential science career away because she didn't take the offer of a Harvard professor to coach her from day one! Don't be foolish, take the mentor test! $\endgroup$ – CuriousOne Oct 10 '14 at 1:01
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    $\begingroup$ Hi @Fireblade. Welcome to Phys.SE. This question fits poorly on Phys.SE for various reasons, e.g., it tends to be too broad and primarily opinion-based. I close it as a duplicate, not because it is necessarily an exact duplicate, but to point in the right direction. $\endgroup$ – Qmechanic Oct 10 '14 at 6:23