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- Best books for mathematical background? 13 answers
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.