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I am a first-year physics graduate student with a deep interest in mathematics. I am specifically interested in algebraic geometry and algebraic topology, and I would like to employ advanced mathematics in my physics research. So far, I haven't chosen an advisor. However, I have been trying to do research on my own without any guidance for a few years and I failed miserably. I do not have any good idea how the best mathematicians do research. I will try my best to summarize one of the problems I am currently facing hoping that someone might help:

I never start by stating some open research problem. I start with the goal of understanding some well-known result or subject. In particular, suppose I want to understand the known ideas about the quantum dynamics of $\mathcal{N}=2$ Super Yang-Mills theory. I try to derive everything starting from scratch e.g. by deriving the SUSY algebra, Lagrangian, and so forth. I try to do this in my own way. I pretend that these results are unknown, and I try to re-derive them. However, it seems I'm not able to go so far in the subject. I'm not sure what the reason is. I believe that $\mathcal{N}=2$ supersymmetric theory is connected to some ideas in mathematics, such as algebraic geometry and supergeometry, so I try to motivate the study of these mathematical subjects using supersymmetric dynamics.

My question is: is this how mathematicians do research? is this approach good? I find that recent papers are impenetrable because they assume I have a lot of background knowledge/ did a lot of computations that I did not do. So I have to study the background. However, when I decide to study the background, it takes me a lot of time because it seems I'm trying to re-invent the wheel, re-deriving everything on my own.

I hope my question is not too vague. Any help would be greatly appreciated.

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    $\begingroup$ You'll get lots of mutually contradictory answers, because people do research in very different ways. I usually begin (like you) by trying to understand some known material but (unlike you) not trying to re-derive everything myself. Rather, I study what others have done but watch for alternative approaches that might (if I'm lucky) be simpler and lead to additional results. (But I know there are also people who definitely start with a specific problem and attack only that.) $\endgroup$
    – Andreas Blass
    Commented Jan 15, 2021 at 18:26
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    $\begingroup$ $\mathcal{N}=2$ and $\mathcal{N}=4$ SYM are extraordinary rich quantum field theories with enormous results in mathematical and theoretical physics. To "derive everything from scratch" in a Bourbaki way I think it's impossible in physics, because she is plagued of heuristic results and bad definitions. I would recommend first to take courses, to read papers and lectures notes, the recent the better. And understand that physics and math work in quite different ways, despite the fact that they have very close ties. $\endgroup$
    – Daniel Castro
    Commented Jan 15, 2021 at 19:06
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    $\begingroup$ You should look for an adviser. It is very unlikely that you can start doing a meaningful research in such subject on your own. $\endgroup$
    – Alexandre Eremenko
    Commented Jan 16, 2021 at 3:16
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    $\begingroup$ Often a good strategy to learn something in math - and I would imagine in physics - is to oscillate back and forth, sometimes quite rapidly, between trying to figure things out yourself and trying to read what others have wrote. Often you will read something, not have any idea what it means, try to prove it yourself, get stuck, then go back, and find that it explains exactly what you need to get unstuck. This is usually more effective than either just reading or just trying things yourself. $\endgroup$
    – Will Sawin
    Commented Jan 17, 2021 at 2:44
  • $\begingroup$ Sorry I cannot formulate it clearly but I have a feeling that this is a very good question with a wrong title. Not vague, just wrong. A suggestion which might also be wrong: make a big list question with the title something like "how would YOU proceed?" $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 20, 2021 at 8:06

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To answer your original question, you could try reading these books:

J. Hadamard The Psychology of Invention in the Mathematical Field (1954)

J.E. Littlewood, "The Mathematician's Art of Work" in Littlewood's miscellany (revised edition, 1986)

I. Stewart Letters to a Young Mathematician (2006)

C. Villani Birth of a Theorem (2015)

However, I suspect that your real question is along the lines of "what's the most effective way of learning algebraic geometry and algebraic topology, so I can use them as tools in my research?"

The resources at https://mathvault.ca/ may be useful for this purpose.

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