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I can talk about math but I can't do math

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> I spoke with one PhD student about this and he told me most graduate student go
> through this and end up looking up answers online,

That's shooting yourself in the foot; I highly advise against it.  If you need feedback, get  it from your peers, adviser and other faculty at your school.  Since they are in live conversation with you, they can solve your problems in a way that leverages your specific knowledge and fills your specific gaps.

> and that even some more post docs might struggle doing problems in textbooks (even > in their area of expertise).

Eeeeh... no.  I mean, some might, but unless the problem in textbook is "prove the entire Poincare-Birkhoff-Witt theorem, no hints" (thanks Howe, I wasted four days on that), postdocs should be able to solve the vast majority of problems in textbooks and monographs of their expertise.

For your situation, the advise is quite standard: find a level of problems that you *can* do consistently and without help, and within that level start *slowly* increasing the difficulty and sophistication.  This may involve some humility and time sink, as the problems you are able to do take you back to the undergraduate.  Write complete solutions, read them back to yourself critically (possibly after a day or two has passed) and try to improve reasoning and exposition.

Find a faculty member that seems non-judgmental on this and ask them for assistance.  It helps, despite what this sub often claims, to tackle some easier competition/putnam problems to push the limits of the more elementary machinery; just don't get too hung up on those.
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These feelings are normal, and also you're not doing enough. You are correct to be uneasy. Knowing math doesn't count unless you can "do" math, meaning that you can use the math for other things. Qualifying and preliminary exams measure exactly that: you need to not just know the math in the sense of being able to remember theorems and use them within the context of the same subject, but you need to be able to see how to use those theorems outside of the immediate context of that subject.

Most of the time you learn how to use a subject by taking the next course which uses the subject. You basically need two courses to learn something: one course that teaches you the thing itself, and another course that shows you where the first thing is used. For example commutative algebra and algebraic geometry go together: algebraic geometry is where commutative algebra is used. In your case, the next class might be probability theory, which uses measure theory, or it might be PDEs, which uses functional analysis.

You don't necessarily have to take the courses in the above order. It's perfectly normal to learn about a subject first and then fill in the background material later, and in fact there are some advantages of doing so since you already know what you would like to use the background material for. You don't need to take courses at all; you can study on your own given enough mathematical maturity. But there is a next topic that you need to follow up with in order to solidify your current knowledge. And then there will be something else needed after that, and so on. Math is hard.

Oh, and I have no idea how I passed my qualifying exam either. You only have to do it once.
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It’s a bit of a dirty little secret that a lotta people (myself included) don’t really learn most of the core graduate material until you study for your qualifying exams, and hence why you cannot do pure math research without a PhD. There’s not really a solution to this other than do every problem in the book. It’s not fun and you will feel very stupid taking hours to solve what should be the easiest problem in the chapter, but it’s just part of the process. Think about how much time and pain it takes kids to learn basic arithmetic and how easy it seems in hindsight. C’est la vie.
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Do you have an intuitive understanding of the concepts or do you learn the theorems and definitions by memorizing?

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