How to understand theorems and their proofs faster?

Question

I am a PhD student works in group theory. I have did the course work required to do research. I am facing one big problem that my learning speed is very slow. I have spent around 6 months on a single research paper to understand that. Is there any way to speedup the learning process?

Question : How to understand theorems and their proofs faster?

Answer 1

I pretty much agree with anyone else who answered to your post: work by examples (Dietrich), what was the motivation/intuition (read about the history of the subject, for example group theory) (Antoine) and yes even David Hilbert admitted being slow understanding mathematics (lhf). So there is hope!
What I found very encouraging is the interaction with human beings: you should surround yourself with mathematicians to whom you can talk, with whom you can communicate. Explain them your problem, ask questions. Even if the person is not knowledgeable in your field, this is even better, since then you really need to explain it - in front of a white or black board - and see if you have understood things yourself! (I once proved something after getting stuck in this way - I talked to my PhD advisor, he did not say a word, was just listening to my explanation on the blackboard and then suddenly - bang, serendipity struck!). Keep up the good work and enthusiasm! Good luck.

Answer 2

Paraphrasing Euclid:

There is no royal road to mathematics.

Understanding takes effort and time.

Answer 3

"Arm" yourself with as much inspiration as you can.

Believe in some importance of what you´re doing, not only for yourself but for many others that could be helped by you.

Try to understand not only the theorem/lemma but also the motivation that led to them.

Find some beauty in that research.

Believe that always something new can be found, no matter how much the field has been researched.

Try to find some concrete examples of some abstract theorems/lemmas.

Answer 4

Of course there is no general answer to this question, as it depends on taste and is "opinion-based". I think, however, that understanding is underestimated, and speed is overestimated, for the question

Question : How to understand theorems and their proofs faster?

One should be able to do a basic example for each theorem and its proof. I know from exams that people study, say, the Sylow theorems and every detail of the proofs, but then fail, if they should name just one Sylow $3$-group in $S_4$. So much for understanding.

Answer 5

It dismays me that I am still wandering around the edge of mathematics (I still get stuck in Rudin's Principles of Mathematical Analysis), so it might not be eligible for me to answer this question. However, through 4 months of effort, I found some basic obstacles, which might help you to dismantle your problem:

• Logic. For me, I can't foster an intuition in the truth table of "If P then Q" and quantifier logic. These are so basic that I didn't realize my difficulties occurred in logic.
• Language \ Linguistics \ Writer's neglect. In my opinion, language is just a faulty duplicate of thoughts. In this situation, I turn to MSE, Google or teachers for help.
• Math itself \ Talents \ Gifts. Like @lhf, understanding takes effort and time.
• Self-blindness. You don't know what you don't know. Be open to any potentially useful information.

Breaking problems into pieces might accelerate learning process.