I would like to know the story of how the proofs of some "hard" theorems were reached. More specifically, how did the author get to the "right" ideas (especially when the proof is long and complicated)? Though I'm not particularly interested in the ideas per se.
I am mainly interested in the field of analysis but any contribution is welcome.
Let me start with the example of Carleson's theorem on the pointwise almost everywhere convergence of Fourier series of $L^2$ functions, which is an example of a theorem that "everybody" thought it wouldn't be true.
I tried for some years and then I forgot about it before it again came back to me. Then, in the beginning of the 1960s, I suddenly realized that I knew exactly why there had to be a counterexample and how one should construct one. Somehow, the trigonometric system is the type of system where it is easiest to provide counterexamples. Then I could prove that my approach was impossible. I found out that this idea would never work; I mean that it couldn’t work. If there were a counterexample for the trigonometric system, it would be an exception to the rule. Then I decided that maybe no one had really tried to prove the converse. From then on it only took two years or so. But it is an interesting example of “to prove something hard, it is extremely important to be convinced of what is right and what is wrong”. You could never do it by alternating between the one and the other because the conviction somehow has to be there.
This is an excerpt from an interview of Carleson by Raussen and Skau.
You might enjoy Cédric Villani's Birth of a Theorem, a first hand account of his journey to fomulate, and prove, the theorem that ultimately won him the Fields Medal. There is also this lecture with the same title (and a Q&A) on Youtube.