There are several stories of great discoveries been made in dreams (see this wonderful wikipedia list for some famous ones).
Unfortunately I have never had the good luck to have a dream which gave me a creative insight to solve a problem. That isn’t to say that I don’t dream about my research. Last night, for instance, I dreamt of classifying two-dimensional quantum phases using a quantum generalisation of the j-invariant and some other invariant which I, in my dream, for some reason wrote as . When I woke up I quickly checked whether any sense could be made of this. As usual, far as I could see, it is total nonsense. Sigh.
I’ve also been rather envious of those who seem to be able to exploit unconscious cognition. When I was doing my PhD, I was mightily impressed by Michael Nielsen who would sometimes pause in the middle of a conversation and exclaim “I now know how to solve problem x!”. I mean, how cool is that!? Alas, it never worked for me. Oh, I do get “aha” moments rather often, but the result is usually complete junk…
I only get results after hard slog. I have to make lots and lots and lots of mistakes and only then, slowly and gradually, the result emerges, reluctantly and complaining all the while, in its final form.
I can’t remember ever really experiencing a dream or an “aha” moment that turned out to be really correct.