What dreams are specific to the academic life?
I have had a dream in which I was giving a lecture or a seminar but realized as I was about to start that I knew nothing about the topic which I was supposed to cover. Huge embarrassment. Of course, some approximation of that has happened to me in real life too, so the amount of fantasy in that dream is quite small, but it is a dream that I never had before becoming a professor, and that I have had several times since then.
Several friends have dreams that as they are giving a talk they suddenly realize that they are naked. Not very original, for sure. People who speak in public know that they must be willing to say something a little bit personal to make their talk interesting, so they are revealing a bit of themselves publicly, thus exposing themselves to possible attack from an unfriendly audience, and the decision to let oneself be vulnerable is scary even if it is by choice. That dream makes perfect sense.
But here is a unique researcher's dream. A friend has had a recurring dream in which he is trying to find a tantalizing error in a proof. He proves a big theorem, but the proof would imply something false, for example, that pi equals three, so he knows that there must be an error somewhere, but where? The dream consists of the frantic search.
And of course, there is the very common, possibly universal dream that happens when we become absorbed in a problem: we start dreaming about it a wake up in the middle of the night with the perfect proof, rush to turn on the light and jot down the main points before they vanish in the fog of dreams of the past, only to find a scribble of vacuous nonsense by our bedside when we wake up the next morning.