First, two tales of hubris and folly. The recursion theorem was a big success with my students. We don't usually teach it in this course, but my students went through the standard material like pie and were hungry for more. Other than a ridiculously easy proof of the undecidability of the halting problem, the recursion theorem yields a slick proof that L_min is not in RE (good luck approaching that one without this tool). Riding a euphoric wave of success, I thought I'd improvise a proof that L_min is not in coRE. I thought I had a clever proof using the fixed-point theorem, but it turned out to be wrong. After spending a couple of days in search of a proof, I turned to mathoverflow (an amazing resource!) where Ryan Williams produced a correct proof.
Second, during the last lecture, a student asked if every undecidable language in RE is complete for RE under mapping reductions. I thought the answer should be true, and tried to prove this during the break. Needless to say, I failed to find a proof. After speaking with Menachem Kojman, I learned that in fact the answer is false; this follows froom the Friedberg-Muchnik theorem.
There are two morals to this story:
1. I was blessed with amazing students this semster.
2. Sometimes (if one is lucky!) innocent-sounding questions that come up in undergraduate lectures turn out to be deep, difficult research problems. By all means attempt to tackle them, but keep in mind this caveat.