[Leszek]

Something I find fascinating is that we know that P != EXPTIME, and that P <= NP <= EXPTIME, but have managed to prove neither P != NP nor NP != EXPTIME. NP has to be somewhere between them but we have no idea where.

[macleginn]

"NP has to be somewhere between them but we have no idea where" – I guess that this state of affairs won't change much even if we prove P != NP?

[dgs_sgd]

I think that's unclear. A constructive proof of P != NP might yield insights into the "gap" between them.