If you run out of (solvable!) problems in your given logic space, just start branching out your space. Until you find yourself in such esoteric spheres, not even your best math co-researcher knows anymore what's happening and vice versa.
Well, that is what mathematicians do, which is why I said "interesting" problems. I mean, I can think of several problems, like "classify projective modules over rings of global dimension 4 that have no zero divisors".
Even popular problems in Langlands like "explicitly find a trace formula for theta groups" will appeal to the fifteen people in the world that can understand what I'm even talking about.
