Though not quite as exciting as a possible proof of the Riemann hypothesis, a paper on realizing the mapping class group of a surface as a subgroup of the group of self-homeomorphisms  caught my eye a couple of days ago. Recall that the set of all homeomorphisms from a surface S to itself form a group that I’ll call Aut(S) or the automorphism group. The mapping class group of S, which I’ll write Mod(S), is the quotient of this group by isotopies of the surface. The quotient construction implies a homomorphism from Aut(S) onto Mod(S). The paper above proves that for a surface of genus at least two, there is no reverse homomorphism from Mod(S) into Aut(S) such that composing the maps produces the identity on Mod(S).
I had though this was already known, but apparently it was only known for genus at least 5 (or 3 if you replace Aut(S) with the group Diff(S) of diffeomorphisms.) The introduction to the paper lists the previously known results. You can ask a similar question for subgroups of Mod(S) as well, i.e. whether there’s a map from the subgroup into Aut(S) or Diff(S) that composes to the identity on that subgroup. For infinite cyclic groups, the answer is an almost immediate yes (just pick any representative for a generator). For finite subgroups, the answer is a much harder to prove yes; this is the Nielsen Realization Theorem (which was proved by Steve Kerckhoff, not by Nielsen). I wonder which infinite, non-cyclic subgroups of Mod(S) can be realized as groups of homeomorphisms?
The proof in the paper examines a certain relation that is discussed in Farb and Margalit’s primer on mapping class groups: Given a separating loop in a surface, you can write a Dehn twist around that loop as a composition of Dehn twists along loops in the interior of one of the complementary components or along loops in the other complementary component. Since both give you a Dehn twist along the same loop, these give you a relation. The authors then use the machinery defined in  (where the “no” answer for genus at least 5 is proved) to show that such a relation cannot exist in Aut(S).
 V. Markovic, Realization of the mapping class group by homeomorphisms. Inventiones Mathematicae
168 , no. 3, 523–566 (2007)