What are some easily-stated recently proven theorems? [closed]

Solution 1:

Catalan‘s conjecture (a.k.a. Mihăilescu’s theorem):
http://en.wikipedia.org/wiki/Catalan%27s_conjecture

Solution 2:

Well, I'm not sure if this has been confirmed yet, but apparently in March, Ciprian Manolescu claims to have refuted the Triangulation Conjecture in dimensions $ \geq 5$. It's not the simplest result to state, but it's not terribly technical (unlike the proof, I imagine). The conjecture essentially states that "every compact topological manifold can be triangulated by a locally finite simplicial complex," in the language of the linked article. Put less rigorously, you can't necessarily take a nice, compact manifold of high dimension and cut it into triangles that fit together like puzzle pieces.