TITLE: Accessible proofs of contemporary results in mathematics
QUESTION [11 upvotes]: Are there strong results in contemporary mathematical research (last 20 years) which have a proof which every mathematician (holding a PhD) can completely understand within a few days? -- If yes, please give examples. If not, please explain the phenomenon.

REPLY [5 votes]: Manjul Bhargava's proof of the 15 theorem (in Quadratic Forms and Their Applications, Contemp. Math. 272, 1999) is strikingly simple, especially when contrasted with Conway and Schneeberger's original, intricate proof.