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.

3$\begingroup$ There are some pretty weak PhDs out there, so on those grounds the answer is "no". But if you mean "a good mathematician who is not an expert", then I would say "yes". $\endgroup$ – Igor Rivin Nov 7 '13 at 13:33

2$\begingroup$ Related: mathoverflow.net/questions/129759/… $\endgroup$ – Waldemar Nov 7 '13 at 13:41

4$\begingroup$ Already 3 votes to close ? I think this is an interesting soft question, and not an exact duplicate of the question 129759 : there is a difference between a proof understandable by an undergraduate, and a proof understandable by most active mathematicians. The former supposes a lowlevel proof (in the sense of not using any sophisticated notion: no manifolds, no Lie Groups, no representations, no complex analysis, etc.) which will be in general, as a matter of compensation, quite technical. Here the OP is asking for nice theorems outside of my field that I could understand quickly... $\endgroup$ – Joël Nov 7 '13 at 14:44

$\begingroup$ Another problem is that it is not clear to me what a "strong result" is. $\endgroup$ – Deane Yang Nov 7 '13 at 14:45

$\begingroup$ Igor, I'm curious about what results you have in mind. $\endgroup$ – Deane Yang Nov 7 '13 at 14:47
There are a number of such results in combinatorics. The Wilf–Zeilberger method is perhaps the "strongest" one that I can think of off the top of my head. It is strong by any reasonable definition of the word. With a more precise definition of "strong" I could probably come up with others.

$\begingroup$ Dear Timothy Chow ,I would say a result is strong if it is published in an A journal, cited and some experts in the field would consider the result as strong. Very Best. $\endgroup$ – Jörg Neunhäuserer Nov 9 '13 at 13:10
My favorite from the last few years is the proof by Hugo DuminilCopin and Stanislav Smirnov that the connective constant of the honeycomb lattice is $\sqrt{2+\sqrt{2}}$: http://arxiv.org/abs/1007.0575
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.
How about the saturation conjecture? It is a quite strong result, and I believe the proof should be accessible within that time, see www.math.rutgers.edu/~asbuch/papers/sat.ps