# Accessible proofs of contemporary results in mathematics

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.

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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". –  Igor Rivin Nov 7 '13 at 13:33
–  Waldemar Nov 7 '13 at 13:41
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 low-level 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... –  Joël Nov 7 '13 at 14:44
Another problem is that it is not clear to me what a "strong result" is. –  Deane Yang Nov 7 '13 at 14:45
Igor, I'm curious about what results you have in mind. –  Deane Yang Nov 7 '13 at 14:47

I think "Primes is in P" is accessible to everyone.

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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.

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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. –  Jörg Neunhäuserer Nov 9 '13 at 13:10

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.

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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

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My favorite from the last few years is the proof by Hugo Duminil-Copin and Stanislav Smirnov that the connective constant of the honeycomb lattice is $\sqrt{2+\sqrt{2}}$: http://arxiv.org/abs/1007.0575

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