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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.

Stefan Kohl
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Jörg Neunhäuserer
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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 07 '13 at 13:33
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    Related: http://mathoverflow.net/questions/129759/modern-mathematical-achievements-accessible-to-undergraduates – Waldemar Nov 07 '13 at 13:41
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    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 07 '13 at 14:44
  • Another problem is that it is not clear to me what a "strong result" is. – Deane Yang Nov 07 '13 at 14:45
  • Igor, I'm curious about what results you have in mind. – Deane Yang Nov 07 '13 at 14:47
  • Dear Deane Yang, 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 – Jörg Neunhäuserer Nov 07 '13 at 16:14
  • Dear Joel, this is exactly the way I would like the question to be understood. – Jörg Neunhäuserer Nov 07 '13 at 16:15
  • Dear Igor, I mean a reasonable mathematician holding a PhD which is not an expert in field. What results do you think about? – Jörg Neunhäuserer Nov 07 '13 at 16:19

5 Answers5

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

GH from MO
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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.

Timothy Chow
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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 09 '13 at 13:10
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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

Johan Wästlund
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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.

Timothy Chow
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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

Per Alexandersson
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