Timeline for Proofs that require fundamentally new ways of thinking
Current License: CC BY-SA 2.5
7 events
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| Nov 19, 2013 at 1:55 | comment | added | user41263 | Regarding Terry Tao's Dec 9 '10 at 17:32 comment: (1) He is referring to joint work of Sun-Chin Chu, answering a conjecture of Hamilton that his Harnack estimate is the same as the positivity of some type of curvature. (2) In my opinion, a direct precedent for Perelman's work is Li and Yau's differential Harnack estimate for the heat equation. also motivating Hamilton's estimate. (3) What's striking about Perelman's work is: (i) The profound synthesis of geometry and analysis, to the point where they are nearly indistinguishable (ii) The high degree of subtlety and complexity of the arguments. | |
| Dec 9, 2010 at 20:54 | comment | added | gowers | Terry's answer illustrates a principle relevant to this question: even if a proof as a whole is too complex to count as a good example, there are quite likely to be steps of the proof that are excellent examples. | |
| Dec 9, 2010 at 17:32 | comment | added | Terry Tao | The other example is when Perelman needed a monotone quantity in order to analyse singularities of the Ricci flow. Here he had this amazing idea to interpret the parabolic Ricci flow as an infinite-dimensional limit of the elliptic Einstein equation, so that monotone quantities from the elliptic theory (specifically, the Bishop-Gromov inequality) could be transported to the parabolic setting. This is a profoundly different perspective on Ricci flow (though there was some precedent in the earlier work of Chow) and it seems unlikely that this quantity would have been discovered otherwise. | |
| Dec 9, 2010 at 17:30 | comment | added | Terry Tao | I think there are at least two aspects of the Perelman-Hamilton theory that fit the bill. One is Hamilton's original realisation that Ricci flow could be used to at least partially resolve the Poincare conjecture (in the case of 3-manifolds that admit a metric with positive Ricci curvature). There was some precedent for using PDE flow methods to attack geometric problems, but I think this was the first serious attempt to attack the manifestly topological Poincare conjecture in that fashion, and was somewhat contrary to the conventional wisdom towards Poincare at the time. [cont.] | |
| Dec 9, 2010 at 16:38 | comment | added | gowers | It would be a better example if the proof were easier to understand ... | |
| Dec 9, 2010 at 16:28 | comment | added | Max Lonysa Muller | @ anonymous: how does Perelman's proof require a fundamental new way of thinking? I haven't read it because I know I won't understand it, but I'm just curious why Perelman's approach is so original. Could you please explain that? | |
| Dec 9, 2010 at 16:06 | history | answered | anonymous | CC BY-SA 2.5 |