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Poincaré conjecture (view source)
Revision as of 08:38, December 22, 2009
, 08:38, December 22, 2009→Perelman's Proof
Perelman's proof of the conjecture relied on a concept called "Ricci flow," which is time-dependent deformation of a space which expands areas which are curved "negatively" (ie, two particles traveling in the same direction diverge) and contracts area which are curved "positively" (ie, two particles traveling in the same direction converge).
Perelman's proof of the conjecture relied on a concept called "Ricci flow," which is time-dependent deformation of a space which expands areas which are curved "negatively" (ie, two particles traveling in the same direction diverge) and contracts area which are curved "positively" (ie, two particles traveling in the same direction converge).
The effect of the Ricci flow has much in common, mathematically at least, with the diffusion of heat in a surface, a concept called [[entropy]].
It has been long understood that Ricci flow on a 3-manifold which satisfies the conditions of the conjecture will result in a three manifold, unless the flow "pinches" the manifold. Perelman was able to show that these "pinches," or singularities, cannot happen.
The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]].
[[Category:Mathematics]]
[[Category:Mathematics]]