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Line 58: − 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).+ − 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.+ − + + + + + + +
Poincaré conjecture (view source)
Revision as of 08:35, December 23, 2009
, 08:35, December 23, 2009→Perelman's Proof
===Perelman's Proof===
===Perelman's Proof===
The proof of the conjecture relies 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).
Two very important insights were necessary to turn the idea of Ricci flow into a proof of the Poincare conjecture. First, and this is the lesser of the insights, it was realized that no matter how complicated a manifold might be, the simple knowledge that a region of a three manifold could be bounded by a loop (or sphere) that could be shrunk to a point allows us to know for CERTAIN that Ricci flow on that region will either make it look like a patch of a three sphere after some finite "run time" for the flow, or "pinch" it beyond recognition (a singularity). This insight is due to Hamilton.
The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]].
The second of these insights, which is the result of Perelman, is whenever the flow pinched the manifold, one could perform surgery on the manifold which would remove the pinch but leave the structure of the manifold unchanged otherwise.
It has been long understood that Ricci flow on a 3-manifold which satisfies the conditions of the conjecture will result in a three sphere, 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]]. For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>. This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold and no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).
Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion, Perelman formulated the concept of "Perelman entropy."
[[Category:Mathematics]]
[[Category:Mathematics]]