Block, SkipCaptcha, Upload, Automoderated users, edit, rollback
14,697
edits
Changes
Jump to navigation
Jump to search
mLine 60:
Line 60: − +
→Perelman's Proof: + acute e
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" (i.e., two particles traveling in the same direction diverge) and contracts area which are curved "positively" (i.e., two particles traveling in the same direction converge).
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" (i.e., two particles traveling in the same direction diverge) and contracts area which are curved "positively" (i.e., 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.
Two very important insights were necessary to turn the idea of Ricci flow into a proof of the Poincaré 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 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.
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.