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Grigori Perelman

Perelman modified [Richard Hamilton] 's program for a proof of the conjecture, in which the central idea is the notion of the [Ricci flow] . Hamilton's basic idea is to formulate a "dynamical process" in which a given three-manifold is geometrically distorted, such that this distortion process is governed by a differential equation analogous to the [heat equation] . The heat equation describes the behavior of scalar quantities such as [temperature] ; it ensures that concentrations of elevated temperature will spread out until a uniform temperature is achieved throughout an object. Similarly, the Ricci flow describes the behavior of a [tensorial quantity] , the [Ricci curvature tensor] . Hamilton's hope was that under the Ricci flow, concentrations of large curvature will spread out until a uniform curvature is achieved over the entire three-manifold. If so, if one starts with any three-manifold and lets the Ricci flow occur, eventually one should in principle obtain a kind of "normal form". According to [William Thurston] , this normal form must take one of a small number of possibilities, each having a different kind of geometry, called [Thurston model geometries] .

This is similar to formulating a dynamical process which gradually "perturbs" a given square matrix, and which is guaranteed to result after a finite time in its [rational canonical form] .

Hamilton's idea had attracted a great deal of attention, but no one could prove that the process would not be impeded by developing "singularities", until Perelman's [eprints] sketched a program for overcoming these obstacles. According to Perelman, a modification of the standard Ricci flow, called Ricci flow with surgery , can systematically excise singular regions as they develop, in a controlled way.

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