Perelman modifies Richard Hamilton’s program to prove the Poincare conjecture. Hamilton’s idea is to formulate a “dynamical process” in which a three-manifold is geometrically distorted governed by a differential equation analogous to the heat equation. The heat equation ensures that concentrations of elevated temperature will spread out until a uniform temperature is achieved. Similarly, the Ricci flow describes the behavior of a tensorial quantity, the Ricci curvature tensor. Hamilton hoped that under the Ricci flow, concentrations of large curvature will spread out until a uniform curvature is achieved over the entire three-manifold. If one starts with any three-manifold and allows the Ricci flow one should obtain a kind of “normal form”.
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