Poincare Conjecture and Ricci Flow | A Million Dollar Problem in Topology

Henri Poincaré posed this conjecture in 1904, asking whether every simply connected closed 3-manifold is homeomorphic to the 3-sphere. Grigori Perelman solved it in 2002-2003, earning a Clay Millennium Prize (which he declined).

Perelman’s 2002-2003 proof combined Hamilton’s Ricci flow with innovative surgery techniques, earning recognition as one of modern mathematics’ greatest achievements. Terrence Tao called it “one of the most impressive recent achievements.”

Perelman introduced surgery procedures to handle singularities that develop during Ricci flow. These singularities threatened to destroy manifolds prematurely, blocking the path to proving extinction implies spherical topology.

Richard Hamilton introduced Ricci flow in 1982 as a way to evolve a Riemannian metric to make manifolds “rounder.” Perelman later used it with surgery to prove the Poincaré conjecture, demonstrating its power for solving topological problems.