Poincaré Conjecture
Every simply connected closed 3-manifold is homeomorphic to the 3-sphere.
Can a blob of clay with no holes always be reshaped into a perfect sphere? The Poincaré conjecture asserts that every closed, simply connected 3-manifold is homeomorphic to S³. Perelman proved this in 2002–2003 using Hamilton's Ricci flow, an equation that deforms geometry toward uniform curvature while performing topological surgery at singularities. The proof also settled Thurston's geometrization conjecture, completing the classification of compact 3-manifolds.
Formula
A closed simply connected 3-manifold is homeomorphic to the 3-sphere.
Hamilton's evolution equation deforms the metric toward uniform curvature — the engine of Perelman's proof.
Thurston's conjecture classifies all compact 3-manifolds; proved by Perelman as a corollary.
Summary
Can a blob of clay with no holes always be reshaped into a perfect sphere? The Poincaré conjecture asserts that every closed, simply connected 3-manifold is homeomorphic to S³. Perelman proved this in 2002–2003 using Hamilton's Ricci flow, an equation that deforms geometry toward uniform curvature while performing topological surgery at singularities. The proof also settled Thurston's geometrization conjecture, completing the classification of compact 3-manifolds.
