TopologySolved
Poincaré Conjecture
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
Solved by Grigori Perelman (2003). He declined the $1M prize.
Overview
The Poincaré Conjecture stated that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. It is the only solved Millennium Prize Problem.
Why It Matters
It transformed 3-manifold topology and validated powerful new tools in geometric analysis and Ricci flow.
Current Status
Proved via Ricci flow with surgery by Grigori Perelman in 2003.
What a Solution Would Require
Solved. The proof uses Hamilton’s Ricci flow program completed by Perelman.