Poincaré conjecture
Geometry and topology
The statement, now a theorem, that every closed simply connected three-dimensional manifold is homeomorphic to the three-sphere.
What is usually left out
Simply connected means every loop can be shrunk to a point. Perelman proved it in preprints of 2002–2003 using Hamilton’s Ricci flow; verification completed by 2006. The Clay Institute awarded its Millennium Prize in 2010 and Perelman declined it — the problem is solved and the prize was awarded, it was simply refused.
Related terms
In the atlas
- Poincaré conjecture — Topology, solved