The problem
Every closed, simply-connected 3-manifold is homeomorphic to the 3-sphere.
The Poincaré Conjecture
historicReferences
History & significance
Posed by Poincaré in 1904 after discovering his own earlier characterization attempt failed. Higher-dimensional analogues fell first (Smale 1961, Freedman 1984 — both Fields work), making dimension three the stubborn holdout: Ricci curvature collapsed, surgeries misbehaved.
The resolution (human proof)
Proved by Grigori Perelman, 2002–2003, in three arXiv preprints proving Thurston's full Geometrization Conjecture via Ricci flow with surgery, overcoming the singularities that stalled Hamilton's programme. Verification occupied three independent teams through 2006. Perelman declined the Fields Medal (2006) and the Clay Millennium Prize (2010) — the only person ever offered either for this problem, declining both. Purely human, purely individual, and possibly the last of its kind.