Perelman turned down a million dollars for proving the Poincaré conjecture
In 2010 the Clay Mathematics Institute offered Grigori Perelman US$1 million for settling a question Henri Poincaré posed in 1904. He refused, saying Richard Hamilton, whose Ricci flow program he had completed, deserved equal credit. As of 2026, no other Millennium Prize Problem has a verified solution.
The question concerns finite spaces that look like ordinary three-dimensional space up close. Poincaré guessed that if every loop in such a space can be shrunk continuously to a point, the space must be a three-dimensional sphere, the boundary of a ball in four dimensions. In technical terms, every closed, connected three-manifold with trivial fundamental group is homeomorphic to the 3-sphere. Efforts to prove it drove much of 20th-century geometric topology.
Two-dimensional surfaces show what the words mean. A ball's surface is closed, without boundary and finite, and connected, in one piece, and any loop on it can be pulled tight to a point. A torus is also closed and connected, but some loops around it cannot be shrunk, so its fundamental group is nontrivial, and since that property survives any continuous reshaping, the two surfaces are topologically different. In two dimensions the analogue of the conjecture follows from a classification of surfaces known in various forms since the 1860s, but higher dimensions lack such a simple catalogue.
The groundwork came in the 19th century. Bernhard Riemann and Enrico Betti introduced Betti numbers, lists of nonnegative integers attached to a manifold, and Riemann showed they fully describe closed connected surfaces. In his 1895 paper Analysis Situs, Poincaré showed this fails in three dimensions: he invented the fundamental group and gave three-dimensional examples sharing Betti numbers but differing in that group. He wondered whether the group alone might suffice, but did not pursue it, remarking only that it would demand lengthy and difficult study.
Perelman posted his papers on the arXiv in 2002 and 2003, developing new Ricci flow techniques that also proved William Thurston's stronger geometrization conjecture, and other mathematicians spent years writing out detailed versions. Science named the proof its Breakthrough of the Year for 2006. Hamilton received the Leroy P. Steele Prize in 2009 and the Shaw Prize in 2011.
Source: Poincaré conjecture
