Grigori Perelman

Grigori Perelman is a Russian mathematician who proved the Poincaré conjecture by completing Richard Hamilton’s program of Ricci flow with surgery.

Ricci flow: A geometric evolution equation that deforms a Riemannian metric in a manner analogous to heat diffusion. Perelman used this process to smooth three-dimensional geometries before analyzing their topology.
Poincaré conjecture: The theorem that every closed, simply connected three-dimensional manifold is homeomorphic to the three-dimensional sphere. This is the conjecture Perelman proved.
Richard S. Hamilton: An American mathematician who introduced Ricci flow and developed it as a tool for studying three-manifolds. Hamilton created the method Perelman extended to prove geometrization.
Clay Mathematics Institute: A private foundation that supports mathematical research and administers the Millennium Prize Problems. The institute awarded Perelman the Poincaré Conjecture Millennium Prize, which he declined.
Ricci flow with surgery: A method that interrupts Ricci flow at singularities, cuts out narrow necks, and continues the evolution. Perelman’s key technical framework controlled singularities while carrying the proof through.
Three-dimensional manifold: A space that locally resembles ordinary three-dimensional Euclidean space at every point. The Poincaré conjecture concerns the global topology of these spaces.
William Thurston: An American mathematician whose geometrization conjecture proposed a classification framework for three-dimensional manifolds. His conjecture supplied the larger program that Perelman’s proof completed.
Fields Medal: An international mathematics award given every four years to recognize outstanding mathematical achievement. Perelman was awarded the 2006 medal for his work on Ricci flow and the Poincaré conjecture, but declined it.
Reduced volume: A scale-sensitive quantity in Ricci flow that is monotone under the evolution. Its monotonicity helped Perelman rule out certain collapsing singularity models.
Simply connected space: A path-connected space in which every loop can be continuously contracted to a point. Simple connectivity is the hypothesis that distinguishes the manifolds in the conjecture.

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Generalized Poincaré conjectureRelated: His proof settled the original three-dimensional case.