Ricci flow

The Ricci flow (/ˈriːtʃi/ REE-chee, Italian: [ˈrittʃi]) is a partial differential equation used in differential geometry and geometric analysis. It is like the heat equation, which is used to study diffusion of heat. The Ricci flow is named for the Ricci tensor, which is part of its definition.

Richard Hamilton used the Ricci flow in the 1980s to prove new results in Riemannian geometry. Later, other mathematicians used it to solve the differentiable sphere conjecture. In the 1990s, Hamilton used the Ricci flow to study William Thurston's geometrization conjecture. In 2002 and 2003, Grigori Perelman made new discoveries about the Ricci flow. These discoveries, along with Hamilton's work, are now seen as a proof of the Thurston conjecture, and of the Poincaré conjecture, which had been an open problem since 1904. This work is a major milestone in the fields of geometry and topology.

Several stages of Ricci flow on a 2D manifold.