Ricci flow

Ricci flow is a process that deforms the metric of a Riemannian manifold in a manner formally analogous to the diffusion of heat, smoothing out irregularities in curvature. Introduced by Richard Hamilton in 1982, the flow evolves a metric according to its Ricci curvature, gradually collapsing positively curved regions and expanding negatively curved ones. Perelman's breakthrough was to show that, despite singularities where the flow collapses regions to points, the topology of the underlying manifold can be recovered by performing surgical cuts at the singular times.

Ricci flow is not merely a technique for proving theorems. It is a physical process: the renormalization group flow of two-dimensional quantum field theories is a Ricci flow, and the evolution of the universe's geometry under certain cosmological models follows similar equations. The mathematics of smoothing space and the physics of scaling reality are the same equation in different clothing.