Ricci flow is a process that deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, smoothing out irregularities in its curvature over time. It played a crucial role in Grigori Perelman's proof of the Poincaré Conjecture, one of the most famous problems in mathematics.

Ricci Flow

Ricci flow

metric of a Riemannian manifold

heat diffusion

Irregularities in curvature

Grigori Perelman's proof confirmed the Poincaré Conjecture, one of the seven Millennium Prize Problems, by employing and advancing Richard S. Hamilton's Ricci flow technique with surgery. His work reshaped modern geometry and topology, demonstrating that every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere.

Grigori Perelman's proof

The Poincaré conjecture posits that any simply connected, closed 3-dimensional manifold is homeomorphic to the 3-dimensional sphere. It was proven by Grigori Perelman in 2003 using Richard S. Hamilton's theory of Ricci flow, marking a significant milestone in the field of topology and earning Perelman the prestigious Clay Millennium Prize, which he declined.

CUSTOMIZE YOUR LEARNING

CUSTOMIZE YOUR LEARNING

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