Richard Hamilton

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Ricci flow

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~ Which problems do d ifferential equations apply to?

~ Is the earth surface flat? Why was it thought to be flat?

The differential equation that was to play a key role in solving the Poincaré conjecture is the Ricci flow equation. It was discovered two times, independently. In physics, by Friedan, 1985 and in mathematics by Richard Hamilton in his 1982 paper. The physicists were working on the renormalization group of quantum field theory, while Hamilton was interested in geometric applications of the Ricci flow equation itself.

On the left-hand side of the Ricci flow equation is a quantity that expresses how the geometry changes with time--the derivative of the metric tensor, as the mathematicians like to say. On the right-hand side is the Ricci tensor, a measure of the extent to which the shape is curved. The Ricci tensor, based on Riemann's theory of geometry (1854), also appears in Einstein's equations for general relativity (1915). Those equations govern the interaction of matter, energy, curvature of space, and the motion of material bodies.

The Ricci flow equation is the analogue, in the geometric context, of Fourier's heat equation. The idea, grosso modo, for its application to geometry is that, just as Fourier's heat equation disperses temperature, the Ricci flow equation disperses curvature. Thus, even if a shape was irregular and distorted, Ricci flow would gradually remove these anomalies, resulting in a very regular shape whose topological nature was evident. Indeed, in 1982 Hamilton showed that for positively curved, simply connected shapes of dimension three (compact three-manifolds) the Ricci flow transforms the shape into one that is ever more like the round three-sphere. In the long run, it becomes almost indistinguishable from this perfect, ideal shape. When the curvature is not strictly positive, however, solutions of the Ricci flow equation behave in a much more complicated way. This is because the equation is nonlinear. While parts of the shape may evolve towards a smoother, more regular state, other parts might develop singularities. This richer behavior posed serious difficulties. But it also held promise: it was conceivable that the formation of singularities could reveal Thurston's decomposition of a shape into its constituent geometric atoms.

~ How does Ricci flow transform irregular shapes?

~ Do all parts of a shape evolve alike?

Hamilton was the driving force in developing the theory of Ricci flow in mathematics, both conceptually and technically. Hamilton had established the Ricci flow equation as a tool with the potential to resolve both conjectures as well as other geometric problems. Nevertheless, serious obstacles barred the way to a proof of the Poincaré conjecture. Notable among these obstacles was lack of an adequate understanding of the formation of singularities in Ricci flow, akin to the formation of black holes in the evolution of the cosmos. Indeed, it was not at all clear how or if formation of singularities could be understood. Despite the new front opened by Hamilton, and despite continued work by others using traditional topological tools for either a proof or a disproof, progress on the conjectures came to a standstill.

Such was the state of affairs in 2000, when John Milnor wrote an article describing the Poincaré conjecture and the many attempts to solve it. At that writing, it was not clear whether the conjecture was true or false, and it was not clear which method might decide the issue. Analytic methods (differential equations) were mentioned in a later version (2004).

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