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Solution of the Poincaré conjecture

The first step is to deform the manifold using the [Ricci flow] . The Ricci flow was used by Richard Hamilton as a way to deform manifolds. He used it to prove that many compact manifolds were [diffeomorphic] to spheres. However, he did not prove they were all diffeomorphic to spheres. The Ricci flow is an imitation of the [heat equation] which describes the way heat flows in a solid. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning.

Hamilton was able to list a number of possible singularities that could form but he was concerned as to whether he had found all possible singularities. He wanted to cut the manifold at the singularities

and paste in caps, and then run the Ricci flow again. But he needed to understand the singularities. Grigori Perelman examined the singularities and discovered they were very simple manifolds: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line.

This was proved using something Perelman called the "Reduced Volume" which is closely related to an [eigenvalue] of a certain " [elliptic equation] ". Eigenvalues are difficult to describe without calculus but they are part of a famous problem: [Can you hear the shape of a drum?] . Essentially an eigenvalue is like a note being played by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the [cigar solution] , which looked like a strand sticking out of a manifold with nothing on the other side. In essence Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.

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