Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Poincare conjecture says that if a 3-dimensional manifold is compact, has no boundary and is simply connected, then it is homeomorphic to a 3-dimensional sphere. The concepts of "manifold", "compact", "no boundary", "simply connected", "homeomorphic" and "3-dimensional sphere" are described below. Perelman (using ideas originally from Hamilton) proved the conjecture by deforming the manifold using something called the Ricci flow (which behaves similarly to the heat equation that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself (like hot mozzarella) towards what are known as singularities.

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