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Solution of the Poincaré conjecture

In November 2002, [Grigori Perelman] posted the first of a series of [eprints] on [arXiv] outlining a solution of the [Poincaré conjecture] . Perelman's proof uses a modified version of a [Ricci flow] program developed by [Richard Hamilton] . On March 18, 2010, the [Clay Mathematics Institute] awarded Perelman the [Millennium Prize] in recognition of his proof.

Description

The Poincaré 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] . Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery") causing the separate pieces to form into ball-like shapes. Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and wondering whether the surgery might need to be repeated infinitely many times.

Explaining the key terms

What is a 3-dimensional sphere ?

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