Who Cares About Poincaré?Million-dollar math problem solved. So what?
The Poincaré Conjecture says, Hey, you've got this alien blob that can ooze its way out of the hold of any lasso you tie around it? Then that blob is just an out-of-shape ball. [Grigory] Perelman and [Columbia University's Richard] Hamilton proved this fact by heating the blob up, making it sing, stretching it like hot mozzarella, and chopping it into a million pieces. In short, the alien ain't no bagel you can swing around with a string through his hole.
Introduction on Poincaré Conjecture (slate.com)
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.[1]
Solution of the Poincaré conjecture - Wikipedia, the free encyclopedia.
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.
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