Editing Poincaré conjecture (section)

== Solution ==
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[[File:Perelman, Grigori (1966).jpg|thumb|right|[[Grigori Perelman]]]]
On November 11, 2002, Russian mathematician [[Grigori Perelman]] posted the first of a series of three [[E-print|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 S. Hamilton]]. In August 2006, Perelman was awarded, but declined, the [[Fields Medal]] (worth $15,000 CAD) for his work on the Ricci flow. On March 18, 2010, the [[Clay Mathematics Institute]] awarded Perelman the $1&nbsp;million [[Millennium Prize Problems|Millennium Prize]] in recognition of his proof.<ref>{{cite web |url=http://www.claymath.org/poincare/ |date=March 18, 2010 |title=Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman |url-status=dead |archive-url=https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/ |archive-date=2010-03-22 |publisher=Clay Mathematics Institute }}</ref><ref>{{cite web |title=Poincaré Conjecture |url=http://www.claymath.org/millennium/poincare-conjecture/ |publisher=Clay Mathematics Institute |access-date=2018-10-04}}</ref> Perelman rejected that prize as well.<ref name="interfax" /><ref name="PhysOrg1">{{cite web |url=https://phys.org/news/2010-07-russian-mathematician-million-prize.html |title=Russian mathematician rejects $1 million prize |publisher=[[Phys.Org]] |author=Malcolm Ritter |date=2010-07-01 |access-date=2011-05-15}}</ref>

Perelman proved the conjecture by deforming the manifold using 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 towards what are known as [[Mathematical singularity|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 establishing that the surgery need not be repeated infinitely many times.

The first step is to deform the manifold using the [[Ricci flow]]. The Ricci flow was defined by Richard S. Hamilton as a way to deform manifolds. The formula for 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. A singularity in a manifold is a place where it is not differentiable: like a corner or a cusp or a pinching. The Ricci flow was only defined for smooth differentiable manifolds. Hamilton used the Ricci flow to prove that some compact manifolds were [[diffeomorphic]] to spheres, and he hoped to apply it to prove the Poincaré conjecture. He needed to understand the singularities.<ref>{{cite book |last=O'Shea |first=Donal |author-link=Donal O'Shea| title=The Surprising Resolution of the Poincaré Conjecture. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein |chapter=The Surprising Resolution of the Poincaré Conjecture |year= 2018 |series=Einstein Studies | volume=14 |pages=401–415 |publisher=Birkhäuser | location=New York, NY | isbn=978-1-4939-7708-6 | doi=10.1007/978-1-4939-7708-6_13 }}</ref>

Hamilton created a list of possible singularities that could form, but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur. Perelman discovered the singularities were all very simple: consider that a cylinder is formed by 'stretching' a circle along a line in another dimension, repeating that process with spheres instead of circles essentially gives the form of the singularities. Perelman proved this using something called the "Reduced Volume", which is closely related to an [[eigenvalue]] of a certain [[Elliptic operator|elliptic equation]].

Sometimes, an otherwise complicated operation reduces to multiplication by a [[Scalar (mathematics)|scalar]] (a number). Such numbers are called eigenvalues of that operation. Eigenvalues are closely related to vibration frequencies and are used in analyzing 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 soliton 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.

Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until, eventually, he is left with a collection of round three-dimensional spheres. Then, he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape, and sees that, despite all the initial confusion, the manifold was, in fact, homeomorphic to a sphere.

One immediate question posed was how one could be sure that infinitely many cuts are not necessary. This was raised due to the cutting potentially progressing forever. Perelman proved this cannot happen by using [[minimal surfaces]] on the manifold. A minimal surface is one on which any local deformation increases area; a familiar example is a [[soap film]] spanning a bent loop of wire. Hamilton had shown that the area of a minimal surface decreases as the manifold undergoes Ricci flow. Perelman verified what happened to the area of the minimal surface when the manifold was sliced. He proved that, eventually, the area is so small that any cut after the area is that small can only be chopping off three-dimensional spheres and not more complicated pieces. This is described as a battle with a [[Lernaean Hydra|Hydra]] by [[Christina Sormani|Sormani]] in Szpiro's book cited below. This last part of the proof appeared in Perelman's third and final paper on the subject.