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Grigori Perelman

It is known that singularities (including those which occur, roughly speaking, after the flow has continued for an infinite amount of time) must occur in many cases. However, any singularity which develops in a finite time is essentially a "pinching" along certain spheres corresponding to the [prime decomposition] of the 3-manifold. Furthermore, any "infinite time" singularities result from certain collapsing pieces of the [JSJ decomposition] . Perelman's work proves this claim and thus proves the geometrization conjecture.

Verification
Since 2003, Perelman's program has attracted increasing attention from the mathematical community. In April 2003, he accepted an invitation to visit [Massachusetts Institute of Technology] , [Princeton University] , [State University of New York at Stony Brook] , [Columbia University] and [New York University] , where he gave a series of talks on his work.Nasar and Gruber.

Three independent groups of scholars have verified that Perelman's papers contain all the essentials for a complete proof of the geometrization conjecture:
1 On 25 May 2006, Bruce Kleiner and John Lott, both of the [University of Michigan] , posted a paper on arXiv that fills in the details of Perelman's proof of the Geometrization conjecture.Bruce Kleiner, John Lott Notes on Perelman's papers [Geometry & Topology] 12 (2008) 2587–2855, , As John Lott said in ICM2006, "It has taken us some time to examine Perelman's work. This is partly due to the originality of Perelman's work and partly to the technical sophistication of his arguments. All indications are that his arguments are correct."

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