In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by Richard Hamilton.

By studying the limit of the manifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group: at large times the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thin piece is a graph manifold, but this extra complication is not necessary for proving just the Poincaré conjecture.

The geometrization theorem has been called Thurston's Monster Theorem, due to the length and difficulty of the proof.

Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture.

In 2003, he proved Thurston's geometrization conjecture.

The elliptization conjecture is a special case of Thurston's geometrization conjecture, which was proved in 2003 by G. Perelman.

The hierarchy played a crucial role in William Thurston's hyperbolization theorem for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds.

This result can be recovered from the combination of Mostow rigidity with Thurston's geometrization theorem.

Not every 3-manifold admits a geometry, but Thurston's geometrization conjecture proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.

Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics.

* Thurston's geometrization conjecture

The spherical manifolds are exactly the manifolds with spherical geometry, one of the 8 geometries of Thurston's geometrization conjecture.

According to Thurston's geometrization conjecture, proved by Perelman, any closed, irreducible, atoroidal 3-manifold with infinite fundamental group is hyperbolic.

The conjecture is implied by Thurston's geometrization conjecture, which was proven by Grigori Perelman in 2003.

This is the fibered part of Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen – Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups.

It was proposed by, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

Here is a statement of Thurston's conjecture:

William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i. e. has a Riemannian metric of constant positive sectional curvature.

Thurston's original motivation for developing this classification was to find geometric structures on mapping tori of the type predicted by the Geometrization conjecture.

The case where S is a torus ( i. e., a surface whose genus is one ) is handled separately ( see torus bundle ) and was known before Thurston's work.

Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton.

Instead, Thurston's data did not indicate such a pronounced minimum of friction for a liquid lubricated journal bearing as was demonstrated by the graphs of Martens and Stribeck.

Thurston's major political achievement was in helping pass the Donation Land Claim Act in 1850.

Beauchamp was educated at Dr. Benjamin Thurston's academy in Barren County, Kentucky until the age of sixteen.

After saving some money, he returned to Thurston's school as a student, and was later employed by the school as an usher.

Besser was so excited by this, he sneaked into Thurston's train after the St. Louis run of the show was over, and was discovered the next day sleeping on top of the lion's cage in Detroit.

Thurston's traveling magic show was the biggest one of all ; it was so large that it needed eight train cars to transport his road show.

In dimension three, the conjecture had an uncertain reputation until the geometrization conjecture put it into a framework governing all 3-manifolds.

* Bruce Kleiner and John W. Lott posted a paper on the arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture.

* John Morgan and Gang Tian posted a paper on the arXiv in July 2006 which gave a detailed proof of just the Poincaré Conjecture ( which is somewhat easier than the full geometrization conjecture ) and expanded this to a book.

Thurston was next led to formulate his geometrization conjecture.

A theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot.

The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces.

Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.

The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.

There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures.

For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the oriented double cover.

One of the numerous consequences of the Thurston-Perelman geometrization theorem is that graph manifolds are precisely the 3-manifolds whose Gromov norm vanishes.

* Huai-Dong Cao and Xi-Ping Zhu published a paper in the June 2006 issue of the Asian Journal of Mathematics with an exposition of the complete proof of the Poincaré and geometrization conjectures.

The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.

In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above.

* Bruce Kleiner and John Lott, Notes on Perelman's Papers ( May 2006 ) ( fills in the details of Perelman's proof of the geometrization conjecture ).

1, 57-78 ( expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture )

For the proof of the conjectures, see the references in the articles on geometrization conjecture or Poincaré conjecture.

He is best known for having discovered the Ricci flow and suggesting the research program that ultimately led to the proof, by Grigori Perelman, of the Thurston geometrization conjecture and the solution of the Poincaré conjecture.