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

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.

* 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.

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.

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.

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.

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.

A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions ; they are listed below and are sometimes called Thurston geometries.

The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, atoroidal, and has infinite fundamental group.

Warning: the JSJ decomposition 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.

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

In particular, the result of geometrization may be a geometry that is not isotropic.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric ; geometrically, it has one of 3 possible geometries: positive curvature / spherical, zero curvature / flat, negative curvature / hyperbolic – and the geometrization conjecture ( now theorem ) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

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.

More generally, geometrization implies that a knot which is neither a satellite knot nor a torus knot is a hyperbolic knot.

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.

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

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.

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

In 1982, Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes " almost round " just before the collapse.

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

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.

Most " small " 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.

Note that the term " uniformization " correctly suggests a kind of smoothing away of irregularities in the geometry, while the term " geometrization " correctly suggests placing a geometry on a smooth manifold.

The Ricci flow does not preserve volume, so to be more careful in applying the Ricci flow to uniformization and geometrization one needs to normalize the Ricci flow to obtain a flow which preserves volume.

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

In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds.

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

* 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 Geometry of 3-Manifolds ( video ) A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.

His early fears of the square marching battalions associated with dictatorships may have led him to oppose any " geometrization " of people and their architecture.

Whether or not Danchin is correct in suggesting that Thompson's resumption of the opium habit also dates from this period is, of course, a matter of conjecture.

Whether it could be as disastrous for American labor as, say, Jimmy Hoffa of the Teamsters, is a matter of conjecture.

Our conjecture is, then, that regardless of the manner in which school lessons are taught, the compulsive child accentuates those elements of each lesson that aid him in systematizing his work.

Because all clades are represented in the southern hemisphere but many not in the northern hemisphere, it is natural to conjecture that there is a common southern origin to them.

In some applications it is useful to be able to compute the Bernoulli numbers B < sub > 0 </ sub > through B < sub > p − 3 </ sub > modulo p, where p is a prime ; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime.

A conjecture is a proposition that is unproven.

In mathematics, a conjecture is an unproven proposition that appears correct.

In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture.

Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results.

For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.

Because many outstanding problems in number theory, such as Goldbach's conjecture are equivalent to solving the halting problem for special programs ( which would basically search for counter-examples and halt if one is found ), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems.

He is remembered today for Goldbach's conjecture.

Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach's conjecture.

In computability theory, the Church – Turing thesis ( also known as the Turing-Church thesis, the Church – Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis ) is a combined hypothesis (" thesis ") about the nature of functions whose values are effectively calculable ; or, in more modern terms, functions whose values are algorithmically computable.

Little is known of his life before he became a bishop ; the assignment of his birth to the year 315 rests on conjecture.

Mathematically, the conjecture states that the maximal Cauchy development of generic compact or asymptotically flat initial data is locally inextendible as a regular Lorentzian manifold.

Whether this formula produces an infinite quantity of Carmichael numbers is an open question ( though it is implied by Dickson's conjecture ).