In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about...

Poincaré conjecture), Fermat's Last Theorem, and others. Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf...

the Poincaré Conjecture and the Geometrization Conjecture". arXiv:math.DG/0612069. Morgan, John W.; Tian, Gang Ricci Flow and the Poincaré Conjecture arXiv:math/0607607...

for solving the Poincaré conjecture, though Perelman declined both awards. The Poincaré conjecture and the spherical space form conjecture are corollaries...

In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold that is a homotopy sphere is a sphere. More precisely...

Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official...

important 3-manifold because of the now-proven Poincaré conjecture. Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks...

problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. Poincaré is regarded...

equivalent to two simpler conjectures: the Poincaré conjecture and the spherical space form conjecture. The elliptization conjecture is a special case of Thurston's...

the Poincaré conjecture, was solved by Grigori Perelman in 2003. However, a generalization called the smooth four-dimensional Poincaré conjecture—that...

smooth Poincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic sphere). The Poincaré conjecture...

are: Geometrization conjecture Hilbert's problems Taniyama's problems List of unsolved problems in mathematics Poincaré conjecture Smale's problems Thurston...

made regarding his work habits while proving the higher-dimensional Poincaré conjecture. He said that his best work had been done "on the beaches of Rio...

claiming a resolution of the renowned Poincaré conjecture, along with the more general geometrization conjecture. His work contained a number of notable...

mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to...

proof was found in 1993. In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal and did not attend the congress. In 2014...

Thurston conjecture and the Poincaré conjecture, regarded as a special case of the former. It should be emphasized that the Poincaré conjecture has been...

hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's theorem, and the Poincaré conjecture. Differential topology considers the properties...

proving the Poincaré and geometrization conjectures in 2003. Perelman was awarded a Millennium Prize for resolving the Poincaré conjecture but declined...

nearly a century, Grigori Perelman proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). William Thurston's geometrization program...

dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005. The resolution of the smooth Poincaré conjecture in dimensions 5 and larger...

counterexample to the Poincaré Conjecture, if one existed. In 2002, Martin Dunwoody posted a claimed proof of the Poincaré Conjecture. Rourke identified...

mathematicians) of some of the circumstances surrounding the proof of the Poincaré conjecture, one of the most important accomplishments of 20th and 21st century...

More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology...

generalization of the Poincaré conjecture to the non-simply connected case. The conjecture is implied by Thurston's geometrization conjecture, which was proven...

Perelman posts the first of three preprint texts with his proof of the Poincaré conjecture. It remains the only of the Millennium Prize Problems in mathematics...

Game complexity List of unsolved problems in mathematics Unique games conjecture Unsolved problems in computer science A nondeterministic Turing machine...

problems in mathematics In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even...

awarded a Fields Medal for his work on the 4-dimensional generalized Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists...

conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture....