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- Poincaré_conjecture abstract "In mathematics, the Poincaré conjecture (/pwɛn.kɑːˈreɪ/ pwen-kar-AY; French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. The analogous conjectures for all higher dimensions had already been proven.After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof built upon the program of Richard Hamilton to use the Ricci flow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method "converged" in three dimensions. Perelman completed this portion of the proof. Several teams of mathematicians verified that Perelman's proof was correct.The Poincaré conjecture, before being proven, was one of the most important open questions in topology. In 2000, it was named one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on March 18, 2010. On July 1, 2010, he turned down the prize saying that he believed his contribution in proving the Poincaré conjecture was no greater than Hamilton's (who first suggested using the Ricci flow for the solution). As of 2015, the Poincaré conjecture is the only solved Millennium problem.On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year", the first time this honor was bestowed in the area of mathematics.".
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- Poincaré_conjecture wikiPageWikiLink 3-manifold.
- Poincaré_conjecture wikiPageWikiLink 3-sphere.
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- Poincaré_conjecture wikiPageWikiLink ArXiv.
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- Poincaré_conjecture wikiPageWikiLink Category:Millennium_Prize_Problems.
- Poincaré_conjecture wikiPageWikiLink Category:Theorems_in_topology.
- Poincaré_conjecture wikiPageWikiLink Characterization_(mathematics).
- Poincaré_conjecture wikiPageWikiLink Christina_Sormani.
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- Poincaré_conjecture wikiPageWikiLink Diffeomorphic.
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- Poincaré_conjecture wikiPageWikiLink Eigenvalue.
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- Poincaré_conjecture wikiPageWikiLink Elliptic_curve.
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- Poincaré_conjecture wikiPageWikiLink Free_product.
- Poincaré_conjecture wikiPageWikiLink Fundamental_group.
- Poincaré_conjecture wikiPageWikiLink Gang_Tian.
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- Poincaré_conjecture wikiPageWikiLink H-cobordism.
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- Poincaré_conjecture wikiPageWikiLink Heat_equation.
- Poincaré_conjecture wikiPageWikiLink Henri_Poincaré.
- Poincaré_conjecture wikiPageWikiLink Homeomorphic.
- Poincaré_conjecture wikiPageWikiLink Homeomorphism.
- Poincaré_conjecture wikiPageWikiLink Homology_(mathematics).
- Poincaré_conjecture wikiPageWikiLink Homology_sphere.
- Poincaré_conjecture wikiPageWikiLink Homotopy.
- Poincaré_conjecture wikiPageWikiLink Homotopy_equivalence.
- Poincaré_conjecture wikiPageWikiLink Homotopy_sphere.
- Poincaré_conjecture wikiPageWikiLink Huai-Dong_Cao.
- Poincaré_conjecture wikiPageWikiLink Hyperbolic_3-manifold.
- Poincaré_conjecture wikiPageWikiLink Ian_Stewart_(mathematician).
- Poincaré_conjecture wikiPageWikiLink In_Our_Time_(BBC_Radio_4).
- Poincaré_conjecture wikiPageWikiLink In_Our_Time_(radio_series).
- Poincaré_conjecture wikiPageWikiLink International_Congress_of_Mathematicians.
- Poincaré_conjecture wikiPageWikiLink J._H._C._Whitehead.
- Poincaré_conjecture wikiPageWikiLink J._W._Milnor.
- Poincaré_conjecture wikiPageWikiLink John_Lott_(mathematician).
- Poincaré_conjecture wikiPageWikiLink John_Milnor.
- Poincaré_conjecture wikiPageWikiLink John_Morgan_(mathematician).
- Poincaré_conjecture wikiPageWikiLink Manifold.
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- Poincaré_conjecture wikiPageWikiLink Mathematical_singularity.
- Poincaré_conjecture wikiPageWikiLink Mathematics.
- Poincaré_conjecture wikiPageWikiLink Melvyn_Bragg.
- Poincaré_conjecture wikiPageWikiLink Michael_Freedman.
- Poincaré_conjecture wikiPageWikiLink Millennium_Prize_Problems.
- Poincaré_conjecture wikiPageWikiLink Millennium_problem.
- Poincaré_conjecture wikiPageWikiLink Milnor.
- Poincaré_conjecture wikiPageWikiLink Minimal_surface.
- Poincaré_conjecture wikiPageWikiLink Minimal_surfaces.
- Poincaré_conjecture wikiPageWikiLink Open_University.
- Poincaré_conjecture wikiPageWikiLink Path_(topology).
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- Poincaré_conjecture wikiPageWikiLink R._H._Bing.
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- Poincaré_conjecture wikiPageWikiLink Ricci_flow.
- Poincaré_conjecture wikiPageWikiLink Richard_Hamilton_(mathematician).
- Poincaré_conjecture wikiPageWikiLink Richard_Hamilton_(professor).
- Poincaré_conjecture wikiPageWikiLink Riemannian_manifold.
- Poincaré_conjecture wikiPageWikiLink Riemannian_metric.
- Poincaré_conjecture wikiPageWikiLink Scalar_(mathematics).
- Poincaré_conjecture wikiPageWikiLink Science_(journal).
- Poincaré_conjecture wikiPageWikiLink Simply_connected.
- Poincaré_conjecture wikiPageWikiLink Simply_connected_space.