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Virtually Fibered Conjecture
In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle. A 3-manifold which has such a finite cover is said to virtually fiber. If ''M'' is a Seifert fiber space, then ''M'' virtually fibers if and only if the rational Euler number of the Seifert fibration or the (orbifold) Euler characteristic of the base space is zero. The hypotheses of the conjecture are satisfied by hyperbolic 3-manifolds. In fact, given that the geometrization conjecture is now settled, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds. The original interest in the virtually fibered conjecture (as well as its weaker cousins, such as the virtually Haken conjecture) stemmed from the fact that any of these conjectures, com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Principles Definition A topological space M is a 3-manifold if it is a second-countable Hausdorff space and if every point in M has a neighbourhood (mathematics), neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, Piecewise linear manifold, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrization Conjecture
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries ( Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by as part of his 24 questions, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quanta Magazine
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science News'' in October 2012, but it was renamed to its current title in July 2013. It was founded by the former ''New York Times'' journalist Thomas Lin, who was the magazine's editor-in-chief until 2024. The two deputy editors are John Rennie and Michael Moyer, formerly of ''Scientific American'', and the art director is Samuel Velasco. In 2024, Samir Patel became the magazine's second editor in chief. Content The articles in the magazine are freely available to read online. ''Scientific American'', ''Wired'', ''The Atlantic'', and ''The Washington Post'', as well as international science publications like '' Spektrum der Wissenschaft'', have reprinted articles from the magazine. In November 2018, MIT Press The MIT Press is the uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenpreis Conjecture
In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any ''K'' greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers which are ''K''- quasiconformal: that is, the covers are arbitrarily close in the Teichmüller metric. A proof was announced by Jeremy Kahn and Vladimir Markovic in January 2011, using their proof of the Surface subgroup conjecture and a newly developed "good pants homology" theory. In June 2012, Kahn and Markovic were given the Clay Research Awards for their work on these two problems by the Clay Mathematics Institute at a ceremony at Oxford University. See also * Surface subgroup conjecture * Virtually Haken conjecture In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is ''virtually Haken''. That is, it has a finite cover (a covering s ... * Virtually fibered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Subgroup Conjecture
In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed manifold, closed, irreducible manifold, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list.Robion Kirby''Problems in low-dimensional topology''/ref> Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds. A mathematical proof, proof of this case was announced in the summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared in the arxiv.org server in October 2009. Their paper was published in the ''Annals of Mathematics'' in 2012. In June 2012, Kahn and Markovic were given the Clay Research Awards by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtually Haken Conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is ''virtually Haken''. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold. After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds. The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968, although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list. A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica. The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ian Agol
Ian Agol (; born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds. Education and career Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and obtained his Ph.D. in 1998 from the University of California, San Diego. At UCSD, his advisor was Michael Freedman and his thesis was ''Topology of Hyperbolic 3-Manifolds''. He is a professor at the University of California, Berkeley and a former professor at the University of Illinois at Chicago. Contributions In 2004, Agol proved the Marden tameness conjecture, a conjecture of Albert Marden. It states that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. The conjecture was also independently proven by Danny Calegari and David Gabai, and implies the Ahlfors measure conjecture. In 2012, he announced a proof of the virtually Haken conjecture, which was pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Institut Henri Poincaré
The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrondissement of Paris, 5th arrondissement of Paris, on the Sainte-Geneviève Hill. History Just after World War I, mathematicians Émile Borel in France and George Birkhoff in the United States persuaded French and American sponsors (Edmond de Rothschild and the Rockefeller Foundation respectively)Rockefeller and the internationalization of mathematics between two World Wars [Texte imprimé][: documents and studies for the social history of mathematics in the 20th century / Reinhard Siegmund-Schultze. - Basel; Boston; Berlin : Birkhäuser, cop. 2001. - 1 vol. (XIII-341 p.) : fig., ill., carte; 24 cm. - (Science networks historical studies; volume 25). - Bibliogr. p. [307]-318. Index. (rel.). - (rel.)] to fund the building of a centre for lect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Wise (mathematician)
Daniel T. Wise (born January 24, 1971) is an American mathematician who specializes in geometric group theory and 3-manifolds. He is a professor of mathematics at McGill University. Wise's conjecture is named after him. Education Daniel Wise obtained his PhD from Princeton University in 1996 supervised by Martin Bridson His thesis was titled ''non-positively curved squared complexes, aperiodic tilings, and non-residually finite groups''. Career and research Wise's research has focused on the role of non-positively curved cube complexes within geometric group theory and their interplay with residual finiteness. His early work was taken to higher dimensions when he introduced with Frédéric Haglund the theory of special cube complexes. In 2009 he announced a solution to the virtually fibered conjecture for cusped hyperbolic 3-manifolds. This was a consequence of his work on the structure of groups with a quasiconvex hierarchy which proved the virtual specialness of a broad cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ArXiv
arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly peer review, peer reviewed. It consists of scientific papers in the fields of mathematics, physics, astronomy, electrical engineering, computer science, quantitative biology, statistics, mathematical finance, and economics, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers are self-archiving, self-archived on the arXiv repository before publication in a peer-reviewed journal. Some publishers also grant permission for authors to archive the peer-reviewed postprint. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014 and two million by the end of 2021. As of November 2024, the submission rate is about 24,000 arti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolization Theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. Statement One form of the hyperbolization theorem states: If ''M'' is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of ''M'' has a complete hyperbolic structure of finite volume. The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique. The conditions that the manifold ''M'' should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtually Haken Conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is ''virtually Haken''. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold. After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds. The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968, although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list. A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica. The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
