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Thurston's 24 Questions
Thurston's 24 questions are a set of mathematical problems in differential geometry posed by American mathematician William Thurston in his influential 1982 paper ''Three-dimensional manifolds, Kleinian groups and hyperbolic geometry'' published in the ''Bulletin of the American Mathematical Society''. These questions significantly influenced the development of geometric topology and related fields over the following decades. History The questions appeared following Thurston's announcement of the geometrization conjecture, which proposed that all compact 3-manifolds could be decomposed into geometric pieces. This conjecture, later proven by Grigori Perelman in 2003, represented a complete classification of 3-manifolds and included the famous Poincaré conjecture as a special case. By 2012, 22 of Thurston's 24 questions had been resolved. Table of problems Thurston's 24 questions are: See also * Geometrization conjecture * Hilbert's problems * Taniyama's problems * List ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C., to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifolds
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 (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 that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, 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 dimensions, and so there is a prevalence of very specialized techniques t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ending Lamination Theorem
In hyperbolic geometry, the ending lamination theorem, originally conjectured by as the eleventh problem out of his twenty-four questions, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold. The ending lamination theorem is a generalization of the Mostow rigidity theorem to hyperbolic manifolds of infinite volume. When the manifold is compact or of finite volume, the Mostow rigidity theorem states that the fundamental group determines the manifold. When the volume is infinite the fundamental group is not enough to determine the manifold: one also needs to know the hyperbolic structure on the surfaces at the "ends" of the manifold, and also the ending laminations on these surfaces. and proved the ending lamination conjecture for Kleinian surface groups. In view of the Tameness theorem this implies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Fuchsian Group
In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that .... If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two. Some authors use "quasi-Fuchsian group" to mean "quasi-Fuchsian group of type 1", in other words the limit set is the whole Jordan curve. This terminology is incompatible with the use of the terms "type 1" and "type 2" for Kleinian groups: all quasi-Fuchsian groups are Kleinian groups of type 2 (even if they are quasi-Fuchsian groups of type 1), as their limit sets are proper subsets of the Riemann sphere. The special case when t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yair Minsky
Yair Nathan Minsky (born in 1962) is an Israeli- American mathematician whose research concerns three-dimensional topology, differential geometry, group theory and holomorphic dynamics. He is a professor at Yale University. He is known for having proved Thurston's ending lamination conjecture and as a student of curve complex geometry. Biography Minsky obtained his Ph.D. from Princeton University in 1989 under the supervision of William Paul Thurston, with the thesis ''Harmonic Maps and Hyperbolic Geometry''. His Ph.D. students include Jason Behrstock, Erica Klarreich, Hossein Namazi and Kasra Rafi. Honors and awards He received a Sloan Fellowship in 1995. He was a speaker at the ICM (Madrid) 2006. He was named to the 2021 class of fellows of the American Mathematical Society "for contributions to hyperbolic 3-manifolds, low-dimensional topology, geometric group theory and Teichmuller theory". He was elected to the American Academy of Arts and Sciences in 2023. Sele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Canary
Richard Douglas Canary (born in 1962) is an American mathematician working mainly on low-dimensional topology. He is a professor at the University of Michigan. Canary obtained his Ph.D. from Princeton University in 1989 under the supervision of William Paul Thurston, with the thesis ''Hyperbolic Structures on 3-Manifolds with Compressible Boundaries''. He received a Sloan Research Fellowship in 1993. In 2015 he became a fellow of the American Mathematical Society, "for contributions to low-dimensional topology and hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ... as well as for service and teaching in mathematics." References External links Canary's home page at the University of Michigan [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky Group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by . Definition Fix some point ''p'' on the Riemann sphere. Each Jordan curve not passing through ''p'' divides the Riemann sphere into two pieces, and we call the piece containing ''p'' the "exterior" of the curve, and the other piece its "interior". Suppose there are 2''g'' disjoint Jordan curves ''A''1, ''B''1,..., ''A''''g'', ''B''''g'' in the Riemann sphere with disjoint interiors. If there are Möbius transformations ''T''''i'' taking the outside of ''A''''i'' onto the inside of ''B''''i'', then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like this. Properties By work of , a finitely generated Kleinian group is Schottky if and only if it is finitely generated, free, has nonempty domain of discontinuity, and all non-trivial elements are loxodromic. A fundamental domain for the action of a Schottky group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tame Manifold
In geometry, a tame manifold is a manifold with a well-behaved compactification. More precisely, a manifold M is called tame if it is homeomorphic to a compact manifold with a closed subset of the boundary removed. The Whitehead manifold In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to \R^3. discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper where ... is an example of a contractible manifold that is not tame. See also * * References * * * Differential geometry Hyperbolic geometry Manifolds {{hyperbolic-geometry-stub ... [...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] |
Hyperbolic Dehn Surgery
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions. Such an operation is often also called hyperbolic Dehn filling, as Dehn surgery proper refers to a "drill and fill" operation on a link which consists of ''drilling'' out a neighborhood of the link and then ''filling'' back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling". We will generally assume that a hyperbolic 3-manifold is complete. Suppose ''M'' is a cusped hyperbolic 3-manifold with ''n'' cusps. ''M'' can be thought of, topologically, as the interior of a compact manifold with toral boundary. Suppose we have chosen a meridian and longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirō Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Hamilton Meeks, III
William Hamilton Meeks, III (born 8 August 1947) is an American mathematician, specializing in differential geometry and minimal surfaces. Biography Meeks studied at the University of California, Berkeley, with a bachelor's degree in 1971, a master's degree in 1974, and a PhD in 1975 with supervisor H. Blaine Lawson and thesis ''The Conformal Structure and Geometry of Triply Periodic Minimal Surfaces in \mathbb R ^ 3 ''. He was an assistant professor in 1975–1977 at the University of California, Los Angeles (UCLA), in 1977–1978 at the Instituto de Matemática Pura e Aplicada (IMPA), and in 1978–1979 at Stanford University. From 1979 to 1983 he was a professor at IMPA. He was from 1983 to 1984 a visiting member of the Institute for Advanced Study and from 1984 to 1986 a professor at Rice University with the academic year 1985–1986 spent as a visiting professor at the University of California, Santa Barbara. From 1986 to 2018 he has been the George David Birkhoff Profe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
