William Paul Thurston (born October 30, 1946) is an American mathematician. Events 637 - Antioch surrenders to the Muslim forces under Rashidun Caliphate after the Battle of Iron bridge. Year 1946 ( MCMXLVI) was a Common year starting on Tuesday (link will display full 1946 calendar of the Gregorian calendar. The United States of America —commonly referred to as the A mathematician is a person whose primary area of study and research is the field of Mathematics. He is a pioneer in the field of low-dimensional topology. In Mathematics, low-dimensional topology is the branch of Topology that studies Manifolds of four or fewer dimensions In 1982, he was awarded the Fields medal for the depth and originality of his contributions to mathematics. The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and He is currently a professor of mathematics and computer science at Cornell University (since 2003). Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their
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Mathematical contributions
Foliations
His early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact. In Mathematics, a foliation is a geometric device used to study manifolds Informally speaking a foliation is a kind of "clothing" worn on a manifold Some of his more significant results include:
- The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one). In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant
- The construction of a continuous family of smooth, codimension one foliations on the three-sphere whose Godbillon-Vey invariant takes every real value.
- With John Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open topology. For the Nobel Prize in Physics 2006 see John C Mather For the Vice-Chancellor of the University of Oxford 1723-28 see John Mather (Vice-Chancellor In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain Topological equivalence redirects here see also Topological equivalence (dynamical systems.
In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that it led to a kind of exodus from the field, where advisors counselled students from going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 [1] ).
The geometrization conjecture
His later work, starting around the late 1970s, revealed that geometry, particularly hyperbolic geometry, played a fundamental role in the theory of 3-manifolds. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert-Weber space. A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant Sectional curvature -1 In Mathematics, Seifert -Weber space is a Closed Hyperbolic 3-manifold. The independent and distinct approaches of Robert Riley and Troels Jorgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure eight knot complement was hyperbolic. In Knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four In Mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative This was the first example of a hyperbolic knot.
Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement. He showed that the figure eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure eight knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. In Mathematics, a normal surface is a Surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. A Dehn surgery is a specific construction used to modify 3-manifolds The process takes as input a 3-manifold together with a link. In Mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a two-sided Incompressible surface A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifold were Haken. These examples were actually hyperbolic and motivated his next revolutionary theorem.
Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem. In Mathematics, hyperbolic Dehn surgery refers to an operation by which one can obtain further Hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold
To complete the picture, Thurston proved a geometrization theorem for Haken manifolds. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures In Mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a two-sided Incompressible surface A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance.
The geometrization theorem has been called Thurston's Monster Theorem, due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions
Thurston was next led to formulate his geometrization conjecture. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. A proof to that conjecture seems to follow from the recent work of Grigori Perelman. Grigori Yakovlevich Perelman (Григорий Яковлевич Перельман born 13 June 1966 in Leningrad, USSR (now St
Orbifold theorem
In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. In the mathematical disciplines of Topology and Geometric group theory, an orbifold (for "orbit-manifold" is a generalization of a Manifold. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Hamilton's work on the Ricci flow. Richard Streit Hamilton (born 1943 is professor of Mathematics at Columbia University. In Differential geometry, the Ricci flow is an intrinsic Geometric flow —a process which deforms the metric of a Riemannian manifold —in this case in
Education and career
He was born in Washington, D.C and received his bachelors degree from New College (now New College of Florida) in 1967. Washington DC ( formally the District of Columbia and commonly referred to as Washington, the District, or simply D New College of Florida is a public Liberal arts college located in Sarasota Florida. For his undergraduate thesis he developed an intuitionist foundation for topology. In the Philosophy of mathematics, intuitionism, or neointuitionism (opposed to Preintuitionism) is an approach to Mathematics as the constructive Following this, he earned a doctorate in mathematics from the University of California, Berkeley, in 1972. The University of California Berkeley (also referred to as Cal, Berkeley and UC Berkeley) is a major research university located in Berkeley His Ph. D. advisor was Morris W. Hirsch and his dissertation was on Foliations of Three-Manifolds which are Circle Bundles.
After completing his Ph. D. , he spent a year at the Institute for Advanced Study, then another year at MIT as Assistant Professor. The Institute for Advanced Study, located in Princeton New Jersey, United States is a center for theoretical research In 1974, he was appointed Professor of Mathematics at Princeton University. Princeton University is a private Coeducational research university located in Princeton, New Jersey. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute. The Mathematical Sciences Research Institute (MSRI, founded in 1982, is a mathematical research institution whose funding sources include the National In 1996, he moved to University of California, Davis. The University of California Davis, commonly known as UC Davis, or just UCD, is a public coeducational university located in the city of Davis, In 2003, he moved again to become Professor of Mathematics at Cornell University.
His Ph. D. students include Richard Canary, David Gabai, William Goldman, Benson Farb, Detlef Hardorp, Craig Hodgson, Steven Kerckhoff, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Oded Schramm, Richard Schwartz, Martin Bridgeman and Jeffrey Weeks. David Gabai is a Mathematician at Princeton University. Intensely focused on Low-dimensional topology and Hyperbolic geometry, he is a leading William Goldman (born 1955 is a professor of Mathematics at the University of Maryland College Park (since 1986 Steven Paul Kerckhoff (born 1952 is a Professor of Mathematics at Stanford University, who works on Hyperbolic 3-manifolds and Teichmüller Oded Schramm ( December 10, 1961 in Jerusalem, Israel – September 1, 2008, Washington State USA (עודד שרם was Richard Schwartz is currently a Professor of Mathematics at Brown University. Jeffrey Renwick Weeks is an American Mathematician. He became a MacArthur Fellow in 1999
Thurston has turned his attention in recent years to mathematical education and bringing mathematics to the general public. He has served as mathematics editor for Quantum Magazine, a youth science magazine, and as head of The Geometry Center. Quantum Magazine was a bimonthly magazine about Science and Math mainly targeted at high school and college students The Geometry Center was a Mathematics research and education center at the University of Minnesota. As director of Mathematical Sciences Research Institute from 1992 to 1997, he initiated a number of programs designed to increase awareness of mathematics among the public. The Mathematical Sciences Research Institute (MSRI, founded in 1982, is a mathematical research institution whose funding sources include the National
In 2005 Thurston won the first AMS Book Prize, for Three-dimensional Geometry and Topology. The prize recognizes an outstanding research book that makes a seminal contribution to the research literature. [2]
Thurston has an Erdős number of 2. The Erdős number (ɛrdøːʃ honoring the late Hungarian mathematician Paul Erdős, is a way of describing the "collaborative distance" between a person
Selected works
- William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978-1981).
- William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5
- William Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math (2) 124 (1986), no. 2, 203--246.
- William Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N. S. ) 6 (1982), 357–381.
- William Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N. S. ) 19 (1988), no. 2, 417--431
- Epstein, David B. A. ; Cannon, James W. ; Holt, Derek F. ; Levy, Silvio V. F. ; Paterson, Michael S. ; Thurston, William P. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp. ISBN 0-86720-244-0
- Eliashberg, Yakov M. ; Thurston, William P. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, RI, 1998. x+66 pp. ISBN 0-8218-0776-5
See also
- Milnor–Thurston kneading theory
- Nielsen-Thurston classification
- Jorgensen-Thurston theorem
- confoliation
- automatic group
- Thurston norm
- Geometrization
- Geometric structure
- Circle packing theorem
External links
- William Thurston at the Mathematics Genealogy Project
- O'Connor, John J. The Milnor–Thurston kneading theory is a mathematical theory which analyzes the iterates of piecewise monotone mappings of an Interval into itself In Mathematics, Thurston's classification theorem characterizes Homeomorphisms of a Compact surface. In Mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint The Mathematics Genealogy Project is a web-based Database that gives an Academic genealogy based on Dissertation supervision relations & Robertson, Edmund F. , “William Thurston”, MacTutor History of Mathematics archive
- Thurston's page at Cornell
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