William Thurston
Encyclopedia
William Paul Thurston is an American
mathematician
. He is a pioneer in the field of low-dimensional topology
. In 1982, he was awarded the Fields Medal
for his contributions to the study of 3-manifolds. He is currently a professor of mathematics and computer science
at Cornell University
(since 2003).
theory, where he had a dramatic impact. His more significant results include:
In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ).
s of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen 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
. 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 surface
s in the knot complement. 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. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds 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.
To complete the picture, Thurston proved a hyperbolization theorem
for Haken manifold
s. 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-manifold
s.
Thurston was next led to formulate his geometrization conjecture
. 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 follows from the recent work of Grigori Perelman
(2002–2003).
structures naturally arose. 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
.
to a homemaker and an aeronautical engineer. He received his bachelors degree from New College (now New College of Florida
) in 1967. For his undergraduate thesis he developed an intuitionist
foundation for topology. Following this, he earned a doctorate in mathematics from the University of California, Berkeley
, in 1972. 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. In 1974, he was appointed Professor of Mathematics at Princeton University
. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute
. In 1996, he moved to University of California, Davis
. In 2003, he moved again to become Professor of Mathematics at Cornell University
.
His Ph.D. students include Richard Canary, Renaud Dreyer, David Gabai
, William Goldman
, Benson Farb
, Detlef Hardorp, Craig Hodgson, Richard Kenyon, Steven Kerckhoff
, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Oded Schramm
, Richard Schwartz
, Martin Bridgeman, William Floyd
and Jeffrey Weeks
. His son, Dylan Thurston, is an assistant professor of mathematics at Barnard College
, Columbia University
.
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
. 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.
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.
Thurston has an Erdős number
of 2.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He is a pioneer in the field of low-dimensional topology
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
. In 1982, he was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
for his contributions to the study of 3-manifolds. He is currently a professor of mathematics and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
(since 2003).
Foliations
His early work, in the early 1970s, was mainly in foliationFoliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
theory, where he had a dramatic impact. His more significant results include:
- The proof that every Haefliger structureHaefliger structureIn mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by . Any foliation on a manifold induces a Haefliger structure, which uniquely determines the foliation.-Definition:...
on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristicEuler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
admits a foliation of codimension one).
- The construction of a continuous family of smooth, codimension one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value.
- With John Mather, he gave a proof that the cohomologyCohomologyIn mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
of the group of homeomorphismHomeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s of a manifold is the same whether the group is considered with its discrete topology or its compact-open topologyCompact-open topologyIn mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
.
In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ).
The geometrization conjecture
His later work, starting around the late 1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifoldHyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
s of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen 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
Hyperbolic link
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry...
. 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
Wolfgang Haken
Wolfgang Haken is a mathematician who specializes in topology, in particular 3-manifolds.In 1976 together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color theorem...
's normal surface
Normal surface
In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad . A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices...
techniques, he classified the incompressible surface
Incompressible surface
In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
s in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
on the figure eight knot resulted in irreducible, non-Haken
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
non-Seifert-fibered
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...
3-manifolds. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds 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
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...
theorem.
To complete the picture, Thurston proved a hyperbolization theorem
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:...
for Haken manifold
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
s. 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-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. 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...
s.
Thurston was next led to formulate his geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
. 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 follows from the recent work of Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
(2002–2003).
Orbifold theorem
In his work on hyperbolic Dehn surgery, Thurston realized that orbifoldOrbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
structures naturally arose. 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
Richard Hamilton (professor)
Richard Streit Hamilton is Davies Professor of mathematics at Columbia University.He received his B.A in 1963 from Yale University and Ph.D. in 1966 from Princeton University. Robert Gunning supervised his thesis...
's work on the Ricci flow
Ricci flow
In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
.
Education and career
Thurston was born in Washington, D.C.Washington, D.C.
Washington, D.C., formally the District of Columbia and commonly referred to as Washington, "the District", or simply D.C., is the capital of the United States. On July 16, 1790, the United States Congress approved the creation of a permanent national capital as permitted by the U.S. Constitution....
to a homemaker and an aeronautical engineer. He received his bachelors degree from New College (now New College of Florida
New College of Florida
New College of Florida is a public liberal arts college located in Sarasota, Florida. It was founded originally as a private institution and is now an autonomous honors college of the State University System of Florida.-History:...
) in 1967. For his undergraduate thesis he developed an intuitionist
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
foundation for topology. Following this, he earned a doctorate in mathematics from the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
, in 1972. 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
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
, then another year at MIT as Assistant Professor. In 1974, he was appointed Professor of Mathematics at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute
Mathematical Sciences Research Institute
The Mathematical Sciences Research Institute , founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions...
. In 1996, he moved to University of California, Davis
University of California, Davis
The University of California, Davis is a public teaching and research university established in 1905 and located in Davis, California, USA. Spanning over , the campus is the largest within the University of California system and third largest by enrollment...
. In 2003, he moved again to become Professor of Mathematics at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
.
His Ph.D. students include Richard Canary, Renaud Dreyer, David Gabai
David Gabai
David Gabai, a mathematician, is currently a professor at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects....
, William Goldman
William Goldman (professor)
William Goldman is a professor of mathematics at the University of Maryland, College Park . He received his Ph.D. in mathematics from the University of California, Berkeley in 1980. He was on the Board of Governors for the The Geometry Center at the University of Minnesota from 1994 to 1996...
, Benson Farb
Benson Farb
Benson Stanley Farb is an American mathematician at the University of Chicago. His research fields include geometric group theory and low-dimensional topology....
, Detlef Hardorp, Craig Hodgson, Richard Kenyon, Steven Kerckhoff
Steven Kerckhoff
Steven Paul Kerckhoff is a professor of mathematics at Stanford University, who works on hyperbolic 3-manifolds and Teichmüller spaces....
, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Oded Schramm
Oded Schramm
Oded Schramm was an Israeli-American mathematician known for the invention of the Schramm–Loewner evolution and for working at the intersection of conformal field theory and probability theory.-Biography:...
, Richard Schwartz
Richard Schwartz
Richard Evan Schwartz is an American mathematician notable for his contributions to geometric group theory and to an area of mathematics known as billiards...
, Martin Bridgeman, William Floyd
William Floyd (mathematician)
William J. Floyd is an American mathematician specializing in topology. He is currently a professor at Virginia Polytechnic Institute and State University....
and Jeffrey Weeks
Jeffrey Weeks (mathematician)
Jeffrey Renwick Weeks is an American mathematician, a geometric topologist and cosmologist.-Biography:Weeks received his B.A. from Dartmouth College in 1978, and his Ph.D. in mathematics from Princeton University in 1985, under the supervision of William Thurston...
. His son, Dylan Thurston, is an assistant professor of mathematics at Barnard College
Barnard College
Barnard College is a private women's liberal arts college and a member of the Seven Sisters. Founded in 1889, Barnard has been affiliated with Columbia University since 1900. The campus stretches along Broadway between 116th and 120th Streets in the Morningside Heights neighborhood in the borough...
, Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
.
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
The Geometry Center
The Geometry Center was a mathematics research and education center at the University of Minnesota. It was established by the National Science Foundation in the late 1980s and closed in 1998. The focus of the Center's work was the use of computer graphics and visualization for research and...
. As director of Mathematical Sciences Research Institute
Mathematical Sciences Research Institute
The Mathematical Sciences Research Institute , founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions...
from 1992 to 1997, he initiated a number of programs designed to increase awareness of mathematics among the public.
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.
Thurston has an Erdős number
Erdos number
The Erdős number describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.The same principle has been proposed for other eminent persons in other fields.- Overview :...
of 2.
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 MathAnnals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
. (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