William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was 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. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.
William Thurston in 1991
William Paul Thurston
October 30, 1946
Washington, D.C., United States
|Died||August 21, 2012 (aged 65)|
Rochester, New York, United States
|Alma mater||New College of Florida|
University of California, Berkeley
|Known for||Thurston's geometrization conjecture|
Thurston's theory of surfaces
Milnor–Thurston kneading theory
|Awards||Fields Medal (1982)|
Oswald Veblen Prize in Geometry (1976)
National Academy of Sciences (1983)
Leroy P. Steele Prize (2012).
University of California, Davis
Mathematical Sciences Research Institute
University of California, Berkeley
|Doctoral advisor||Morris Hirsch|
|Doctoral students||Richard Canary|
His early work, in the early 1970s, was mainly in foliation 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 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 )
His later work, starting around the mid-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-manifolds 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 surfaces 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 manifolds. 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.
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. The conjecture was proved by Grigori Perelman in 2002–2003.
In his work on hyperbolic Dehn surgery, Thurston realized that orbifold 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 Richard S. Hamilton's work on the Ricci flow.
Thurston was born in Washington, D.C. to a homemaker and an aeronautical engineer. He received his bachelor's 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 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. He and his first wife, née Rachel Findley, had three children: Dylan, Nathaniel, and Emily. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute. In 1996, his wife Julian, who had earlier been his Ph.D. student at Princeton University, made a career switch to veterinary medicine, and began her studies at the UC Davis School of Veterinary Medicine. Bill and Julian moved to Davis, California, where Bill became Professor of Mathematics at UC Davis. In 2000, their first child Jade was born, and in 2003 their second child Liam was born. Bill and Julian had visited Ithaca in 1997 for a family celebration for his mother's 80th birthday. They were enchanted by the beauty of Ithaca, and in 2003 the family moved to Ithaca, New York, where Bill became Professor of Mathematics at Cornell University.
His Ph.D. students include Martin Bridgeman, Danny Calegari, Richard Canary, Suhyoung Choi, Renaud Dreyer, Julian Thurston (aka Karen Barris), David Gabai, William Goldman, Benson Farb, Matt Grayson, Sergio Fenley, Detlef Hardorp, Craig Hodgson, Christopher Jerdonek, Richard Kenyon, Steven Kerckhoff, Silvio Levy, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Nicolau Saldanha, Oded Schramm, Richard Schwartz, William Floyd, Biao Wang and Jeffrey Weeks. His son Dylan Thurston is a professor of mathematics at Indiana University.
In later years Thurston widened his attention to include mathematical education and bringing mathematics to the general public. He has served as mathematics editor for Quantum Magazine, a youth science magazine, and was one of the founders 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 and his family had been in the process of moving back to Davis, California, where he was to rejoin the mathematics faculty at UC Davis while his wife completed her veterinary medical degree. Thurston died before he could make the move to California. He had remained with his brother George in Rochester, New York, while his family went ahead of him to California to get settled, waiting for him to gain better physical strength for making the cross-country trip to California to join them. Bill's health declined rapidly, and the family returned to Rochester to be with him during his final days.
In Thurston's last days, he sometimes used American Sign Language to communicate with his children, Liam and Jade. Bill and Julian had spent a year studying ASL when Jade was an infant, and the family had become somewhat fluent. He also communicated by writing on one of his many pads of paper. One of his last written messages was, "Treasure Island," and this reference remains mysterious to his family.
Allen Edward Hatcher (born October 23, 1944) is an American topologist.Benson Farb
Benson Stanley Farb (born October 25, 1967) is an American mathematician at the University of Chicago. His research fields include geometric group theory and low-dimensional topology.
A native of Norristown, Pennsylvania, Farb earned his bachelor's degree from Cornell University. In 1994, he obtained his doctorate from Princeton University, under supervision of William Thurston. He has advised over 30 students, including Andrew Putman.
He is married to Amie Wilkinson, professor of mathematics at University of Chicago.In 2012 he became a fellow of the American Mathematical Society.
He has authored four books.Earthquake map
In hyperbolic geometry, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by William Thurston (1986).Ending lamination theorem
In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston (1982), 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.
Minsky & preprint 2003, published 2010 and Brock et al. proved the ending lamination conjecture for Kleinian surface groups. In view of the Tameness theorem this implies the ending lamination conjecture for all finitely generated Kleinian groups, from which the general case of ELT follows.Geometrization conjecture
In mathematics, Thurston's geometrization conjecture 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 William Thurston (1982), 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 announced a proof in the 1980s and since then several complete proofs have appeared in print.
Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.
There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.Gromov norm
In mathematics, the Gromov norm (or simplicial volume) of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class.It is named after Mikhail Gromov, who with William Thurston, proved that the Gromov norm of a finite volume hyperbolic n-manifold is proportional to the hyperbolic volume. Thurston also used the Gromov norm to prove that hyperbolic volume decreases under hyperbolic Dehn surgery.Hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ) chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.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. A hyperbolic knot is a hyperbolic link with one component.
As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links.
As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3-manifolds.Hyperbolic volume
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.Milnor–Thurston kneading theory
The Milnor–Thurston kneading theory is a mathematical theory which analyzes the iterates of piecewise monotone mappings of an interval into itself. The emphasis is on understanding the properties of the mapping that are invariant under topological conjugacy.
The theory had been developed by John Milnor and William Thurston in two widely circulated and influential Princeton preprints from 1977 that were revised in 1981 and finally published in 1988. Applications of the theory include piecewise linear models, counting of fixed points, computing the total variation, and constructing an invariant measure with maximal entropy.Morris Hirsch
Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley.
A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of Edwin Spanier and Stephen Smale. His thesis was entitled Immersions of Manifolds. In 2012 he became a fellow of the American Mathematical Society.Hirsch had 23 doctoral students, including William Thurston, William Goldman, and Mary Lou Zeeman.Orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.
Groups representable in this notation include the point groups on the sphere (), the frieze groups and wallpaper groups of the Euclidean plane (), and their analogues on the hyperbolic plane ().Oswald Veblen Prize in Geometry
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen. The Veblen Prize is now worth US$5000, and is awarded every three years.
The first seven prize winners were awarded for works in topology. James Harris Simons and William Thurston were the first ones to receive it for works in geometry (for some distinctions, see geometry and topology).Pseudo-Anosov map
In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation introduced by William Thurston, who also coined the term "pseudo-Anosov diffeomorphism" when he proved his classification of diffeomorphisms of a surface.Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot.
Paul A. Smith (1939, remark after theorem 4) showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have fixed point set equal to a circle, and asked in (Eilenberg 1949, Problem 36) if the fixed point set can be knotted. Friedhelm Waldhausen (1969) proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case was described by John Morgan and Hyman Bass (1984) and depended on several major advances in 3-manifold theory, in particular the work of William Thurston on hyperbolic structures on 3-manifolds, and results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some additional help from Bass, Cameron Gordon, Peter Shalen, and Rick Litherland.
Deane Montgomery and Leo Zippin (1954) gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture is false in the topological (rather than the smooth or PL) category. Charles Giffen (1966) showed that the analogue of the Smith conjecture in higher dimensions is false: the fixed point set of a periodic diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2.Thurston, New York
Thurston is a town in Steuben County, New York, United States. The population was 1,309 at the 2000 census. The town is named after early landowner William Thurston.
The Town of Thurston is the east-central part of the county, northwest of Corning, New York.Thurston–Bennequin number
In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a Legendrian knot is defined as the writhe of the diagram minus the number of right cusps. It is named after William Thurston and Daniel Bennequin.
The maximum Thurston–Bennequin number over all Legendrian representatives of a knot is a topological knot invariant.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.
Floyd received a PhD in Mathematics from Princeton University 1978 under the direction of William Thurston.
Recipients of the Oswald Veblen Prize in Geometry