William Paul Thurston (October 30, 1946August 21, 2012) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. He was a pioneer in the field of

low-dimensional topology In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot th ...

and was awarded the

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

in 1982 for his contributions to the study of

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

s. Thurston was a professor of mathematics at

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

University of California, Davis The University of California, Davis (UC Davis, UCD, or Davis) is a public land-grant research university near Davis, California. Named a Public Ivy, it is the northernmost of the ten campuses of the University of California system. The institut ...

, and

Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...

. He was also a director of the

Mathematical Sciences Research Institute The Simons Laufer Mathematical Sciences Institute (SLMath), formerly the Mathematical Sciences Research Institute (MSRI), is an independent nonprofit mathematical research institution on the University of California campus in Berkeley, Califo ...

Early life and education

William Thurston was born in

Washington, D.C. ) , image_skyline = , image_caption = Clockwise from top left: the Washington Monument and Lincoln Memorial on the National Mall, United States Capitol, Logan Circle, Jefferson Memorial, White House, Adams Morgan, ...

to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital

strabismus Strabismus is a vision disorder in which the eyes do not properly align with each other when looking at an object. The eye that is focused on an object can alternate. The condition may be present occasionally or constantly. If present during a ...

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 In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...

foundation for topology. Following this, he received a doctorate in mathematics from the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

under

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 ...

, with his thesis ''Foliations of Three-Manifolds which are Circle Bundles'' in 1972.

Career

After completing his Ph.D., Thurston spent a year at the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

, then another year at the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...

as an assistant professor. In 1974, Thurston was appointed a full professor at

. He returned to

Berkeley Berkeley most often refers to: *Berkeley, California, a city in the United States **University of California, Berkeley, a public university in Berkeley, California * George Berkeley (1685–1753), Anglo-Irish philosopher Berkeley may also refer ...

in 1991 to be a professor (1991-1996) and was also director of the

(MSRI) from 1992 to 1997. He was on the faculty at

UC Davis The University of California, Davis (UC Davis, UCD, or Davis) is a public land-grant research university near Davis, California. Named a Public Ivy, it is the northernmost of the ten campuses of the University of California system. The institut ...

from 1996 until 2003, when he moved to

. Thurston was an early adopter of computing in pure mathematics research. He inspired Jeffrey Weeks to develop the

SnapPea SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version as part of his doctoral thesis, supervised ...

computing program. During Thurston's directorship at MSRI, the institute introduced several innovative educational programs that have since become standard for research institutes. His Ph.D. students include

Danny Calegari Danny Matthew Cornelius Calegari is a mathematician who is currently a professor of mathematics at the University of Chicago. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory. Educ ...

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 W ...

David Gabai David Gabai is an American mathematician and the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects. Biography David Ga ...

William Goldman William Goldman (August 12, 1931 – November 16, 2018) was an American novelist, playwright, and screenwriter. He first came to prominence in the 1950s as a novelist before turning to screenwriting. He won Academy Awards for his screenplays '' ...

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. Early life A native of Norristown, Pennsylvania, Farb earned ...

, Richard Kenyon, Steven Kerckhoff,

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 ...

, Igor Rivin,

Oded Schramm Oded Schramm ( he, עודד שרם; December 10, 1961 – September 1, 2008) was an Israeli-American mathematician known for the invention of the Schramm–Loewner evolution (SLE) and for working at the intersection of conformal field theory ...

, Richard Schwartz,

William Floyd William Floyd (December 17, 1734 – August 4, 1821) was an American Founding Father, wealthy farmer, and political leader from New York. Floyd served as a delegate to the Continental Congress and was a signer of the Continental Association and ...

, and Jeffrey Weeks.

Research

Foliations

His early work, in the early 1970s, was mainly in

foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...

theory. 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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

admits a foliation of

codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the ...

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 N. Mather, he gave a proof that the

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

of the group of

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

s of a manifold is the same whether the group is considered with its

discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

or its

compact-open topology In 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 ...

. In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that it led to an 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 mid-1970s, revealed that

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

played a far more important role in the general theory of

s than was previously realised. Prior to Thurston, there were only a handful of known examples of

hyperbolic 3-manifold In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...

s of finite volume, such as the

Seifert–Weber space In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discover ...

. 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 The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...

was

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

. This was the first example of a

hyperbolic knot Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

. Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the

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 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'' (see figure). A triangle cuts off a vertex of the tetrahedron wh ...

techniques, he classified the

incompressible surface In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the surface of a suitcase is compressible, because ...

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 In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

, non- Haken non-

Seifert-fibered A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for a ...

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 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 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 Thurston's geometrization theor ...

for

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. Sometimes one considers only orientable Haken manifolds, in w ...

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 hyperbolization theorem for Haken manifolds 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

s. Thurston was next led to formulate his

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-dimens ...

. 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 Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...

in 2002–2003.

Density conjecture

Thurston and

Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...

generalized

Lipman Bers Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also kn ...

' density conjecture from singly degenerate Kleinian surface groups to all finitely generated

Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...

s in the late 1970s and early 1980s. The conjecture states that every finitely generated Kleinian group is an algebraic limit of

geometrically finite In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite group ...

Kleinian groups, and was independently proven by Ohshika and Namazi–Souto in 2011 and 2012 respectively.

Orbifold theorem

In his work on hyperbolic Dehn surgery, Thurston realized that

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 which is locally a finite group quotient of a Euclidean space. D ...

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 In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...

Awards and honors

In 1976, Thurston and

James Harris Simons James Harris Simons (; born 25 April 1938) is an American mathematician, billionaire hedge fund manager, and philanthropist. He is the founder of Renaissance Technologies, a quantitative hedge fund based in East Setauket, New York. He and his f ...

shared the

Oswald Veblen Prize in Geometry __NOTOC__ 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 ...

. Thurston received the

in 1982 for "revolutioniz ng hestudy of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry" and "contribut ng heidea that a very large class of closed

s carry a hyperbolic structure." In 2005, Thurston won the first

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

Book Prize, for ''Three-dimensional Geometry and Topology''. The prize "recognizes an outstanding research book that makes a seminal contribution to the research literature". He was awarded the 2012

Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

by the American Mathematical Society for seminal contribution to research. The citation described his work as having "revolutionized 3-manifold theory".

Personal life

Thurston and his first wife, Rachel Findley, had three children: Dylan, Nathaniel, and Emily. Thurston had two children with his second wife, Julian Muriel Thurston: Hannah Jade and Liam. His son Dylan Thurston is also a mathematician. Thurston died on August 21, 2012 in

Rochester, New York Rochester () is a City (New York), city in the U.S. state of New York (state), New York, the county seat, seat of Monroe County, New York, Monroe County, and the fourth-most populous in the state after New York City, Buffalo, New York, Buffalo, ...

, of a sinus mucosal

melanoma Melanoma, also redundantly known as malignant melanoma, is a type of skin cancer that develops from the pigment-producing cells known as melanocytes. Melanomas typically occur in the skin, but may rarely occur in the mouth, intestines, or eye ( ...

that was diagnosed in 2011.Department mourns loss of friend and colleague, Bill Thurston
, Cornell University

Selected publications

* William Thurston, '' The geometry and topology of three-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, New Jersey, 1997. x+311 pp. * William Thurston, ''Hyperbolic structures on 3-manifolds''. I. Deformation of acylindrical manifolds. 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, Massachusetts, 1992. xii+330 pp. Reviews of ''Word Processing in Groups'': B. N. Apanasov, ;

Gilbert Baumslag Gilbert Baumslag (April 30, 1933 – October 20, 2014) was a Distinguished Professor at the City College of New York, with joint appointments in mathematics, computer science, and electrical engineering. He was director of thCenter for Algorithms ...

, ''Bull. AMS'', doi:10.1090/S0273-0979-1994-00481-1; D. E. Cohen, ''Bull LMS'', doi:10.1112/blms/25.6.614; Richard M. Thomas, * Eliashberg, Yakov M.; Thurston, William P. ''Confoliations''. University Lecture Series, 13. American Mathematical Society, Providence, Rhode Island and Providence Plantations, 1998. x+66 pp. * William Thurston
''On proof and progress in mathematics''
Bull. Amer. Math. Soc. (N.S.) 30 (1994) 161–177 * William P. Thurston
"Mathematical education"
Notices of the AMS 37:7 (September 1990) pp 844–850

References

External links

* * *
Thurston's page at Cornell

Etienne Ghys : La géométrie et la mode
{{DEFAULTSORT:Thurston, William 1946 births 2012 deaths New College of Florida alumni University of California, Berkeley alumni Topologists Differential geometers Members of the United States National Academy of Sciences 20th-century American mathematicians 21st-century American mathematicians Fields Medalists Princeton University faculty University of California, Berkeley faculty Cornell University faculty University of California, Davis faculty Institute for Advanced Study visiting scholars Mathematicians from Washington, D.C.