Editing William Thurston

{{short description|American mathematician (1946–2012)}}
{{Infobox scientist
| name              = William Thurston
| image             = William Thurston.jpg
| caption           = Thurston in 1991
| birth_name        = William Paul Thurston
| birth_date        = {{birth date|1946|10|30}}
| birth_place       = [[Washington, D.C.]], U.S.
| death_date        = {{death date and age|2012|08|21|1946|10|30}}
| death_place       = [[Rochester, New York|Rochester]], New York, U.S.
| field             = [[Mathematics]]
| work_institutions = [[Cornell University]]<br />[[University of California, Davis]]<br />[[Mathematical Sciences Research Institute]]<br />[[University of California, Berkeley]]<br />[[Princeton University]]<br />[[Massachusetts Institute of Technology]]<br />[[Institute for Advanced Study]]
| alma_mater        = [[New College of Florida]]<br />[[University of California, Berkeley]]
| thesis_title      = Foliations of three-manifolds which are circle bundles
| thesis_year       = 1972
| doctoral_advisor  = [[Morris Hirsch]]
| doctoral_students = [[Danny Calegari]]<br />[[Richard Canary]]<br />[[Benson Farb]]<br />[[David Gabai]]<br />[[William Goldman (mathematician)|William Goldman]]<br />[[Richard Kenyon]]<br />[[Steven Kerckhoff]]<br />[[Yair Minsky]]<br />[[Igor Rivin]]<br />[[Oded Schramm]]<br />[[Richard Schwartz (mathematician)|Richard Schwartz]]<br /> [[Jeffrey Weeks (mathematician)|Jeffrey Weeks]]
| prizes            = [[Fields Medal]] (1982)<br />[[Oswald Veblen Prize in Geometry]] (1976)<br />[[Alan T. Waterman Award]] (1979)<br />[[National Academy of Sciences]] (1983)<br />[[Doob Prize]] (2005)<br />[[Leroy P. Steele Prize]] (2012).
| known_for         = [[Thurston's geometrization conjecture]]<br />[[Thurston's 24 questions]]<br/>[[Thurston's theory of surfaces]]<br />[[Milnor–Thurston kneading theory]] <br /> [[Orbifold]]
}}
'''William Paul Thurston''' (October 30, 1946{{spaced ndash}}August 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-manifold]]s.

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 ({{nee|Martt}}), a seamstress, and Paul Thurston, an aeronautical engineer.<ref name="notices-1">{{cite journal |first1=David |last1=Gabai |author1-link=David Gabai |first2=Steven |last2=Kerckhoff |author2-link=Steven Kerckhoff |url=http://www.ams.org/notices/201511/rnoti-p1318.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.ams.org/notices/201511/rnoti-p1318.pdf |archive-date=2022-10-09 |url-status=live |title=William P. Thurston, 1946–2012 |journal=[[Notices of the American Mathematical Society]] |year=2015 |volume=62 |issue=11 |pages=1318–1332|doi=10.1090/noti1300 }}</ref> William Thurston suffered from congenital [[strabismus]] as a child, causing issues with depth perception.<ref name="notices-1"/> His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones.<ref name="notices-1"/>

He received his bachelor's degree from [[New College of Florida|New College]] in 1967 as part of its inaugural class.<ref name="notices-1"/><ref>{{Cite web|url=http://www.news.cornell.edu/stories/2012/08/mathematics-innovator-william-thurston-dies-65|title=World-renowned mathematician William Thurston dies at 65|first=Susan|last=Kelley|date=Aug 24, 2012|accessdate=2023-01-11}}</ref> For his undergraduate thesis, he developed an [[intuitionism|intuitionist]] foundation for topology.<ref>See p. 3 in {{cite arXiv |last1=Laudenbach |first1=François |last2=Papadopoulos| first2=Athanase  |eprint=1912.03115 |title=W. P. Thurston and French mathematics |class=math.GT |date=2019 }}</ref> Following this, he received a doctorate in mathematics from the [[University of California, Berkeley]] under [[Morris Hirsch]], with his thesis ''Foliations of Three-Manifolds which are Circle Bundles'' in 1972.<ref name="notices-1"/><ref>{{Cite web|url=http://www.genealogy.math.ndsu.nodak.edu/id.php?id=11749|title = William Thurston – the Mathematics Genealogy Project}}</ref>

==Career==
After completing his Ph.D., Thurston spent a year at the [[Institute for Advanced Study]],<ref name="notices-1"/><ref>{{cite web|url=http://www.ias.edu/people/cos/ |title=Institute for Advanced Study: A Community of Scholars |publisher=Ias.edu |access-date=2013-09-06}}</ref> then another year at the [[Massachusetts Institute of Technology]] as an assistant professor.<ref name="notices-1"/>

In 1974, Thurston was appointed a full professor at [[Princeton University]].<ref name="notices-1"/><ref name=NYT/> He returned to [[University of California, Berkeley|Berkeley]] in 1991 to be a professor (1991-1996) and was also director of the [[Mathematical Sciences Research Institute]] (MSRI) from 1992 to 1997.<ref name="notices-1"/><ref name="AMS-web-obit"/> He was on the faculty at [[University of California, Davis|UC Davis]] from 1996 until 2003, when he moved to [[Cornell University]].<ref name="notices-1"/> 

Thurston was an early adopter of computing in pure mathematics research.<ref name="notices-1"/> He inspired [[Jeffrey Weeks (mathematician)|Jeffrey Weeks]] to develop the [[SnapPea]] computing program.<ref name="notices-1"/>

During Thurston's directorship at MSRI, the institute introduced several innovative educational programs that have since become standard for research institutes.<ref name="notices-1"/>

His Ph.D. students include [[Danny Calegari]], [[Richard Canary]], [[David Gabai]], [[William Goldman (professor)|William Goldman]], [[Benson Farb]], [[Richard Kenyon]], [[Steven Kerckhoff]], [[Yair Minsky]], [[Igor Rivin]], [[Oded Schramm]], [[Richard Schwartz (mathematician)|Richard Schwartz]], [[William Floyd (mathematician)|William Floyd]], and [[Jeffrey Weeks (mathematician)|Jeffrey Weeks]].<ref>{{Cite web|url=http://www.genealogy.ams.org/id.php?id=11749|title = William Thurston – the Mathematics Genealogy Project}}</ref>

== Research ==
=== Foliations ===
{{Expand section|date=June 2008}}
His early work, in the early 1970s, was mainly in [[foliation]] 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]] admits a foliation of [[codimension]] one).
* The construction of a continuous family of smooth, codimension-one foliations on the [[n-sphere|three-sphere]] whose [[Godbillon–Vey invariant]]s (after [[Claude Godbillon]] and Jacques Vey) take every real value.
* With [[John N. Mather]], he gave a proof that the [[cohomology]] of the group of [[homeomorphism]]s of a manifold is the same whether the group is considered with its [[discrete space|discrete topology]] or its [[compact-open topology]].

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,<ref>{{Cite web|url=http://blogs.scientificamerican.com/observations/the-mathematical-legacy-of-william-thurston-1946-2012/|title = The Mathematical Legacy of William Thurston (1946–2012)| work=Scientific American Blog Network | last1=Lamb | first1=Evelyn }}</ref> because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6<ref name=bams>{{cite journal | last = Thurston  | first = William P.  |date=April 1994  | title = On Proof and Progress in Mathematics | journal = [[Bulletin of the American Mathematical Society]]  | volume = 30  | issue = 2  | pages =161–177  | arxiv = math/9404236  | doi = 10.1090/S0273-0979-1994-00502-6 | bibcode = 1994math......4236T }}</ref>).

=== The geometrization conjecture ===
{{Main|Geometrization conjecture}}
{{See also|Thurston's 24 questions}}
{{Refimprove section|date=March 2022}}
His later work, starting around the mid-1970s, revealed that [[hyperbolic geometry]] played a far more important role in the general theory of [[3-manifold]]s than was previously realised.  Prior to Thurston, there were only a handful of known examples of [[hyperbolic 3-manifold]]s of finite volume, such as the [[Seifert–Weber space]].  The independent and distinct approaches of [[Robert Riley (mathematician)|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 (mathematics)|figure-eight knot]] [[knot complement|complement]] was [[hyperbolic link|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 [[manifold decomposition|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 [[Wolfgang Haken|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 surgery|Dehn surgeries]] on the figure-eight knot resulted in [[irreducibility (mathematics)|irreducible]], non-[[Haken manifold|Haken]] non-[[Seifert fiber space|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 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 hyperbolization theorem for Haken manifolds has been called ''Thurston's Monster Theorem,'' due to the length and difficulty of the proof.<ref>{{Cite journal |last=Friedl |first=Stefan |date=2014-12-01 |title=Thurston's Vision and the Virtual Fibering Theorem for 3-Manifolds |url=https://link.springer.com/article/10.1365/s13291-014-0102-x |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |language=en |volume=116 |issue=4 |pages=223–241 |doi=10.1365/s13291-014-0102-x |issn=1869-7135}}</ref>  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. The conjecture was proved by [[Grigori Perelman]] in 2002–2003.<ref>{{cite arXiv|last=Perelman|first=Grisha|date=2003-03-10|title=Ricci flow with surgery on three-manifolds|eprint=math/0303109}}</ref><ref>{{Cite journal|last1=Kleiner|first1=Bruce|last2=Lott|first2=John|date=2008-11-06|title=Notes on Perelman's papers|arxiv=math/0605667|journal=Geometry & Topology|volume=12|issue=5|pages=2587–2855|doi=10.2140/gt.2008.12.2587|issn=1364-0380|doi-access=free}}</ref>

=== Density conjecture ===
Thurston and [[Dennis Sullivan]] generalized [[Lipman Bers]]' [[Density theorem for Kleinian groups|density conjecture]] from singly degenerate Kleinian surface groups to all [[finitely generated group|finitely generated]] [[Kleinian group]]s in the late 1970s and early 1980s.<ref name="Namazi-Souto"/><ref name="Ohshika"/> The conjecture states that every finitely generated Kleinian group is an algebraic limit of [[Geometric finiteness|geometrically finite]] Kleinian groups, and was independently proven by Ohshika and Namazi–Souto in 2011 and 2012 respectively.<ref name="Namazi-Souto">{{cite journal |last1=Namazi |first1=Hossein |last2=Souto |first2=Juan |title=Non-realizability and ending laminations: Proof of the density conjecture |year=2012 |journal=[[Acta Mathematica]] |volume=209 |issue=2 |pages=323–395 |doi=10.1007/s11511-012-0088-0 |s2cid=10138438 |issn=0001-5962 |doi-access=free }}</ref><ref name="Ohshika">{{cite journal | last1=Ohshika | first1=Ken'ichi | title=Realising end invariants by limits of minimally parabolic, geometrically finite groups | url=http://www.msp.warwick.ac.uk/gt/2011/15-02/p023.xhtml | year=2011 | journal=[[Geometry and Topology]] | issn=1364-0380 | volume=15 | issue=2 | pages=827–890 | doi=10.2140/gt.2011.15.827 | arxiv=math/0504546 | s2cid=14463721 | access-date=March 24, 2022 | archive-date=May 25, 2014 | archive-url=https://web.archive.org/web/20140525195019/http://www.msp.warwick.ac.uk/gt/2011/15-02/p023.xhtml | url-status=live }}</ref>

=== Orbifold theorem ===
{{Unreferenced section|date=March 2022}}
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#3-dimensional orbifolds|orbifold theorem]], an extension of his geometrization theorem to the setting of 3-orbifolds.<ref>{{Cite book |last=Thurston |first=William P. |title=Collected works of William P. Thurston with commentary. Vol. II. 3-manifolds, complexity and geometric group theory. |publisher=American Mathematical Society |year=2022 |isbn=9781470468347 |pages=147–151}}</ref>  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]].

==Awards and honors==
In 1976, Thurston and [[James Harris Simons]] shared the [[Oswald Veblen Prize in Geometry]].<ref name="notices-1"/>

Thurston received the [[Fields Medal]] in 1982 for "revolutioniz[ing] [the] study of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry" and "contribut[ing] [the] idea that a very large class of closed [[3-manifold]]s carry a hyperbolic structure."<ref>{{cite web |url=http://www.math.cornell.edu/spb/index.php|title=William P. Thurston, 1946–2012|date=30 August 2012|access-date=18 August 2014}}</ref><ref name="82reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1982/index.html |title=Fields Medals and Nevanlinna Prize 1982 |publisher=International Mathematical Union |website=mathunion.org}}</ref>

In 2005, Thurston won the first [[American Mathematical Society]] [[Joseph L. Doob Prize|Book Prize]], for ''[[The_geometry_and_topology_of_three-manifolds|Three-dimensional Geometry and Topology]]''.
The prize "recognizes an outstanding research book that makes a seminal contribution to the research literature".<ref name=amsbookprize>{{cite web | title=William P. Thurston Receives 2005 AMS Book Prize|url=http://www.ams.org/ams/press/book-thurston.html| access-date=2008-06-26}}</ref> He was awarded the 2012 [[Leroy P. Steele Prize]] by the American Mathematical Society for seminal contribution to research. The citation described his work as having "revolutionized 3-manifold theory".<ref name=prizebooklet>{{cite web | title=AMS prize booklet 2012|url=http://www.ams.org/profession/prizebooklet-2012.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.ams.org/profession/prizebooklet-2012.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

==Personal life==
Thurston and his first wife, Rachel Findley, had three children: Dylan, Nathaniel, and Emily.<ref name=NYT /> Dylan was a [[Mathematical Olympiad Program|MOSP]] participant (1988–90)<ref>{{cite web |title=YEAR 1990 |url=https://www.maa.org/sites/default/files/pdf/AMC/usamo/e8-usamo/e8-1-usamoarchive/listings-ua/1990.pdf |website=USAMO Archive |access-date=30 January 2023}}</ref> and is a mathematician at [[Indiana University Bloomington]].<ref>{{cite book |editor1-last=Thurston |editor1-first=Dylan P. |title=What's Next? The Mathematical Legacy of William P. Thurston |date=2020 |publisher=Princeton University Press |isbn=978-0-691-16776-3}}</ref> Thurston had two children with his second wife, Julian Muriel Thurston: Hannah Jade and Liam.<ref name=NYT />

Thurston died on August 21, 2012, in [[Rochester, New York]], of a sinus mucosal [[melanoma]] that was diagnosed in 2011.<ref name=NYT>{{cite news
| newspaper = [[New York Times]]
| url       = https://www.nytimes.com/2012/08/23/us/william-p-thurston-theoretical-mathematician-dies-at-65.html
| page      = B15
| date      = August 23, 2012 <!-- date and page are for print edition, online article date is August 22 -->
| title     = William P. Thurston, Theoretical Mathematician, Dies at 65
| author    = Leslie Kaufman}}</ref><ref name=cornell-obit>"[http://www.math.cornell.edu/News/2012-2013/thurston.html Department mourns loss of friend and colleague, Bill Thurston]", Cornell University</ref><ref name="AMS-web-obit">{{cite web |url=http://www.ams.org/news?news_id=1602 |title=William P. Thurston, 1946-2012 |website=[[American Mathematical Society]] |date=August 22, 2012 |access-date=March 25, 2022}}</ref>

== 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.&nbsp;{{isbn|0-691-08304-5}}
* William Thurston, ''Hyperbolic structures on 3-manifolds''. I. Deformation of acylindrical manifolds. [[Annals of Mathematics|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.&nbsp;{{isbn|0-86720-244-0}}<ref>Reviews of ''Word Processing in Groups'': B. N. Apanasov, {{zbl|0764.20017}}; [[Gilbert Baumslag]], ''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, {{mr|1161694}}</ref>
* Eliashberg, Yakov M.; Thurston, William P. ''Confoliations''. University Lecture Series, 13. American Mathematical Society, Providence, Rhode Island and Providence Plantations, 1998. x+66 pp.&nbsp;{{isbn|0-8218-0776-5}}
* William Thurston, [https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf ''On proof and progress in mathematics'']. Bull. Amer. Math. Soc. (N.S.) 30 (1994) 161–177
* William P. Thurston, [https://arxiv.org/abs/math.HO/0503081 "Mathematical education"].  Notices of the AMS 37:7 (September 1990) pp 844–850

== See also ==
{{Div col}}
* [[Automatic group]]
* [[Cannon–Thurston map]]
* [[Circle packing theorem]]
* [[Hyperbolic volume]]
* [[Hyperbolic Dehn surgery]]
* [[Thurston boundary]]
* [[Milnor–Thurston kneading theory]]
* [[Misiurewicz point|Misiurewicz–Thurston points]]
* [[Nielsen–Thurston classification]]
* [[Normal surface]]
* [[Orbifold notation]]
* [[Thurston norm]]
* [[Thurston's 24 questions]]
* [[Thurston's double limit theorem]]
* [[Thurston elliptization conjecture]]
* [[Thurston's geometrization conjecture]]
* [[Domino tiling#Thurston's height condition|Thurston's height condition]]
* [[Orbifold#3-dimensional orbifolds|Thurston's orbifold theorem]]
* [[Earthquake theorem]]
{{colend}}

== References ==
{{Reflist}}

== Further reading ==
* Gabai, David; Kerckhoff, Steve (Coordinating Editors). "[http://www.ams.org/publications/journals/notices/201601/rnoti-p31.pdf William P. Thurston, 1946–2012]" (part 2), ''[[Notices of the American Mathematical Society]]'', January 2015, Volume 63, Number 1, pp.&nbsp;31–41.

== External links ==
* {{Commons category inline}}
{{Wikiquote}}
* {{MathGenealogy |id=11749}}
* {{MacTutor Biography|id=Thurston}}
* [http://www.math.cornell.edu/People/Faculty/thurston.html Thurston's page at Cornell]
* [http://www.math.cornell.edu/News/2012-2013/thurston.html Tribute and remembrance page at Cornell]
* [https://www.youtube.com/watch?v=-FUGmiqD408 Etienne Ghys : La géométrie et la mode]
* {{cite web|title=Landau Lectures {{!}} Prof. Thurston {{!}} Part 1 {{!}} 1995/6|date=April 8, 2014|publisher=Hebrew University of Jerusalem|website=YouTube|url=https://www.youtube.com/watch?v=o6SucT2Zzys}}
* {{cite web|title=Landau Lectures {{!}} Prof. Thurston {{!}} Part 2 {{!}} 1995/6|date=April 8, 2014|publisher=Hebrew University of Jerusalem|website=YouTube|url=https://www.youtube.com/watch?v=w30xG_vGwoc}}
* {{cite web|title=Landau Lectures {{!}} Prof. Thurston {{!}} Part 3 {{!}} 1995/6|date=April 8, 2014|publisher=Hebrew University of Jerusalem|website=YouTube|url=https://www.youtube.com/watch?v=PXsqdiS-eRA}}
* {{cite web|title=The Mystery of 3-Manifolds - William Thurston|date=November 27, 2011|publisher=PoincareDuality|website=YouTube|url=https://www.youtube.com/watch?v=4jdmkUQDWtQ}} 2010 Clay Research Conference
* {{cite web|author=Goldman, William|author-link=William Goldman (mathematician)|title=William Thurston: A Mathematical Perspective|date=May 9, 2013|publisher=UMD Mathematics|website=YouTube|url=https://www.youtube.com/watch?v=apgWp5Tz5j0}} William Goldman (U. of Maryland), Collloquium, Department of Mathematics, Howard University, 25 January 2013

{{Fields medalists}}
{{Veblen Prize recipients}}

{{Authority control}}

{{DEFAULTSORT:Thurston, William}}
[[Category:1946 births]]
[[Category:2012 deaths]]
[[Category:New College of Florida alumni]]
[[Category:University of California, Berkeley alumni]]
[[Category:American topologists]]
[[Category:Differential geometers]]
[[Category:Hyperbolic geometers]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:20th-century American mathematicians]]
[[Category:21st-century American mathematicians]]
[[Category:Fields Medalists]]
[[Category:Princeton University faculty]]
[[Category:University of California, Berkeley faculty]]
[[Category:Cornell University faculty]]
[[Category:University of California, Davis faculty]]
[[Category:Institute for Advanced Study visiting scholars]]
[[Category:Mathematicians from Washington, D.C.]]