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==List of Fields medalists== In certain years, the Fields medalists have been officially cited for particular mathematical achievements, while in other years such specificities have not been given. However, in every year that the medal has been awarded, noted mathematicians have lectured at the [[International Congress of Mathematicians]] on each medalist's body of work. In the following table, official citations are quoted when possible (namely for the years 1958, 1998, and every year since 2006). For the other years through 1986, summaries of the ICM lectures, as written by Donald Albers, [[Gerald L. Alexanderson]], and [[Constance Reid]], are quoted.<ref>Albers, Donald J.; Alexanderson, G. L.; Reid, Constance. ''International mathematical congresses. An illustrated history 1893–1986''. Rev. ed. including ICM 1986. Springer-Verlag, New York, 1986</ref> In the remaining years (1990, 1994, and 2002), part of the text of the ICM lecture itself has been quoted. The upcoming awarding of the Fields Medal at the 2026 International Congress of the International Mathematical Union is planned to take place in [[Philadelphia]].<ref>{{cite web |title=ICM 2026 |url=https://www.mathunion.org/icm/icm-2026 |website=International Mathematical Union |access-date=2 June 2024}}</ref> {| class="wikitable sortable" style="margin: 1ex auto 1ex auto" |- ! Year ! [[International Congress of Mathematicians#List of Congresses|ICM]] location ! colspan=2|Medalists<ref name=IMU>{{cite web|url=http://www.mathunion.org/general/prizes/fields/prizewinners|title=The Fields Medalists, chronologically listed|publisher=[[International Mathematical Union]] (IMU)|date=8 May 2008|access-date=25 March 2009|archive-date=26 December 2018|archive-url=https://web.archive.org/web/20181226015744/https://www.mathunion.org/imu-awards/fields-medal|url-status=live}}</ref> ! Affiliation<br />(when awarded) ! Affiliation<br />(current/last) ! Reasons |- | rowspan=2| 1936 | rowspan=2| [[Oslo]], Norway | [[File:Lars Ahlfors - MFO.jpg|60px]] | {{sortname|Lars|Ahlfors}} | [[University of Helsinki]], Finland | [[Harvard University]], US<ref>{{cite web |url=https://www.ams.org/notices/199802/comm-krantz.pdf |title=Lars Valerian Ahlfors (1907–1996) |website=Ams.org |access-date=31 March 2017 |archive-date=3 March 2022 |archive-url=https://web.archive.org/web/20220303211929/https://www.ams.org/notices/199802/comm-krantz.pdf |url-status=live }}</ref><ref>{{cite web|url=http://www.math.harvard.edu/history/ahlfors/|title=Lars Ahlfors (1907–1996)|date=7 November 2004|publisher=Harvard University, Dept. of Math.|access-date=19 August 2014|archive-date=23 November 2014|archive-url=https://web.archive.org/web/20141123144942/http://www.math.harvard.edu/history/ahlfors/|url-status=live}}</ref> | "Awarded medal for research on covering surfaces related to [[Riemann surface]]s of [[inverse function]]s of entire and [[meromorphic function]]s. Opened up new fields of analysis."<ref name="36reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1936/index.html |title=Fields Medals 1936 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=31 July 2020 |archive-url=https://web.archive.org/web/20200731102503/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1936/index.html |url-status=live }}</ref> |- | [[File:Portrait of Jesse Douglas in c. 1932.jpg|60px]] | {{sortname|Jesse|Douglas}} | [[Massachusetts Institute of Technology]], US | [[City College of New York]], US<ref>{{cite encyclopedia|url=https://www.britannica.com/EBchecked/topic/170174/Jesse-Douglas|title=Jesse Douglas|date=28 May 2010|encyclopedia=Encyclopædia Britannica|access-date=19 August 2014|archive-date=3 September 2014|archive-url=https://web.archive.org/web/20140903211356/http://www.britannica.com/EBchecked/topic/170174/Jesse-Douglas|url-status=live}}</ref><ref>{{cite web |url=http://wdb.ugr.es/~geometry/seminar/files/talks/MMicallef20130207.pdf |title=The work of Jesse Douglas on Minimal Surfaces |author1=Mario J. Micallef |author2=J. Gray |website=Wdb.ugr.es |access-date=31 March 2017 |archive-url=https://web.archive.org/web/20141006070205/http://wdb.ugr.es/~geometry/seminar/files/talks/MMicallef20130207.pdf |archive-date=6 October 2014 |url-status=dead }}</ref> | "Did important work on the [[Plateau problem]] which is concerned with finding [[minimal surface]]s connecting and determined by some fixed boundary."<ref name="36reasons" /> |- |rowspan=2| 1950 |rowspan=2| [[Cambridge, Massachusetts|Cambridge]], US | [[File:LaurentSchwartz.jpg|60px]] | {{sortname|Laurent|Schwartz}} | [[University of Nancy]], France | [[University of Paris VII]], France<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/Biographies/Schwartz.html|title=Laurent Moise Schwartz|date=24 June 2007|publisher=School of Mathematics and Statistics University of St Andrews, Scotland|access-date=19 August 2014|archive-date=6 October 2014|archive-url=https://web.archive.org/web/20141006092731/http://www-history.mcs.st-and.ac.uk/Biographies/Schwartz.html|url-status=live}}</ref><ref>{{cite book |last=Schwartz |first=Laurent |year=2001 |title=Un mathématicien aux prises avec le siècle |trans-title=A Mathematician Grappling with His Century |url=https://www.springer.com/birkhauser/history+of+science/book/978-3-7643-6052-8 |archive-url=https://archive.today/20140821114917/http://www.springer.com/birkhauser/history+of+science/book/978-3-7643-6052-8 |url-status=dead |archive-date=21 August 2014 |location=AMS |publisher=Birkhäuser |isbn=978-3-0348-7584-4 |access-date=21 August 2014 }}</ref> | "Developed the [[theory of distributions]], a new notion of generalized function motivated by the [[Dirac delta-function]] of theoretical physics."<ref name="50reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1950/index.html |title=Fields Medals 1950 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074621/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1950/index.html |url-status=live }}</ref> |- | [[File:Atle Selberg.jpg|60px]] | {{sortname|Atle|Selberg}} | [[Institute for Advanced Study]], US | [[Institute for Advanced Study]], US<ref>{{cite web |url=https://www.ams.org/notices/200906/rtx090600692p-corrected.pdf |title=Remembering Atle Selberg, 1917–2007 |website=Ams.org |access-date=31 March 2017 |archive-date=23 November 2021 |archive-url=https://web.archive.org/web/20211123172218/http://www.ams.org/notices/200906/rtx090600692p-corrected.pdf |url-status=live }}</ref> | "Developed generalizations of the [[Brun's sieve|sieve methods of Viggo Brun]]; achieved major results on zeros of the [[Riemann zeta function]]; gave an elementary proof of the [[prime number theorem]] (with P. Erdős), with a generalization to prime numbers in an arbitrary arithmetic progression."<ref name="50reasons" /> |- |rowspan=2| 1954 |rowspan=2| [[Amsterdam]], Netherlands | [[File:Kodaira Kunihiko.jpg|60px]] | {{sortname|Kunihiko|Kodaira}} | | [[Princeton University]], US, [[University of Tokyo]], Japan and [[Institute for Advanced Study]], US<ref name="mathunion.org">{{cite web |url=http://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1954.1/ICM1954.1.ocr.pdf |title=Proceedings of the International Congress of Mathematicians |date=1954 |website=Mathunion.org |access-date=31 March 2017 |archive-date=29 March 2022 |archive-url=https://web.archive.org/web/20220329075759/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1954.1/ICM1954.1.ocr.pdf |url-status=live }}</ref> | [[University of Tokyo]], Japan<ref>{{cite web |url=https://www.ams.org/notices/199803/comm-obit-spencer.pdf |title=Kunihiko Kodaira (1915–1997) |author=Donald C. Spencer |website=Ams.org |access-date=31 March 2017 |archive-date=23 February 2022 |archive-url=https://web.archive.org/web/20220223055632/https://www.ams.org/notices/199803/comm-obit-spencer.pdf |url-status=live }}</ref> | "Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically to [[algebraic varieties]]. He demonstrated, by [[sheaf cohomology]], that such varieties are [[Hodge manifold]]s."<ref name="54reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1954/index.html |title=Fields Medals 1954 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074624/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1954/index.html |url-status=live }}</ref> |- | [[File:Jean-Pierre Serre, Erlangen 1997.JPG|60px]] | {{sortname|Jean-Pierre|Serre}} | [[University of Nancy]], France | [[Collège de France]], France<ref>{{cite web |url=http://www.math.rug.nl/~top/lectures/delft.pdf |title=Jean-Pierre Serre |website=Math.rug.nl |access-date=31 March 2017 |archive-date=16 June 2016 |archive-url=https://web.archive.org/web/20160616160010/http://www.math.rug.nl/~top/lectures/delft.pdf |url-status=live }}</ref><ref>{{cite encyclopedia|url=https://www.britannica.com/EBchecked/topic/535878/Jean-Pierre-Serre|title=Jean-Pierre Serre|date=5 February 1997|encyclopedia=Encyclopædia Britannica|access-date=19 August 2014|archive-date=6 October 2014|archive-url=https://web.archive.org/web/20141006104956/http://www.britannica.com/EBchecked/topic/535878/Jean-Pierre-Serre|url-status=live}}</ref> | "Achieved major results on the [[homotopy group]]s of spheres, especially in his use of the method of [[spectral sequence]]s. Reformulated and extended some of the main results of complex variable theory in terms of [[sheaf (mathematics)|sheaves]]."<ref name="54reasons" /> |- |rowspan=2| 1958 |rowspan=2| [[Edinburgh]], UK | | {{sortname|Klaus|Roth}} | [[University College London]], UK | [[Imperial College London]], UK<ref>{{Harvnb|McKinnon Riehm|Hoffman|2011|p=212}}</ref> | "for solving a famous problem of number theory, namely, the determination of the exact exponent in the Thue-Siegel inequality"<ref name="hopf">[[Heinz Hopf|H. Hopf]]. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1958/ICM1958.ocr.pdf Proceedings of the International Congress of Mathematicians 1958.] {{Webarchive|url=https://web.archive.org/web/20220408074808/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1958/ICM1958.ocr.pdf |date=8 April 2022 }} Report of the Inaugural Session. p. liv</ref> |- | [[File:René Thom.jpeg|60px]] | {{sortname|René|Thom}} | [[University of Strasbourg]], France | [[Institut des Hautes Études Scientifiques]], France<ref>{{cite web |url=http://www.robertnowlan.com/pdfs/Thom,%20Rene.pdf |title=René Thom |website=Robertnowlan.com |access-date=31 March 2017 |archive-url=https://web.archive.org/web/20160527024340/http://www.robertnowlan.com/pdfs/Thom,%20Rene.pdf |archive-date=27 May 2016 |url-status=dead }}</ref> | "for creating the theory of 'Cobordisme' which has, within the few years of its existence, led to the most penetrating insight into the topology of differentiable manifolds."<ref name="hopf" /> |- |rowspan=2| 1962 |rowspan=2| [[Stockholm]], Sweden | [[File:Lars Hörmander.jpg|60px]] | {{sortname|Lars|Hörmander}} | [[Stockholm University]], Sweden | [[Lund University]], Sweden<ref>{{cite web |url=http://smai.emath.fr/IMG/pdf/matapli100_Hormander.pdf |title=A tribute to Lars Hörmander |website=Smai.emath.fr |access-date=31 March 2017 |archive-date=4 June 2016 |archive-url=https://web.archive.org/web/20160604080024/http://smai.emath.fr/IMG/pdf/matapli100_Hormander.pdf |url-status=live }}</ref> | "Worked in [[partial differential equation]]s. Specifically, contributed to the general theory of linear differential operators. The questions go back to one of [[Hilbert's problems]] at the 1900 congress."<ref name="62reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1962/index.html |title=Fields Medals 1962 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074729/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1962/index.html |url-status=live }}</ref> |- | [[File:John Milnor, 1993 Oct (portioned).jpg|60px]] | {{sortname|John|Milnor}} | [[Princeton University]], US | [[Stony Brook University]], US<ref>{{cite web|url=http://www.math.sunysb.edu/~jack/|title=John W. Milnor|date=5 March 1997|publisher=Stony Brook University|access-date=17 August 2014|archive-date=30 August 2014|archive-url=https://web.archive.org/web/20140830025217/http://www.math.sunysb.edu/%7Ejack/|url-status=live}}</ref> | "Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of [[differential topology]]."<ref name="62reasons" /> |- |rowspan=4| 1966 |rowspan=4| [[Moscow]], USSR | [[File:Atiyah-Hirzebruch crop.jpeg|60px]] | {{sortname|Michael|Atiyah}} | [[University of Oxford]], UK | [[University of Edinburgh]], UK<ref>{{cite web |language=es |url=http://upcommons.upc.edu/video/bitstream/2099.2/946/17/Poster05-AbelPrize-CV.pdf |title=Sir Michael F. Atiyah : The Abel Prize |website=Upcommons.upc.edu |access-date=31 March 2017 |archive-date=16 June 2013 |archive-url=https://web.archive.org/web/20130616140748/http://upcommons.upc.edu/video/bitstream/2099.2/946/17/Poster05-AbelPrize-CV.pdf |url-status=live }}</ref> | "Did joint work with Hirzebruch in [[K-theory]]; proved jointly with Singer the [[Atiyah–Singer index theorem|index theorem]] of elliptic operators on complex manifolds; worked in collaboration with Bott to prove a fixed point theorem related to the '[[Lefschetz fixed-point theorem|Lefschetz formula]]'."<ref name="66reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1966/index.html |title=Fields Medals 1966 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=22 March 2019 |archive-url=https://web.archive.org/web/20190322225223/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1966/index.html |url-status=live }}</ref> |- | | {{sortname|Paul|Cohen}} | [[Stanford University]], US | [[Stanford University]], US<ref>{{cite web |url=http://historicalsociety.stanford.edu/pdfmem/Cohen_P.pdf |title=Memorial Resolution – Paul Cohen (1934–2007) |date=2011 |publisher=[[Stanford]] Historical Society |access-date=24 August 2014 |url-status=dead |archive-url=https://web.archive.org/web/20150105133237/http://historicalsociety.stanford.edu/pdfmem/Cohen_P.pdf |archive-date=5 January 2015 }}</ref> | "Used technique called "[[Forcing (mathematics)|forcing]]" to prove the independence in set theory of the axiom of choice and of the generalized [[continuum hypothesis]]. The latter problem was the first of Hilbert's problems of the 1900 Congress."<ref name="66reasons" /> |- | [[File:Alexander Grothendieck.jpg|60px]] | {{sortname|Alexander|Grothendieck}} | [[Institut des Hautes Études Scientifiques]], France | [[Centre National de la Recherche Scientifique]], France<ref>{{cite web |url=http://www.math.ucdenver.edu/~jloats/StudentCELEBS/Grothendieck_Trenkamp.pdf |title=Alexander Grothendieck |website=Math.ucdenver.edu |access-date=31 March 2017 |archive-date=20 October 2016 |archive-url=https://web.archive.org/web/20161020015323/http://www.math.ucdenver.edu/~jloats/StudentCELEBS/Grothendieck_Trenkamp.pdf |url-status=dead }}</ref> | "Built on work of Weil and Zariski and effected fundamental advances in [[algebraic geometry]]. He introduced the idea of K-theory (the Grothendieck groups and rings). Revolutionized [[homological algebra]] in his celebrated ‘[[Grothendieck's Tôhoku paper|''Tôhoku'' paper]]’."<ref name="66reasons" /> |- | [[File:Stephen Smale2.jpg|60px]] | {{sortname|Stephen|Smale}} | [[University of California, Berkeley]], US | [[City University of Hong Kong]], Hong Kong<ref>{{cite web|url=http://www6.cityu.edu.hk/ma/people/profile/smales.htm|title=Prof. Stephen SMALE (史梅爾)|date=5 April 2012|publisher=City University of Hong Kong|access-date=18 August 2014|archive-date=9 November 2014|archive-url=https://web.archive.org/web/20141109090333/http://www6.cityu.edu.hk/ma/people/profile/smales.htm|url-status=live}}</ref> | "Worked in differential topology where he proved the [[generalized Poincaré conjecture]] in dimension <math>n \geq 5</math>: Every closed, n-dimensional manifold homotopy-equivalent to the n-dimensional sphere is homeomorphic to it. Introduced the method of [[Handlebody|handle-bodies]] to solve this and related problems."<ref name="66reasons" /> |- |rowspan=4| 1970 |rowspan=4| [[Nice]], France | [[File:Alan Baker.jpg|60px]] | {{sortname|Alan|Baker|Alan Baker (mathematician)}} | [[University of Cambridge]], UK | [[Trinity College, Cambridge]], UK<ref>{{cite web|url=http://www.heidelberg-laureate-forum.org/blog/laureate/alan-baker|title=The Laureates|date=25 September 2013|publisher=Heidelberg Laureate Forum Foundation (HLFF)|access-date=16 August 2014|archive-date=18 October 2014|archive-url=https://web.archive.org/web/20141018234635/http://www.heidelberg-laureate-forum.org/blog/laureate/alan-baker/|url-status=live}}</ref> | "[[Baker's theorem|Generalized]] the [[Gelfond-Schneider theorem]] (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified."<ref name="70reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1970/index.html |title=Fields Medals 1970 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074621/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1970/index.html |url-status=live }}</ref> |- | [[File:Hironaka heisuke 1976.jpg|60px]] | {{sortname|Heisuke|Hironaka}} | [[Harvard University]], US | [[Kyoto University]], Japan<ref>{{cite web |url=https://www.ams.org/notices/200509/fea-hironaka.pdf |title=Interview with Heisuke Hironaka |website=Ams.org |access-date=31 March 2017 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074623/https://www.ams.org/notices/200509/fea-hironaka.pdf |url-status=live }}</ref><ref>{{cite web|url=http://www.kurims.kyoto-u.ac.jp/en/emeritus.html|title=Professor Emeritus|date=26 May 2007|publisher=Research Institute for Mathematical Sciences, Kyoto, Japan|access-date=16 August 2014|archive-date=5 April 2022|archive-url=https://web.archive.org/web/20220405035547/https://www.kurims.kyoto-u.ac.jp/en/emeritus.html|url-status=dead}}</ref> | "Generalized work of Zariski who had proved for dimension ≤ 3 the theorem concerning the [[resolution of singularities]] on an [[algebraic variety]]. Hironaka proved the results in any dimension."<ref name="70reasons" /> |- | | {{sortname|Sergei|Novikov|Sergei Novikov (mathematician)}} | [[Moscow State University]], USSR | [[Steklov Mathematical Institute]], Russia [[Moscow State University]], Russia [[University of Maryland-College Park]], US<ref>{{cite web |url=http://www.mi.ras.ru/~snovikov/998.pdf |title=Interview with Sergey P. Novikov |website=Mi.ras.ru |access-date=31 March 2017 |archive-date=14 May 2016 |archive-url=https://web.archive.org/web/20160514032252/http://www.mi.ras.ru/~snovikov/998.pdf |url-status=live }}</ref><ref>{{cite web|url=http://www.mi.ras.ru/~snovikov/index.html|title=Novikov, Sergei Petrovich|date=1 January 2012|publisher=Russian Academy of Science|access-date=20 August 2014|archive-date=26 July 2014|archive-url=https://web.archive.org/web/20140726083130/http://www.mi.ras.ru/~snovikov/index.html|url-status=live}}</ref> | "Made important advances in topology, the most well-known being his proof of the topological invariance of the [[Pontryagin class]]es of the differentiable manifold. His work included a study of the cohomology and homotopy of [[Thom space]]s."<ref name="70reasons" /> |- | [[File:John Griggs Thompson (cropped).jpg|60px]] | {{sortname|John G.|Thompson}} | [[University of Cambridge]], UK | [[University of Cambridge]], UK [[University of Florida]], US<ref>{{cite web |url=http://www.abelprize.no/c53860/binfil/download.php?tid=53792 |format=PDF |title=John Griggs Thompson |website=Abelprize.no |access-date=31 March 2017 |archive-date=11 June 2016 |archive-url=https://web.archive.org/web/20160611151241/http://www.abelprize.no/c53860/binfil/download.php?tid=53792 |url-status=dead }}</ref> | "Proved jointly with [[Walter Feit|W. Feit]] that [[Feit–Thompson theorem|all non-cyclic finite simple groups have even order]]. The extension of this work by Thompson determined the minimal simple finite groups, that is, the simple finite groups whose proper subgroups are solvable."<ref name="70reasons" /> |- |rowspan=2| 1974 |rowspan=2| [[Vancouver]], Canada | [[File:Enrico Bombieri.jpg|60px]] | {{sortname|Enrico|Bombieri}} | [[University of Pisa]], Italy | [[Institute for Advanced Study]], US<ref>{{cite book |editor1-last=Bartocci |editor1-first=Claudio |editor2-last=Betti |editor2-first=Renato |editor3-last=Guerraggio |editor3-first=Angelo |display-editors = 3 |editor4-last=Lucchetti |editor4-first=Roberto Lucchetti |title=Vite Mathematiche |trans-title=Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles |edition=2011 |publisher=Springer |date=2011 |pages=2013–2014 |isbn=978-3642136054 }}</ref> | "Major contributions in the primes, in [[univalent function]]s and the local [[Bieberbach conjecture]], in theory of functions of several complex variables, and in theory of partial differential equations and minimal surfaces – in particular, to the solution of [[Bernstein's problem]] in higher dimensions."<ref name="74reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1974/index.html |title=Fields Medals 1974 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074703/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1974/index.html |url-status=live }}</ref> |- | [[File:Mumford2.jpg|60px]] | {{sortname|David|Mumford}} | [[Harvard University]], US | [[Brown University]], US<ref>{{cite web|url=http://www.dam.brown.edu/people/facultypage.mumford.html|title=David Mumford|publisher=The Division of Applied Mathematics, Brown University|access-date=18 August 2014|archive-date=6 October 2014|archive-url=https://web.archive.org/web/20141006072947/http://www.dam.brown.edu/people/facultypage.mumford.html|url-status=live}}</ref> | "Contributed to problems of the existence and structure of [[Moduli space|varieties of moduli]], varieties whose points parametrize isomorphism classes of some type of geometric object. Also made several important contributions to the theory of [[algebraic surface]]s."<ref name="74reasons" /> |- |rowspan=4| 1978 |rowspan=4| [[Helsinki]], Finland | [[File:Pierre Deligne (2005) (cropped).jpg|60px]] | {{sortname|Pierre|Deligne}} | [[Institut des Hautes Études Scientifiques]], France | [[Institute for Advanced Study]], US<ref>{{cite web |url=http://www.abelprize.no/c57681/binfil/download.php?tid=57756 |format=PDF |title=Pierre Deligne |website=Abelprize.no |access-date=31 March 2017 |archive-date=3 March 2016 |archive-url=https://web.archive.org/web/20160303181225/http://www.abelprize.no/c57681/binfil/download.php?tid=57756 |url-status=dead }}</ref> | "Gave solution of the three [[Weil conjectures]] concerning generalizations of the [[Riemann hypothesis]] to finite fields. His work did much to unify algebraic geometry and algebraic number theory."<ref name="78reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1978/index.html |title=Fields Medals 1978 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=10 October 2020 |archive-url=https://web.archive.org/web/20201010144934/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1978/index.html |url-status=live }}</ref> |- | [[File:Charles Fefferman.jpg|60px]] | {{sortname|Charles|Fefferman}} | [[Princeton University]], US | [[Princeton University]], US<ref>{{cite web |url=https://web.math.princeton.edu/WebCV/FeffermanCV.pdf |title=CV : Charles Fefferman |access-date=31 March 2017 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074624/https://web.math.princeton.edu/WebCV/FeffermanCV.pdf |url-status=live }}</ref> | "Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results."<ref name="78reasons" /> |- | [[File:Grigory Margulis 1978 (portioned).jpg|60px]] | {{sortname|Grigory|Margulis}} | [[Moscow State University]], USSR | [[Yale University]], US<ref>{{cite web|url=http://users.math.yale.edu/public_html/People/gam3.html|title=Yale Mathematics Department: Gregory A. Margulis|access-date=16 March 2015|archive-date=5 January 2015|archive-url=https://web.archive.org/web/20150105132947/http://users.math.yale.edu/public_html/People/gam3.html|url-status=live}}</ref> | "Provided innovative analysis of the structure of [[Lie group]]s. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups."<ref name="78reasons" /> |- |[[File:DQuillen.jpg|60x60px]] | {{sortname|Daniel|Quillen}} | [[Massachusetts Institute of Technology]], US | [[University of Oxford]], UK<ref>{{cite journal |url=https://www.ams.org/notices/201210/rtx121001392p.pdf |title=Daniel Quillen |first1=Eric |last1=Friedlander |first2=Daniel |last2=Grayson |journal=[[Notices of the AMS]] |volume=59 |issue=10 |pages=1392–1406 |doi=10.1090/noti903 |doi-access=free |date=November 2012 |access-date=31 March 2017 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074623/https://www.ams.org/notices/201210/rtx121001392p.pdf |url-status=live }}</ref> | "The prime architect of the higher [[algebraic K-theory]], a new tool that successfully employed geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory."<ref name="78reasons" /> |- |rowspan=3| 1982 |rowspan=3| [[Warsaw]], Poland | [[File:Alain Connes.jpg|60px]] | {{sortname|Alain|Connes}} | [[Institut des Hautes Études Scientifiques]], France | [[Institut des Hautes Études Scientifiques]], France [[Collège de France]], France [[Ohio State University]], US<ref>{{cite web|url=http://www.alainconnes.org/en/|title=Alain Connes|date=25 May 2012|access-date=18 August 2014|archive-date=29 August 2014|archive-url=https://web.archive.org/web/20140829154109/http://www.alainconnes.org/en/|url-status=live}}</ref> | "Contributed to the theory of [[operator algebra]]s, particularly the general classification and structure theorem of factors of type III, classification of automorphisms of the hyperfinite factor, classification of injective factors, and applications of the theory of [[C*-algebra]]s to foliations and differential geometry in general."<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 |access-date=7 April 2019 |archive-date=3 December 2018 |archive-url=https://web.archive.org/web/20181203152158/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1982/index.html |url-status=live }}</ref> |- | [[File:William Thurston.jpg|60px]] | {{sortname|William|Thurston}} | [[Princeton University]], US | [[Cornell University]], US<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|archive-date=21 August 2016|archive-url=https://web.archive.org/web/20160821013938/http://www.math.cornell.edu/spb/index.php|url-status=live}}</ref> | "Revolutionized study of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry. Contributed idea that a very large class of closed [[3-manifold]]s carry a hyperbolic structure."<ref name="82reasons" /> |- | [[File:Shing-Tung Yau at Harvard.jpg|60px]] | {{sortname|Shing-Tung|Yau}} | [[Institute for Advanced Study]], US | [[Tsinghua University]], China<ref>{{cite web |url=http://www.doctoryau.com/yau_cv.pdf |title=CV : Shing-Tung Yau |website=Doctoryau.com |access-date=31 March 2017 |archive-url=https://web.archive.org/web/20171025135744/http://www.doctoryau.com/yau_cv.pdf |archive-date=25 October 2017 |url-status=usurped }}</ref> | "Made contributions in differential equations, also to the [[Calabi conjecture]] in algebraic geometry, to the [[Positive energy theorem|positive mass conjecture]] of general relativity theory, and to real and complex [[Monge–Ampère equation]]s."<ref name="82reasons" /> |- |rowspan=3| 1986 |rowspan=3| [[Berkeley, California|Berkeley]], US | [[File:Simon Donaldson.jpg|60px]] | {{sortname|Simon|Donaldson}} | [[University of Oxford]], UK | [[Imperial College London]], UK<ref>{{cite web|url=http://wwwf.imperial.ac.uk/~skdona/|title=Simon Donaldson (Royal Society Research Professor)|date=16 January 2008|publisher=Department of Mathematics, Imperial College, Queen's Gate, London|access-date=16 August 2014|archive-date=15 September 2014|archive-url=https://web.archive.org/web/20140915032606/http://wwwf.imperial.ac.uk/~skdona/|url-status=live}}</ref> [[Stony Brook University]], US<ref>{{cite web|url=http://scgp.stonybrook.edu/people/faculty/bios/simon-donaldson|title=Simon Donaldson|access-date=16 March 2015|archive-date=20 March 2015|archive-url=https://web.archive.org/web/20150320072154/http://scgp.stonybrook.edu/people/faculty/bios/simon-donaldson|url-status=live}}</ref> | "Received medal primarily for his work on topology of [[four-manifold]]s, especially for showing that there is a differential structure on euclidian four-space which is [[Exotic R4|different]] from the usual structure."<ref name="86reasons">{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1986/index.html |title=Fields Medals and Nevanlinna Prize 1986 |publisher=International Mathematical Union |website=mathunion.org |access-date=7 April 2019 |archive-date=22 March 2019 |archive-url=https://web.archive.org/web/20190322225314/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1986/index.html |url-status=live }}</ref><ref name="IMU Fields Medal 1986">{{cite web | title=Fields Medals 1986 | website=International Mathematical Union (IMU) | date=1986 | url=https://www.mathunion.org/imu-awards/fields-medal/fields-medals-1986 | access-date=2024-01-24 | archive-date=24 January 2024 | archive-url=https://web.archive.org/web/20240124060355/https://www.mathunion.org/imu-awards/fields-medal/fields-medals-1986 | url-status=live }}</ref> |- | [[File:Gerd Faltings MFO.jpg|60px]] | {{sortname|Gerd|Faltings}} | [[Princeton University]], US | [[Max Planck Institute for Mathematics]], Germany<ref>{{cite web|url=http://www.heidelberg-laureate-forum.org/blog/laureate/gerd-faltings/|title=The Laureates|date=6 October 2013|publisher=Heidelberg Laureate Forum Foundation (HLFF)|access-date=16 August 2014|archive-url=https://web.archive.org/web/20141006065705/http://www.heidelberg-laureate-forum.org/blog/laureate/gerd-faltings/|archive-date=6 October 2014|url-status=dead}}</ref> | "Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the [[Mordell Conjecture]]."<ref name="86reasons" /> |- | [[File:Michael Freedman, circa 1970s (re-scanned C; headshot).jpg|60px]] | {{sortname|Michael|Freedman}} | [[University of California, San Diego]], US | [[Microsoft Station Q]], US<ref>{{cite web|url=http://celebratio.org/cmmedia/essaypdf/19_main_3.pdf|title=Michael H. Freedman|author=Rob Kirby|date=2012|work=celebratio.org|url-status=dead|archive-url=https://web.archive.org/web/20141006102901/http://celebratio.org/cmmedia/essaypdf/19_main_3.pdf|archive-date=6 October 2014}}</ref> | "Developed new methods for topological analysis of [[four-manifold]]s. One of his results is a proof of the [[Generalized Poincaré conjecture|four-dimensional Poincaré Conjecture]]."<ref name="86reasons" /> |- |rowspan=4| 1990 |rowspan=4| [[Kyoto]], Japan | | {{sortname|Vladimir|Drinfeld}} | [[B Verkin Institute for Low Temperature Physics and Engineering]], USSR<ref>{{cite encyclopedia|url=https://www.britannica.com/EBchecked/topic/171669/Vladimir-Gershonovich-Drinfeld|title=Vladimir Gershonovich Drinfeld|date=19 August 2009|encyclopedia=Encyclopædia Britannica|access-date=2 September 2014|archive-date=10 August 2014|archive-url=https://web.archive.org/web/20140810212856/http://www.britannica.com/EBchecked/topic/171669/Vladimir-Gershonovich-Drinfeld|url-status=live}}</ref> | [[University of Chicago]], US<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/Biographies/Drinfeld.html|title=Vladimir Gershonovich Drinfeld|date=18 August 2009|publisher=School of Mathematics and Statistics, University of St Andrews, Scotland|access-date=16 August 2014|archive-date=6 October 2014|archive-url=https://web.archive.org/web/20141006123607/http://www-history.mcs.st-and.ac.uk/Biographies/Drinfeld.html|url-status=live}}</ref> | "Drinfeld's main preoccupation in the last decade [are] Langlands' program and quantum groups. In both domains, Drinfeld's work constituted a decisive breakthrough and prompted a wealth of research."<ref>Yuri Ivanovich Manin. ''On the mathematical work of Vladimir Drinfeld''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.1/ICM1990.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1990. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074657/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.1/ICM1990.1.ocr.pdf |date=8 April 2022 }} pp. 3–7</ref> |- | [[File:Vaughan Jones p1190550.jpg|60px]] | {{sortname|Vaughan|Jones}} | [[University of California, Berkeley]], US | [[University of California, Berkeley]], US<ref>{{cite web|url=http://math.berkeley.edu/~vfr/vita/|title=Curriculum Vitae: Vaughan F. R. Jones|publisher=University of California, Berkeley|date=10 November 2001|access-date=16 August 2014|url-status=dead|archive-url=https://web.archive.org/web/20130806082228/http://math.berkeley.edu/~vfr/vita|archive-date=6 August 2013}}</ref> [[Vanderbilt University]], US<ref>{{cite web|last=Salisbury|first=David|title=Fields Medalist joins Vanderbilt faculty|url=http://news.vanderbilt.edu/2011/04/vaughan-jones/|publisher=Vanderbilt University|access-date=17 May 2011|date=6 April 2011|archive-date=14 April 2011|archive-url=https://web.archive.org/web/20110414130456/http://news.vanderbilt.edu/2011/04/vaughan-jones/|url-status=live}}</ref> | "Jones discovered an astonishing relationship between [[von Neumann algebra]]s and geometric topology. As a result, he found a new polynomial invariant for knots and links in 3-space."<ref>Joan S. Birman. ''The work of Vaughan F. R. Jones''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.1/ICM1990.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1990. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074657/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.1/ICM1990.1.ocr.pdf |date=8 April 2022 }} pp. 9–18</ref> |- | [[File:Shigefumi Mori.jpg|60px]] | {{sortname|Shigefumi|Mori}} | [[Kyoto University]], Japan | [[Kyoto University]], Japan<ref>{{cite web|url=http://www.heidelberg-laureate-forum.org/blog/laureate/shigefumi-mori/|title=The Laureates|publisher=Heidelberg Laureate Forum Foundation (HLFF)|date=10 April 2014|access-date=16 August 2014|archive-url=https://web.archive.org/web/20140815120453/http://www.heidelberg-laureate-forum.org/blog/laureate/shigefumi-mori/|archive-date=15 August 2014|url-status=dead}}</ref> | "The most profound and exciting development in algebraic geometry during the last decade or so was [...] ''[[minimal model program|Mori's Program]]'' in connection with the classification problems of algebraic varieties of dimension three." "Early in 1979, Mori brought to algebraic geometry a completely new excitement, that was his proof of Hartshorne's conjecture."<ref>Heisuke Hironaka. ''On the work of Shigefumi Mori''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.1/ICM1990.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1990. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074657/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.1/ICM1990.1.ocr.pdf |date=8 April 2022 }} pp. 19–25</ref> |- | [[File:Edward Witten at Harvard.jpg|60px]] | {{sortname|Edward|Witten}} | [[Institute for Advanced Study]], US | [[Institute for Advanced Study]], US<ref>{{cite web |url=http://www.sns.ias.edu/~witten/CurrentCV.pdf |title=Edward Witten – Vita |date=2011 |access-date=26 October 2011 |url-status=dead |archive-url=https://web.archive.org/web/20120204111241/http://www.sns.ias.edu/~witten/CurrentCV.pdf |archive-date=4 February 2012 }}</ref> | "Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems."<ref>{{cite web |url=http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |title=On the Work of Edward Witten |author=Michael Atiyah |website=Mathunion.org |access-date=31 March 2017 |url-status=dead |archive-url=https://web.archive.org/web/20170301004342/http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |archive-date=1 March 2017 }}</ref> |- |rowspan=4| 1994 |rowspan=4| [[Zürich]], Switzerland | [[File:Jean Bourgain (vertical crop).jpg|60px]] | {{sortname|Jean|Bourgain}} | [[Institut des Hautes Études Scientifiques]], France | [[Institute for Advanced Study]], US<ref>{{cite web |url=https://www.math.ias.edu/files/bourgain/CVBourgain.pdf |title=CV : Jean Bourgain |website=Math.ias.edu |access-date=31 March 2017 |archive-date=30 May 2017 |archive-url=https://web.archive.org/web/20170530041057/https://www.math.ias.edu/files/bourgain/CVBourgain.pdf |url-status=live }}</ref> | "Bourgain's work touches on several central topics of mathematical analysis: the geometry of [[Banach space]]s, convexity in high dimensions, harmonic analysis, ergodic theory, and finally, nonlinear partial differential equations from mathematical physics."<ref>Luis Caffarelli. ''The work of Jean Bourgain''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1994. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074735/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf |date=8 April 2022 }} pp. 3–5</ref> |- | [[File:Pierre-Louis Lions par Philippe Binant.jpg|60px]] | {{sortname|Pierre-Louis|Lions}} | [[University of Paris 9]], France | [[Collège de France]], France [[École polytechnique]], France<ref>{{cite web|url=http://www.college-de-france.fr/site/pierre-louis-lions/biographie.htm|title=Collège de France|website=College-de-france.fr|date=16 December 2013|access-date=18 August 2014|archive-date=15 September 2013|archive-url=https://web.archive.org/web/20130915103327/http://www.college-de-france.fr/site/pierre-louis-lions/biographie.htm|url-status=live}}</ref> | "His contributions cover a variety of areas, from probability theory to partial differential equations (PDEs). Within the PDE area he has done several beautiful things in nonlinear equations. The choice of his problems have always been motivated by applications."<ref>S. R. S. Varadhan. ''The work of Pierre-Louis Lions''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1994. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074735/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf |date=8 April 2022 }} pp. 6–10</ref> |- | [[File:Jean-Christophe Yoccoz.jpg|60px]] | {{sortname|Jean-Christophe|Yoccoz}} | [[Paris-Sud 11 University]], France | [[Collège de France]], France<ref>{{cite web|url=http://www.college-de-france.fr/site/jean-christophe-yoccoz/biographie.htm|title=Collège de France|website=College-de-france.fr|date=16 December 2013|access-date=18 August 2014|archive-date=5 January 2015|archive-url=https://web.archive.org/web/20150105141948/http://www.college-de-france.fr/site/jean-christophe-yoccoz/biographie.htm|url-status=live}}</ref> | "Yoccoz obtained a very enlightening proof of [[Brjuno number|Bruno's theorem]], and he was able to prove the converse [...] Palis and Yoccoz obtained a complete system of C<sup>∞</sup> conjugation invariants for Morse-Smale diffeomorphisms."<ref>Adrien Douady. ''Presentation de Jean-Christophe Yoccoz''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1994. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074735/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf |date=8 April 2022 }} pp. 11–16</ref> |- | [[File:EfimIZelmanov.jpg|60px]] | {{sortname|Efim|Zelmanov}} | [[University of Wisconsin-Madison]] [[University of Chicago]], US | [[Steklov Mathematical Institute]], Russia, [[University of California, San Diego]], US<ref>{{cite web |url=http://www.ime.usp.br/~grishkov/papers/CV.Zelm..pdf |title=CV : Efim Zelmanov |website=Ime.usp.br |access-date=31 March 2017 |archive-date=2 July 2017 |archive-url=https://web.archive.org/web/20170702045759/https://www.ime.usp.br/~grishkov/papers/CV.Zelm..pdf |url-status=live }}</ref> | "For the solution of the restricted [[Burnside problem]]."<ref>Walter Feit. ''On the Work of Efim Zelmanov''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1994. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074735/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf |date=8 April 2022 }} pp. 17–24</ref> |- |rowspan=4| 1998 |rowspan=4| [[Berlin]], Germany | [[File:Richard Borcherds.jpg|60px]] | {{sortname|Richard|Borcherds}} | [[University of California, Berkeley]], US [[University of Cambridge]], UK | [[University of California, Berkeley]], US<ref>{{cite web|url=http://www.heidelberg-laureate-forum.org/blog/laureate/richard-borcherds/|title=The Laureates|publisher=Heidelberg Laureate Forum Foundation (HLFF)|date=10 April 2014|access-date=16 August 2014|archive-url=https://web.archive.org/web/20141006103635/http://www.heidelberg-laureate-forum.org/blog/laureate/richard-borcherds/|archive-date=6 October 2014|url-status=dead}}</ref> | "For his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie algebras, the [[Monstrous_moonshine#Borcherds'_proof|proof of the Conway–Norton moonshine conjecture]] and the discovery of a new class of automorphic infinite products."<ref name="1998cer">''Opening ceremony''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1998.1/ICM1998.1.ocr.pdf Proceedings of the International Congress of Mathematicians 1998. Volume I] {{Webarchive|url=https://web.archive.org/web/20220422051649/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1998.1/ICM1998.1.ocr.pdf |date=22 April 2022 }} pp. 46–48</ref> |- | [[File:Gowers timothy 2011.jpg|60px]] | {{sortname|Timothy|Gowers}} | [[University of Cambridge]], UK | [[University of Cambridge]], UK<ref>{{cite encyclopedia|url=https://www.britannica.com/EBchecked/topic/240293/William-Timothy-Gowers|title=William Timothy Gowers|encyclopedia=Encyclopædia Britannica|date=28 March 2009|access-date=16 August 2014|archive-date=6 October 2014|archive-url=https://web.archive.org/web/20141006080937/http://www.britannica.com/EBchecked/topic/240293/William-Timothy-Gowers|url-status=live}}</ref> | "For his contributions to functional analysis and combinatorics, developing a new vision of infinite-dimensional geometry, including the solution of two of Banach's problems and the discovery of the so called Gowers' dichotomy: every infinite dimensional Banach space contains either a subspace with many symmetries (technically, with an unconditional basis) or a subspace every operator on which is Fredholm of index zero."<ref name="1998cer" /> |- | [[File:Maxim Kontsevich 1994 (headshot).jpg|60px]] | {{sortname|Maxim|Kontsevich}} | [[Institut des Hautes Études Scientifiques]], France [[Rutgers University]], US | [[Institut des Hautes Études Scientifiques]], France [[Rutgers University]], US<ref>{{cite web|url=https://www.ihes.fr/~maxim/CVAnglais.html|title=CV Maxim Kontsevich|publisher=[[Institut des Hautes Études Scientifiques]]|date=22 November 2009|access-date=16 August 2014|url-status=dead|archive-url=https://web.archive.org/web/20141010040126/http://www.ihes.fr/~maxim/CVAnglais.html|archive-date=10 October 2014}}</ref> | "For his contributions to algebraic geometry, topology, and mathematical physics, including the proof of Witten's conjecture of intersection numbers in moduli spaces of stable curves, construction of the universal Vassiliev invariant of knots, and formal quantization of Poisson manifolds."<ref name="1998cer" /> |- | [[File:Curtis T. McMullen, 1994 Aug-Sept (re-scanned, headshot).jpg|60px]] | {{sortname|Curtis T.|McMullen}} | [[Harvard University]], US | [[Harvard University]], US<ref>{{cite web |url=http://abel.math.harvard.edu/~ctm/vita/resume.pdf |title=CV : Curtis T McMullen |website=Abel.math.harvard.edu |access-date=31 March 2017 |archive-date=10 February 2017 |archive-url=https://web.archive.org/web/20170210173438/http://abel.math.harvard.edu/~ctm/vita/resume.pdf |url-status=live }}</ref> | "For his contributions to the theory of holomorphic dynamics and geometrization of three-manifolds, including proofs of Bers' conjecture on the density of cusp points in the boundary of the Teichmüller space, and Kra's theta-function conjecture."<ref name="1998cer" /> |- |rowspan=2| 2002 |rowspan=2| [[Beijing]], China | [[File:Laurent Lafforgue.png|60px]] | {{sortname|Laurent|Lafforgue}} | [[Institut des Hautes Études Scientifiques]], France | [[Institut des Hautes Études Scientifiques]], France<ref>{{cite web|url=https://www.ihes.fr/~lafforgue/cv.html|title=Curriculum Vitae|publisher=ihes|date=6 December 2005|access-date=19 August 2014|archive-date=10 March 2022|archive-url=https://web.archive.org/web/20220310232627/http://laurentlafforgue.org/cv.html|url-status=live}}</ref> | "Laurent Lafforgue has been awarded the Fields Medal for his proof of the [[Langlands correspondence]] for the full linear groups GLr (r≥1) over function fields of positive characteristic."<ref>Gérard Laumon. ''The work of Laurent Lafforgue''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2002.1/ICM2002.1.ocr.pdf Proceedings of the International Congress of Mathematicians 2002. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074623/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2002.1/ICM2002.1.ocr.pdf |date=8 April 2022 }} pp. 91–97</ref> |- | [[File:VladimirVoevodsky.jpg|60px]] | {{sortname|Vladimir|Voevodsky}} | [[Institute for Advanced Study]], US | [[Institute for Advanced Study]], US<ref>{{cite web |url=http://www.math.ias.edu/~vladimir/Site3/home_files/mitcv12.pdf |title=CV : Vladimir Voevodsky |website=Math.ias.edu |access-date=31 March 2017 |archive-date=2 April 2016 |archive-url=https://web.archive.org/web/20160402041701/http://www.math.ias.edu/~vladimir/Site3/home_files/mitcv12.pdf |url-status=live }}</ref> | "He defined and developed motivic cohomology and the A1-homotopy theory, provided a framework for describing many new cohomology theories for algebraic varieties; he proved the [[Milnor conjecture (K-theory)|Milnor conjecture]]s on the K-theory of fields."<ref>Christophe Soulé. ''The work of Vladimir Voevodsky''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2002.1/ICM2002.1.ocr.pdf Proceedings of the International Congress of Mathematicians 2002. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074623/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2002.1/ICM2002.1.ocr.pdf |date=8 April 2022 }} pp. 99–103</ref> |- |rowspan=4| 2006 |rowspan=4| [[Madrid]], Spain | [[File:Andrei Okounkov 1999 (re-scanned, portioned).jpg|60px]] | {{sortname|Andrei|Okounkov}} | [[Princeton University]], US | [[Columbia University]], US<ref>{{cite web|url=http://www.math.columbia.edu/people/directory/?dname=Okounkov&did=69|title=Department of Mathematics|date=20 December 2012|publisher=Columbia University, Department of Mathematics|access-date=19 August 2014|archive-date=6 October 2014|archive-url=https://web.archive.org/web/20141006102403/http://www.math.columbia.edu/people/directory/?dname=Okounkov&did=69|url-status=live}}</ref> [[University of California, Berkeley]], US<ref>{{cite web|url=https://math.berkeley.edu/people/faculty/andrei-okounkov|access-date=August 22, 2022|title=Andrei Okounkov|publisher=Berkeley Mathematics|website=math.berkeley.edu|archive-date=22 February 2023|archive-url=https://web.archive.org/web/20230222214335/https://math.berkeley.edu/people/faculty/andrei-okounkov|url-status=live}}</ref> | "For his contributions bridging probability, representation theory and algebraic geometry."<ref name="2006cer">''Opening ceremony''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2006.1/ICM2006.1.ocr.pdf Proceedings of the International Congress of Mathematicians 2006. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074623/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2006.1/ICM2006.1.ocr.pdf |date=8 April 2022 }} p. 36</ref> |- | [[File:Grigori Perelman, 1993 (re-scanned) (cropped).jpg|60px]] | {{sortname|Grigori|Perelman}} ''(declined)'' | None | [[St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences]], Russia<ref>{{cite encyclopedia|url=https://www.britannica.com/EBchecked/topic/1272681/Grigori-Perelman|date=28 May 2008|encyclopedia=Encyclopædia Britannica|access-date=19 August 2014|title=Grigori Perelman | Biography & Facts|archive-date=16 September 2014|archive-url=https://web.archive.org/web/20140916012923/http://www.britannica.com/EBchecked/topic/1272681/Grigori-Perelman|url-status=live}}</ref> | "For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the [[Ricci flow]]."<ref name="2006cer" /> |- | [[File:Terence Tao.jpg|60px]] | {{sortname|Terence|Tao}} | [[University of California, Los Angeles]], US | [[University of California, Los Angeles]], US<ref>{{cite web|url=https://www.math.ucla.edu/~tao/preprints/cv.html|title=Vitae and Bibliography for Terence Tao|date=16 March 2010|publisher=UCLA Dept. of Math.|access-date=19 August 2014|archive-date=8 April 2000|archive-url=https://web.archive.org/web/20000408183915/https://www.math.ucla.edu/~tao/preprints/cv.html|url-status=live}}</ref> | "For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory."<ref name="2006cer" /> |- | [[File:Wendelin Werner.jpg|60px]] | {{sortname|Wendelin|Werner}} | [[Paris-Sud 11 University]], France | [[ETH Zurich]], Switzerland<ref>{{cite web|url=http://www.math.ethz.ch/~wewerner/|title=Wendelin WERNER|date=18 September 2013|publisher=ETH Zurich|access-date=19 August 2014|archive-date=6 October 2014|archive-url=https://web.archive.org/web/20141006131413/http://www.math.ethz.ch/~wewerner/|url-status=live}}</ref> | "For his contributions to the development of [[Schramm–Loewner evolution|stochastic Loewner evolution]], the geometry of two-dimensional [[Brownian motion]], and [[conformal field theory]]."<ref name="2006cer" /> |- |rowspan=4| 2010 |rowspan=4| [[Hyderabad, India|Hyderabad]], India | [[File:Elon Lindenstrauss MFO.jpg|60px]] | {{sortname|Elon|Lindenstrauss}} | [[Hebrew University of Jerusalem]], Israel [[Princeton University]], US | [[Hebrew University of Jerusalem]], Israel<ref>{{cite web|url=http://www.huji.ac.il/huji/nobel/lindE.htm|title=Nobel at HU|date=5 July 2011|publisher=The Hebrew University of Jerusalem|access-date=16 August 2014|archive-date=14 August 2014|archive-url=https://web.archive.org/web/20140814071745/http://www.huji.ac.il/huji/nobel/lindE.htm|url-status=live}}</ref> | "For his results on measure rigidity in [[ergodic theory]], and their applications to number theory."<ref name="2010cer">''Opening ceremony''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2010.1/ICM2010.1.pdf Proceedings of the International Congress of Mathematicians 2010. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074655/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2010.1/ICM2010.1.pdf |date=8 April 2022 }} p. 23</ref> |- | [[File:Ngo Bau Chau MFO.jpg|60px]] | [[Ngô Bảo Châu]] | [[Paris-Sud 11 University]], France [[Institute for Advanced Study]], US | [[University of Chicago]], US Institute for Advanced Study, US<ref>{{cite web|url=http://www.heidelberg-laureate-forum.org/blog/laureate/bao-chau-ngo/|title=Ngô Bảo Châu › Heidelberg Laureate Forum|access-date=16 March 2015|archive-url=https://web.archive.org/web/20150207021714/http://www.heidelberg-laureate-forum.org/blog/laureate/bao-chau-ngo/|archive-date=7 February 2015|url-status=dead}}</ref> | "For his proof of the [[Fundamental lemma (Langlands program)|Fundamental Lemma]] in the theory of [[automorphic form]]s through the introduction of new algebro-geometric methods."<ref name="2010cer" /> |- | [[File:Stanislav Smirnov2005 2.JPG|60px]] | {{sortname|Stanislav|Smirnov}} | [[University of Geneva]], Switzerland | [[University of Geneva]], Switzerland [[St. Petersburg State University]], Russia<ref>{{cite web|url=http://www.unige.ch/~smirnov/|title=Home Page of Stanislav Smirnov|access-date=16 March 2015|archive-date=19 June 2013|archive-url=https://web.archive.org/web/20130619221150/http://www.unige.ch/~smirnov/|url-status=live}}</ref> | "For the proof of conformal invariance of percolation and the planar [[Ising model]] in statistical physics."<ref name="2010cer" /> |- | [[File:Cedric Villani at his office 2015 n2 (cropped).jpg|60px]] | {{sortname|Cédric|Villani}} | [[École Normale Supérieure de Lyon]], France [[Institut Henri Poincaré]], France | [[Lyon University]], France [[Institut Henri Poincaré]], France<ref>{{cite web |url=http://cedricvillani.org/wp-content/uploads/2012/10/cv-2012.pdf |title=CV : Cedric Villani |website=Cedricvillani.org |access-date=31 March 2017 |archive-url=https://web.archive.org/web/20160623224603/http://cedricvillani.org/wp-content/uploads/2012/10/cv-2012.pdf |archive-date=23 June 2016 |url-status=dead }}</ref> | "For his proofs of nonlinear [[Landau damping]] and convergence to equilibrium for the [[Boltzmann equation]]."<ref name="2010cer" /> |- |rowspan=4| 2014 |rowspan=4| [[Seoul]], South Korea | [[File:Artur Ávila crop.jpg|60px]] | {{sortname|Artur|Avila}} | [[University of Paris VII]], France [[CNRS]], France [[Instituto Nacional de Matemática Pura e Aplicada]], Brazil | [[University of Zurich]], Switzerland [[Instituto Nacional de Matemática Pura e Aplicada]], Brazil | "For his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle."<ref name="2014cer">''Opening ceremony''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2014.1/ICM2014.1.pdf Proceedings of the International Congress of Mathematicians 2014. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074729/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2014.1/ICM2014.1.pdf |date=8 April 2022 }} p. 23</ref> |- | [[File:Manjul Bhargava.jpg|60px]] | {{sortname|Manjul|Bhargava}} | [[Princeton University]], US | [[Princeton University]], US<ref>{{cite web |url=http://www2.maths.ox.ac.uk/cmi/library/research_award/Bhargava/CV.pdf |title=CV : Manjul Bhargava |website=2.maths.ox.ac.uk |access-date=31 March 2017 |archive-date=22 December 2014 |archive-url=https://web.archive.org/web/20141222225037/http://www2.maths.ox.ac.uk/cmi/library/research_award/Bhargava/CV.pdf |url-status=live }}</ref><ref>{{cite web |url=http://www.mathunion.org/fileadmin/IMU/Prizes/2014/news_release_bhargava.pdf |title=The Work of Manjul Bhargava |website=Mathunion.org |access-date=31 March 2017 |url-status=dead |archive-url=https://web.archive.org/web/20170713133635/http://www.mathunion.org/fileadmin/IMU/Prizes/2014/news_release_bhargava.pdf |archive-date=13 July 2017 }}</ref><ref>{{cite web|url=https://www.math.princeton.edu/directory/faculty|title=Faculty|date=8 May 2012|publisher=The Princeton University, Department of Mathematics|access-date=19 December 2014|url-status=dead|archive-url=https://web.archive.org/web/20141225194217/http://www.math.princeton.edu/directory/faculty|archive-date=25 December 2014}}</ref> | "For developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of [[elliptic curve]]s."<ref name="2014cer" /> |- | [[File:Professor Martin Hairer FRS.jpg|60px]] | {{sortname|Martin|Hairer}} | [[University of Warwick]], UK | [[École Polytechnique Fédérale de Lausanne]], Switzerland [[Imperial College London]], UK | "For his outstanding contributions to the theory of stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations."<ref name="2014cer" /> |- | [[File:Maryam-mirzakhani (cropped).jpg|60px]] | {{sortname|Maryam|Mirzakhani}} | [[Stanford University]], US | [[Stanford University]], US<ref>{{cite web|url=http://www2.maths.ox.ac.uk/cmi/library/annual_report/ar2008/08Interview.pdf|title=Interview with Research Fellow Maryam Mirzakhani|access-date=24 August 2014|archive-date=27 August 2014|archive-url=https://web.archive.org/web/20140827222953/http://www2.maths.ox.ac.uk/cmi/library/annual_report/ar2008/08Interview.pdf|url-status=live}}</ref><ref>{{cite web|url=http://mathematics.stanford.edu/people/faculty-lecturers/|title=Department of Mathematics|date=22 January 2009|publisher=Stanford University|access-date=19 December 2014|archive-date=21 December 2014|archive-url=https://web.archive.org/web/20141221152037/http://mathematics.stanford.edu/people/faculty-lecturers/|url-status=live}}</ref> | "For her outstanding contributions to the dynamics and geometry of [[Riemann surface]]s and their moduli spaces."<ref name="2014cer" /> |- |rowspan=4| 2018 |rowspan=4| [[Rio de Janeiro]], Brazil | [[File:Birkar (cropped01).jpg|60px]] | {{sortname|Caucher|Birkar}} | [[University of Cambridge]], UK | [[University of Cambridge]], UK | "For the proof of the boundedness of [[Fano varieties]] and for contributions to the [[minimal model program]]."<ref name="2018cer">''Opening ceremonies''. [https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2018/ICM-2018-vol1-ver1-eb.pdf Proceedings of the International Congress of Mathematicians 2018. Volume I] {{Webarchive|url=https://web.archive.org/web/20220408074623/https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2018/ICM-2018-vol1-ver1-eb.pdf |date=8 April 2022 }} pp. 13–16</ref> |- | [[File:Alessio Figalli (cropped).jpg|60px]] | {{sortname|Alessio|Figalli}} | [[Swiss Federal Institute of Technology Zurich]], Switzerland | [[Swiss Federal Institute of Technology Zurich]], Switzerland | "For contributions to the theory of [[optimal transport]] and its applications in [[partial differential equation]]s, [[metric geometry]] and [[probability]]."<ref name="2018cer" /> |- | [[File:Peter Scholze (cropped).jpg|60px]] | {{sortname|Peter|Scholze}} | [[University of Bonn]], Germany | [[University of Bonn]], Germany | "For having transformed arithmetic algebraic geometry over [[p-adic field]]s."<ref name="2018cer" /> |- | [[File:Akshay Venkatesh 2014 (cropped).jpg|60px]] | {{sortname|Akshay|Venkatesh}} | [[Stanford University]], US | [[Institute for Advanced Study]], US<ref>{{cite web|url=https://www.ias.edu/press-releases/faculty-appointee-akshay-venkatesh-awarded-2018-fields-medal|title=Faculty Appointee Akshay Venkatesh Awarded 2018 Fields Medal|date=August 2018|access-date=2 August 2018|archive-date=24 January 2022|archive-url=https://web.archive.org/web/20220124080025/https://www.ias.edu/news/press-releases/venkatesh-fields|url-status=live}}</ref> |"For his synthesis of [[analytic number theory]], [[Dynamical system|homogeneous dynamics]], [[topology]], and [[representation theory]], which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects."<ref name="2018cer" /> |- |rowspan=4| 2022 |rowspan=4| [[Helsinki]], Finland{{efn|ICM 2022 was originally planned to be held in [[Saint Petersburg]], Russia, but was moved online following the [[2022 Russian invasion of Ukraine]]. The award ceremony for the Fields Medals and prize winner lectures took place in Helsinki, Finland and were live-streamed.<ref>{{Cite web |title=Decision of the Executive Committee of the IMU on the upcoming ICM 2022 and IMU General Assembly |url=https://www.mathunion.org/fileadmin/IMU/ICM2022/ICM_2022_statement.pdf |access-date=5 July 2022 |archive-date=2 April 2022 |archive-url=https://web.archive.org/web/20220402092339/https://www.mathunion.org/fileadmin/IMU/ICM2022/ICM_2022_statement.pdf |url-status=live }}</ref><ref>{{cite web|url=https://www.mathunion.org/icm/virtual-icm-2022|title=Virtual ICM 2022|website=International Mathematical Union|access-date=5 July 2022|archive-date=1 June 2023|archive-url=https://web.archive.org/web/20230601054456/https://www.mathunion.org/icm/virtual-icm-2022|url-status=live}}</ref>}} | [[File:Hugo Duminil-Copin in Oberwolfach (cropped).jpg|60px]] | {{sortname|Hugo|Duminil-Copin}} | [[Institut des Hautes Études Scientifiques]], France [[University of Geneva]], Switzerland <ref name="unige.ch">{{cite web | url=https://www.unige.ch/sciences/math/fields2022/en | title=Hugo Duminil-Copin – Fields Medal 2022 – UNIGE | date=28 June 2022 | access-date=5 July 2022 | archive-date=6 July 2022 | archive-url=https://web.archive.org/web/20220706023552/https://www.unige.ch/sciences/math/fields2022/en | url-status=live }}</ref> | [[Institut des Hautes Études Scientifiques]], France [[University of Geneva]], Switzerland <ref name="unige.ch"/> | "For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four."<ref name="2022fields">{{cite web|url=https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2022|title=Fields Medals 2022|website=International Mathematical Union|access-date=5 July 2022|archive-date=5 July 2022|archive-url=https://web.archive.org/web/20220705072608/https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2022|url-status=live}}</ref> |- | [[File:June Huh - ICM 2018.jpg|60px]] | {{sortname|June|Huh}} | [[Princeton University]], US | [[Princeton University]], US | "For bringing the ideas of [[Hodge theory]] to combinatorics, the proof of the Dowling–Wilson conjecture for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for [[matroid]]s, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture."<ref name="2022fields" /> |- | [[File:James Maynard MFO 2013.jpg|60px]] | {{sortname|James|Maynard|James Maynard (mathematician)}} | [[University of Oxford]], UK | [[University of Oxford]], UK | "For contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in [[Diophantine approximation]]."<ref name="2022fields" /> |- | [[File:Maryna Vazovska MFO 2013 crop.jpg|60px]] | {{sortname|Maryna|Viazovska}} | [[École Polytechnique Fédérale de Lausanne]], Switzerland | [[École Polytechnique Fédérale de Lausanne]], Switzerland | "For the proof that the [[E8_lattice#Sphere_packings_and_kissing_numbers|<math>E_8</math> lattice]] provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in [[Fourier analysis]]."<ref name="2022fields" /><ref>{{cite web |date=July 5, 2022 |first1=Thomas |last1=Lin |first2=Erica |last2=Klarreich |title=Ukrainian Mathematician Maryna Viazovska Wins Fields Medal |url=https://www.quantamagazine.org/ukrainian-mathematician-maryna-viazovska-wins-fields-medal-20220705/ |access-date=18 July 2022 |archive-date=5 July 2022 |archive-url=https://web.archive.org/web/20220705072102/https://www.quantamagazine.org/ukrainian-mathematician-maryna-viazovska-wins-fields-medal-20220705/ |url-status=live }}</ref> |}
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