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{{Short description|Mathematics award}}
]]]
{{Distinguish|Field's metal}}
A prize awarded to up to four ]s (under forty) at each ] since 1936 and regularly since ] at the initiative of the Canadian mathematican ].
{{Use American English|date=July 2018}}
The purpose is to give recognition and support to young mathematical researchers having already made important contributions.
{{Use dmy dates|date=September 2021}}
{{Infobox award
| name = Fields Medal
| image = FieldsMedalFront.jpg
| caption = The obverse of the Fields Medal
| awarded_for = Outstanding contributions in mathematics attributed to young scientists
| presenter = ]
| reward = {{CA$}}15,000
| year = {{Start date and age|1936}}
| year2 = {{end date|2022}}
| website = {{official URL}}
|image2=FieldsMedalBack.jpg|caption2=The reverse of the medal}}


The '''Fields Medal''' is a prize awarded to two, three, or four ]s under 40 years of age at the ] of the ] (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician ].<ref name="Fields Institute About Us">{{cite web |url=http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html |title=About Us: The Fields Medal |publisher=The Fields Institute, ] |access-date=21 August 2010 |archive-date=1 April 2022 |archive-url=https://web.archive.org/web/20220401044832/http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html |url-status=live }}</ref>
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The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been ],<ref name=":0">{{Cite journal|last=Ball|first=Philip|title=Iranian is first woman to nab highest prize in maths|url=https://www.nature.com/news/iranian-is-first-woman-to-nab-highest-prize-in-maths-1.15686|journal=Nature|language=en|doi=10.1038/nature.2014.15686|year=2014|s2cid=180573813|doi-access=free|access-date=29 March 2018|archive-date=8 October 2019|archive-url=https://web.archive.org/web/20191008040735/https://www.nature.com/news/iranian-is-first-woman-to-nab-highest-prize-in-maths-1.15686|url-status=live}}</ref><ref name=":2">{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/Honours/FieldsMedal.html|title=Fields Medal|website=www-history.mcs.st-andrews.ac.uk|access-date=29 March 2018|archive-date=26 May 2021|archive-url=https://web.archive.org/web/20210526200633/https://mathshistory.st-andrews.ac.uk/Honours/FieldsMedal/|url-status=live}}</ref><ref name=":3">{{Cite web|url=https://www.uchicago.edu/about/accolades/14/|title=Fields Medal|website=The University of Chicago|language=en|access-date=29 March 2018|archive-date=7 April 2019|archive-url=https://web.archive.org/web/20190407140029/https://www.uchicago.edu/about/accolades/14/|url-status=live}}</ref> although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria.<ref>{{Cite journal |last=Klainerman |first=Sergiu |date=2015 |title=Is the Fields Medal the Nobel Prize of Mathematics? |url=https://www.ams.org/notices/201504/rnoti-p327.pdf |journal=Notices of the American Mathematical Society |volume=62 |issue=4 |pages=327 |issn=0002-9920 |access-date=28 June 2022 |archive-date=28 June 2022 |archive-url=https://web.archive.org/web/20220628180534/https://www.ams.org/notices/201504/rnoti-p327.pdf |url-status=live }}</ref> According to the annual Academic Excellence Survey by ], the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide,<ref>{{Cite web|url=http://www.shanghairanking.com/subject-survey/awards.html|title=Top Award, ShanghaiRanking Academic Excellence Survey 2017 {{!}} Shanghai Ranking – 2017|website=Shanghairanking.com|access-date=29 March 2018|archive-date=17 October 2020|archive-url=https://web.archive.org/web/20201017012526/http://www.shanghairanking.com/subject-survey/awards.html|url-status=dead}}</ref> and in another reputation survey conducted by ] in 2013–14, the Fields Medal came closely after the ] as the second most prestigious international award in mathematics.<ref>{{cite book|author=IREG Observatory on Academic Ranking and Excellence|title=IREG List of International Academic Awards|publisher=]|location=Brussels|url=http://ireg-observatory.org/en/pdfy/IREG-list-academic-awards-EN.pdf|access-date=3 March 2018|archive-date=12 March 2019|archive-url=https://web.archive.org/web/20190312125455/http://ireg-observatory.org/en/pdfy/IREG-list-academic-awards-EN.pdf|url-status=dead}}</ref><ref>{{cite journal|last1=Zheng|first1=Juntao|last2=Liu|first2=Niancai|title=Mapping of important international academic awards|journal=Scientometrics|date=2015|volume=104|issue=3|pages=763–791|doi=10.1007/s11192-015-1613-7|s2cid=25088286}}</ref>
'''See also:'''
* ]
* ], another prize for outstanding mathematicians
* ], for other mathematicians


The prize includes a monetary award which, since 2006, has been {{CA$|link=yes}}15,000.<ref name="BBC">{{cite news |title= Maths genius turns down top prize |url= http://news.bbc.co.uk/2/hi/science/nature/5274040.stm |publisher= BBC |date= 22 August 2006 |access-date= 22 August 2006 |archive-date= 15 August 2010 |archive-url= https://web.archive.org/web/20100815015937/http://news.bbc.co.uk/2/hi/science/nature/5274040.stm |url-status= live }}</ref><ref> {{Webarchive|url=https://web.archive.org/web/20130523181501/http://www.jpost.com/Israel/Article.aspx?id=185366 |date=23 May 2013 }}, '']''</ref> Fields was instrumental in establishing the award, designing the medal himself, and funding the monetary component, though he died before it was established and his plan was overseen by ].<ref name="Fields Institute About Us" />
'''External link:'''

* IMU Fields medal page: http://elib.zib.de/IMU/medals/
The medal was first awarded in 1936 to Finnish mathematician ] and American mathematician ], and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions. In 2014, the Iranian mathematician ] became the first female Fields Medalist.<ref name="Farsnews">{{cite news|url=http://english.farsnews.ir/newstext.aspx?nn=13930523000321|title=President Rouhani Congratulates Iranian Woman for Winning Math Nobel Prize|date=14 August 2014|access-date=14 August 2014|publisher=]|archive-date=26 December 2018|archive-url=https://web.archive.org/web/20181226060915/http://en.farsnews.ir/newstext.aspx?nn=13930523000321%0A|url-status=dead}}</ref><ref>{{cite web|url=http://www.mathunion.org/general/prizes/2014|publisher=International Mathematical Union|title=IMU Prizes 2014|access-date=12 August 2014|archive-date=26 December 2018|archive-url=https://web.archive.org/web/20181226060901/https://www.mathunion.org/general/prizes/2014%0A|url-status=dead}}</ref><ref>{{Cite news|url=https://www.theguardian.com/world/2017/jul/16/maryam-mirzakhani-iranian-newspapers-break-hijab-taboo-in-tributes#img-2|title=Maryam Mirzakhani: Iranian newspapers break hijab taboo in tributes|last=Dehghan|first=Saeed Kamali Dehghan|date=16 July 2017|work=The Guardian|access-date=18 July 2017|language=en-GB|issn=0261-3077|archive-date=18 July 2017|archive-url=https://web.archive.org/web/20170718033745/https://www.theguardian.com/world/2017/jul/16/maryam-mirzakhani-iranian-newspapers-break-hijab-taboo-in-tributes#img-2|url-status=live}}</ref> In total, 64 people have been awarded the Fields Medal.

The most recent group of Fields Medalists received their awards on 5 July 2022 in an online event which was live-streamed from Helsinki, Finland. It was originally meant to be held in ], Russia, but was moved following the ].

==Conditions of the award==
The Fields Medal has for a long time been regarded as the most prestigious award in the field of mathematics and is often ].<ref name=":0" /><ref name=":2" /><ref name=":3" /> Unlike the Nobel Prize, the Fields Medal is only awarded every four years. The Fields Medal also has an age limit: a recipient must be under age 40 on 1 January of the year in which the medal is awarded. The under-40 rule is based on Fields's desire that "while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others."<ref>{{Harvnb|McKinnon Riehm|Hoffman|2011|p=183}}</ref> Moreover, an individual can only be awarded one Fields Medal; winners are ineligible to be awarded future medals.<ref>{{cite web |url=http://darboux.mathunion.org/fileadmin/IMU/Prizes/Fields/Statutes%20Fields%20Medal.pdf |title=Rules for the Fields Medal |website=mathunion.org |access-date=1 May 2018 |archive-date=2 May 2018 |archive-url=https://web.archive.org/web/20180502070440/http://darboux.mathunion.org/fileadmin/IMU/Prizes/Fields/Statutes%20Fields%20Medal.pdf |url-status=live }}</ref>

First awarded in 1936, 64 people have won the medal as of 2022.<ref>{{cite web |title=Fields Medal |url=https://www.mathunion.org/imu-awards/fields-medal |publisher=International Mathematical Union |access-date=14 September 2020 |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> With the exception of two PhD holders in physics (] and ]),<ref>{{cite web |title=Edward Witten |url=https://www.worldsciencefestival.com/participants/dr-edward-witten/ |publisher=World Science Festival |access-date=14 September 2020 |archive-date=8 April 2022 |archive-url=https://web.archive.org/web/20220408074621/https://www.worldsciencefestival.com/participants/dr-edward-witten/ |url-status=live }}</ref> only people with a PhD in mathematics have won the medal.<ref>{{cite web |last1=Kollár |first1=János |title=Is there a curse of the Fields medal? |url=https://web.math.princeton.edu/~kollar/FromMyHomePage/fm-essay.pdf |publisher=] |date=2014 |access-date=14 September 2020 |archive-date=9 March 2022 |archive-url=https://web.archive.org/web/20220309061236/http://web.math.princeton.edu/~kollar/FromMyHomePage/fm-essay.pdf |url-status=live }}</ref>

==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 ] 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, ], and ], 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 ].<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
! ] 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=] (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| ], Norway

| ]
| {{sortname|Lars|Ahlfors}}
| ], Finland
| ], 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 ]s of ]s of entire and ]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>

|-

| ]
| {{sortname|Jesse|Douglas}}
| ], US
| ], 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 ] which is concerned with finding ]s connecting and determined by some fixed boundary."<ref name="36reasons" />

|-
|rowspan=2| 1950
|rowspan=2| ], US

| ]
| {{sortname|Laurent|Schwartz}}
| ], France
| ], 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 ], a new notion of generalized function motivated by the ] 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>

|-

| ]
| {{sortname|Atle|Selberg}}
| ], US
| ], 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 ]; achieved major results on zeros of the ]; gave an elementary proof of the ] (with P. Erdős), with a generalization to prime numbers in an arbitrary arithmetic progression."<ref name="50reasons" />

|-
|rowspan=2| 1954
|rowspan=2| ], Netherlands

| ]
| {{sortname|Kunihiko|Kodaira}}
| | ], US, ], Japan and ], 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>
| ], 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 ]. He demonstrated, by ], that such varieties are ]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>

|-

| ]
| {{sortname|Jean-Pierre|Serre}}
| ], 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 ]s of spheres, especially in his use of the method of ]s. Reformulated and extended some of the main results of complex variable theory in terms of ]."<ref name="54reasons" />

|-
|rowspan=2| 1958
|rowspan=2| ], UK

|
| {{sortname|Klaus|Roth}}
| ], UK
| ], 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">]. {{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>

|-

| ]
| {{sortname|René|Thom}}
| ], France
| ], 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| ], Sweden

| ]
| {{sortname|Lars|Hörmander}}
| ], Sweden
| ], 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 ]s. Specifically, contributed to the general theory of linear differential operators. The questions go back to one of ] 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>

|-

| ]
| {{sortname|John|Milnor}}
| ], US
| ], 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 ]."<ref name="62reasons" />

|-

|rowspan=4| 1966
|rowspan=4| ], USSR

| ]
| {{sortname|Michael|Atiyah}}
| ], UK
| ], 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 ]; proved jointly with Singer the ] of elliptic operators on complex manifolds; worked in collaboration with Bott to prove a fixed point theorem related to the ']'."<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}}
| ], US
| ], US<ref>{{cite web |url=http://historicalsociety.stanford.edu/pdfmem/Cohen_P.pdf |title=Memorial Resolution – Paul Cohen (1934–2007) |date=2011 |publisher=] 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 "]" to prove the independence in set theory of the axiom of choice and of the generalized ]. The latter problem was the first of Hilbert's problems of the 1900 Congress."<ref name="66reasons" />

|-

| ]
| {{sortname|Alexander|Grothendieck}}
| ], France
| ], 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 ]. He introduced the idea of K-theory (the Grothendieck groups and rings). Revolutionized ] in his celebrated ‘]’."<ref name="66reasons" />

|-

| ]
| {{sortname|Stephen|Smale}}
| ], US
| ], 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 ] 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 ] to solve this and related problems."<ref name="66reasons" />

|-

|rowspan=4| 1970
|rowspan=4| ], France
| ]
| {{sortname|Alan|Baker|Alan Baker (mathematician)}}
| ], UK
| ], 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>
| "] the ] (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>

|-

| ]
| {{sortname|Heisuke|Hironaka}}
| ], US
| ], 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 ] on an ]. Hironaka proved the results in any dimension."<ref name="70reasons" />

|-

|
| {{sortname|Sergei|Novikov|Sergei Novikov (mathematician)}}
| ], USSR
| ], Russia
], Russia
], 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 ]es of the differentiable manifold. His work included a study of the cohomology and homotopy of ]s."<ref name="70reasons" />

|-

| ]
| {{sortname|John G.|Thompson}}
| ], UK
| ], UK
], 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 ] that ]. 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| ], Canada

| ]
| {{sortname|Enrico|Bombieri}}
| ], Italy
| ], 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 ]s and the local ], in theory of functions of several complex variables, and in theory of partial differential equations and minimal surfaces – in particular, to the solution of ] 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>

|-

| ]
| {{sortname|David|Mumford}}
| ], US
| ], 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 ], varieties whose points parametrize isomorphism classes of some type of geometric object. Also made several important contributions to the theory of ]s."<ref name="74reasons" />

|-
|rowspan=4| 1978
|rowspan=4| ], Finland

| ]
| {{sortname|Pierre|Deligne}}
| ], France
| ], 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 ] concerning generalizations of the ] 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>

|-

| ]
| {{sortname|Charles|Fefferman}}
| ], US
| ], 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" />

|-

| ]
| {{sortname|Grigory|Margulis}}
| ], USSR
| ], 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 ]s. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups."<ref name="78reasons" />

|-

|
| {{sortname|Daniel|Quillen}}
| ], US
| ], 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=] |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 ], 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| ], Poland

| ]
| {{sortname|Alain|Connes}}
| ], France
| ], France
], France
], 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 ]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 ]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>

|-

| ]
| {{sortname|William|Thurston}}
| ], US
| ], 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 ]s carry a hyperbolic structure."<ref name="82reasons" />

|-

| ]
| {{sortname|Shing-Tung|Yau}}
| ], US
| ], 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 ] in algebraic geometry, to the ] of general relativity theory, and to real and complex ]s."<ref name="82reasons" />

|-
|rowspan=3| 1986
|rowspan=3| ], US

| ]
| {{sortname|Simon|Donaldson}}
| ], UK
| ], 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> ], 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 ]s, especially for showing that there is a differential structure on euclidian four-space which is ] 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>

|-

| ]
| {{sortname|Gerd|Faltings}}
| ], US
| ], 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 ]."<ref name="86reasons" />

|-

| ]
| {{sortname|Michael|Freedman}}
| ], US
| ], 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 ]s. One of his results is a proof of the ]."<ref name="86reasons" />

|-
|rowspan=4| 1990
|rowspan=4| ], Japan

|
| {{sortname|Vladimir|Drinfeld}}
| ], 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>
| ], 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 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''. {{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>

|-

| ]
| {{sortname|Vaughan|Jones}}
| ], US
| ], 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>
], 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 ]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''. {{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>

|-

| ]
| {{sortname|Shigefumi|Mori}}
| ], Japan
| ], 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 '']'' 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''. {{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>

|-

| ]
| {{sortname|Edward|Witten}}
| ], US
| ], 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| ], Switzerland

| ]
| {{sortname|Jean|Bourgain}}
| ], France
| ], 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 ]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''. {{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>
|-

| ]
| {{sortname|Pierre-Louis|Lions}}
| ], France
| ], France
], 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''. {{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>

|-

| ]
| {{sortname|Jean-Christophe|Yoccoz}}
| ], 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 ], 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''. {{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>

|-

| ]
| {{sortname|Efim|Zelmanov}}
| ] ], US
| ], Russia,
], 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 ]."<ref>Walter Feit. ''On the Work of Efim Zelmanov''. {{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| ], Germany

| ]
| {{sortname|Richard|Borcherds}}
| ], US
], UK
| ], 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 ] and the discovery of a new class of automorphic infinite products."<ref name="1998cer">''Opening ceremony''. {{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>

|-

| ]
| {{sortname|Timothy|Gowers}}
| ], UK
| ], 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" />

|-

| ]
| {{sortname|Maxim|Kontsevich}}
| ], France
], US
| ], France
], US<ref>{{cite web|url=https://www.ihes.fr/~maxim/CVAnglais.html|title=CV Maxim Kontsevich|publisher=]|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" />

|-

| ]
| {{sortname|Curtis T.|McMullen}}
| ], US
| ], 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| ], China

| ]
| {{sortname|Laurent|Lafforgue}}
| ], France
| ], 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 ] for the full linear groups GLr (r≥1) over function fields of positive characteristic."<ref>Gérard Laumon. ''The work of Laurent Lafforgue''. {{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>

|-

| ]
| {{sortname|Vladimir|Voevodsky}}
| ], US
| ], 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 ]s on the K-theory of fields."<ref>Christophe Soulé. ''The work of Vladimir Voevodsky''. {{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| ], Spain

| ]
| {{sortname|Andrei|Okounkov}}
| ], US
| ], 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>
], 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''. {{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>

|-

| ]
| {{sortname|Grigori|Perelman}} ''(declined)''
| None
| ], 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 &#124; 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 ]."<ref name="2006cer" />

|-

| ]
| {{sortname|Terence|Tao}}
| ], US
| ], 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" />

|-

| ]
| {{sortname|Wendelin|Werner}}
| ], France
| ], 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 ], the geometry of two-dimensional ], and ]."<ref name="2006cer" />

|-
|rowspan=4| 2010
|rowspan=4| ], India

| ]
| {{sortname|Elon|Lindenstrauss}}
| ], Israel
], US
| ], 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 ], and their applications to number theory."<ref name="2010cer">''Opening ceremony''. {{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>

|-

| ]
| ]
| ], France
], US
| ], 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 ] in the theory of ]s through the introduction of new algebro-geometric methods."<ref name="2010cer" />

|-

| ]
| {{sortname|Stanislav|Smirnov}}
| ], Switzerland
| ], Switzerland
], 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 ] in statistical physics."<ref name="2010cer" />

|-

| ]
| {{sortname|Cédric|Villani}}
| ], France
], France
| ], France
], 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 ] and convergence to equilibrium for the ]."<ref name="2010cer" />

|-
|rowspan=4| 2014
|rowspan=4| ], South Korea

| ]
| {{sortname|Artur|Avila}}
| ], France
], France
], Brazil
| ], Switzerland
], 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''. {{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>

|-

| ]
| {{sortname|Manjul|Bhargava}}
| ], US
| ], 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 ]s."<ref name="2014cer" />

|-

| ]
| {{sortname|Martin|Hairer}}
| ], UK
| ], 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" />

|-

| ]
| {{sortname|Maryam|Mirzakhani}}
| ], US
| ], 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 ]s and their moduli spaces."<ref name="2014cer" />

|-

|rowspan=4| 2018
|rowspan=4| ], Brazil
| ]
| {{sortname|Caucher|Birkar}}
| ], UK
| ], UK
| "For the proof of the boundedness of ] and for contributions to the ]."<ref name="2018cer">''Opening ceremonies''. {{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>

|-

| ]
| {{sortname|Alessio|Figalli}}
| ], Switzerland
| ], Switzerland
| "For contributions to the theory of ] and its applications in ]s, ] and ]."<ref name="2018cer" />

|-

| ]
| {{sortname|Peter|Scholze}}
| ], Germany
| ], Germany
| "For having transformed arithmetic algebraic geometry over ]s."<ref name="2018cer" />

|-

| ]
| {{sortname|Akshay|Venkatesh}}
| ], US
| ], 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 ], ], ], and ], which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects."<ref name="2018cer" />
|-
|rowspan=4| 2022
|rowspan=4| ], Finland{{efn|ICM 2022 was originally planned to be held in ], Russia, but was moved online following the ]. 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>}}
| ]
| {{sortname|Hugo|Duminil-Copin}}
| ], France
], 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>
| ], France
], 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>

|-
| ]
| {{sortname|June|Huh}}
| ], US
| ], US
| "For bringing the ideas of ] to combinatorics, the proof of the Dowling–Wilson conjecture for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for ]s, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture."<ref name="2022fields" />

|-
| ]
| {{sortname|James|Maynard|James Maynard (mathematician)}}
| ], UK
| ], UK
| "For contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in ]."<ref name="2022fields" />

|-
| ]
| {{sortname|Maryna|Viazovska}}
| ], Switzerland
| ], Switzerland
| "For the proof that the ] provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in ]."<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>
|}

==Landmarks==
The medal was first awarded in 1936 to the Finnish mathematician ] and the American mathematician ], and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions.

In 1954, ] became the youngest winner of the Fields Medal, at 27.<ref>{{cite journal | last1 = Nawlakhe | first1 = Anil | last2 = Nawlakhe | first2 = Ujwala | last3 = Wilson | first3 = Robin | author3-link = Robin Wilson (mathematician) | date = July 2011 | department = Stamp Corner | doi = 10.1007/s00283-011-9244-1 | issue = 4 | journal = ] | pages = 70 | title = Fields Medallists | volume = 33| s2cid = 189866710 | doi-access = free }}</ref> He retains this distinction.<ref>{{Cite web |last=Raikar |first=Sanat Pai |date=2023-05-08 |title=Fields Medal |url=https://www.britannica.com/science/Fields-Medal |access-date=2023-06-22 |website=] |language=en |archive-date=7 February 2022 |archive-url=https://web.archive.org/web/20220207104118/https://www.britannica.com/science/Fields-Medal |url-status=live }}</ref>

In 1966, ] boycotted the ICM, held in Moscow, to protest Soviet military actions taking place in Eastern Europe.<ref>{{cite journal|last=Jackson|first=Allyn|date=October 2004|title=As If Summoned from the Void: The Life of Alexandre Grothendieck|url=https://www.ams.org/notices/200410/fea-grothendieck-part2.pdf|journal=]|volume=51|issue=9|page=1198|access-date=26 August 2006|archive-date=25 August 2006|archive-url=https://web.archive.org/web/20060825030446/http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf|url-status=live}}</ref> ], founder and director of the ], attended and accepted Grothendieck's Fields Medal on his behalf.<ref>{{cite web|url=https://www.ams.org/samplings/this-math-month/thismathmonth-aug|title=This Mathematical Month – August|publisher=]|archive-url=https://web.archive.org/web/20100811071525/http://www.ams.org/samplings/this-math-month/thismathmonth-aug|archive-date=11 August 2010|url-status=dead}}</ref>

In 1970, ], because of restrictions placed on him by the Soviet government, was unable to travel to the congress in ] to receive his medal.<ref>{{cite web |last1=Raikar |first1=Sanat Pai |title=Fields Medal |date=15 December 2023 |url=https://www.britannica.com/science/Fields-Medal |access-date=7 February 2022 |archive-date=7 February 2022 |archive-url=https://web.archive.org/web/20220207104118/https://www.britannica.com/science/Fields-Medal |url-status=live }}</ref>

In 1978, ], because of restrictions placed on him by the Soviet government, was unable to travel to the congress in ] to receive his medal. The award was accepted on his behalf by ], who said in his address: "I cannot but express my deep disappointment—no doubt shared by many people here—in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration."<ref> {{Webarchive|url=https://web.archive.org/web/20191019123826/http://www-history.mcs.st-andrews.ac.uk/Biographies/Margulis.html |date=19 October 2019 }}, School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 27 August 2006.</ref>

In 1982, the congress was due to be held in ] but had to be rescheduled to the next year, because of ] introduced in Poland on 13 December 1981. The awards were announced at the ninth General Assembly of the IMU earlier in the year and awarded at the 1983 Warsaw congress.<ref>{{Cite web |title=1982 ICM - Warsaw |url=https://mathshistory.st-andrews.ac.uk/ICM/ICM_Warsaw_1983/ |access-date=2024-06-05 |website=Maths History |language=en}}</ref>

In 1990, ] became the first ] to win the award.<ref name="National Science Foundation">{{cite web | title=The National Medal of Science 50th Anniversary | website=National Science Foundation | url=https://www.nsf.gov/news/special_reports/medalofscience50/witten.jsp | access-date=2022-08-30 | archive-date=22 June 2022 | archive-url=https://web.archive.org/web/20220622125848/https://www.nsf.gov/news/special_reports/medalofscience50/witten.jsp | url-status=live }}</ref>

In 1998, at the ICM, ] was presented by the chair of the Fields Medal Committee, ], with the first-ever IMU silver plaque in recognition of his proof of ]. ] referred to the plaque as a "quantized Fields Medal". Accounts of this award frequently make reference that at the time of the award Wiles was over the age limit for the Fields medal.<ref> {{webarchive|url=https://web.archive.org/web/20080827194548/http://wwwa.britannica.com/eb/article-9090319|date=27 August 2008}}, Encyclopædia Britannica. Retrieved 27 August 2006.</ref> Although Wiles was slightly over the age limit in 1994, he was thought to be a favorite to win the medal; however, a gap (later resolved by ] and Wiles) in the proof was found in 1993.<ref>, 2002 International Congress of Mathematicians. Retrieved 27 August 2006. {{webarchive|url=https://web.archive.org/web/20070927193530/http://www.icm2002.org.cn/general/prize/medal/1998.htm|date=27 September 2007}}</ref><ref>{{cite journal |url=https://www.ams.org/notices/199810/comm-fields.pdf |journal=] |date=November 1998 |volume=45 |issue=10 |pages=1359 |title=Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals |access-date=28 April 2021 |archive-date=23 April 2021 |archive-url=https://web.archive.org/web/20210423161336/http://www.ams.org/notices/199810/comm-fields.pdf |url-status=live }}</ref>

In 2006, ], who proved the ], refused his Fields Medal<ref name="BBC" /> and did not attend the congress.<ref>{{cite magazine|url=http://www.newyorker.com/fact/content/articles/060828fa_fact2|title=Manifold Destiny: A legendary problem and the battle over who solved it.|last1=Nasar|first1=Sylvia|date=21 August 2006|magazine=]|access-date=24 August 2006|archive-url=https://web.archive.org/web/20060831020716/http://www.newyorker.com/fact/content/articles/060828fa_fact2|archive-date=31 August 2006|url-status=live|last2=Gruber|first2=David}}</ref>

In 2014, ] became the first Iranian as well as the first woman to win the Fields Medal, and ] became the first ] and ] became the first person of Indian origin to do so.<ref name="CF">{{Cite book|url=http://unesdoc.unesco.org/images/0023/002315/231519e.pdf|title=A Complex Formula: Girls and Women in Science, Technology, Engineering and Mathematics in Asia|last=UNESCO|publisher=Paris, UNESCO|year=2015|isbn=978-92-9223-492-8|pages=23|access-date=3 May 2017|archive-date=15 November 2017|archive-url=https://web.archive.org/web/20171115233123/http://unesdoc.unesco.org/images/0023/002315/231519e.pdf|url-status=live}}</ref><ref>{{Cite news|url=https://www.theguardian.com/world/2017/jul/16/maryam-mirzakhani-iranian-newspapers-break-hijab-taboo-in-tributes#img-2|title=Maryam Mirzakhani: Iranian newspapers break hijab taboo in tributes|author=Saeed Kamali Dehghan|date=16 July 2017|work=]|access-date=18 July 2017|language=en-GB|issn=0261-3077|archive-date=18 July 2017|archive-url=https://web.archive.org/web/20170718033745/https://www.theguardian.com/world/2017/jul/16/maryam-mirzakhani-iranian-newspapers-break-hijab-taboo-in-tributes#img-2|url-status=live}}</ref>

In 2022, ] became the first Ukrainian to win the Fields Medal, and ] became the first person of Korean ancestry to do so.<ref name="DZH">{{Cite web |title=Друга жінка в історії: українка Марина В'язовська отримала престижну математичну нагороду |url=https://life.pravda.com.ua/society/2022/07/5/249420/ |access-date=2022-07-11 |website=Українська правда _Життя |archive-date=13 July 2022 |archive-url=https://web.archive.org/web/20220713211953/https://life.pravda.com.ua/society/2022/07/5/249420/ |url-status=live }}</ref><ref name="The Korea Times 2022">{{cite web | title=June Huh becomes 1st scholar of Korean descent to win Fields Medal | website=The Korea Times | date=5 July 2022 | url=https://www.koreatimes.co.kr/www/culture/2024/05/135_332224.html | access-date=5 July 2022 | archive-date=5 July 2022 | archive-url=https://web.archive.org/web/20220705083700/https://m.koreatimes.co.kr/pages/article.asp?newsIdx=332224 | url-status=live }}</ref>

==Medal==
]
The medal was designed by Canadian sculptor ].<ref>{{cite web |url=http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html |title=Fields Institute – The Fields Medal |publisher=Fields.utoronto.ca |date=9 August 1932 |access-date=21 August 2010 |archive-date=1 April 2022 |archive-url=https://web.archive.org/web/20220401044832/http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html |url-status=live }}</ref> It is made of 14KT gold, has a diameter of 63.5mm, and weighs 169g.<ref>{{cite web |url=https://www.mathunion.org/imu-awards/fields-medal |title=Fields Medal |publisher=International Mathematical Union |date=2022 |access-date=July 7, 2022 |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>

* On the obverse is ] and a quote attributed to 1st century AD poet ], which reads in Latin: <span style="font-variant: all-small-caps;">{{lang|la|Transire suum pectus mundoque potiri|italic=no}}</span> ("To surpass one's understanding and master the world").<ref>{{Cite journal|last=Riehm|first=C.|date=2002|title=The early history of the Fields Medal|url=https://www.ams.org/notices/200207/comm-riehm.pdf|journal=Notices of the AMS|volume=49|issue=7|pages=778–782|quote=The Latin inscription from the Roman poet Manilius surrounding the image may be translated 'To pass beyond your understanding and make yourself master of the universe.' The phrase comes from Manilius's Astronomica 4.392 from the first century A.D. (p. 782).|access-date=28 April 2021|archive-date=26 October 2006|archive-url=https://web.archive.org/web/20061026000014/http://www.ams.org/notices/200207/comm-riehm.pdf|url-status=live}}</ref><ref>{{Cite web|date=5 February 2015|title=The Fields Medal|url=http://www.fields.utoronto.ca/about/fields-medal|access-date=23 April 2021|website=Fields Institute for Research in Mathematical Sciences|language=en|archive-date=23 April 2021|archive-url=https://web.archive.org/web/20210423094533/http://www.fields.utoronto.ca/about/fields-medal|url-status=live}}</ref> The year number 1933 is written in Roman numerals and contains an error (MC'''N'''XXXIII rather than MC'''M'''XXXIII).<ref>{{cite book|first=Eberhard|last=Knobloch|chapter=Generality and Infinitely Small Quantities in Leibniz's Mathematics: The Case of his Arithmetical Quadrature of Conic Sections and Related Curves|title=Infinitesimal Differences: Controversies between Leibniz and his Contemporaries|editor-first1=Ursula|editor-last1=Goldenbaum|editor-first2=Douglas|editor-last2=Jesseph|publisher=]|date=2008}}</ref> In capital Greek letters the word Ἀρχιμηδους, or "of Archimedes," is inscribed.
* On the reverse is the inscription:

:: <span style="display: inline-block; font-variant: all-small-caps; line-height:1; text-align: center">{{lang|la|Congregati<br/>ex toto orbe<br/>mathematici<br/>ob scripta insignia<br/>tribuere|italic=no}}</span>

Translation: "Mathematicians gathered from the entire world have awarded for outstanding writings."

In the background, there is the representation of Archimedes' ], with the carving illustrating his theorem ], behind an olive branch. (This is the mathematical result of which Archimedes was reportedly most proud: Given a sphere and a circumscribed cylinder of the same height and diameter, the ratio between their volumes is equal to {{frac|2|3}}.)

The rim bears the name of the prizewinner.<ref name="Fields Institute for Research in Mathematical Sciences 2015">{{cite web | title=The Fields Medal | website=Fields Institute for Research in Mathematical Sciences | date=2015-02-05 | url=http://www.fields.utoronto.ca/about/fields-medal | access-date=2022-08-30 | archive-date=23 April 2021 | archive-url=https://web.archive.org/web/20210423094533/http://www.fields.utoronto.ca/about/fields-medal | url-status=live }}</ref>

==Female recipients==

The Fields Medal has had two female recipients, ] from ] in 2014, and ] from ] in 2022.<ref name="CF" /><ref name="DZH" />
<!-- There has been news coverage of this fact / gender imbalance more generally -- could be covered here in a paragraph or two. -->

==In popular culture==

The Fields Medal gained some recognition in popular culture due to references in the 1997 film, '']''. In the movie, Gerald Lambeau (]) is an ] professor who won the award prior to the events of the story. Throughout the film, references made to the award are meant to convey its prestige in the field.<ref>{{Cite news|date=2018-08-01|title=Maths gives its 'Nobel Prize' to an Australian — here's why it matters|language=en-AU|work=ABC News|url=https://www.abc.net.au/news/2018-08-01/fields-medals-to-be-awarded-to-best-mathematical-minds/10049510|access-date=2021-12-20|archive-date=8 April 2022|archive-url=https://web.archive.org/web/20220408074618/https://www.abc.net.au/news/2018-08-01/fields-medals-to-be-awarded-to-best-mathematical-minds/10049510|url-status=live}}</ref>

==See also==
{{Portal|Mathematics}}
{{div col|colwidth=20em}}
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
{{div col end}}

==Notes==
{{notelist}}

==References==
{{Reflist|30em}}

==Further reading==
{{Refbegin}}
* {{Cite book |last1= McKinnon Riehm |first1= Elaine |last2= Hoffman |first2= Frances |year= 2011 |title= Turbulent Times in Mathematics: The Life of J.C. Fields and the History of the Fields Medal |location= Providence, RI |publisher= ] |isbn= 978-0-8218-6914-7 }}
* {{Cite book |last= Monastyrsky |first= Michael |year= 1998 |title= Modern Mathematics in the Light of the Fields Medal |location= Wellesley, MA |publisher= A. K. Peters |isbn= 1-56881-083-0 }}
* {{Cite journal |last=Tropp |first= Henry S. |year= 1976 |title= The Origins and History of the Fields Medal |journal= Historia Mathematica |volume= 3 |issue= 2 |pages= 167–181 |doi= 10.1016/0315-0860(76)90033-1 |doi-access= }}.
{{Refend}}

==External links==
{{Commons category|Fields medal}}
* {{Official website}}
* at britannica.com

{{Fields medalists|state=expanded}}
{{IMUPrizes}}
{{International mathematical activities}}
{{Authority control}}

]
]
]
]

Latest revision as of 00:32, 21 December 2024

Mathematics award Not to be confused with Field's metal.

Award
Fields Medal
The obverse of the Fields Medal
Awarded forOutstanding contributions in mathematics attributed to young scientists
Presented byInternational Mathematical Union
Reward(s)CA$15,000
First awarded1936; 88 years ago (1936)
Last awarded2022 (2022)
Websitemathunion.org/imu-awards/fields-medal Edit this at Wikidata
The reverse of the 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 honours the Canadian mathematician John Charles Fields.

The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics.

The prize includes a monetary award which, since 2006, has been CA$15,000. Fields was instrumental in establishing the award, designing the medal himself, and funding the monetary component, though he died before it was established and his plan was overseen by John Lighton Synge.

The medal was first awarded in 1936 to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions. In 2014, the Iranian mathematician Maryam Mirzakhani became the first female Fields Medalist. In total, 64 people have been awarded the Fields Medal.

The most recent group of Fields Medalists received their awards on 5 July 2022 in an online event which was live-streamed from Helsinki, Finland. It was originally meant to be held in Saint Petersburg, Russia, but was moved following the 2022 Russian invasion of Ukraine.

Conditions of the award

The Fields Medal has for a long time been regarded as the most prestigious award in the field of mathematics and is often described as the Nobel Prize of Mathematics. Unlike the Nobel Prize, the Fields Medal is only awarded every four years. The Fields Medal also has an age limit: a recipient must be under age 40 on 1 January of the year in which the medal is awarded. The under-40 rule is based on Fields's desire that "while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others." Moreover, an individual can only be awarded one Fields Medal; winners are ineligible to be awarded future medals.

First awarded in 1936, 64 people have won the medal as of 2022. With the exception of two PhD holders in physics (Edward Witten and Martin Hairer), only people with a PhD in mathematics have won the medal.

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. 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.

Year ICM location Medalists Affiliation
(when awarded)
Affiliation
(current/last)
Reasons
1936 Oslo, Norway Lars Ahlfors University of Helsinki, Finland Harvard University, US "Awarded medal for research on covering surfaces related to Riemann surfaces of inverse functions of entire and meromorphic functions. Opened up new fields of analysis."
Jesse Douglas Massachusetts Institute of Technology, US City College of New York, US "Did important work on the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary."
1950 Cambridge, US Laurent Schwartz University of Nancy, France University of Paris VII, France "Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics."
Atle Selberg Institute for Advanced Study, US Institute for Advanced Study, US "Developed generalizations of the 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."
1954 Amsterdam, Netherlands Kunihiko Kodaira Princeton University, US, University of Tokyo, Japan and Institute for Advanced Study, US University of Tokyo, Japan "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 manifolds."
Jean-Pierre Serre University of Nancy, France Collège de France, France "Achieved major results on the homotopy groups of spheres, especially in his use of the method of spectral sequences. Reformulated and extended some of the main results of complex variable theory in terms of sheaves."
1958 Edinburgh, UK Klaus Roth University College London, UK Imperial College London, UK "for solving a famous problem of number theory, namely, the determination of the exact exponent in the Thue-Siegel inequality"
René Thom University of Strasbourg, France Institut des Hautes Études Scientifiques, France "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."
1962 Stockholm, Sweden Lars Hörmander University of Stockholm, Sweden Lund University, Sweden "Worked in partial differential equations. Specifically, contributed to the general theory of linear differential operators. The questions go back to one of Hilbert's problems at the 1900 congress."
John Milnor Princeton University, US Stony Brook University, US "Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology."
1966 Moscow, USSR Michael Atiyah University of Oxford, UK University of Edinburgh, UK "Did joint work with Hirzebruch in K-theory; proved jointly with Singer the index theorem of elliptic operators on complex manifolds; worked in collaboration with Bott to prove a fixed point theorem related to the 'Lefschetz formula'."
Paul Cohen Stanford University, US Stanford University, US "Used technique called "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."
Alexander Grothendieck Institut des Hautes Études Scientifiques, France Centre National de la Recherche Scientifique, France "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 ‘Tôhoku paper’."
Stephen Smale University of California, Berkeley, US City University of Hong Kong, Hong Kong "Worked in differential topology where he proved the generalized Poincaré conjecture in dimension n 5 {\displaystyle n\geq 5} : Every closed, n-dimensional manifold homotopy-equivalent to the n-dimensional sphere is homeomorphic to it. Introduced the method of handle-bodies to solve this and related problems."
1970 Nice, France Alan Baker University of Cambridge, UK Trinity College, Cambridge, UK "Generalized the Gelfond-Schneider theorem (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified."
Heisuke Hironaka Harvard University, US Kyoto University, Japan "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."
Sergei Novikov Moscow State University, USSR Steklov Mathematical Institute, Russia

Moscow State University, Russia University of Maryland-College Park, US

"Made important advances in topology, the most well-known being his proof of the topological invariance of the Pontryagin classes of the differentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces."
John G. Thompson University of Cambridge, UK University of Cambridge, UK

University of Florida, US

"Proved jointly with W. Feit that 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."
1974 Vancouver, Canada Enrico Bombieri University of Pisa, Italy Institute for Advanced Study, US "Major contributions in the primes, in univalent functions 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."
David Mumford Harvard University, US Brown University, US "Contributed to problems of the existence and structure of 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 surfaces."
1978 Helsinki, Finland Pierre Deligne Institut des Hautes Études Scientifiques, France Institute for Advanced Study, US "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."
Charles Fefferman Princeton University, US Princeton University, US "Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results."
Grigory Margulis Moscow State University, USSR Yale University, US "Provided innovative analysis of the structure of Lie groups. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups."
Daniel Quillen Massachusetts Institute of Technology, US University of Oxford, UK "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."
1982 Warsaw, Poland Alain Connes Institut des Hautes Études Scientifiques, France Institut des Hautes Études Scientifiques, France

Collège de France, France Ohio State University, US

"Contributed to the theory of operator algebras, 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*-algebras to foliations and differential geometry in general."
William Thurston Princeton University, US Cornell University, US "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-manifolds carry a hyperbolic structure."
Shing-Tung Yau Institute for Advanced Study, US Tsinghua University, China "Made contributions in differential equations, also to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge–Ampère equations."
1986 Berkeley, US Simon Donaldson University of Oxford, UK Imperial College London, UK Stony Brook University, US "Received medal primarily for his work on topology of four-manifolds, especially for showing that there is a differential structure on euclidian four-space which is different from the usual structure."
Gerd Faltings Princeton University, US Max Planck Institute for Mathematics, Germany "Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture."
Michael Freedman University of California, San Diego, US Microsoft Station Q, US "Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincaré Conjecture."
1990 Kyoto, Japan Vladimir Drinfeld B Verkin Institute for Low Temperature Physics and Engineering, USSR University of Chicago, US "Drinfeld's main preoccupation in the last decade Langlands' program and quantum groups. In both domains, Drinfeld's work constituted a decisive breakthrough and prompted a wealth of research."
Vaughan Jones University of California, Berkeley, US University of California, Berkeley, US

Vanderbilt University, US

"Jones discovered an astonishing relationship between von Neumann algebras and geometric topology. As a result, he found a new polynomial invariant for knots and links in 3-space."
Shigefumi Mori Kyoto University, Japan Kyoto University, Japan "The most profound and exciting development in algebraic geometry during the last decade or so was 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."
Edward Witten Institute for Advanced Study, US Institute for Advanced Study, US "Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems."
1994 Zürich, Switzerland Jean Bourgain Institut des Hautes Études Scientifiques, France Institute for Advanced Study, US "Bourgain's work touches on several central topics of mathematical analysis: the geometry of Banach spaces, convexity in high dimensions, harmonic analysis, ergodic theory, and finally, nonlinear partial differential equations from mathematical physics."
Pierre-Louis Lions University of Paris 9, France Collège de France, France

École polytechnique, France

"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."
Jean-Christophe Yoccoz Paris-Sud 11 University, France Collège de France, France "Yoccoz obtained a very enlightening proof of Bruno's theorem, and he was able to prove the converse Palis and Yoccoz obtained a complete system of C conjugation invariants for Morse-Smale diffeomorphisms."
Efim Zelmanov University of Wisconsin-Madison University of Chicago, US Steklov Mathematical Institute, Russia,

University of California, San Diego, US

"For the solution of the restricted Burnside problem."
1998 Berlin, Germany Richard Borcherds University of California, Berkeley, US

University of Cambridge, UK

University of California, Berkeley, US "For his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie algebras, the proof of the Conway–Norton moonshine conjecture and the discovery of a new class of automorphic infinite products."
Timothy Gowers University of Cambridge, UK University of Cambridge, UK "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."
Maxim Kontsevich Institut des Hautes Études Scientifiques, France

Rutgers University, US

Institut des Hautes Études Scientifiques, France

Rutgers University, US

"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."
Curtis T. McMullen Harvard University, US Harvard University, US "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."
2002 Beijing, China Laurent Lafforgue Institut des Hautes Études Scientifiques, France Institut des Hautes Études Scientifiques, France "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."
Vladimir Voevodsky Institute for Advanced Study, US Institute for Advanced Study, US "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 conjectures on the K-theory of fields."
2006 Madrid, Spain Andrei Okounkov Princeton University, US Columbia University, US

University of California, Berkeley, US

"For his contributions bridging probability, representation theory and algebraic geometry."
Grigori Perelman (declined) None St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences, Russia "For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow."
Terence Tao University of California, Los Angeles, US University of California, Los Angeles, US "For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory."
Wendelin Werner Paris-Sud 11 University, France ETH Zurich, Switzerland "For his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory."
2010 Hyderabad, India Elon Lindenstrauss Hebrew University of Jerusalem, Israel

Princeton University, US

Hebrew University of Jerusalem, Israel "For his results on measure rigidity in ergodic theory, and their applications to number theory."
Ngô Bảo Châu Paris-Sud 11 University, France

Institute for Advanced Study, US

University of Chicago, US

Institute for Advanced Study, US

"For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods."
Stanislav Smirnov University of Geneva, Switzerland University of Geneva, Switzerland

St. Petersburg State University, Russia

"For the proof of conformal invariance of percolation and the planar Ising model in statistical physics."
Cédric Villani École Normale Supérieure de Lyon, France

Institut Henri Poincaré, France

Lyon University, France

Institut Henri Poincaré, France

"For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation."
2014 Seoul, South Korea 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."
Manjul Bhargava Princeton University, US Princeton University, US "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 curves."
Martin Hairer University of Warwick, UK 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."
Maryam Mirzakhani Stanford University, US Stanford University, US "For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."
2018 Rio de Janeiro, Brazil 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."
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 equations, metric geometry and probability."
Peter Scholze University of Bonn, Germany University of Bonn, Germany "For having transformed arithmetic algebraic geometry over p-adic fields."
Akshay Venkatesh Stanford University, US Institute for Advanced Study, US "For his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects."
2022 Helsinki, Finland Hugo Duminil-Copin Institut des Hautes Études Scientifiques, France

University of Geneva, Switzerland

Institut des Hautes Études Scientifiques, France

University of Geneva, Switzerland

"For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four."
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 matroids, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture."
James Maynard 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."
Maryna Viazovska École Polytechnique Fédérale de Lausanne, Switzerland École Polytechnique Fédérale de Lausanne, Switzerland "For the proof that the E 8 {\displaystyle E_{8}} lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis."

Landmarks

The medal was first awarded in 1936 to the Finnish mathematician Lars Ahlfors and the American mathematician Jesse Douglas, and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions.

In 1954, Jean-Pierre Serre became the youngest winner of the Fields Medal, at 27. He retains this distinction.

In 1966, Alexander Grothendieck boycotted the ICM, held in Moscow, to protest Soviet military actions taking place in Eastern Europe. Léon Motchane, founder and director of the Institut des Hautes Études Scientifiques, attended and accepted Grothendieck's Fields Medal on his behalf.

In 1970, Sergei Novikov, because of restrictions placed on him by the Soviet government, was unable to travel to the congress in Nice to receive his medal.

In 1978, Grigory Margulis, because of restrictions placed on him by the Soviet government, was unable to travel to the congress in Helsinki to receive his medal. The award was accepted on his behalf by Jacques Tits, who said in his address: "I cannot but express my deep disappointment—no doubt shared by many people here—in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration."

In 1982, the congress was due to be held in Warsaw but had to be rescheduled to the next year, because of martial law introduced in Poland on 13 December 1981. The awards were announced at the ninth General Assembly of the IMU earlier in the year and awarded at the 1983 Warsaw congress.

In 1990, Edward Witten became the first physicist to win the award.

In 1998, at the ICM, Andrew Wiles was presented by the chair of the Fields Medal Committee, Yuri I. Manin, with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized Fields Medal". Accounts of this award frequently make reference that at the time of the award Wiles was over the age limit for the Fields medal. Although Wiles was slightly over the age limit in 1994, he was thought to be a favorite to win the medal; however, a gap (later resolved by Taylor and Wiles) in the proof was found in 1993.

In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal and did not attend the congress.

In 2014, Maryam Mirzakhani became the first Iranian as well as the first woman to win the Fields Medal, and Artur Avila became the first South American and Manjul Bhargava became the first person of Indian origin to do so.

In 2022, Maryna Viazovska became the first Ukrainian to win the Fields Medal, and June Huh became the first person of Korean ancestry to do so.

Medal

The reverse of the Fields Medal

The medal was designed by Canadian sculptor R. Tait McKenzie. It is made of 14KT gold, has a diameter of 63.5mm, and weighs 169g.

  • On the obverse is Archimedes and a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri ("To surpass one's understanding and master the world"). The year number 1933 is written in Roman numerals and contains an error (MCNXXXIII rather than MCMXXXIII). In capital Greek letters the word Ἀρχιμηδους, or "of Archimedes," is inscribed.
  • On the reverse is the inscription:
Congregati
ex toto orbe
mathematici
ob scripta insignia
tribuere

Translation: "Mathematicians gathered from the entire world have awarded for outstanding writings."

In the background, there is the representation of Archimedes' tomb, with the carving illustrating his theorem On the Sphere and Cylinder, behind an olive branch. (This is the mathematical result of which Archimedes was reportedly most proud: Given a sphere and a circumscribed cylinder of the same height and diameter, the ratio between their volumes is equal to 2⁄3.)

The rim bears the name of the prizewinner.

Female recipients

The Fields Medal has had two female recipients, Maryam Mirzakhani from Iran in 2014, and Maryna Viazovska from Ukraine in 2022.

In popular culture

The Fields Medal gained some recognition in popular culture due to references in the 1997 film, Good Will Hunting. In the movie, Gerald Lambeau (Stellan Skarsgård) is an MIT professor who won the award prior to the events of the story. Throughout the film, references made to the award are meant to convey its prestige in the field.

See also

Notes

  1. 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.

References

  1. ^ "About Us: The Fields Medal". The Fields Institute, University of Toronto. Archived from the original on 1 April 2022. Retrieved 21 August 2010.
  2. ^ Ball, Philip (2014). "Iranian is first woman to nab highest prize in maths". Nature. doi:10.1038/nature.2014.15686. S2CID 180573813. Archived from the original on 8 October 2019. Retrieved 29 March 2018.
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Further reading

  • McKinnon Riehm, Elaine; Hoffman, Frances (2011). Turbulent Times in Mathematics: The Life of J.C. Fields and the History of the Fields Medal. Providence, RI: American Mathematical Society. ISBN 978-0-8218-6914-7.
  • Monastyrsky, Michael (1998). Modern Mathematics in the Light of the Fields Medal. Wellesley, MA: A. K. Peters. ISBN 1-56881-083-0.
  • Tropp, Henry S. (1976). "The Origins and History of the Fields Medal". Historia Mathematica. 3 (2): 167–181. doi:10.1016/0315-0860(76)90033-1..

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