# Fields Medal

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

Fields Medal | |
---|---|

The obverse of the Fields Medal | |

Awarded for | Outstanding contributions in mathematics attributed to young scientists |

Country | Varies |

Presented by | International Mathematical Union (IMU) |

Reward(s) | CA$15,000 |

First awarded | 1936 |

Last awarded | 2018 |

Website | Mathunion.org |

The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the mathematician's Nobel Prize,[1][2][3] although there are several key differences, including frequency of award, number of awards, and age limits. 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,[4] 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.[5][6]

The prize comes with a monetary award which, since 2006, has been CA$15,000.[7][8] The name of the award is in honour of Canadian mathematician John Charles Fields.[9] Fields was instrumental in establishing the award, designing the medal itself, and funding the monetary component.[9]

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.[10][11][12] In all, sixty people have been awarded the Fields Medal.

The most recent group of Fields Medalists received their awards on 1 August 2018 at the opening ceremony of the IMU International Congress, held in Rio de Janeiro, Brazil.[13] The medal belonging to one of the four joint winners, Caucher Birkar, was stolen shortly after the event.[14] The ICM presented Birkar with a replacement medal a few days later.[15]

## 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.[1][2][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. This is similar to restrictions applicable to the Clark Medal in economics. 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."[16] Moreover, an individual can only be awarded one Fields Medal; laureates are ineligible to be awarded future medals.[17]

The monetary award is much lower than the 8 million Swedish kronor given with each Nobel prize as of 2014.[18] Other major awards in mathematics, such as the Abel Prize and the Chern Medal, have larger monetary prizes compared to the Fields Medal.

## Fields medalists

Year | ICM location | Medalists[19] | Nationality (when awarded) |
Affiliation (when awarded) |
Affiliation (current/last) |
Reasons |
---|---|---|---|---|---|---|

1936 | Oslo, Norway | Lars Ahlfors | Finland | University of Helsinki, Finland | Harvard University, US[20][21] | "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."[22] |

Jesse Douglas | United States | Massachusetts Institute of Technology, US | City College of New York, US[23][24] | "Did important work on the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary."[22] | ||

1950 | Cambridge, US | Laurent Schwartz | France | University of Nancy, France | University of Paris VII, France[25][26] | "Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics."[27] |

Atle Selberg | Norway | Institute for Advanced Study, US | Institute for Advanced Study, US[28] | "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."[27] | ||

1954 | Amsterdam, Netherlands | Kunihiko Kodaira | Japan | University of Tokyo, Japan and Institute for Advanced Study, US[29] | University of Tokyo, Japan[30] | "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."[31] |

Jean-Pierre Serre | France | University of Nancy, France | Collège de France, France[32][33] | "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."[31] | ||

1958 | Edinburgh, UK | Klaus Roth | United Kingdom | University College London, UK | Imperial College London, UK[34] | "Solved in 1955 the famous Thue-Siegel problem concerning the approximation to algebraic numbers by rational numbers and proved in 1952 that a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdős and Turán of 1935)."[35] |

René Thom | France | University of Strasbourg, France | Institut des Hautes Études Scientifiques, France[36] | "In 1954 invented and developed the theory of cobordism in algebraic topology. This classification of manifolds used homotopy theory in a fundamental way and became a prime example of a general cohomology theory."[35] | ||

1962 | Stockholm, Sweden | Lars Hörmander | Sweden | University of Stockholm, Sweden | Lund University, Sweden[37] | "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."[38] |

John Milnor | United States | Princeton University, US | Stony Brook University, US[39] | "Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology."[38] | ||

1966 | Moscow, USSR | Michael Atiyah | United Kingdom | University of Oxford, UK | University of Edinburgh, UK[40] | "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'."[41] |

Paul Joseph Cohen | United States | Stanford University, US | Stanford University, US[42] | "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."[41] | ||

Alexander Grothendieck | None | Institut des Hautes Études Scientifiques, France | Centre National de la Recherche Scientifique, France[43] | "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’."[41] | ||

Stephen Smale | United States | University of California, Berkeley, US | City University of Hong Kong, Hong Kong[44] | "Worked in differential topology where he proved the generalized Poincaré conjecture in dimension n≥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."[41] | ||

1970 | Nice, France | Alan Baker | United Kingdom | University of Cambridge, UK | Trinity College, Cambridge, UK[45] | "Generalized the Gelfond-Schneider theorem (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified."[46] |

Heisuke Hironaka | Japan | Harvard University, US | Kyoto University, Japan[47][48] | "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."[46] | ||

Sergei Novikov | Soviet Union | Moscow State University, USSR | Steklov Mathematical Institute, Russia
Moscow State University, Russia University of Maryland-College Park, US[49][50] |
"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."[46] | ||

John G. Thompson | United States | University of Cambridge, UK | University of Cambridge, UK | "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."[46] | ||

1974 | Vancouver, Canada | Enrico Bombieri | Italy | University of Pisa, Italy | Institute for Advanced Study, US[52] | "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."[53] |

David Mumford | United States | Harvard University, US | Brown University, US[54] | "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."[53] | ||

1978 | Helsinki, Finland | Pierre Deligne | Belgium | Institut des Hautes Études Scientifiques, France | Institute for Advanced Study, US[55] | "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."[56] |

Charles Fefferman | United States | Princeton University, US | Princeton University, US[57] | "Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results."[56] | ||

Grigori Margulis | Soviet Union | Moscow State University, USSR | Yale University, US[58] | "Provided innovative analysis of the structure of Lie groups. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups."[56] | ||

Daniel Quillen | United States | Massachusetts Institute of Technology, US | University of Oxford, UK[59] | "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."[56] | ||

1982 | Warsaw, Poland | Alain Connes | France | Institut des Hautes Études Scientifiques, France | Institut des Hautes Études Scientifiques, France
Collège de France, France Ohio State University, US[60] |
"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."[61] |

William Thurston | United States | Princeton University, US | Cornell University, US[62] | "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."[61] | ||

Shing-Tung Yau | None | Institute for Advanced Study, US | Harvard University, US[63] | "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."[61] | ||

1986 | Berkeley, US | Simon Donaldson | United Kingdom | University of Oxford, UK | Imperial College London, UK[64] Stony Brook University, US[65] | "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."[66] |

Gerd Faltings | West Germany | Princeton University, US | Max Planck Institute for Mathematics, Germany[67] | "Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture."[66] | ||

Michael Freedman | United States | University of California, San Diego, US | Microsoft Station Q, US[68] | "Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincaré Conjecture."[66] | ||

1990 | Kyoto, Japan | Vladimir Drinfeld | Soviet Union | B Verkin Institute for Low Temperature Physics and Engineering, USSR[69] | University of Chicago, US[70] | "For his work on quantum groups and for his work in number theory." |

Vaughan F. R. Jones | New Zealand | University of California, Berkeley, US | University of California, Berkeley, US,[71] | "For his discovery of an unexpected link between the mathematical study of knots – a field that dates back to the 19th century – and statistical mechanics, a form of mathematics used to study complex systems with large numbers of components." | ||

Shigefumi Mori | Japan | Kyoto University, Japan | Kyoto University, Japan[73] | "For the proof of Hartshorne’s conjecture and his work on the classification of three-dimensional algebraic varieties." | ||

Edward Witten | United States | Institute for Advanced Study, US | Institute for Advanced Study, US[74] | " Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems."[75] | ||

1994 | Zurich, Switzerland | Jean Bourgain | Belgium | Institut des Hautes Études Scientifiques, France | Institute for Advanced Study, US[76] | "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 | France | University of Paris 9, France | Collège de France, France
École polytechnique, France[77] |
"... Such nonlinear partial differential equation simply do not have smooth or even C1 solutions existing after short times. ... The only option is therefore to search for some kind of "weak" solution. This undertaking is in effect to figure out how to allow for certain kinds of "physically correct" singularities and how to forbid others. ... Lions and Crandall at last broke open the problem by focusing attention on viscosity solutions, which are defined in terms of certain inequalities holding wherever the graph of the solution is touched on one side or the other by a smooth test function." | ||

Jean-Christophe Yoccoz | France | Paris-Sud 11 University, France | Collège de France, France[78] | "Proving stability properties – dynamic stability, such as that sought for the solar system, or structural stability, meaning persistence under parameter changes of the global properties of the system." | ||

Efim Zelmanov | Russia | University of California, San Diego, US | Steklov Mathematical Institute, Russia, | "For his solution to the restricted Burnside problem." | ||

1998 | Berlin, Germany | Richard Borcherds | United Kingdom | University of California, Berkeley, US | University of California, Berkeley, US[80] | "For his work on the introduction of vertex algebras, the proof of the Moonshine conjecture and for his discovery of a new class of automorphic infinite products." |

Timothy Gowers | United Kingdom | University of Cambridge, UK | University of Cambridge, UK[81] | "William Timothy Gowers has provided important contributions to functional analysis, making extensive use of methods from combination theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully." | ||

Maxim Kontsevich | Russia | Institut des Hautes Études Scientifiques, France | Institut des Hautes Études Scientifiques, France | "Contributions to four problems of geometry." | ||

Curtis T. McMullen | United States | Harvard University, US | Harvard University, US[83] | "He has made important contributions to various branches of the theory of dynamical systems, such as the algorithmic study of polynomial equations, the study of the distribution of the points of a lattice of a Lie group, hyperbolic geometry, holomorphic dynamics and the renormalization of maps of the interval." | ||

2002 | Beijing, China | Laurent Lafforgue | France | Institut des Hautes Études Scientifiques, France | Institut des Hautes Études Scientifiques, France[84] | "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." |

Vladimir Voevodsky | Russia | Institute for Advanced Study, US | Institute for Advanced Study, US[85] | "He defined and developed motivic cohomology and the A1-homotopy theory of algebraic varieties; he proved the Milnor conjectures on the K-theory of fields." | ||

2006 | Madrid, Spain | Andrei Okounkov | Russia | Princeton University, US | Columbia University, US[86] | "For his contributions bridging probability, representation theory and algebraic geometry." |

Grigori Perelman (declined) |
Russia | None | St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences, Russia[87] | "For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." | ||

Terence Tao | Australia | University of California, Los Angeles, US | University of California, Los Angeles, US[88] | "For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory." | ||

Wendelin Werner | France | Paris-Sud 11 University, France | ETH Zurich, Switzerland[89] | "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 | Israel | Hebrew University of Jerusalem, Israel | Hebrew University of Jerusalem, Israel[90] | "For his results on measure rigidity in ergodic theory, and their applications to number theory." |

Ngô Bảo Châu | Vietnam, France | Paris-Sud 11 University, France | University of Chicago, US
Vietnam Institute for Advanced Study, Vietnam[91] |
"For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods." | ||

Stanislav Smirnov | Russia | University of Geneva, Switzerland | University of Geneva, Switzerland | "For the proof of conformal invariance of percolation and the planar Ising model in statistical physics." | ||

Cédric Villani | France | École Normale Supérieure de Lyon, France
Institut Henri Poincaré, France |
Lyon University, France
Institut Henri Poincaré, France[93] |
"For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation." | ||

2014 | Seoul, South Korea | Artur Avila | Brazil, France | University of Paris VII, France
CNRS, France Instituto Nacional de Matemática Pura e Aplicada, Brazil |
University of Zurich, Switzerland | "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."[94] |

Manjul Bhargava | Canada, United States | Princeton University, US | Princeton University, US[95][96][97] | "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."[94] | ||

Martin Hairer | Austria | 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."[94] | ||

Maryam Mirzakhani | Iran | Stanford University, US | Stanford University, US[98][99] | "For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."[94] | ||

2018 | Rio de Janeiro, Brazil | Caucher Birkar | Iran, United Kingdom | 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."[100] |

Alessio Figalli | Italy | 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."[100] | ||

Peter Scholze | Germany | University of Bonn, Germany | University of Bonn, Germany | "For transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories."[100] | ||

Akshay Venkatesh | Australia | Stanford University, US | Stanford University, 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."[100] | ||

## 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.[102] Léon Motchane, founder and director of the Institut des Hautes Études Scientifiques, attended and accepted Grothendieck's Fields Medal on his behalf.[103]

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."[104]

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.[105] 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.[106][107]

In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal[7] and did not attend the congress.[108]

In 2014, Maryam Mirzakhani became the first woman as well as the first Iranian 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.[109][110][111][112] President Rouhani congratulated Mirzakhani for this notable success.[10]

## Medal

The medal was designed by Canadian sculptor R. Tait McKenzie.[113]

- On the obverse is Archimedes and a quote attributed to him which reads in Latin: "Transire suum pectus mundoque potiri" ("Rise above oneself and grasp the world"). The date is written in Roman numerals and contains an error ("MCNXXXIII" rather than "MCMXXXIII").[114] In capital Greek letters the word ΑΡXIMHΔΟΥΣ, or "of Archimedes".
- On the reverse is the inscription (in Latin):

*CONGREGATI**EX TOTO ORBE**MATHEMATICI**OB SCRIPTA INSIGNIA**TRIBUERE*

Translation: "Mathematicians gathered from the entire world have awarded [understood but not written: 'this prize'] 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 ⅔.)

The rim bears the name of the prizewinner.

## Female recipients

In terms of the most prestigious awards in STEM fields, only a small proportion have been awarded to women. The Fields Medal has been obtained only one time by a woman, Maryam Mirzakhani, in 2014, out of a total of (currently) 60 medalists.[115][116]

## See also

## References

- Ball, Philip (2014). "Iranian is first woman to nab highest prize in maths".
*Nature*. doi:10.1038/nature.2014.15686. - "Fields Medal".
*www-history.mcs.st-andrews.ac.uk*. Retrieved 29 March 2018. - "Fields Medal".
*The University of Chicago*. Retrieved 29 March 2018. - "Top Award, ShanghaiRanking Academic Excellence Survey 2017 | Shanghai Ranking – 2017".
*Shanghairanking.com*. Retrieved 29 March 2018. - IREG Observatory on Academic Ranking and Excellence.
*IREG List of International Academic Awards*(PDF). Brussels: IREG Observatory on Academic Ranking and Excellence. Retrieved 3 March 2018. - Zheng, Juntao; Liu, Niancai (2015). "Mapping of important international academic awards".
*Scientometrics*.**104**(3): 763–791. doi:10.1007/s11192-015-1613-7. - "Maths genius turns down top prize". BBC. 22 August 2006. Retrieved 22 August 2006.
- "Israeli wins 'Nobel' of Mathematics",
*The Jerusalem Post* - "About Us: The Fields Medal". The Fields Institute, University of Toronto. Retrieved 21 August 2010.
- "President Rouhani Congratulates Iranian Woman for Winning Math Nobel Prize". Fars News Agency. 14 August 2014. Retrieved 14 August 2014.
- "IMU Prizes 2014". International Mathematical Union. Retrieved 12 August 2014.
- correspondent, Saeed Kamali Dehghan Iran (16 July 2017). "Maryam Mirzakhani: Iranian newspapers break hijab taboo in tributes".
*The Guardian*. ISSN 0261-3077. Retrieved 18 July 2017. - "Scientific Program: Program at a glance".
*ICM 2018 event website*. - Philips, Don (1 August 2018). "World's most prestigious maths medal is stolen minutes after professor wins it".
*The Guardian*. Retrieved 1 August 2018. - ICM announcement, August 4, 2018.
- McKinnon Riehm & Hoffman 2011, p. 183
- "Rules for the Fields Medal" (PDF).
*mathunion.org*. - "The Nobel Prize Amounts".
*Nobelprize.org*. Nobel Foundation. Retrieved 13 August 2014. - "The Fields Medalists, chronologically listed". International Mathematical Union (IMU). 8 May 2008. Retrieved 25 March 2009.
- "Lars Valerian Ahlfors (1907–1996)" (PDF).
*Ams.org*. Retrieved 31 March 2017. - "Lars Ahlfors (1907–1996)". Harvard University, Dept. of Math. 7 November 2004. Retrieved 19 August 2014.
- "Fields Medals 1936".
*mathunion.org*. International Mathematical Union. - "Jesse Douglas".
*Encyclopædia Britannica*. 28 May 2010. Retrieved 19 August 2014. - Mario J. Micallef; J. Gray. "The work of Jesse Douglas on Minimal Surfaces" (PDF).
*Wdb.ugr.es*. Archived from the original (PDF) on 6 October 2014. Retrieved 31 March 2017. - "Laurent Moise Schwartz". School of Mathematics and Statistics University of St Andrews, Scotland. 24 June 2007. Retrieved 19 August 2014.
- Schwartz, Laurent (1 February 2001).
*Un mathématicien aux prises avec le siècle*[*A Mathematician Grappling with His Century*]. AMS: Birkhäuser. ISBN 978-3-0348-7584-4. Archived from the original on 21 August 2014. Retrieved 21 August 2014. - "Fields Medals 1950".
*mathunion.org*. International Mathematical Union. - "Remembering Atle Selberg, 1917–2007" (PDF).
*Ams.org*. Retrieved 31 March 2017. - "Proceedings of the International Congress of Mathematicians" (PDF).
*Mathunion.org*. 1954. Retrieved 31 March 2017. - Donald C. Spencer. "Kunihiko Kodaira (1915–1997)" (PDF).
*Ams.org*. Retrieved 31 March 2017. - "Fields Medals 1954".
*mathunion.org*. International Mathematical Union. - "Jean-Pierre Serre" (PDF).
*Math.rug.nl*. Retrieved 31 March 2017. - "Jean-Pierre Serre".
*Encyclopædia Britannica*. 5 February 1997. Retrieved 19 August 2014. - McKinnon Riehm & Hoffman 2011, p. 212
- "Fields Medals 1958".
*mathunion.org*. International Mathematical Union. - "René Thom" (PDF).
*Robertnowlan.com*. Archived from the original (PDF) on 27 May 2016. Retrieved 31 March 2017. - "A tribute to Lars Hörmander" (PDF).
*Smai.emath.fr*. Retrieved 31 March 2017. - "Fields Medals 1962".
*mathunion.org*. International Mathematical Union. - "John W. Milnor". Stony Brook University. 5 March 1997. Retrieved 17 August 2014.
- "Sir Michael F. Atiyah : The Abel Prize" (PDF).
*Upcommons.upc.edu*(in Spanish). Retrieved 31 March 2017. - "Fields Medals 1966".
*mathunion.org*. International Mathematical Union. - "Archived copy" (PDF). Archived from the original (PDF) on 5 January 2015. Retrieved 24 August 2014.CS1 maint: archived copy as title (link)
- "Alexander Grothendieck" (PDF).
*Math.ucdenver.edu*. Retrieved 31 March 2017. - "Prof. Stephen SMALE (史梅爾)". City University of Hong Kong. 5 April 2012. Retrieved 18 August 2014.
- "The Laureates". Heidelberg Laureate Forum Foundation (HLFF). 25 September 2013. Retrieved 16 August 2014.
- "Fields Medals 1970".
*mathunion.org*. International Mathematical Union. - "Interview with Heisuke Hironaka" (PDF).
*Ams.org*. Retrieved 31 March 2017. - "No title". Research Institute for Mathematical Sciences, Kyoto, Japan. 26 May 2007. Retrieved 16 August 2014.
- "Interview with Sergey P. Novikov" (PDF).
*Mi.ras.ru*. Retrieved 31 March 2017. - "Novikov, Sergei Petrovich". Russian Academy of Science. 1 January 2012. Retrieved 20 August 2014.
- "John Griggs Thompson" (PDF).
*Abelprize.no*. Retrieved 31 March 2017. - BARTOCCI, CLAUDIO; Betti, Renato; Guerraggio, Angelo; et al., eds. (2011).
*Vite Mathematiche*[*Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles*] (2011 ed.). Springer. pp. 2013–2014. ISBN 978-3642136054. - "Fields Medals 1974".
*mathunion.org*. International Mathematical Union. - "David Mumford=12 May 2006". The Division of Applied Mathematics, Brown University. Retrieved 18 August 2014.
- "Pierre Deligne" (PDF).
*Abelprize.no*. Retrieved 31 March 2017.< - "Fields Medals 1978".
*mathunion.org*. International Mathematical Union. - "CV : Charles Fefferman" (PDF).
*Web.math.princeton.edu*. Retrieved 31 March 2017.< - "Yale Mathematics Department: Gregory A. Margulis". Retrieved 16 March 2015.
- "fea-memorial-quillen.dvi" (PDF).
*Ams.org*. Retrieved 31 March 2017. - "Alain Connes". 25 May 2012. Retrieved 18 August 2014.
- "Fields Medals and Nevanlinna Prize 1982".
*mathunion.org*. International Mathematical Union. - "William P. Thurston, 1946–2012". 30 August 2012. Retrieved 18 August 2014.
- "CV : Shing-Tung Yau" (PDF).
*Doctoryau.com*. Archived from the original (PDF) on 25 October 2017. Retrieved 31 March 2017. - "Simon Donaldson (Royal Society Research Professor)". Department of Mathematics, Imperial College, Queen's Gate, London. 16 January 2008. Retrieved 16 August 2014.
- "Simon Donaldson". Retrieved 16 March 2015.
- "Fields Medals and Nevanlinna Prize 1986".
*mathunion.org*. International Mathematical Union. - "The Laureates". Heidelberg Laureate Forum Foundation (HLFF). 6 October 2013. Archived from the original on 6 October 2014. Retrieved 16 August 2014.
- ROB KIRBY. "Michael H. Freedman" (PDF). Archived from the original on 6 October 2014.CS1 maint: BOT: original-url status unknown (link)
- "Vladimir Gershonovich Drinfeld".
*Encyclopædia Britannica*. 19 August 2009. Retrieved 2 September 2014. - "Vladimir Gershonovich Drinfeld". School of Mathematics and Statistics, University of St Andrews, Scotland. 18 August 2009. Retrieved 16 August 2014.
- "Curriculum Vitae: Vaughan F. R. Jones". University of California, Berkeley. 10 November 2001. Archived from the original on 6 August 2013. Retrieved 16 August 2014.
- Salisbury, David (6 April 2011). "Fields Medalist joins Vanderbilt faculty". Vanderbilt University. Retrieved 17 May 2011.
- "The Laureates". Heidelberg Laureate Forum Foundation (HLFF). 10 April 2014. Archived from the original on 15 August 2014. Retrieved 16 August 2014.
- "Archived copy" (PDF). Archived from the original (PDF) on 4 February 2012. Retrieved 26 October 2011.CS1 maint: archived copy as title (link)
- Michael Atiyah. "On the Work of Edward Witten" (PDF).
*Mathunion.org*. Archived from the original (PDF) on 1 March 2017. Retrieved 31 March 2017. - "CV : Jean Bourgain" (PDF).
*Math.ias.edu*. Retrieved 31 March 2017. - "Collège de France".
*College-de-france.fr*. 16 December 2013. Retrieved 18 August 2014. - "Collège de France".
*College-de-france.fr*. 16 December 2013. Retrieved 18 August 2014. - "CV : Efim Zelmanov" (PDF).
*Ime.usp.br*. Retrieved 31 March 2017. - "The Laureates". Heidelberg Laureate Forum Foundation (HLFF). 10 April 2014. Archived from the original on 6 October 2014. Retrieved 16 August 2014.
- "William Timothy Gowers".
*Encyclopædia Britannica*. 28 March 2009. Retrieved 16 August 2014. - "CURRICULUM VITAE". ihes. 22 November 2009. Archived from the original on 10 October 2014. Retrieved 16 August 2014.
- "CV : Curtis T McMullen" (PDF).
*Abel.math.harvard.edu*. Retrieved 31 March 2017. - "Curriculum Vitae". ihes. 6 December 2005. Retrieved 19 August 2014.
- "CV : Vladimir Voevodsky" (PDF).
*Math.ias.edu*. Retrieved 31 March 2017. - "Department of Mathematics". University of Columbia, Department of Mathematics. 20 December 2012. Retrieved 19 August 2014.
- "Encyclopædia Britannica".
*Encyclopædia Britannica*. 28 May 2008. Retrieved 19 August 2014. - "Vitae and Bibliography for Terence Tao". UCLA Dept. of Math. 16 March 2010. Retrieved 19 August 2014.
- "Wendelin WERNER". ETH Zurich. 18 September 2013. Retrieved 19 August 2014.
- "Nobel at HU". The Hebrew University of Jerusalem. 5 July 2011. Retrieved 16 August 2014.
- "Ngô Bảo Châu › Heidelberg Laureate Forum". Archived from the original on 7 February 2015. Retrieved 16 March 2015.
- "Home Page of Stanislav Smirnov". Retrieved 16 March 2015.
- "CV : Cedric Villani" (PDF).
*Cedricvillani.org*. Archived from the original (PDF) on 23 June 2016. Retrieved 31 March 2017. - "Fields Medals 2014".
*mathunion.org*. International Mathematical Union. - "CV : Manjul Bhargava" (PDF).
*2.maths.ox.ac.uk*. Retrieved 31 March 2017. - "The Work of Manjul Bhargava" (PDF).
*Mathunion.org*. Archived from the original (PDF) on 13 July 2017. Retrieved 31 March 2017. - "Faculty". The Princeton University, Department of Mathematics. 8 May 2012. Archived from the original on 25 December 2014. Retrieved 19 December 2014.
- "Interview with Research Fellow Maryam Mirzakhani" (PDF).
- "Department of Mathematics". Stanford University. 22 January 2009. Retrieved 19 December 2014.
- "Fields Medals 2018".
*mathunion.org*. International Mathematical Union. - "Faculty Appointee Akshay Venkatesh Awarded 2018 Fields Medal".
- Jackson, Allyn (October 2004). "As If Summoned from the Void: The Life of Alexandre Grothendieck" (PDF).
*Notices of the American Mathematical Society*.**51**(9): 1198. Retrieved 26 August 2006. - "This Mathematical Month – August". American Mathematical Society. Archived from the original on 11 August 2010.
- Margulis biography, School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 27 August 2006.
- Wiles, Andrew John Archived 27 August 2008 at the Wayback Machine, Encyclopædia Britannica. Retrieved 27 August 2006.
- Fields Medal Prize Winners (1998), 2002 International Congress of Mathematicians. Retrieved 27 August 2006. Archived 27 September 2007 at the Wayback Machine
- Notices of the AMS, November 1998. Vol. 45, No. 10, p. 1359.
- Nasar, Sylvia; Gruber, David (21 August 2006). "Manifold Destiny: A legendary problem and the battle over who solved it".
*The New Yorker*. Archived from the original on 31 August 2006. Retrieved 24 August 2006.CS1 maint: BOT: original-url status unknown (link) (dead link). dated 14 December 2006; accessed 14 August 2014 - The Work of Maryam Mirzakhani, IMU Press Release 2014, accessed 30 September 2014
- UNESCO (2015).
*A Complex Formula: Girls and Women in Science, Technology, Engineering and Mathematics in Asia*(PDF). Paris, UNESCO. p. 23. ISBN 978-92-9223-492-8. - "IMU Prizes 2014". International Mathematical Union. Retrieved 12 August 2014.
- correspondent, Saeed Kamali Dehghan Iran (16 July 2017). "Maryam Mirzakhani: Iranian newspapers break hijab taboo in tributes".
*The Guardian*. ISSN 0261-3077. Retrieved 18 July 2017. - "Fields Institute – The Fields Medal". Fields.utoronto.ca. 9 August 1932. Retrieved 21 August 2010.
- EBERHARD KNOBLOCH. Generality and Infinitely Small Quantities in Leibniz's Mathematics: The Case of his Arithmetical Quadrature of Conic Sections and Related Curves. In Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Edited by Ursula Goldenbaum and Douglas Jesseph. Walter de Gruyter, 2008
- IMU. 2014.
*The Work of Maryam Mirzakhani*. Press Release (Accessed 30 September 2014) - UNESCO (2015).
*A Complex Formula: Girls and Women in Science, Technology, Engineering and Mathematics in Asia*(PDF). Paris, UNESCO. p. 23. ISBN 978-92-9223-492-8.

## 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..