# Fields Medal

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

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 Fields Medal is widely regarded as the highest honor a mathematician can receive, and has been described by many as the mathematician's "Nobel Prize".[1][2][3] According to Academic Excellence Survey by ARWU, Fields Medal is 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 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 woman Fields Medallist.[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]

## Conditions of the award

The Fields Medal is often described as the "Nobel Prize of Mathematics" and for a long time has been regarded as the most prestigious award in the field of mathematics.[1][2][3] Unlike the Nobel Prize, however, 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' 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."[14] Moreover, an individual can only be awarded one Fields Medal; laureates are ineligible to be awarded future medals.[15] This is in contrast with the Nobel Prize which can be, and has been awarded to an individual or an entity more than once, whether in the same category (John Bardeen and Frederick Sanger), or in different categories (Marie Curie and Linus Pauling).

The monetary award is much lower than the 8,000,000 Swedish kronor (roughly 1,400,000 Canadian dollars)[16] given with each Nobel prize as of 2014.[17] 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[18] Affiliation
(when awarded)
Affiliation
(current/last)
Citation
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 of 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 University of Tokyo, Japan and Institute for Advanced Study, US[26] 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 "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 Erdös and Turán of 1935)."
René Thom University of Strasbourg, France Institut des Hautes Études Scientifiques, France "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."
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 Joseph 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: 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."
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."
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 Pontrjagin classes of the differentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces."
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."
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."
Grigori 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."
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 Harvard University, US "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[57] University of Chicago, US "For his work on quantum groups and for his work in number theory."
Vaughan F. R. Jones University of California, Berkeley, US University of California, Berkeley, US,

Vanderbilt University, US

"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 Kyoto University, Japan Kyoto University, Japan "for the proof of Hartshorne’s conjecture and his work on the classification of three-dimensional algebraic varieties."
Edward Witten Institute for Advanced Study, US Institute for Advanced Study, US "proof in 1981 of the positive energy theorem in general relativity"[63]
1994 Zurich, 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[6

"... 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 Paris-Sud 11 University, France Collège de France, France "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 University of California, San Diego, US Steklov Mathematical Institute, Russia,

University of California, San Diego, US

"For his solution to 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 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 University of Cambridge, UK University of Cambridge, UK "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 Institut des Hautes Études Scientifiques, France

Rutgers University, US

Institut des Hautes Études Scientifiques, France

Rutgers University, US

"contributions to four problems of geometry"
Curtis T. McMullen Harvard University, US Harvard University, US "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 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."

Vladimir Voevodsky Institute for Advanced Study, US Institute for Advanced Study, US " 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 Princeton University, US Columbia University, 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

University of Chicago, US

Vietnam Institute for Advanced Study, Vietnam

"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 Paris VII, France

CNRS, France

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 his proof of the boundedness of Fano varieties and for contributions to the minimal model program".[87]
Alessio Figalli Swiss Federal Institute of Technology Zurich, Switzerland Swiss Federal Institute of Technology Zurich, Switzerland "for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability"
Peter Scholze 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."
Akshay Venkatesh 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."

## Landmarks

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

In 1966, Alexander Grothendieck boycotted the ICM, held in Moscow, to protest Soviet military actions taking place in Eastern Europe.[89] Léon Motchane, founder and director of the Institut des Hautes Études Scientifiques attended and accepted Grothendieck's Fields Medal on his behalf.[90]

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 this 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[7] and did not attend the congress.[95]

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

## Medal

The reverse of the Fields Medal

The medal was designed by Canadian sculptor R. Tait McKenzie.

 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"). 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 a 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.