Fields Medal

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Fields Medal, obverse

The Fields Medal , officially named International Medal for Outstanding Discoveries in Mathematics (German: "International Medal for Outstanding Discoveries in Mathematics"), is one of the highest awards a mathematician can receive. It is named after its founder, the Canadian mathematician John Charles Fields (1864–1932), and was first awarded in 1936. Since 1950 it has been awarded every four years by the International Mathematical Union (IMU) on the occasion of the International Congress of Mathematicians (ICM) to two to four mathematicians who are younger than 40 and who have distinguished themselves in the field of mathematical research in a special way (see above formally defined since 1966). The award comes with a cash prize of 15,000 Canadian dollars . At the ICM, two other prizes are awarded at the same time: the Carl Friedrich Gauß Prize for contributions to applied mathematics and the Nevanlinna Prize for contributions to theoretical computer science .

Award principles

John Charles Fields, who gave the medal its name

The selection committee appointed by the IMU Executive Committee, whose members, with the exception of the chairman, remain secret until the award ceremony, has the task of selecting at least two, but preferably four, recipients who represent a variety of areas in mathematics. The founder of the award, John Charles Fields , considered the basic principles behind the award to be the solution of a difficult problem and the formulation of a new theory that would expand the areas of application of mathematics.

Medal recipients must be under 40 at the beginning of the year in which they are awarded. The rule, formalized in 1966 and later specified more precisely, goes back to the expectation formulated at the establishment of Fields, “ that […] while it was in recognition of work already done, it was intended at the same time to be an encouragement for further achievement on the part of the recipients [...] ”(German:“ that, even if it was in recognition of work that had already been done, it was also intended as an incentive for further services on the part of the recipient ”).

This prevented, for example, the award to Andrew Wiles (* 1953), who only partially succeeded in proving the modularity theorem (from which the great Fermatian theorem follows) in 1993 and fully in 1995. Instead, Wiles received a special award from the IMU at the ICM 1998 in Berlin , combined with a silver plaque. Mathematicians born at the beginning of the 20th century such as Kolmogorow , Cartan , Weil , Leray , Pontryagin , Chern and Whitney were also excluded due to the age restriction, since the award was not awarded between 1936 and 1950.

Comparison with Nobel Prize

The Fields Medal is often viewed as an equal replacement for a nonexistent Nobel Prize in Mathematics because of its longstanding prestige . With the Abel Prize , founded in 2002, there is a newer counterpart, which is more similar to the Nobel Prizes due to the lack of an age limit, the annual award, the considerably higher prize money and the Scandinavian selection committee.

The medal


The medal minted by the Royal Canadian Mint is made of 14-carat gold and has a diameter of 63.5 mm. The design was created in 1933 by the Canadian sculptor Robert Tait McKenzie (1867-1938). On the front the head of Archimedes is depicted, next to it are the inscription ΑΡΧΙΜΗΔΟΥΣ (Greek 'from Archimedes'), the ancient motto TRANSIRE SVVM PECTVS MVNDOQVE POTIRI (Latin “Transcend your own mind and take over the world”) and the initials RTM by the artist with the misspelled Roman numeral MCNXXXIII for 1933 (correct would be MCMXXXIII). The reverse bears the inscription CONGREGATI / EX TOTO ORBE / MATHEMATICI / OB SCRIPTA INSIGNIA / TRIBVERE (Latin: "The mathematicians who came together from all over the world awarded [the medal] for excellent writing"), behind which is a laurel branch in front of a diagram of a cylinder inscribed Sphere that is said to have been engraved on Archimedes' tombstone. The name of the winner is embossed on the edge.


Tao, Werner, Okunkow at the awarding of the Fields Medal in Madrid (2006)

The mathematician John Charles Fields was President of the Organizing Committee of the ICM in 1924 in Toronto , Canada . The committee had a surplus of about 2,700 Canadian dollars when the planning was completed and decided to use 2,500 of that to honor two distinguished mathematicians at an upcoming congress. When Fields died in 1932, he bequeathed the proposed foundation $ 47,000 Canadian dollars. The medal became known by his name, contrary to his express wish that it should be international and impersonal and therefore not associated with any name. The prize money was initially 1,500 Canadian dollars and rose to 3,000 in 1983, to 6,000 in 1986 and to 15,000 in 1990. Fields was less committed to the criteria and gave the committee a lot of freedom: The prize was to be awarded as recognition of work already done and as an encouragement for further achievement . It was important to Fields to avoid international rivalries that overshadowed the International Congress of Mathematicians at the time.

The first two Fields medals were awarded in 1936, the first selection committee included Birkhoff , Carathéodory , Cartan , Severi and Takagi . An anonymous foundation has made it possible since 1966 to award the Fields Medal to up to four mathematicians. In 1990 Edward Witten was the first and so far only physicist to receive the award. In 2014 the first and so far only woman, Maryam Mirzakhani , received the award. She died of cancer in 2017 at the age of 40.

The criteria changed over time. Initially, the medals were not so much awarded to the most important mathematicians, but rather to little recognized ones, whose potential was considered to be the highest. In 1950, it was not André Weil who received the medal, but Laurent Schwartz (relatively unknown compared to Weil, but favored by the committee chairman Harald Bohr ). Friedrich Hirzebruch did not receive the medal in 1958 mainly because, in the opinion of the committee chairman Heinz Hopf, he was already established. It was only at the ICM 1966 that, following a proposal by Georges de Rham , an agreement was reached on an age limit of 40 years, as this came closest to the age distribution of those who had been awarded in the year of the award.

The mathematician Grigori Perelman , an expert in the field of the Ricci River , was to receive the award in 2006 for his proof of the Poincaré conjecture published in 2002, but was the only one so far to reject the award.

By 2018, a total of 59 mathematicians had been awarded the Fields Medal in 19 awards. Seven medals were awarded two, three awards three and nine awards four medals each. Jesse Douglas , who was honored at the first award ceremony , was the first Fields Medalist to die. Since the death of Klaus Friedrich Roth in November 2015, the now 93-year-old Jean-Pierre Serre has been the oldest still living bearer. This makes him older than any Fields medal holder who has since died. Atle Selberg holds the record of the maximum age at death with 90 years and 53 days, closely followed by Klaus Friedrich Roth with 90 years and 12 days. The award winner Maryam Mirzakhani died the earliest, 72 days after her 40th birthday. The other 17 award winners who had already passed away were at least 52 years old.

Jean-Pierre Serre , who received the award in 1954 at the age of 27, is the youngest recipient at the time of the award. The youngest wearer is 32-year-old Peter Scholze , followed by 36-year-old Alessio Figalli . In addition to these two, there are only two other award winners who are still under 40. Six medalists received their medals in the year they turned 40, so they maxed out. 28, almost half, got the medal in a year in which they were older than 36. It was the last possible time for them to be awarded a Fields Medal.

Award winners

year Loan location Award winners birth
Reason for the award (area),
special features
1936 Oslo Lars V. Ahlfors (Finland) 1907 1996 Methods for researching the Riemann surfaces of the functions inverse to whole and meromorphic functions ( function theory )
Jesse Douglas (USA) 1897 1965 Work on the plateau problem ( calculus of variations , theory of minimal areas ). Was at the presentation of Wiener Norbert represented
1950 Cambridge, (USA) Laurent Schwartz (France) 1915 2002 Development of the theory of distributions ( functional analysis )
Atle Selberg (Norway) 1917 2007 Generalization of the sieving methods by Viggo Brun , results for the zeros of the Riemann function and, in parallel to Paul Erdős , elementary proof and generalization of the prime number theorem ( number theory )
1954 Amsterdam Kunihiko Kodaira (Japan) 1915 1997 Results in the theory of harmonic integrals , numerous applications to Kahler manifolds and especially algebraic varieties and proof by means of sheaf cohomology that these are Hodge manifolds ( algebraic topology , Hodge theory )
Jean-Pierre Serre (France) 1926 Results on the homotopy groups of spheres using spectral sequences , reformulation and expansion of results from function theory with the term sheaf ( algebraic topology , algebraic geometry )
1958 Edinburgh Klaus Friedrich Roth (UK) 1925 2015 Proof of the Thue-Siegel-Roth theorem and a conjecture by Erdős and Turán that every sequence of natural numbers with a density greater than zero contains three elements in arithmetic progression ( number theory )
René Thom (France) 1923 2002 Development of the theory of cobordisms for the classification of manifolds using homotopy theory , example of a general cohomology theory ( algebraic topology )
1962 Stockholm Lars Hörmander (Sweden) 1931 2012 Work on partial differential equations , especially contributions to the general theory of linear and hypoelliptic differential operators (theory of differential operators )
John Milnor (USA) 1931 Proof that a seven-dimensional sphere can carry different differentiable structures , thereby opening up the research area of differential topology ( topology , differential geometry )
1966 Moscow Michael Atiyah (UK) 1929 2019 With Hirzebruch work on the K-theory , with Singer proof of the Atiyah-Singer index theorem , with Bott proof of the Atiyah-Bott fixed point theorem ( algebraic topology , differential geometry )
Paul Cohen (USA) 1934 2007 Proof of the independence of the axiom of choice and the generalized continuum hypothesis from Zermelo-Fraenkel set theory with the help of the forcing technique, thus solving Hilbert's first problem ( mathematical logic )
Alexander Grothendieck (France) 1928 2014 Introduction of schemes for the further abstraction of sheaves , spectral sequences and other things, idea of ​​the K-theory , innovations in homological algebra ( algebraic geometry , category theory ). Did not show up for the award for political reasons
Stephen Smale (USA) 1930 Proof of the Poincaré conjecture for dimensions n  ≥ 5: every n -dimensional closed manifold that is homotopy-equivalent to the n -dimensional sphere is homeomorphic to this , contributions to the theory of dynamic systems ( topology )
1970 Nice Alan Baker (UK) 1939 2018 Work on Diophantine equations , generalization of Gelfond-Schneider's theorem , thereby proving further numbers as transcendent ( number theory )
Heisuke Hironaka (Japan) 1931 Generalization of a result by Zariski for the resolution of singularities of algebraic varieties for dimensions less than or equal to three to any dimensions ( algebraic geometry )
Sergei Novikov (USSR) 1938 Proof of the topological invariance of the rational Pontryagin classes of differentiable manifolds , investigations on the cohomology and homotopy of Thom spaces ( algebraic topology ). Was not allowed to attend the award ceremony in Nice
John G. Thompson (USA) 1932 With Feit proof of the Feit-Thompson theorem that every group of odd order is solvable , and classification of the finite simple groups whose real subgroups are solvable ( group theory )
1974 Vancouver Enrico Bombieri (Italy) 1940 Work on the distribution of prime numbers in arithmetic sequences , on simple functions , the local Bieberbach conjecture , functions of several complex variables, partial differential equations and Bernstein's problem over minimal areas in higher dimensions ( number theory , function theory )
David Mumford (UK) 1937 Contributions to the question of the existence and structure of module varieties , varieties whose points parameterize the isomorphism classes of a type of geometric objects, and work on algebraic surfaces ( algebraic geometry )
1978 Helsinki Pierre Deligne (Belgium) 1944 Proof of three conjectures by Weil on generalizations of the Riemann conjecture to finite fields , contribution to the union of algebraic geometry and algebraic number theory ( algebraic geometry , algebraic number theory )
Charles Fefferman (USA) 1949 Contribution to function theory in higher dimensions by discovering the correct generalizations of classical results in lower dimensions ( function theory )
Grigori Margulis (USSR) 1946 Research into the structure of Lie groups , especially the discrete subgroups with finite covolume (lattice), and others ( combinatorics , differential geometry , ergodic theory , dynamic systems , Lie theory ). Was not allowed to travel to Helsinki for the awards ceremony
Daniel Quillen (USA) 1940 2011 Construction of the higher algebraic K-theory , with whose geometrical and topological methods problems in algebra , especially ring and module theory , can be formulated and solved, parallel to Suslin's proof of the Quillen-Suslin theorem ( K-theory , abstract algebra )
1982 (1983) Warsaw Alain Connes (France) 1947 Contributions to the theory of operator algebras , especially the classification of factors of type III , the automorphisms of the hyperfinite factor and the injective factors as well as the application of C * -algebras to scrolls and differential geometry , cyclic cohomology ( functional analysis , differential geometry )
William Thurston (USA) 1946 2012 New methods in two- and three-dimensional topology that show the interplay between analysis, topology and geometry, and the idea that many closed manifolds have a hyperbolic structure, Thurston hypothesis ( topology , differential geometry )
Shing-Tung Yau (China, USA since 1990) 1949 Contributions to differential equations , to the Calabi conjecture in algebraic geometry , with Schoen's proof of the positive energy theorem in general relativity , work on the real and complex Monge-Ampère equations ( algebraic geometry , mathematical physics )
1986 Berkeley Simon Donaldson (UK) 1957 Work on the topology of four-dimensional manifolds , especially the proof that different differential structures exist for four-dimensional Euclidean space , Donaldson invariants ( differential topology )
Gerd Faltings (Federal Republic of Germany) 1954 Proof of Mordell's conjecture that there are only finitely many rational points on an algebraic curve with gender greater than one ( algebraic geometry , number theory )
Michael Freedman (USA) 1951 New methods for the topological investigation of four-dimensional manifolds , especially the proof of the Poincaré conjecture in four dimensions and the classification of compact, simply connected four-dimensional manifolds ( topology )
1990 Kyoto Vladimir Drinfeld (USSR) 1954 Contributions to the Langlands program , discovery of quantum groups , deformations of Lie groups abstracted to Hopf algebras similar to the deformation of classical mechanics to quantum mechanics ( number theory , theory of algebraic groups , Lie theory )
Vaughan FR Jones (USA) 1952 Discovery of new node invariants in the investigation of certain Von Neumann algebras including proof of an index theorem ( topology , theory of operator algebras )
Shigefumi Mori (Japan) 1951 Proof of the Hartshorne conjecture , work on the classification of three-dimensional algebraic varieties ( algebraic geometry )
Edward Witten (USA) 1951 Easier proof of the positive energy theorem in general relativity with the help of supersymmetry , connection of supersymmetry with Morse theory , discovery of topological quantum field theories ( mathematical physics )
1994 Zurich Jean Bourgain (Belgium) 1954 2018 Contributions to the geometry of Banach spaces , convexity in high-dimensional spaces, harmonic analysis , ergodic theory and theory of nonlinear evolution equations ( functional analysis , theory of nonlinear partial differential equations )
Pierre-Louis Lions (France) 1956 With Crandall development of the viscosity method , work on the Boltzmann equation and on variation problems (theory of nonlinear partial differential equations )
Jean-Christophe Yoccoz (France) 1957 2016 Contributions to the problem of the small denominator from celestial mechanics with a solution in a special case (theory of dynamic systems )
Efim Zelmanov (Russia) 1955 Solution of the restricted Burnside problem , before that contributions to the theory of Lie algebras and Jordan algebras ( group theory , Lie theory , commutative algebra )
1998 Berlin Richard Borcherds (UK) 1959 Introduction of vertex algebras , proof of the moonshine conjecture about a relation of the monster group to the j-function and discovery of a new class of automorphic infinite products ( algebra , theory of automorphic forms , mathematical physics )
Timothy Gowers (UK) 1963 Contributions to the theory of Banach spaces , simpler proof of a Szemerédi theorem ( functional analysis , combinatorics )
Maxim Konzewitsch (Russia) 1964 Section theory on the module space of algebraic curves , construction of node invariants and a quantization of Poisson manifolds , method for counting rational algebraic curves ( mathematical physics , algebraic geometry , topology )
Curtis McMullen (USA) 1958 Clarification of a question about the iterative approximate solution of polynomial equations , work on the Mandelbrot set and the Julia sets , contribution to Thurston's program to introduce hyperbolic structures on three-dimensional manifolds ( complex dynamics , hyperbolic geometry )
2002 Beijing Laurent Lafforgue (France) 1966 Contributions to the Langlands program ( number theory )
Vladimir Voivodsky (Russia) 1966 2017 Proof of the Milnor conjecture , new cohomology theories for algebraic varieties ( K-theory , algebraic geometry , topology )
2006 Madrid Andrei Okunkow (Russia) 1969 Articles combining probability theory , representation theory and algebraic geometry
Grigori Perelman (Russia) 1966 Insights into the analytical and geometric structure of the Ricci River , from which the proof of the geometrization conjecture , which was still under review at that time, results, from which the Poincaré conjecture follows ( differential geometry , topology ). Didn't accept the award.
Terence Tao (Australia) 1975 Contributions to partial differential equations , combinatorics , Fourier analysis and additive number theory
Wendelin Werner (France) 1968 Contributions to the Schramm-Loewner development , the geometry of the two-dimensional Brownian motion and the conformal field theory
2010 Hyderabad Elon Lindenstrauss (Israel) 1970 Results on dimensional rigidity in ergodic theory and its applications in number theory
Ngô Bảo Châu (Vietnam, France) 1972 Proof of the fundamental lemma in the Langlands program by developing new algebro-geometric methods
Stanislav Smirnov (Russia) 1970 Proof of the conformal invariance of percolation theory as well as the planar Ising model in statistical physics
Cédric Villani (France) 1973 Proof of the nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation
2014 Seoul Artur Ávila (Brazil, France) 1979 Fundamental contributions to dynamic systems with the renormalization group as a unifying principle
Manjul Bhargava (Canada) 1974 Contributions to number theory , development of powerful new methods in the geometry of numbers, for example in a new interpretation and extension of the composition laws of quadratic forms of Gauss and bounds for the averaged rank of elliptic curves
Martin Hairer (Austria) 1975 Contributions to stochastic partial differential equations and especially the development of a regularity structure for them
Maryam Mirzakhani (Iran) 1977 2017 Contributions to (hyperbolic) geometry in connection with modular spaces of Riemann surfaces ( Teichmüller spaces ) and their dynamics
2018 Rio de Janeiro Caucher Birkar (UK, Iran) 1978 Proof of the limitedness of Fano varieties and contributions to the program of minimal models initiated by Shigefumi Mori in the birational classification of algebraic varieties in more than three dimensions
Alessio Figalli (Italy) 1984 Contributions to the theory of optimal transport and its application to partial differential equations , probability theory and metric geometry
Peter Scholze (Germany) 1987 Introduction perfektoider rooms to treat arithmetic- algebraic geometry over p-adic bodies with applications to Galois representations and for developing new cohomology theories
Akshay Venkatesh (Australia, India) 1981 Synthesis of analytical number theory , homogeneous dynamics, topology and representation theory and the resultant solution of long open conjectures about the equal distribution of number theoretic objects


Caucher Birkar , one of the 2018 award winners, had the medal stolen shortly after it was awarded, but it was replaced.

Prize Committee

The award committees usually consist of nine mathematicians who switch from ICM to ICM, with only the chairman of the current committee being announced before the award ceremony. The chairman is usually the president of the IMU and the committee members are determined by the executive committee of the IMU. Members of the committee were:


  • Henry S. Tropp: The Origins and History of the Fields Medal. Historia Mathematica 3, May 1976, pp. 167-181 (English).
  • Michael Atiyah , Daniel Iagolnitzer (Eds.): Fields medallists' lectures. World Scientific / Singapore University Press, Singapore 1997, ISBN 981-02-3102-4 (English, French).
  • Michail Monastyrski: Modern mathematics in the light of the Fields medals. AK Peters, Wellesley 1998, ISBN 1-56881-065-2 (English).
  • Carl Riehm: The Early History of the Fields Medal . ( PDF ; 373 kB), Notices of the AMS 49, August 2002, pp. 778–782 (English).
  • EM Riehm, F. Hoffman: Turbulent Times in Mathematics: The Life of JC Fields and the History of the Fields Medal. American Mathematical Society & Fields Institute, 2011.
  • Guillermo P. Curbera: Interlude. Awards of the ICM. In: Mathematicians of the world, unite! AK Peters, Wellesley 2009, ISBN 978-1-56881-330-1 , pp. 109-123 (English).
  • Elaine McKinnon Riehm: The Fields Medal: Serendipity and JL Synge. (PDF; 2.3 MB), Fields Notes 10, May 2010, pp. 1–2 (English).

Web links

Commons : Fields Medal  - Collection of images, videos and audio files
 Wikinews: Fields Medal  - in the news

Individual evidence

  1. a b Michael Monastyrsky: Some Trends in Modern Mathematics and the Fields Medal ( PDF file, 97 kB), CMS Notes 33, March 2001, pp. 3–5, and April 2001, pp. 11–13 (English).
  2. Physical Medal , description of the material facts (English), accessed on August 1, 2018.
  3. Marcus Manilius : M. Manilii astronomicon liber quartus. Line 392, 1st century AD (Latin).
  4. Michael Barany: The Fields Medal should return to its roots. In: Nature . Volume 553, 2018, pp. 271-273.
  5. Léon Motchane , President of IHES , where Grothendieck was, received them for him.
  6. World's most prestigious maths medal is stolen minutes after professor wins it. Article in The Guardian on August 1, 2018, accessed August 3, 2018.
  7. Top math laureate gets new medal after prize stolen. ( Memento of August 3, 2018 in the Internet Archive ). In: August 3, 2018, accessed August 3, 2018.
  8. ^ Fields Medal - Former Prize Committees. In: International Mathematical Union, accessed August 1, 2018 .