The Fields Medal, officially known as the International Medal for Outstanding Discoveries in Mathematics, is an award granted to two, three, or four mathematicians under 40 years of age every four years at the International Congress of the International Mathematical Union (IMU) to recognize outstanding mathematical achievement for existing work and the promise of future achievement.
The award’s name honors the renowned Canadian mathematician John Charles Fields (1863–1932).
The Executive Committee of the International Mathematical Union chooses the Fields Medal Committee, which the IMU President typically chairs. It then needs to select at least two, with a strong preference for four, Fields Medalists and have regard for representing a diversity of mathematical fields. A candidate’s 40th birthday must not fall before the 1st of January of the year of the Congress at which they are awarded the Fields Medal.
Although the Fields Medal is often referred to as the mathematical equivalent of the Nobel Prize, there are several significant differences between the two, including the award criteria, frequency of the award, number of awards, age limits, and monetary value.
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History of Fields Medal
The Fields Medal originated from surplus funds that John Charles Fields, a professor of mathematics at the University of Toronto, raised as the organizer and president of the 1924 International Congress of Mathematicians in Toronto. After printing the conference proceedings, the Committee of the International Congress had US Dollars 2,700 left and voted to put aside 2,500 US Dollars for establishing two medals that would be awarded later.
In contrast to Fields’s explicit request, the proposed awards came to be known as the Fields Medals after an endowment from his estate. The first two Fields Medals were awarded in 1936, and an anonymous donation increased the number of prize medals starting in 1966. Medalists currently also receive a small cash award of USD 1,500. Since 1982, a related award, the Rolf Nevanlinna Prize, has also been presented at every International Congress of Mathematicians. It is awarded to a young mathematician for notable work related to the mathematical aspects of information science.
The International Mathematical Union’s executive committee appoints Fields Medal and Nevanlinna Prize committees, which can suggest candidates to the secretary of the International Mathematical Union in writing. Since 1936, the medals have been presented at every International Congress of Mathematicians. Except for two Ph.D. holders in physics (Edward Witten and Martin Hairer), only individuals with a Ph.D. in mathematics have won the medal.
Also see: Abel Prize Winners
Fields Medal Award Winners in 2022
- Hugo Duminil-Copin: For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four.
- June Huh: 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: 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: For the proof that the $E_8$ lattice provides the densest packing of identical spheres in 8 dimensions and further contributions to related extremal problems and interpolation problems in Fourier analysis.
All Fields Medal Winners
|2022||Hugo Duminil-Copin||For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four.|
|2022||June Huh||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.|
|2022||James Maynard||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.|
|2022||Maryna Viazovska||For the proof that the $E_8$ lattice provides the densest packing of identical spheres in 8 dimensions and further contributions to related extremal problems and interpolation problems in Fourier analysis.|
|2018||Caucher Birkar||For the proof of the boundedness of Fano varieties and for contributions to the minimal model program.|
|2018||Alessio Figalli||For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability.|
|2018||Peter Scholze||For having transformed arithmetic algebraic geometry over p-adic fields.|
|2018||Akshay Venkatesh||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.|
|2014||Artur Avila||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.|
|2014||Manjul Bhargava||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.|
|2014||Martin Hairer||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.|
|2014||Maryam Mirzakhani||For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.|
|2010||Elon Lindenstrauss||For his results on measure rigidity in ergodic theory, and their applications to number theory.|
|2010||Ngô Bảo Châu||For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebra-geometric methods.|
|2010||Stanislav Smirnov||For the proof of conformal invariance of percolation and the planar Ising model in statistical physics.|
|2010||Cédric Villani||For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.|
|2006||Andrei Okounkov||For his contributions bridging probability, representation theory and algebraic geometry.|
|2006||Grigori Perelman (declined)||For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.|
|2006||Terence Tao||For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.|
|2006||Wendelin Werner||For his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.|
|2002||Laurent Lafforgue||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.|
|2002||Vladimir Voevodsky||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.|
|1998||Richard Borcherds||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.|
|1998||Timothy Gowers||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.|
|1998||Maxim Kontsevich||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.|
|1998||Curtis T. McMullen||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.|
|1994||Jean Bourgain||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.|
|1994||Pierre-Louis Lions||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.|
|1994||Jean-Christophe Yoccoz||Yoccoz obtained a very enlightening proof of Bruno’s theorem, and he was able to prove the converse|
|1994||Efim Zelmanov||For the solution of the restricted Burnside problem.|
|1990||Vladimir Drinfeld||Drinfeld’s main preoccupation in the last decade|
|1990||Vaughan Jones||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.|
|1990||Shigefumi Mori||The most profound and exciting development in algebraic geometry during the last decade or so was|
|1990||Edward Witten||Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems.|
|1986||Simon Donaldson||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.|
|1986||Gerd Faltings||Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture.|
|1986||Michael Freedman||Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincaré Conjecture.|
|1982||Alain Connes||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.|
|1982||William Thurston||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.|
|1982||Shing-Tung Yau||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.|
|1978||Pierre Deligne||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.|
|1978||Charles Fefferman||Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results.|
|1978||Grigory Margulis||Provided innovative analysis of the structure of Lie groups. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups.|
|1978||Daniel Quillen||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.|
|1974||Enrico Bombieri||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.|
|1974||David Mumford||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.|
|1970||Alan Baker||Generalized the Gelfond-Schneider theorem (the solution to Hilbert’s seventh problem). From this work he generated transcendental numbers not previously identified.|
|1970||Heisuke Hironaka||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.|
|1970||Sergei Novikov||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.|
|1970||John G. Thompson||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.|
|1966||Michael Atiyah||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’.|
|1966||Paul Cohen||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.|
|1966||Alexander Grothendieck||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’.|
|1966||Stephen Smale||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.|
|1962||Lars Hörmander||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.|
|1962||John Milnor||Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology.|
|1958||Klaus Roth||for solving a famous problem of number theory, namely, the determination of the exact exponent in the Thue-Siegel inequality|
|1958||René Thom||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.|
|1954||Kunihiko Kodaira||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.|
|1954||Jean-Pierre Serre||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.|
|1950||Laurent Schwartz||Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics.|
|1950||Atle Selberg||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.|
|1936||Lars Ahlfors||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.|
|1936||Jesse Douglas||Did important work on the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary.|
- 1936 – The Fields Medal was first awarded to the American mathematician Jesse Douglas and the Finnish mathematician Lars Ahlfors. Since 1950, it has been awarded every four years to honor younger mathematical researchers.
- 1954 – The 27-year-old Jean-Pierre Serre became the youngest winner of the Fields Medal. He retains that distinction to this day.
- 1966 – Alexander Grothendieck boycotted the International Congress of Mathematicians (held in Moscow) to protest Soviet military actions in Eastern Europe. Therefore, Léon Motchane, the founder and director of the Institut des Hautes Études Scientifiques, attended and accepted Grothendieck’s Fields Medal on his behalf.
- 1970 – Sergei Novikov could not travel to the Congress in Nice to receive his Fields Medal because of restrictions the Soviet government put on him.
- 1978 – Grigory Margulis could not travel to the Congress in Helsinki and receive his Fields Medal due to similar restrictions the Soviet government placed on him. Jacques Tits accepted the award was accepted on his behalf.
- 1982 – The Congress was planned to be held in Warsaw but had to be rescheduled to the following year because of the introduction of martial law in Poland on the 13th of December, 1981. Earlier that year, the awards were announced at the ninth General Assembly of the IMU and awarded at the 1983 Warsaw Congress.
- 1990 – Edward Witten became the first physicist to win the Fields Medal.
- 1998 – At the International Congress of Mathematicians, the chair of the Fields Medal Committee, Yuri I. Manin, presented Andrew Wiles with the first-ever IMU silver plaque to honor his proof of Fermat’s Last Theorem. Don Zagier called this plaque a “quantized Fields Medal.” Although Wiles was slightly over the age limit for the Fields Medal then, many felt he was a favorite to win the medal.
- 2006 – Grigori Perelman, who proved the Poincaré conjecture, rejected his Fields Medal and chose not to attend the Congress.
- 2014 – Maryam Mirzakhani became the first Iranian and the first woman to win the Fields Medal. Also, Manjul Bhargava became the first person of Indian origin, and Artur Avila became the first South American to do so.
- 2022 – Maryna Viazovska became the first Ukrainian to win the Fields Medal, and June Huh became the first person of Korean origin to do so.
Terms for Winning the Fields Medal
Unlike the Nobel Prize, the Fields Medal is only awarded every four years. Also, there is an age limit – the recipient must be under 40 on the 1st of January of the year the medal is awarded. It is based on Fields’s wish 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.”
Also, a person can only receive one Fields Medal in their lifetime; winners are ineligible to be awarded medals in the future.
The Medal of The Fields Medal
Photo Source: By Stefan Zachow for the International Mathematical Union; retouched by King of Hearts – Public Domain
Canadian sculptor R. Tait McKenzie designed the Fields Medal. It is made of 14kt gold, weighs 169 grams, and has a diameter of 63.5mm.
Its obverse features Archimedes (287-212 BC) and a quote attributed to 1st century AD poet Manilius, reading the following in Latin: “Transire suum pectus mundoque potiri” (“Rise above oneself and grasp the world”).
It also has the year number 1933 written in Roman numerals and contains an error (“MCNXXXIII” instead of “MCMXXXIII”).
ΑΡXIMHΔΟΥΣ, or “of Archimedes,” is present in Greek capital letters.
On the medal’s reverse is the following Latin inscription:
EX TOTO ORBE
OB SCRIPTA INSIGNIA
It translates to: “Mathematicians gathered from the entire world have awarded (this prize) for outstanding writings.“
The background represents Archimedes’ tomb, with the carving illustrating his theorem On the Sphere and Cylinder behind an olive branch.
The medal’s rim bears the name of the awardee.
The Fields Medal indicates current fertile areas of mathematical research because the winners have usually made contributions that opened up entire fields or integrated technical ideas and tools from various disciplines.
Many winners worked in highly abstract and integrative fields such as algebraic topology and geometry. To some extent, this trend reflects the influence and power of the French consortium of mathematicians, writing under the name of Nicolas Bourbaki since 1939. In their multivolume Éléments de mathématiques, they sought a rigorous, modern, and comprehensive treatment of all mathematics and mathematical foundations.
Nevertheless, individuals have also received medals for their work in more classical mathematics and mathematical physics fields, including solutions to problems that David Hilbert enunciated at the International Congress of Mathematicians in Paris in 1900.
Interestingly, there are large clusters of Fields Medalists within certain research institutions. In fact, almost half of the medalists have held appointments at the Institute for Advanced Study, Princeton, N.J., U.S.
How often is the Fields Medal awarded?
The Fields Medal is awarded every four years on the occasion of the International Congress of Mathematicians for outstanding achievements in mathematics.
What is the age limit to receive the Fields Medal?
The Fields Medal is awarded to mathematicians who are not over 40 years of age.
What is the Fields Medal made of?
The Fields Medal is made of gold and displays the head of Archimedes along with a quotation attributed to him – “Transire suum pectus mundoque potiri” (“Rise above oneself and grasp the world”).
Which country has the most Fields Medals?
Currently, the US has the most Fields Medals (15), followed by the UK (8).
Have any Indians won the Fields Medal?
So far, there are two awardees of Indian origin — Akshay Venkatesh of the Institute for Advanced Study at Princeton (2018) and Manjul Bhargava of the Department of Mathematics at Princeton University (2014).