Fields Medal

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Not to be confused with Field's metal. Award

Fields Medal
The obverse of the Fields Medal
Description	Outstanding contributions in mathematics attributed to young scientists
Country	Varies
Presented by	International Mathematical Union (IMU)
Reward(s)	CA$15,000
First awarded	1936; 88 years ago (1936)
Last awarded	2018 (2018)
Website	Mathunion.org

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 regarded as one of the highest honors a mathematician can receive, and has been described as the mathematician's Nobel Prize, 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, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics.

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

The medal was first awarded in 1936 to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and it has been awarded every four years since 1950. Its purpose is to give recognition and support to younger mathematical researchers who have made major contributions. In 2014, the Iranian mathematician Maryam Mirzakhani became the first female Fields Medalist. In 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. The medal belonging to one of the four joint winners, Caucher Birkar, was stolen shortly after the event. The ICM presented Birkar with a replacement medal a few days later.

Conditions of the award

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

First awarded in 1936, 60 people have won the medal as of 2018. With the exception of one Ph.D. holder in physics (Edward Witten), only people with a Ph.D. in mathematics have won the medal.

Fields medalists

Year	ICM location	Medalists	Affiliation (when awarded)	Affiliation (current/last)	Reasons
1936	Oslo, Norway	Lars Ahlfors	University of Helsinki, Finland	Harvard University, US	"Awarded medal for research on covering surfaces related to Riemann surfaces of inverse functions of entire and meromorphic functions. Opened up new fields of analysis."
1936	Oslo, Norway	Jesse Douglas	Massachusetts Institute of Technology, US	City College of New York, US	"Did important work on the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary."
1950	Cambridge, US	Laurent Schwartz	University of Nancy, France	University of Paris VII, France	"Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics."
1950	Cambridge, US	Atle Selberg	Institute for Advanced Study, US	Institute for Advanced Study, US	"Developed generalizations of the sieve methods of Viggo Brun; achieved major results on zeros of the Riemann zeta function; gave an elementary proof of the prime number theorem (with P. Erdős), with a generalization to prime numbers in an arbitrary arithmetic progression."
1954	Amsterdam, Netherlands	Kunihiko Kodaira	Princeton University, US, University of Tokyo, Japan and Institute for Advanced Study, US	University of Tokyo, Japan	"Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically to algebraic varieties. He demonstrated, by sheaf cohomology, that such varieties are Hodge manifolds."
1954	Amsterdam, Netherlands	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 Erdős and Turán of 1935)."
1958	Edinburgh, UK	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."
1962	Stockholm, Sweden	John Milnor	Princeton University, US	Stony Brook University, US	"Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology."
1966	Moscow, USSR	Michael Atiyah	University of Oxford, UK	University of Edinburgh, UK	"Did joint work with Hirzebruch in K-theory; proved jointly with Singer the index theorem of elliptic operators on complex manifolds; worked in collaboration with Bott to prove a fixed point theorem related to the 'Lefschetz formula'."
		Paul Cohen	Stanford University, US	Stanford University, US	"Used technique called "forcing" to prove the independence in set theory of the axiom of choice and of the generalized continuum hypothesis. The latter problem was the first of Hilbert's problems of the 1900 Congress."
		Alexander Grothendieck	Institut des Hautes Études Scientifiques, France	Centre National de la Recherche Scientifique, France	"Built on work of Weil and Zariski and effected fundamental advances in algebraic geometry. He introduced the idea of K-theory (the Grothendieck groups and rings). Revolutionized homological algebra in his celebrated ‘Tôhoku paper’."
		Stephen Smale	University of California, Berkeley, US	City University of Hong Kong, Hong Kong	"Worked in differential topology where he proved the generalized Poincaré conjecture in dimension n≥5: Every closed, n-dimensional manifold homotopy-equivalent to the n-dimensional sphere is homeomorphic to it. Introduced the method of handle-bodies to solve this and related problems."
1970	Nice, France	Alan Baker	University of Cambridge, UK	Trinity College, Cambridge, UK	"Generalized the Gelfond-Schneider theorem (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified."
		Heisuke Hironaka	Harvard University, US	Kyoto University, Japan	"Generalized work of Zariski who had proved for dimension ≤ 3 the theorem concerning the resolution of singularities on an algebraic variety. Hironaka proved the results in any dimension."
		Sergei Novikov	Moscow State University, USSR	Steklov Mathematical Institute, Russia Moscow State University, Russia University of Maryland-College Park, US	"Made important advances in topology, the most well-known being his proof of the topological invariance of the Pontryagin classes of the differentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces."
		John G. Thompson	University of Cambridge, UK	University of Cambridge, UK University of Florida, US	"Proved jointly with W. Feit that all non-cyclic finite simple groups have even order. The extension of this work by Thompson determined the minimal simple finite groups, that is, the simple finite groups whose proper subgroups are solvable."
1974	Vancouver, Canada	Enrico Bombieri	University of Pisa, Italy	Institute for Advanced Study, US	"Major contributions in the primes, in univalent functions and the local Bieberbach conjecture, in theory of functions of several complex variables, and in theory of partial differential equations and minimal surfaces – in particular, to the solution of Bernstein's problem in higher dimensions."
1974	Vancouver, Canada	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."
		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."
		Daniel Quillen	Massachusetts Institute of Technology, US	University of Oxford, UK	"The prime architect of the higher algebraic K-theory, a new tool that successfully employed geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory."
1982	Warsaw, Poland	Alain Connes	Institut des Hautes Études Scientifiques, France	Institut des Hautes Études Scientifiques, France Collège de France, France Ohio State University, US	"Contributed to the theory of operator algebras, particularly the general classification and structure theorem of factors of type III, classification of automorphisms of the hyperfinite factor, classification of injective factors, and applications of the theory of C*-algebras to foliations and differential geometry in general."
		William Thurston	Princeton University, US	Cornell University, US	"Revolutionized study of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry. Contributed idea that a very large class of closed 3-manifolds carry a hyperbolic structure."
		Shing-Tung Yau	Institute for Advanced Study, US	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	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	" Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems."
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	"... 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 Wisconsin-Madison University of Chicago, 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."
2002	Beijing, China	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 Institute for Advanced Study, US	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 Zurich, Switzerland Instituto Nacional de Matemática Pura e Aplicada, Brazil	"For his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle."
		Manjul Bhargava	Princeton University, US	Princeton University, US	"For developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves."
		Martin Hairer	University of Warwick, UK	Imperial College London, UK	"For his outstanding contributions to the theory of stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations."
		Maryam Mirzakhani	Stanford University, US	Stanford University, US	"For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."
2018	Rio de Janeiro, Brazil	Caucher Birkar	University of Cambridge, UK	University of Cambridge, UK	"For the proof of the boundedness of Fano varieties and for contributions to the minimal model program."
		Alessio Figalli	Swiss Federal Institute of Technology Zurich, Switzerland	Swiss Federal Institute of Technology Zurich, Switzerland	"For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability."
		Peter Scholze	University of Bonn, Germany	University of Bonn, Germany	"For 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	Institute for Advanced Study, US	"For his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects."

Landmarks

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

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

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

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

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

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

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

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

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

In 2014, Maryam Mirzakhani became the first 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. President Rouhani congratulated Mirzakhani for this notable success.

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 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.

Notes

References

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