Fields MedalFrom Wikipedia, the free encyclopediaNot to be confused with Field's metal.Fields Medal

The obverse of the Fields Medal

Awarded forOutstanding contributions in mathematics attributed to young scientists

CountryVaries

Presented byInternational Mathematical Union (IMU)

RewardC$15,000

First awarded1936

Last awarded2014

Official websitewww.mathunion.org/general/prizes/fields/details

The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, 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 sometimes viewed as the highest honour a mathematician can receive.[1][2] The Fields Medal and the Abel Prize have often been described as the "mathematician's Nobel Prize" (but different at least for the age restriction).The prize comes with a monetary award, which since 2006 has been C$15,000 (in Canadian dollars).[3][4] The colloquial name is in honour of Canadian mathematician John Charles Fields.[5] Fields was instrumental in establishing the award, designing the medal itself, and funding the monetary component.[5]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 Maryam Mirzakhani became the first woman as well as the first Iranian, and Artur Avila became the first mathematician from Latin America to be awarded a Fields Medal.[6][7]Contents 1 Conditions of the award 2 Fields medalists 3 Landmarks 4 Fields Medals by Affiliation 5 The medal 6 See also 7 References 8 Further reading 9 External linksConditions of the awardThe Fields Medal is often described as the "Nobel Prize of Mathematics" and for a long time was regarded as the most prestigious award in the field of mathematics.[8] However, in contrast to the Nobel Prize, the Fields Medal is awarded only every four years. The Fields Medal also has an age limit: a recipient must be under age 40 until 1 January of the year in which the medal is awarded (similar 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."[9]The monetary award is much lower than the 8,000,000 Swedish kronor (roughly 1,400,000 Canadian dollars)[10] given with each Nobel prize as of 2014.[11] Other major awards in mathematics, such as the Abel Prize and the Chern Medal, have a larger monetary prizes, comparable to the Nobel.Fields medalistsYearICM LocationMedalists[12]Affiliation (When Awarded)Current/Last AffiliationCitation

1936Oslo, NorwayLars AhlforsUniversity of Helsinki, FinlandHarvard University, US[13][14]"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 DouglasMassachusetts Institute of Technology, USCity College of New York, US[15][16]"Did important work of the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary."

1950Cambridge, USLaurent SchwartzUniversity of Nancy, FranceUniversity of Paris VII, France[17][18]"Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics."

Atle SelbergInstitute for Advanced Study, USInstitute for Advanced Study, US[19]"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. Erds), with a generalization to prime numbers in an arbitrary arithmetic progression."

1954Amsterdam, NetherlandKunihiko KodairaUniversity of Tokyo[20] and Princeton University[20]University of Tokyo, Japan[21]"Achieved major results in the theory of harmonic integrals and numerous applications to Khlerian and more specifically to algebraic varieties. He demonstrated, by sheaf cohomology, that such varieties are Hodge manifolds."

Jean-Pierre SerreUniversity of Nancy, FranceCollge de France, France[22][23]"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."

1958Edinburgh, UKKlaus RothUniversity College London, UKImperial College London, UK[24]"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 Erds and Turn of 1935)."

Ren ThomUniversity of Strasbourg, FranceInstitut des Hautes tudes Scientifiques, France[25]"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."

1962Stockholm, SwedenLars HrmanderUniversity of Stockholm, SwedenLund University, Sweden[26]"Worked in partial diffential 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 MilnorPrinceton University, USStony Brook University, US[27]"Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology."

1966Moscow, USSRMichael AtiyahUniversity of Oxford, UKUniversity of Edinburgh, UK[28]"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 CohenStanford University, USStanford University, US[29]"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 GrothendieckInstitut des Hautes tudes Scientifiques, FranceCentre National de la Recherche Scientifique, France[30]"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 "Tohoku paper""

Stephen SmaleUniversity of California, Berkeley, USCity University of Hong Kong, Hong Kong[31]"Worked in differential topology where he proved the generalized Poincar 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."

1970Nice, FranceAlan BakerUniversity of Cambridge, UKTrinity College, Cambridge, UK[32]"Generalized the Gelfond-Schneider theorem (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified."

Heisuke HironakaHarvard University, USKyoto University, Japan[33][34]"Generalized work of Zariski who had proved for dimension"

John G. ThompsonUniversity of Cambridge, UKUniversity of Cambridge, UK University of Florida, US[35]"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 NovikovMoscow State University, USSRSteklov Mathematical Institute, Russia Moscow State University, RussiaUniversity of Maryland-College Park, US[36][37]"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."

1974Vancouver, CanadaEnrico BombieriUniversity of Pisa, ItalyInstitute for Advanced Study, US[38]"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 MumfordHarvard University, USBrown University, US[39]"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."

1978Helsinki, FinlandPierre DeligneInstitut des Hautes tudes Scientifiques, FranceInstitute for Advanced Study, US[40]"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 FeffermanPrinceton University, USPrinceton University, US[41]"Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results."

Daniel QuillenMassachusetts Institute of Technology, USUniversity of Oxford, UK[42]"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 MargulisMoscow State University, USSRYale University, US[43]"Provided innovative analysis of the structure of Lie groups. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups."

1982Warsaw, PolandAlain ConnesInstitut des Hautes tudes Scientifiques, FranceInstitut des Hautes tudes Scientifiques, France Collge de France, FranceOhio State University, US[44]"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 ThurstonPrinceton University, USCornell University, US[45]"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 YauInstitute for Advanced Study, USHarvard University, US[46]"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-Ampre equations."

1986Berkeley, USSimon DonaldsonUniversity of Oxford, UKImperial College London, UK[47]"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 FaltingsPrinceton University, USMax Planck Institute for Mathematics, Germany[48]"Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture."

Michael FreedmanUniversity of California, San Diego, USMicrosoft Station Q, US[49]"Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincar Conjecture."

1990Kyoto, JapanVladimir DrinfeldB Verkin Institute for Low Temperature Physics and Engineering, Ukraine,[50]University of Chicago, US[51]"For his work on quantum groups and for his work in number theory."

Vaughan F. R. JonesUniversity of California, Berkeley, USUniversity of California, Berkeley, US,[52] Vanderbilt University, US[53]"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 MoriKyoto University, JapanKyoto University, Japan[54]"for the proof of Hartshornes conjecture and his work on the classification of three-dimensional algebraic varieties."

Edward WittenInstitute for Advanced Study, USInstitute for Advanced Study, US[55]"proof in 1981 of the positive energy theorem in general relativity"[56]

1994Zurich, SwitzerlandJean BourgainInstitut des Hautes tudes Scientifiques, FranceInstitute for Advanced Study, France[57]"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 LionsUniversity of Paris 9, FranceCollge de France, France cole polytechnique, France[58]"... 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 YoccozParis-Sud 11 University, FranceCollge de France, France[59]"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 ZelmanovUniversity of California, San Diego, USSteklov Mathematical Institute, Russia, University of California, San Diego, US[60]"For his solution to the restricted Burnside problem."

1998Berlin, GermanyRichard BorcherdsUniversity of California, Berkeley, US University of Cambridge, UKUniversity of California, Berkeley, US[61]"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 GowersUniversity of Cambridge, UKUniversity of Cambridge, UK[62]"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 KontsevichInstitut des Hautes tudes Scientifiques, France Rutgers University, USInstitut des Hautes tudes Scientifiques, France Rutgers University, US[63]"contributions to four problems of geometry"

Curtis T. McMullenHarvard University, USHarvard University, US[64]"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."

2002Beijing, ChinaLaurent LafforgueInstitut des Hautes tudes Scientifiques, FranceInstitut des Hautes tudes Scientifiques, France[65]"Laurent Lafforgue has been awarded the Fields Medal for his proof of the Langlands correspondence for the full linear groups GLr (r1) over function fields."

Vladimir VoevodskyInstitute for Advanced Study, USInstitute for Advanced Study, US[66]" 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"

2006Madrid, SpainAndrei OkounkovPrinceton University, USColumbia University, US[67]"for his contributions bridging probability, representation theory and algebraic geometry"

Grigori PerelmanNoneSteklov Mathematical Institute, Russia[68]"for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow"

Terence TaoUniversity of California, Los Angeles, USUniversity of California, Los Angeles, US[69]"for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory "

Wendelin WernerParis-Sud 11 University, FranceETH Zurich, Switzerland[70]"for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory"

2010Hyderabad, IndiaElon LindenstraussHebrew University of Jerusalem, Israel Princeton University, USHebrew University of Jerusalem, Israel[71]"For his results on measure rigidity in ergodic theory, and their applications to number theory."

Ng Bo ChuParis-Sud 11 University, France Institute for Advanced Study, USParis-Sud 11 University, France University of Chicago, USVietnam Institute for Advanced Study, Vietnam[72]"For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods"

Stanislav SmirnovUniversity of Geneva, SwitzerlandUniversity of Geneva, Switzerland[73]"For the proof of conformal invariance of percolation and the planar Ising model in statistical physics"

Cdric Villanicole Normale Suprieure de Lyon, France Institut Henri Poincar, FranceLyon University, France Institut Henri Poincar, France[74]"For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation."

2014Seoul, South KoreaArtur AvilaUniversity of Paris VII, France CNRS, FranceInstituto Nacional de Matemtica Pura e Aplicada, BrazilUniversity of Paris VII, France CNRS, FranceInstituto Nacional de Matemtica Pura e Aplicada, Brazil[75]"is awarded a Fields Medal 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 BhargavaPrinceton University, USPrinceton University, US[76][77]"is awarded a Fields Medal 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 HairerUniversity of Warwick, UKUniversity of Warwick, UK[78][79][80][81]"is awarded a Fields Medal 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 MirzakhaniStanford University, USStanford University, US[82][83]"is awarded the Fields Medal for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."

2018Rio de Janeiro, Brazil[84]n/an/an/an/a

LandmarksIn 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.[85] Lon Motchane, founder and director of the Institut des Hautes tudes Scientifiques attended and accepted Grothendieck's Fields Medal on his behalf.[86]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."[87]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 13 Dec 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 and so far only 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.[88] 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.[89][90]In 2006, Grigori Perelman, who proved the Poincar conjecture, refused his Fields Medal[3] and did not attend the congress.[91]In 2014, Maryam Mirzakhani became the first woman and Artur Avila the first South American to win the Fields Medal, .Fields Medals by AffiliationUpon appointment, the Fields medalists were working in the following institutions:AffiliationMedal(s)

Princeton University8

Institut des Hautes tudes Scientifiques6

University of Paris6

Institute for Advanced Study5

Harvard University3

University of California, Berkeley3

University of Cambridge3

Massachusetts Institute of Technology2

Oxford University2

Stanford University2

University of California, San Diego2

CNRS1

Hebrew University of Jerusalem1

Institut Henri Poincar1

Instituto Nacional de Matemtica Pura e Aplicada1

Kyoto University1

Moscow State University1

Rutgers University1

University College London1

University of California, Los Angeles1

University of Geneva1

University of Helsinki1

University of Kharkiv1

University of Nancy1

University of Pisa1

University of Stockholm1

University of Strasbourg1

University of Warwick1

University of Tokyo1

cole Normale Suprieure de Lyon1

None (independent researcher)1

The medal

The reverse of the Fields MedalThe medal was designed by Canadian sculptor R. Tait McKenzie.[92] On the obverse is Archimedes and a quote attributed to Marcus Manilius 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").[93] On the reverse is the inscription (in Latin):CONGREGATIEX TOTO ORBEMATHEMATICIOB SCRIPTA INSIGNIATRIBUERETranslation: "Mathematicians gathered from the entire world have awarded [understood '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 the 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 .)The rim bears the name of the prizewinner.