Fields Medal (mathematics)

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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.

Photo of the obverse of a Fields Medal made by Stefan Zachow for the International Mathematical Union (IMU), showing a bas relief of Archimedes (as identified by the Greek text).

The Fields Medal is regarded as one of the highest honours 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/American mathematician Maryam Mirzakhani became the first female Fields Medalist, and in 2022 the Ukrainian/Swiss mathematician Maryna Viazovska was the second one to win the medal.

In all, 64 people (two women) have been awarded the Fields Medal (by 2022).

Fields medalists:

• 1936: Lars Ahlfors (Finland/USA) (1907-1996) "For research on covering surfaces related to Riemann surfaces of inverse functions of entire and meromorphic functions. Opened up new fields of analysis" & Jesse Douglas (USA) (1897-1965), "Did important work on the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary."
• 1950: Laurent-Moïse Schwartz (France) (1915-2002), "Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics" & Atle Selberg (Norway/USA) (1917-2007), "Developed generalisations 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 generalisation to prime numbers in an arbitrary arithmetic progression."
• 1954: Kunihiko Kodaira (Japan/USA) (1915-1997), "Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically to algebraic varieties. He demonstrated, by sheaf cohomology, that such varieties are Hodge manifolds" & Jean-Pierre Serre (France) (b. 1926), "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: Klaus Friedrich Roth FRS (UK)(1925-2015), "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)" & René Thom (France) (1923-2002), "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: Lars Hörmander (Sweden) (1931-2012), "Worked in partial differential equations. Specifically, contributed to the general theory of linear differential operators. The questions go back to one of Hilbert's problems at the 1900 congress." & John Milnor (USA) (b. 1931), "Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology."
• 1966: Sir Michael Francis Atiyah OM FRS FRSE FMedSci FAA FREng (UK) (1929-2019), "Did joint work with Hirzebruch in K-theory; proved jointly with Singer the index theorem of elliptic operators on complex manifolds; worked in collaboration with Bott to prove a fixed point theorem related to the 'Lefschetz formula'." & Paul Joseph Cohen (USA) (1934-2007), "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 (France) (1928-2014), "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 (USA/Hong Kong) (b. 1930), "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: Alan Baker FRS (UK) (1939-2018), "Generalized the Gelfond-Schneider theorem (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified.", & Heisuke Hironaka (Japan/USA) (b. 1931), "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 Petrovich Novikov (Russia) (b. 1938), "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 Griggs Thompson (USA/UK) (b. 1932), "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: Enrico Bombieri (Italy/USA) (b. 1940), "Major contributions in the primes, in univalent functions and the local Bieberbach conjecture, in theory of functions of several complex variables, and in theory of partial differential equations and minimal surfaces – in particular, to the solution of Bernstein's problem in higher dimensions." & David Mumford (USA) (b. 1937), "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: Pierre René, Viscount Deligne (Belgium-born/France/USA) (b. 1944), "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 Louis Fefferman (USA) (b. 1949), "Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results." & Grigori Aleksandrovich Margulis (Russia/USA) (b. 1946), "Provided innovative analysis of the structure of Lie groups. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups." & Daniel Gray "Dan" Quillen (USA) (1940-2011), "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: Alain Connes (France) (b. 1947), "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 (USA) (1946-2012), "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 (Chinese-born/USA) (b. 1949), "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: Sir Simon Kirwan Donaldson FRS (UK) (b. 1957), "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 (Germany) (b. 1954), "Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture." &0 Michael Freedman (USA) (b. 1951), "Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincaré Conjecture."
• 1990: Vladimir Gershonovich Drinfeld (Russia/USA) (b. 1954), "For his work on quantum groups and for his work in number theory." & Sir Vaughan Frederick Randal Jones KNZM FRS FRSNZ FAA (New Zealand/USA) (b. 1952), "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 (Japan) (b. 1951), "For the proof of Hartshorne’s conjecture and his work on the classification of three-dimensional algebraic varieties." & Edward Witten (USA) (b. 1951) "Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems."
• 1994: Jean, Baron Bourgain (Belgium/France) (1954-2018), "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 (France) (b. 1956), "... 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 (France) (b. 1957), "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 Isaakovich Zelmanov (Russia/USA) (b. 1955), "For his solution to the restricted Burnside problem."
• 1998: Richard Ewen Borcherds (UK/USA) (b. 1959), "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." & Sir William Timothy Gowers, FRS (UK) (b. 1963) "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 Lvovich Kontsevich (Russia/France) (b. 1964), "Contributions to four problems of geometry." & Curtis Tracy McMullen (USA) (b. 1958), "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: Laurent Lafforgue (France) (b. 1966), "For his proof of the Langlands correspondence for the full linear groups GLr (r≥1) over function fields." & Vladimir Alexandrovich Voevodsky (Russia/USA) (1966-2017), "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: Andrei Okounkov (Russia/USA) (b. 1969), "For his contributions bridging probability, representation theory and algebraic geometry." & Grigori Perelman (declined) (Russia) (b. 1966) "For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." & Chi-Shen Tao FAA FRS (Australia/USA) (1975), "For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory." & Wendelin Werner (France/Switzerland) (b. 1968), "For his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory."
• 2010: Elon Lindenstrauss (Israel) (b. 1970) "For his results on measure rigidity in ergodic theory, and their applications to number theory." & Ngô Bảo Châu (Vietnam/France/USA) (b. 1972), "For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods." & Stanislav Smirnov (Russia/Switzerland) (b. 1970), "For the proof of conformal invariance of percolation and the planar Ising model in statistical physics." & Cédric Villani (France) (b. 1973), "For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation."
• 2014: Artur Avila Cordeiro de Melo (Brazil/France/Switzerland) (b. 1979), "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 FRS (Canada/USA) (b. 1974), "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 (Austria/UK) (b. 1975), "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 (Iran/USA) (1977-2017), "For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."
• 2018: Caucher Birkar (Iran/UK) (b. 1978), "For the proof of the boundedness of Fano varieties and for contributions to the minimal model program." & Alessio Figalli (Italy) (b. 1984), "For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability." & Peter Scholze (Germany) (b. 1987), "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 (Australia/USA) (b. 1981), "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."
• 2022: Hugo Duminil-Copin (France) (b. 1985) "For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four." & June Huh (Korea/USA) (b. 1983) "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 (United Kingdom) (b. 1987) "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 (Ukraine/Switzerland) (b. 1984) "For the proof that the E8 lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis."