# Definition:Fields Medal

## Definition

The Fields Medal is the highest award of the International Mathematical Union, for "oustanding achievement in mathematics".

It is a gold medal, awarded once every $4$ years at its International Congress, using the funds remaining after the financing of the Congress in Toronto in $1924$.

It is awarded to no fewer than $2$ and no more than $4$ mathematicians, usually under $40$ years of age.

The medals stem from a bequest by John Charles Fields, who wanted to provide an award comparable to the Nobel Prize in stature.

It was his intention to stress the international nature of mathematics, and stated that there should not be attached to the medals:

the name of any country, institution or person.

The first Fields Medals were awarded in $1936$, but were interrupted because of the Second World War.

They were resumed in $1950$.

The award is administered by a Board of Trustees set up by the University of Toronto, and awarded by a committee of mathematicians appointed by the ICM.

## Recipients

Recipients of the Fields Medal have been as follows:

### 1936

Recipient(s) Citation
Lars Valerian 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.
Jesse Douglas Did important work on the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary.

Location of conference: Oslo, Norway

### 1950

Recipient(s) Citation
Laurent-Moïse Schwartz Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics.
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.

Location of conference: Cambridge, Massachusetts, USA

### 1954

Recipient(s) Citation
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.
Jean-Pierre Albert Achille 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.

Location of conference: Amsterdam, Netherlands

### 1958

Recipient(s) Citation
Klaus Friedrich Roth For solving a famous problem of number theory, namely, the determination of the exact exponent in the Thue-Siegel inequality.
René Frédéric 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.

Location of conference: Edinburgh, Scotland, United Kingdom

### 1962

Recipient(s) Citation
Lars Valter 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.
John Willard Milnor Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology.

Location of conference: Stockholm, Sweden

### 1966

Recipient(s) Citation
Michael Francis 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'.
Paul Joseph 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.
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’.
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.

Location of conference: Moscow, Russia

### 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. 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. Sergei Petrovich 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. John Griggs Thompson Proved jointly with Walter 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.

Location of conference: Nice, France

### 1974

Recipient(s) Citation
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.
David Bryant 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.

Location of conference: Helsinki, Finland

### 1978

Recipient(s) Citation
Pierre René 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.
Charles Louis Fefferman Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results.
Grigory Aleksandrovich Margulis 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 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.

Location of conference: Vancouver, British Columbia, Canada

### 1982

Recipient(s) Citation
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.
William Paul 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.
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.

Location of conference: Warsaw, Poland

### 1986

Recipient(s) Citation
Simon Kirwan 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.
Gerd Faltings Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture.
Michael Hartley Freedman Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincaré Conjecture.

Location of conference: Berkeley, California, USA

### 1990

Recipient(s) Citation
Vladimir Gershonovich Drinfeld Drinfeld's main preoccupation in the last decade [are] Langlands' program and quantum groups. In both domains, Drinfeld's work constituted a decisive breakthrough and prompted a wealth of research.
Vaughan Frederick Randal 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.
Shigefumi Mori The most profound and exciting development in algebraic geometry during the last decade or so was [...] Mori's Program in connection with the classification problems of algebraic varieties of dimension three. Early in 1979, Mori brought to algebraic geometry a completely new excitement, that was his proof of Hartshorne's conjecture.
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.

Location of conference: Kyoto, Japan

### 1994

Recipient(s) Citation
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.
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.
Jean-Christophe Yoccoz Yoccoz obtained a very enlightening proof of Bruno's theorem, and he was able to prove the converse [...] Palis and Yoccoz obtained a complete system of $C^\infty$ conjugation invariants for Morse-Smale diffeomorphisms.
Efim Isaakovich Zelmanov For the solution of the restricted Burnside problem.

Location of conference: Zürich, Switzerland

### 1998

Recipient(s) Citation
Richard Ewen 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.
William 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.
Maxim Lvovich 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.
Curtis Tracy 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.

A silver plate was awarded to Andrew Wiles as a special tribute.

Location of conference: Berlin, Germany

### 2002

Recipient(s) Citation
Laurent Lafforgue Laurent Lafforgue has been awarded the Fields Medal for his proof of the Langlands correspondence for the full linear groups $\map {GLr} {r \ge 1}$ over function fields of positive characteristic.
Vladimir Alexandrovich 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.

Location of conference: Beijing, China

### 2006

Recipient(s) Citation
Andrei Yuryevich Okounkov For his contributions bridging probability, representation theory and algebraic geometry.
Grigori Yakovlevich Perelman For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.
Terence Chi-Shen Tao For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.
Wendelin Werner For his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.

Location of conference: Madrid, Spain

### 2010

Recipient(s) Citation
Elon Lindenstrauss For his results on measure rigidity in ergodic theory, and their applications to number theory.
Ngô Bảo Châu For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebra-geometric methods.
Stanislav Konstantinovich Smirnov For the proof of conformal invariance of percolation and the planar Ising model in statistical physics.
Cédric Patrice Thierry Villani For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.

Location of conference: Hyderabad, India

### 2014

Recipient(s) Citation
Artur Avila Cordeiro de Melo 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 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 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 For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.

Location of conference: Seoul, South Korea

### 2018

Recipient(s) Citation
Caucher Birkar For the proof of the boundedness of Fano varieties and for contributions to the minimal model program.
Alessio Figalli For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability.
Peter Scholze For having transformed arithmetic algebraic geometry over p-adic fields.
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.

### 2022

Recipient(s) Citation
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 Sergiivna 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.

## Source of Name

This entry was named for John Charles Fields.