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World of Mathematics

Since 1936 · quadrennial

Fields Medal. The mathematics Nobel for work done young.

Awarded every four years at the International Congress of Mathematicians to up to four mathematicians under the age of 40. Established by John Charles Fields and funded by a bequest from his estate. The most prestigious honour for early- to mid-career mathematical achievement.

  1. 2022

    • June Huh

      Korean-American

      Combinatorics and algebraic geometry; log-concavity conjectures via Hodge theory.

    • Hugo Duminil-Copin

      French

      Phase transitions in statistical physics; long-range Ising model.

    • Maryna Viazovska

      Ukrainian

      Sphere packings in dimensions 8 and 24.

    • James Maynard

      British

      Prime gaps, analytic number theory.

  2. 2018

    • Caucher Birkar

      Iranian-British

      Fano varieties and the minimal model program.

    • Alessio Figalli

      Italian

      Optimal transport, Monge-Ampère regularity.

    • Peter Scholze

      German

      Perfectoid spaces in p-adic geometry.

    • Akshay Venkatesh

      Australian

      Analytic number theory, automorphic forms, homological algebra.

  3. 2014

    • Artur Avila

      Brazilian

      Dynamical systems, renormalisation.

    • Manjul Bhargava

      Canadian-American

      Number theory; Bhargava cubes, composition laws.

    • Martin Hairer

      Austrian

      Theory of regularity structures for singular stochastic PDEs.

    • Maryam Mirzakhani

      Iranian

      Dynamics and geometry of Riemann surfaces and their moduli spaces, and the earthquake flow on moduli space. The first woman to win the Fields Medal in the prize's 78-year history. Died of cancer in 2017, aged 40.

  4. 2010

    • Elon Lindenstrauss

      Israeli

      Measure rigidity in ergodic theory and applications to number theory.

    • Ngô Bảo Châu

      Vietnamese-French

      Proof of the fundamental lemma in the Langlands program.

    • Stanislav Smirnov

      Russian

      Conformal invariance in 2D lattice models.

    • Cédric Villani

      French

      Optimal transport, Landau damping, Boltzmann equation.

  5. 2006

    • Andrei Okounkov

      Russian

      Probability, representation theory, algebraic geometry interactions.

    • Grigori Perelman

      Russian

      Proof of the Poincaré conjecture — a century-old open problem — using Hamilton's Ricci flow technique. Declined both the Fields Medal and the $1M Clay Millennium Prize. Withdrew from mathematics shortly afterwards.

    • Terence Tao

      Australian-American

      Partial differential equations, combinatorics, harmonic analysis.

    • Wendelin Werner

      French

      Schramm-Loewner evolution, conformally invariant random processes.

  6. 2002

    • Laurent Lafforgue

      French

      Proof of the Langlands correspondence for GL(n) over function fields.

    • Vladimir Voevodsky

      Russian

      Motivic cohomology; foundations of univalent type theory.

  7. 1998

    • Richard Borcherds

      British

      Vertex operator algebras, monstrous moonshine.

    • Timothy Gowers

      British

      Functional analysis and combinatorics.

    • Maxim Kontsevich

      Russian

      Deformation quantization, mirror symmetry, motivic integration.

    • Curtis T. McMullen

      American

      Complex dynamics, renormalisation theory.

  8. 1994

    • Jean Bourgain

      Belgian

      Banach space theory, ergodic theory, number theory, PDE.

    • Pierre-Louis Lions

      French

      Viscosity solutions for nonlinear PDE.

    • Jean-Christophe Yoccoz

      French

      Dynamical systems and MLC conjecture.

    • Efim Zelmanov

      Russian-American

      Restricted Burnside problem in group theory.

  9. 1990

    • Vladimir Drinfeld

      Ukrainian-Soviet

      Quantum groups, Drinfeld modules.

    • Vaughan Jones

      New Zealand

      Jones polynomial in knot theory, von Neumann algebras.

    • Shigefumi Mori

      Japanese

      Minimal model program in algebraic geometry.

    • Edward Witten

      American

      Mathematical physics, topological QFT. Only physicist to win.

  10. 1986

    • Simon Donaldson

      British

      Exotic smooth structures on R⁴ via Yang-Mills.

    • Gerd Faltings

      German

      Proof of the Mordell conjecture.

    • Michael Freedman

      American

      Topology of 4-manifolds, topological Poincaré conjecture.

  11. 1982

    • Alain Connes

      French

      Classification of factors in operator algebras; noncommutative geometry.

    • William Thurston

      American

      Geometrisation of 3-manifolds.

    • Shing-Tung Yau

      Chinese-American

      Calabi conjecture; Calabi-Yau manifolds.

  12. 1978

    • Pierre Deligne

      Belgian

      Proof of the last Weil conjecture.

    • Charles Fefferman

      American

      Complex analysis in several variables.

    • Grigory Margulis

      Soviet

      Arithmeticity of lattices in semisimple Lie groups.

    • Daniel Quillen

      American

      Higher algebraic K-theory.

  13. 1974

    • Enrico Bombieri

      Italian

      Analytic number theory; Bombieri-Vinogradov theorem.

    • David Mumford

      British-American

      Moduli spaces in algebraic geometry.

  14. 1970

    • Alan Baker

      British

      Transcendental number theory; Baker's theorem.

    • Heisuke Hironaka

      Japanese

      Resolution of singularities in characteristic zero.

    • Sergei Novikov

      Soviet

      Topology of manifolds, foliations.

    • John G. Thompson

      American

      Classification of finite simple groups.

  15. 1966

    • Michael Atiyah

      British

      K-theory and, jointly with Isadore Singer, the index theorem — a result connecting the analytical index of an elliptic operator to a topological invariant. One of the most consequential theorems of the century; later bridged mathematics and theoretical physics.

    • Paul Cohen

      American

      Proved that the continuum hypothesis is independent of the standard axioms of set theory — that is, neither provable nor disprovable from them. Invented the technique of forcing, now a standard tool in set theory. Settled the first problem on Hilbert's 1900 list.

    • Alexander Grothendieck

      Stateless

      Rebuilt the foundations of algebraic geometry with the theory of schemes, étale cohomology, and topoi. Declined to attend the Moscow ceremony in protest at the Soviet treatment of dissidents. Left mathematics entirely in 1970; lived as a hermit until his death in 2014.

    • Stephen Smale

      American

      Topology of manifolds, Poincaré conjecture in dim ≥ 5.

  16. 1962

    • Lars Hörmander

      Swedish

      Pseudodifferential operators in PDE theory.

    • John Milnor

      American

      Exotic smooth structures on the 7-sphere.

  17. 1958

    • Klaus Roth

      German-British

      Diophantine approximation (Thue–Siegel–Roth theorem).

    • René Thom

      French

      Cobordism theory, singularity theory.

  18. 1954

    • Kunihiko Kodaira

      Japanese

      Harmonic integrals, classification of algebraic surfaces.

    • Jean-Pierre Serre

      French

      Topology, algebraic geometry, number theory. Youngest winner ever at 27.

  19. 1950

    • Laurent Schwartz

      French

      Theory of distributions.

    • Atle Selberg

      Norwegian

      Elementary proof of the prime number theorem.

  20. 1936

    • Lars Ahlfors

      Finnish

      Riemann surfaces, complex analysis.

    • Jesse Douglas

      American

      Solution to the Plateau problem.

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