FIELDS MEDAL WINNERS
1936 - 1982

Fields Medal - History & Background

FIELDS MEDAL WINNERS IN 1982

ALAIN CONNES - France - Alain Connes was awarded the Fields Medal in 1982 for his pioneering work in operator algebras and their applications to non-commutative geometry. He developed the theory of Von Neumann algebras and introduced key concepts like the Connes Cyclic Cohomology, providing deep insights into topology, geometry and quantum physics. His groundbreaking ideas have influenced diverse fields, including number theory, dynamical systems and mathematical physics, reshaping modern mathematical thought.

WILLIAM THURSTON - USA - William Thurston (1946 – 2012) was awarded the Fields Medal in 1982 for his contributions to low-dimensional topology and geometric structures on 3-manifolds. His geometrization conjecture, which classifies 3-manifolds using eight model geometries, transformed the field and later led to Perelman's proof of the Poincaré conjecture. Thurston’s insights into foliations, Teichmüller theory and hyperbolic geometry profoundly influenced topology, geometry and mathematical physics, reshaping the study of three-dimensional spaces.

SHING-TUNG YAU - China - Shing-Tung Yau (1949 – Present) was awarded the Fields Medal in 1982 for his contributions to differential geometry and geometric analysis. His proof of the Calabi conjecture led to the development of Calabi-Yau manifolds, which became fundamental in string theory. He also proved the positive mass theorem in general relativity, linking geometry with physics. Yau’s work profoundly influenced complex geometry, topology, and mathematical physics, shaping modern geometric analysis.

FIELDS MEDAL WINNERS IN 1978

PIERRE DELIGNE - Belgium - Pierre Deligne was awarded the Fields Medal in 1978 for his work in algebraic geometry and number theory, particularly for proof of the Weil conjectures, which revolutionized the understanding of zeta functions of algebraic varieties over finite fields. His work extended and refined Grothendieck’s methods, introducing powerful cohomological techniques.

CHARLES FEFFERMAN - USA - Charles Fefferman was awarded the Fields Medal in 1978 for his deep and innovative contributions to mathematical analysis, particularly in Fourier analysis, partial differential equations and several complex variables. His work on the fine structure of functions and singular integrals led to breakthroughs in smoothness and approximation theory. Fefferman also made major advances in fluid dynamics and quantum mechanics, reshaping modern analysis with new techniques and perspectives.

DANIEL QUILLEN - USA - Daniel Quillen was awarded the Fields Medal in 1978 for his revolutionary work in algebraic K-theory, where he introduced homotopical and categorical methods to solve deep problems in topology and algebra. His Quillen model categories and application of simplicial techniques provided a powerful framework for understanding algebraic and geometric structures. His insights connected K-theory with geometry, topology and number theory, which greatly influenced modern mathematics.

GRIGORY MARGULIS - Russia - Grigory Margulis was awarded the Fields Medal in 1978 for his groundbreaking work on Lie groups, ergodic theory and discrete subgroups, particularly for proof of the Oppenheim conjecture and development of superrigidity and arithmeticity theorems for lattices in semisimple Lie groups. His use of ergodic theory in geometry and number theory revolutionized the field, influencing areas such as dynamical systems, representation theory and spectral theory.

FIELDS MEDAL WINNERS IN 1974

ENRICO BOMBIERI - Italy - Enrico Bombieri (1940 - Present) was awarded the Fields Medal in 1974 for his outstanding contributions to number theory, analysis and algebraic geometry. His work advanced the understanding of the large sieve method in analytic number theory and solved long-standing problems such as proving the Mordell conjecture for function fields. Bombieri's insights bridged multiple mathematical disciplines, making significant impacts on Diophantine approximations, prime number theory, and geometric properties of algebraic varieties.

DAVID MUMFORD - USA - David Mumford was awarded the Fields Medal in 1974 for his pioneering work in algebraic geometry, particularly in developing the theory of moduli spaces of algebraic curves. His contributions to the geometric invariant theory and the classification of algebraic surfaces expanded the field’s foundations. Mumford’s insights linked geometry with topology and number theory, thereby influencing modern mathematics for further research in both pure and applied mathematics.

FIELDS MEDAL WINNERS IN 1970

ALAN BAKER - UK - Alan Baker (1939 - 2018) was awarded the Fields Medal in 1970 for his great contributions to transcendental number theory and Diophantine equations. His work extended the Gelfond-Schneider theorem by developing general methods for dealing with linear forms in logarithms of algebraic numbers. This advanced the understanding of effective bounds in number theory and resolved long-standing problems, including the finiteness of solutions to certain equations.

HEISUKE HIRONAKA - Japan - Heisuke Hironaka (1931 - Present) was awarded the Fields Medal in 1970 for his groundbreaking work in algebraic geometry, specifically for proving the ‘resolution of singularities’ of algebraic varieties over a field of characteristic zero, a fundamental problem in the field that had been open for many years.

His breakthrough on the ‘resolution of singularities’ is considered his most significant contribution in mathematics. This breakthrough significantly advanced the field of algebraic geometry by providing a way to understand and analyze complex geometric shapes by ‘smoothing out’ their singularities.

SERGEI NOVIKOV - Russia - Sergei Novikov (1938 - 2024) was awarded the Fields Medal in 1970 for his groundbreaking work in topology, particularly the study of manifolds. He made significant contributions to the understanding of homotopy groups of spheres and the topology of high-dimensional manifolds. Novikov's proof of the topological invariance of the rational Pontryagin classes was a major achievement, resolving fundamental questions in differential topology and influencing both mathematics and theoretical physics.

JOHN G THOMPSON - USA - John G. Thompson was awarded the Fields Medal in 1970 for his contributions to finite group theory, particularly on the classification of finite simple groups. His collaboration with Walter Feit in proving that all nonabelian finite simple groups have even order was a milestone in algebra. Thompson also developed key techniques that significantly advanced the classification project, laying the foundation for modern group theory.

FIELDS MEDAL WINNERS IN 1966

MICHAEL ATIYAH - UK - Michael Atiyah (1929 - 2019) was awarded the Fields Medal in 1966 for his path breaking contributions in geometry and topology, particularly the work on K-theory and Atiyah-Singer Index Theorem. These advances linked algebraic topology, differential geometry and theoretical Physics, solving long-standing problems and inspiring new research fields. His insights bridged pure mathematics with quantum theory, leaving a lasting impact on mathematical sciences.

PAUL JOSEPH COHEN - USA - Paul Joseph Cohen (1934 - 2007) was awarded the Fields Medal in 1966 for proving that Continuum Hypothesis is independent from the other axioms of set theory, essentially demonstrating that it cannot be proven or disproven within the standard framework of set theory, which is considered a landmark achievement in set theory.

ALEXANDER GROTHENDIECK - France - Alexander Grothendieck (1928 – 2014) was awarded the Fields Medal in 1966 for his work in algebraic geometry, specifically for revolutionizing the field by introducing new concepts and techniques using category theory and topology, which allowed mathematicians to apply geometric methods to problems in number theory.

His work significantly reshaped algebraic geometry, providing a new framework for studying geometric objects through the lens of abstract algebra, which opened up new avenues of research in mathematics.

FIELDS MEDAL WINNERS IN 1962

LARS HORMANDER - Sweden - Lars Hörmander (1931 - 2012) was awarded the Fields Medal in 1962 for his great work in the field of partial differential equations, specifically for his contributions to the theory of linear differential operators. This includes his characterization of hypo-ellipticity for constant coefficients and his geometrical explanation of the Lewy non-solvability phenomenon.

Lars Hormander had introduced important concepts like pseudo-differential operators and Fourier integral operators, and his work is considered a major advancement in the modern understanding of partial differential equations.

JOHN MILNOR - USA - John Milnor (1931 – Present) was awarded the Fields Medal in 1962 for his path breaking work in differential topology, and specifically for proving that a 7-dimensional sphere can have multiple differential structures. This finding opened up a new field of study within Mathematics, which is considered one of his most significant contributions to the field.

FIELDS MEDAL WINNERS IN 1958

KLAUS ROTH - UK - Klaus Roth (1925 – 2015) was awarded the Fields Medal in 1958 for his groundbreaking work in number theory, specifically for proving "Roth's Theorem" on Diophantine approximation. This advancement solved a major long-standing open problem regarding how well algebraic numbers can be approximated by rational numbers, which is considered a significant contribution.

RENÉ THOM - France - René Thom (1923 - 2002) was awarded the Fields Medal in 1958 for his significant contributions to the field of topology, particularly for laying the foundation of his groundbreaking concept of ‘Cobordism’ theory.

While Thom is widely known for developing ‘Catastrophe’ theory later in his career, it was his earlier work on topology that earned him the Fields Medal. The catastrophe theory, beyond pure Mathematics enabled newer applications in fields like Biology and Physics.

FIELDS MEDAL WINNERS IN 1954

KUNIHIKO KODAIRA -Japan - Kunihiko Kodaira (1915 - 1997) was awarded the Fields Medal in 1954 for his pioneering contributions to algebraic geometry and complex manifolds. He linked differential geometry and algebraic geometry, developed techniques for classifying complex surfaces, and studied their deformations.

Kodaira’s work on harmonic integrals and embedding theorems had profound implications for mathematics and physics. His methods transformed the understanding of complex structures and topology, leaving a lasting impact on modern geometry.

JEAN-PIERRE SERRE - France - Jean-Pierre Serre (1926 - Present) was awarded the Fields Medal in 1954 for his groundbreaking work in algebraic topology, particularly his contributions to the study of homotopy groups of spheres and the development of sheaf theory, which helped unify various branches of mathematics and paved the way for further advancements in algebraic geometry and number theory.

FIELDS MEDAL WINNERS IN 1950

LAURENT SCHWARTZ - France - Laurent Schwartz (1915 -2002) was awarded the Fields Medal in 1950 for his groundbreaking work on the "theory of distributions," which essentially provides a well-defined meaning to objects like the Dirac delta function, significantly advancing the field of mathematical analysis and finding applications in various areas like partial differential equations and potential theory. Schwartz's theory of distributions had a profound impact in the study of differential equations and functional analysis.

ATLE SELBERG - Norway - Atle Selberg (1917 - 2007) was awarded the Fields Medal in 1950 for his groundbreaking work in analytic number theory, specifically for his elementary proof of the Prime Number Theorem, his research on the zeros of the Riemann zeta function (demonstrating a positive proportion lie on the critical line), and his contributions to sieve methods, including the Selberg sieve, which are generalizations of Eratosthenes' method for finding prime numbers.

FIELDS MEDAL WINNERS IN 1936

LARS AHLFORS - Finland - Lars Ahlfors (1907-1996) was awarded the Fields Medal in 1936 for his significant contributions to the field of complex analysis, particularly his innovative methods for analyzing Riemann surfaces, including his work on conformal mappings and the theory of covering surfaces, which led to what is now known as "Ahlfors theory."

His work on Riemann surfaces, including his "Ahlfors finiteness theorem" and "Ahlfors five-disk theorem," provided new insights into the behavior of complex functions on these surfaces.

JESSE DOUGLAS - USA - Jesse Douglas (1897 - 1965) was awarded the Fields Medal in 1936 for solving the Plateau problem in geometry. The Plateau problem, also known as the soap bubble problem, asks if a closed curve can always span a surface with the least area. It was first posed in 1760 by Joseph-Louis Lagrange and Leonhard Euler. Douglas published his solution in 1931 in Transactions of the American Mathematical Society.

Fields Medal - History & Background
Fields Medal Winners 2006 - 2022
Fields Medal Winners 1986 - 2002
Fields Medal Winners 1936 - 1982