Fields Medal

Hugo Duminil-Copin,

France

For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four.

June Huh,

United States of America

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 of Great Britain and Northern Ireland

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.解析的整数論に貢献し，素数の構造理解とディオファントス近似の理解に大きな進歩をもたらした[22]。

Maryna Viazovska,

Ukraine

For the proof that theE8{\displaystyle 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.球充填問題を8次元と24次元で解決したことや，フーリエ解析における極値および補間問題への更なる貢献が評価[22]。

Maryna Viazovska,

Ukraine

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Caucher Birkar,

United Kingdom of Great Britain and Northern Ireland

For the proof of the boundedness ofFano varietiesand for contributions to theminimal model program.

Alessio Figalli,

Italy

For contributions to the theory ofoptimal transportand its applications inpartial differential equations,metric geometryandprobability.

Peter Scholze,

Germany

For transformingarithmetic algebraic geometryoverp-adic fieldsthrough his introduction ofperfectoid spaces, with application toGalois representations, and for the development of newcohomology theories.

Akshay Venkatesh,

Australia

For his synthesis ofanalytic number theory, homogeneous dynamics,topology, andrepresentation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.

Maryam Mirzakhani,

Iran (Islamic Republic of)

for her outstanding contributions to the dynamics and geometry ofRiemann surfacesand theirmoduli spaces.

Artur Avila,

Brazil

for his profound contributions todynamical systemstheory have changed the face of the field, using the powerful idea of renormalization as a unifying principle.

Manjul Bhargava,

Canada

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 ofelliptic curves.

Martin Hairer,

Austria

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.

Elon Lindenstrauss,

Israel

For his results on measure rigidity inergodic theory, and their applications tonumber theory.

Stanislav Smirnov,

Russian Federation

For the proof of conformal invariance ofpercolationand the planarIsing modelinstatistical physics.

Ngô Bảo Châu,

Viet Nam

For his proof of the Fundamental Lemma in the theory ofautomorphic formsthrough the introduction of new algebro-geometric methods.

Cédric Villani,

France

For his proofs of nonlinear Landau damping and convergence to equilibrium for theBoltzmann equation.

Terence Tao,

Australia

for his contributions topartial differential equations,combinatorics,harmonic analysisand additive number theory

Grigori Perelman,

Russian Federation

for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of theRicci flow

Andrei Okounkov,

Russian Federation

for his contributions bridgingprobability,representation theoryandalgebraic geometry

Wendelin Werner,

France

for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensionalBrownian motion, andconformal field theory

Laurent Lafforgue,

France

Laurent Lafforgue has been awarded the Fields Medal for his proof of theLanglands correspondencefor thefull linear groupsGLr(r≥1) overfunction fields.

Vladimir Voevodsky,

Russian Federation

he defined and developed motivic cohomology and the A1-homotopy theory ofalgebraic varieties; he proved theMilnor conjectureson theK-theoryof fields

Richard E. Borcherds,

United Kingdom of Great Britain and Northern Ireland

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

William Timothy Gowers,

United Kingdom of Great Britain and Northern Ireland

William Timothy Gowers has provided important contributions tofunctional analysis, making extensive use of methods fromcombination 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 Kontsevich,

Russian Federation

contributions to four problems of geometry

Curtis T. Mcmullen,

United States of America

He has made important contributions to various branches of the theory ofdynamical systems, such as the algorithmic study of polynomial equations, the study of the distribution of the points of a lattice of aLie group,hyperbolic geometry,holomorphic dynamicsand the renormalization of maps of the interval.

Jean Bourgain, 1954年 - 2018年）

Belgium

Bourgain's work touches on several central topics of mathematical analysis: the geometry ofBanach spaces, convexity in high dimensions,harmonic analysis,ergodic theory, and finally, nonlinearpartial differential equationsfrommathematical physics.

Pierre-Louis Lions,

France

... such nonlinearpartial differential equationsimply 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 onviscosity 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 smoothtest function

Jean-Christophe Yoccoz,

France

proving stability properties - dynamic stability, such as that sought for the solar system, orstructural stability, meaning persistence under parameter changes of the global properties of the system.

Efim Zelmanov,

Russian Federation

For his solution to the restricted Burnside problem.

Vladimir Drinfeld, 1954年 - ）

Soviet Union

For his work onquantum groupsand for his work innumber theory.

Vaughan F. R. Jones,

New Zealand

for his discovery of an unexpected link between the mathematical study ofknots– a field that dates back to the 19th century – andstatistical mechanics, a form of mathematics used to study complex systems with large numbers of components.

Shigefumi Mori, 1951年 -）

Japan

for the proof of Hartshorne’s conjecture and his work on the classification of three-dimensionalalgebraic varieties.

Edward Witten, 1951年 - ）

United States of America

proof in 1981 of the positive energy theorem ingeneral relativity

Simon K. Donaldson,

United Kingdom of Great Britain and Northern Ireland

Received medal primarily for his work ontopologyof 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

Using methods ofarithmetic algebraic geometry, he received medal primarily for his proof of theMordell Conjecture.

Michael H. Freedman,

United States of America

Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensionalPoincaré Conjecture.

Alain Connes,

France

Contributed to thetheory 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 ofC*-algebrasto foliations and differential geometry in general.

William P. Thurston, 1946年 - 2012年）

United States of America

Revolutionized study oftopologyin 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, 1949年 - ）

United States of America

Made contributions indifferential equations, also to theCalabi conjectureinalgebraic geometry, to the positive mass conjecture ofgeneral relativity theory, and to real and complex Monge-Ampère equations.

Pierre René Deligne,

Belgium

Gave solution of the threeWeil conjecturesconcerning generalizations of theRiemann hypothesisto finite fields. His work did much to unifyalgebraic geometryandalgebraic number theory.

Charles Louis Fefferman,

United States of America

Contributed several innovations that revised the study of multidimensionalcomplex analysisby finding correct generalizations of classical (low-dimensional) results.

Gregori Aleksandrovi（t?）ch Margulis,

Soviet Union

Provided innovative analysis of the structure ofLie groups. His work belongs tocombinatorics,differential geometry,ergodic theory,dynamical systems, and Lie groups.

Daniel G. Quillen,

United States of America

The prime architect of the higheralgebraic K-theory, a new tool that successfully employed geometric and topological methods and ideas to formulate and solve major problems in algebra, particularlyring theoryandmodule theory.

Enrico Bombieri,

Italy

Major contributions in theprimes, inunivalent functionsand the localBieberbach conjecture, intheory of functionsof several complex variables, and in theory ofpartial differential equationsand minimal surfaces - in particular, to the solution of Bernstein's problem in higher dimensions.

David Bryant Mumford,

United Kingdom of Great Britain and Northern Ireland

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 ofalgebraic surfaces.

Alan Baker,

United Kingdom of Great Britain and Northern Ireland

Generalized theGelfond-Schneider theorem(the solution to Hilbert's seventh problem). From this work he generatedtranscendental numbersnot previously identified.

Heisuke Hironaka, 1931年 - ）

Japan

Generalized work ofZariskiwho had proved for dimension ≤3 the theorem concerning the resolution of singularities on analgebraic variety. Hironaka proved the results in any dimension.

Sergei Novikov,

Soviet Union

Made important advances in topology, the most well-known being his proof of the topological invariance of thePontrjagin classesof thedifferentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces.

John Griggs Thompson,

United States of America

Proved jointly with W. Feit that all non-cyclic finitesimple groupshave even order. The extension of this work by Thompson determined the minimal simple finite groups, that is, the simple finite groups whose proper subgroups aresolvable.

Michael Francis Atiyah,

United Kingdom of Great Britain and Northern Ireland

Did joint work withHirzebruchinK-theory; proved jointly withSingertheindex theoremof elliptic operators oncomplex manifolds; worked in collaboration withBottto prove afixed point theoremrelated to the "Lefschetz formula".

Paul Joseph Cohen,

United States of America

Used technique called "forcing" to prove the independence inset theoryof theaxiom of choiceand of thegeneralized continuum hypothesis. The latter problem was the first ofHilbert's problemsof the 1900 Congress.

Alexander Grothendieck,

Built on work ofWeilandZariskiand effected fundamental advances inalgebraic geometry. He introduced the idea ofK-theory(theGrothendieck groupsand rings). Revolutionizedhomological algebrain his celebrated "Tohokupaper"

Stephen Smale,

United States of America

Worked indifferential topologywhere he proved the generalizedPoincaré conjecturein dimension n≥5: Every closed, n-dimensional manifoldhomotopy-equivalentto then-dimensional sphereishomeomorphicto it. Introduced the method ofhandle-bodiesto solve this and related problems.

Lars Hörmander,

Sweden

Worked inpartial differential equations. Specifically, contributed to the general theory of linear differential operators. The questions go back to one ofHilbert's problemsat the 1900 congress.

John Willard Milnor, 1931年 - ）

United States of America

Proved that a 7-dimensionalspherecan have several differential structures; this led to the creation of the field ofdifferential topology.

Klaus Friedrich Roth,

United Kingdom of Great Britain and Northern Ireland

Solved in 1955 the famous Thue-Siegel problem concerning theapproximationtoalgebraic numbersbyrational numbersand proved in 1952 that a sequence with no three numbers inarithmetic progressionhas zero density (a conjecture ofErdösand Turán of 1935).

René Thom,

France

In 1954 invented and developed the theory of cobordism inalgebraic topology. This classification of manifolds usedhomotopy theoryin a fundamental way and became a prime example of a general cohomology theory.

Kunihiko Kodaira,

Japan

Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically toalgebraic varieties. He demonstrated, by sheaf cohomology, that such varieties areHodge manifolds.

Jean-Pierre Serre,

France

Achieved major results on thehomotopy groupsof 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 ofsheaves.

Laurent Schwartz,

France

Developed the theory ofdistributions, a new notion of generalized function motivated by theDirac delta-functionoftheoretical physics.

Atle Selberg,

Norway

Developed generalizations of thesieve methodsofViggo Brun; achieved major results on zeros of theRiemann zeta function; gave an elementary proof of theprime number theorem(withP. Erdős), with a generalization to prime numbers in an arbitraryarithmetic progression.

Lars Valerian Ahlfors,

Finland

Awarded medal for research on covering surfaces related toRiemann surfacesof inverse functions of entire andmeromorphic functions. Opened up new fields of analysis.

Jesse Douglas,

United States of America

Did important work of the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary.