|公式サイト||International Mathematical Union (IMU) Details|
名前の読み（名前の綴り、生年 - 没年）、国籍、受賞理由（英語）の順。国籍は受賞時の国名で記す。
|「||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.||」|
|「||Did important work of the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary.||」|
|「||Developed the theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics.||」|
|「||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.||」|
|「||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.||」|
|「||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.||」|
|「||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 Erdös and Turán of 1935).||」|
|「||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.||」|
|「||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.||」|
|「||Proved that a 7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology.||」|
|「||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".||」|
|「||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.||」|
|「||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 "Tohoku paper"||」|
|「||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.||」|
|「||Generalized the Gelfond-Schneider theorem (the solution to Hilbert's seventh problem). From this work he generated transcendental numbers not previously identified.||」|
|「||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.||」|
|「||Made important advances in topology, the most well-known being his proof of the topological invariance of the Pontrjagin classes of the differentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces.||」|
|「||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.||」|
|「||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.||」|
|「||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.||」|
|「||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.||」|
|「||Contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalizations of classical (low-dimensional) results.||」|
|「||Provided innovative analysis of the structure of Lie groups. His work belongs to combinatorics, differential geometry, ergodic theory, dynamical systems, and Lie groups.||」|
|「||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.||」|
|「||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.||」|
|「||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.||」|
|「||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.||」|
|「||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.||」|
|「||Using methods of arithmetic algebraic geometry, he received medal primarily for his proof of the Mordell Conjecture.||」|
|「||Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensional Poincaré Conjecture.||」|
|「||For his work on quantum groups and for his work in number theory.||」|
|「||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.||」|
|「||for the proof of Hartshorne’s conjecture and his work on the classification of three-dimensional algebraic varieties.||」|
|「||proof in 1981 of the positive energy theorem in general relativity||」|
|「||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.||」|
|「||... such nonlinear partial differential equation simply 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 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||」|
|「||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.||」|
|「||For his solution to the restricted Burnside problem.||」|
|「||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 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.||」|
|「||contributions to four problems of geometry||」|
|「||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.||」|
|「||Laurent Lafforgue has been awarded the Fields Medal for his proof of the Langlands correspondence for the full linear groups
GLr (r≥1) over function fields.
|「||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||」|
|「||for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory||」|
|「||for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow||」|
|「||for his contributions bridging probability, representation theory and algebraic geometry||」|
|「||for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory||」|
|「||For his results on measure rigidity in ergodic theory, and their applications to number theory.||」|
|「||For the proof of conformal invariance of percolation and the planar Ising model in statistical physics.||」|
|「||For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods.||」|
|「||For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.||」|
|「||for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.||」|
|「||for his profound contributions to dynamical systems theory have changed the face of the field, using the powerful idea of renormalization as a unifying principle.||」|
|「||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.||」|
|「||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.||」|
|「||For the proof of the boundedness of Fano varieties and for contributions to the minimal model program.||」|
|「||For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability.||」|
|「||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.||」|
|「||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.||」|
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- ^ Tropp 1976, p. 181.
- ^ Riehm 2002, p. 781.
- ^ Curbera 2009, p. .
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- ^ 世界大百科事典 第2版『フィールズ賞』
- ^ a b モナスティルスキー 2013.
- ^ Curbera 2009, p. .
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- ^ “ICM2010におけるフィールズ賞を含むIMU各賞の受賞者について”. 日本数学会. 2015年4月27日閲覧。
- ^ モナスティルスキー 2013, p. 19.
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- ^ モナスティルスキー 2013, p. 34.
- ^ 渡辺 & 上総 1985.
- ^ ICM 2010, p. .
- ^ International Congress of Mathematicians (2014年). “Awards”. 2017年11月13日閲覧。
- ^ International Mathematical Union (IMU) (2018年). “Fields Medals 2018”. 2018年8月2日閲覧。
- Atiyah, Michael; Iagolnitzer, Daniel (1997). Fields Medalists' Lectures. World Scientific Series in 20th Century Mathematics. 5. World Scientific Publishing. doi:10.1142/3445. ISBN 981-02-3117-2. MR1622945.
- Barany, Michael J. (2015), “The myth and the medal”, Notices Amer. Math. Soc. 62: 15–20, MR3308164, Zbl 1338.01009
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- Riehm, Elaine McKinnon; Hoffman, Frances (2011). Turbulent Times in Mathematics: The Life of J.C. Fields and the History of the Fields Medal. AMS. doi:10.1090/mbk/080. ISBN 978-0-8218-6914-7. MR2850575. Zbl 1247.01047.
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