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.
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.
クラウス・フリードリッヒ・ロス （Klaus Friedrich Roth, 1925年 - 2015年） イギリス（ヴロツワフ生まれ。当時はドイツ領、現在はポーランド領）
「
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.
マイケル・フランシス・アティヤ（Michael Francis Atiyah, 1929年 - 2019年） イギリス
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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.
」
広中平祐（Heisuke Hironaka, 1931年 - ） 日本
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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.
」
チャールズ・ルイス・フェファーマン（Charles Louis Fefferman, 1949年 - ）アメリカ合衆国
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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.
」
ウィリアム・サーストン（William P. Thurston, 1946年 - 2012年）アメリカ合衆国
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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.
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.
」
森重文 （Shigefumi Mori, 1951年 -） 日本
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for the proof of Hartshorne’s conjecture and his work on the classification of three-dimensional algebraic varieties.
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.
リチャード・ボーチャーズ （Richard E. Borcherds, 1959年 - ） イギリス（南アフリカ出身）
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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.
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.
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 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 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.
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.
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。
For the proof that the 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。
Curbera, Guillermo P. (2009). Mathematicians of the World, Unite!: The International Congress of Mathematicians—A Human Endeavor. A. K. Peters. p. 110–118. doi:10.1201/b10584. ISBN 978-1-56881-330-1. MR2499757. Zbl 1166.01001. https://books.google.co.jp/books?id=9uDqBgAAQBAJ
Monastyrsky, Michael (1997). Modern Mathematics in the Light of the Fields Medals. A. K. Peters. ISBN 1-56881-065-2. MR1427488. Zbl 0874.01014
Tropp, Henry S. (1976). “The origins and history of the Fields Medal”. Historia Math.3: 167-181. doi:10.1016/0315-0860(76)90033-1. MR0505005. Zbl 0326.01007.
R. Bhatia et al., ed (2010). Proceedings of the International Congress of Mathematicians: Hyderabad 2010. I. World Scientific Publishing. ISBN 978-981-4324-31-1. MR2840854. https://books.google.co.jp/books?id=GFE1vx2pynMC
関連文献
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
Riehm, Elaine McKinnon (2010), The Fields Medal: Serendipity and J. L. Synge, Fields Notes10: 1–2.
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. https://books.google.co.jp/books?id=h8S_AwAAQBAJ
lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis.
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