# フィールズ賞

フィールズ賞
フィールズ・メダルの表面。アルキメデスの肖像と銘文 ラテン語: TRANSIRE SUUM PECTUS MUNDOQUE POTIRI（己を高め、世界を捉えよ[1]）が刻まれている。カナダの彫刻家によるデザインで[2]、カナダ王室造幣局で鋳造されている[3]。受賞者の名前は縁に刻まれる[4]

フィールズ賞（フィールズしょう）は、若い数学者のすぐれた業績を顕彰し、その後の研究を励ますことを目的に、カナダ数学者ジョン・チャールズ・フィールズ (John Charles Fields, 1863年 - 1932年) の提唱によって1936年に作られたのことである[5][6][7]

## 概要

4年に一度開催される国際数学者会議 (ICM) において、顕著な業績を上げた40歳以下[注釈 1]の数学者（2名以上4名以下）に授与される[5]。ICMで同時に授与される賞としては、ネヴァンリンナ賞ガウス賞チャーン賞などがある。

フィールズ賞は、フィールズ賞選考委員会で決められる[5]グリゴリー・ペレルマンは、2019年時点では受賞を辞退したただ一人の人物である。

### 他の贈賞との比較

2002年には、ノーベル賞により性格の近いアーベル賞が設立された。フィールズ賞とアーベル賞の両方を受賞した人物も存在する。

さらに2014年には、ノーベル賞をも超越する莫大な賞金額を誇る数学ブレイクスルー賞が創設された。

## 受賞者の一覧

1936年オスロ
 「 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. 」
1950年ケンブリッジ
 「 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. 」
1954年アムステルダム
 「 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. 」
1958年エディンバラ
 「 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. 」
1962年ストックホルム
 「 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. 」
1966年モスクワ
 「 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. 」
1970年ニース
 「 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. 」
1974年バンクーバー
 「 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. 」
1978年ヘルシンキ
 「 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. 」
1982年ワルシャワ
 「 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. 」
1986年バークレー
 「 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. 」
1990年京都
 「 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 」
1994年チューリッヒ
 「 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. 」
1998年ベルリン
 「 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. 」
2002年北京
 「 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 」
2006年マドリード
 「 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 」
2010年ハイデラバード[18]
 「 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. 」
2014年ソウル[19]
 「 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. 」
2018年リオデジャネイロ[20]
 「 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. 」
2022年（オンライン開催[注釈 3][21]
 「 For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four. 」
 「 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[22]。 」
 「 For the proof that the ${\displaystyle E_{8}}$ 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]。 」

アメリカ合衆国 14
フランス 14
ロシア
ソビエト連邦を含む）
9
イギリス 9

ウクライナ ベルギー
ドイツ
イラン
イタリア
2
オーストラリア
オーストリア
ブラジル
カナダ
フィンランド
イスラエル
ノルウェー
ニュージーランド
スウェーデン
ベトナム
1

## 脚注

### 注釈

1. ^ 正確な規定は次の通り：受賞年の1月1日より前に40歳の誕生日を迎えたものは候補となれない[8]。ただし厳格な規定が成文化されたのは1966年のICMにおいてである[9]
2. ^ たとえばアティヤはインタビューでそれらの名声と効果の違いについて比較し述べている[17]。また日本でフィールズ賞をとることは、ノーベル賞をとるようなものだが、英国では誰も気にとめてくれないなど、国によって扱われ方に違いがあることにも言及している。
3. ^ フィールズ賞の授賞式はヘルシンキで行われた。

### 出典

1. ^ モナスティルスキー 2013, p. 27.
2. ^ Tropp 1976, p. 181.
3. ^ Riehm 2002, p. 781.
4. ^ Curbera 2009, p. 111.
5. ^ a b c Fields Medal”. 国際数学連合. 2021年12月6日閲覧。
6. ^ 世界大百科事典 第2版『フィールズ賞』
7. ^ a b
8. ^ Curbera 2009, p. 110.
9. ^ Barany 2015, p. 17.
10. ^ “2006 Fields Medals awarded” (PDF). Notices of the American Mathematical Society (American Mathematical Society) 53 (9): 1037-1044. (10 2006) 2021年12月6日閲覧。.
11. ^ 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 (2010年8月19日). 2021年12月6日閲覧。
12. ^ a b Fields Medals Are Concentrated in Mathematical ‘Families’”. SCIENTIFIC AMERICAN. 2021年12月5日閲覧。
13. ^ モナスティルスキー 2013, p. 19.
14. ^ a b 「数学のノーベル賞」で女性初受賞、イラン出身の米大教授”. AFPBB News (2014年8月13日). 2021年12月6日閲覧。
15. ^ モナスティルスキー 2013, p. 34.
16. ^
17. ^ マイケル・F. アティヤ 『数学とは何か アティヤ 科学・数学論集』志賀浩二 訳、朝倉書店、2010年11月25日、148-149頁。ISBN 978-4-254-10247-5
18. ^ ICM 2010, p. 23.
19. ^ International Congress of Mathematicians (2014年). “Awards”. 2021年12月6日閲覧。
20. ^ International Mathematical Union (IMU) (2018年). “Fields Medals 2018”. 2021年12月6日閲覧。
21. ^ International Mathematical Union (IMU) (2022年). “Fields Medals 2022”. 2022年7月5日閲覧。
22. ^ a b 【ざっくり分かる】フィールズ賞2022 ，どんな人がどんな理由で受賞した？”. 2022年7月30日閲覧。