勒内·托姆 René Thom

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勒内·托姆 René Thom
René Thom.jpeg
勒内·托姆在尼斯, 1970年
Born1923年9月2日
法国蒙贝利亚
Died2002年10月25日
法国伊维特河畔布尔斯
Nationality法国
Alma mater巴黎高等师范学校
Known forDold–Thom theorem
Thom isomorphism
Pontryagin–Thom construction
Thom–Porteous formula
Awards菲尔兹奖 1958年
Scientific career
Fields数学
Institutions斯特拉斯堡大学，约瑟夫·傅里叶大学，高等科学研究所
Thesis球形纤维丛和斯廷罗德平方 (1951年)
Doctoral advisor亨利·嘉当
Doctoral students戴维·特罗特曼


René Frédéric Thom (模板:IPA-fr; 2 September 1923 – 25 October 2002) was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). He received the Fields Medal in 1958.

René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). He received the Fields Medal in 1958.

勒内·托姆René Frédéric Thom（法语：[ʁənetɔm]；1923年9月2日至2002年10月25日）是法国数学家。他以拓扑学家著称，特别是在研究所谓的 奇点理论Singularity theory领域。在广大学术界以及教育程度高的大众眼中，他作为 突变理论Catastrophe theory的创始人而举世闻名，后来该理论由埃里克·克里斯托弗·塞曼Erik Christopher Zeeman深入研究。1958年，他获得了菲尔兹奖。


Biography 个人简介

René Thom was born in Montbéliard, Doubs. He was educated at the Lycée Saint-Louis and the École Normale Supérieure, both in Paris. He received his PhD in 1951 from the University of Paris. His thesis, titled Espaces fibrés en sphères et carrés de Steenrod (Sphere bundles and Steenrod squares), was written under the direction of Henri Cartan. The foundations of cobordism theory, for which he received the Fields Medal at the International Congress of Mathematicians in Edinburgh in 1958, were already present in his thesis.

René Thom was born in Montbéliard, Doubs. He was educated at the Lycée Saint-Louis and the École Normale Supérieure, both in Paris. He received his PhD in 1951 from the University of Paris. His thesis, titled Espaces fibrés en sphères et carrés de Steenrod (Sphere bundles and Steenrod squares), was written under the direction of Henri Cartan. The foundations of cobordism theory, for which he received the Fields Medal at the International Congress of Mathematicians in Edinburgh in 1958, were already present in his thesis.

勒内·托姆出生于杜省的蒙贝利亚。他曾就读于巴黎的圣路易中学LycéeSaint-Louis和巴黎高等师范学校École Normale Supérieure，并于1951年在巴黎大学获得博士学位。在亨利·嘉当Henri Cartan的指导下，他完成了他的论文“ Espaces fibrés en sphères et carrés de Steenrod”（球形纤维丛和斯廷罗德平方）。1958年，勒内·托姆因他提出的配边理论基础，在爱丁堡国际数学家大会上获得菲尔兹奖。该理论其实曾在他早期的论文中提出过。


After a fellowship in the United States, he went on to teach at the Universities of Grenoble (1953–1954) and Strasbourg (1954–1963), where he was appointed Professor in 1957. In 1964, he moved to the Institut des Hautes Études Scientifiques, in Bures-sur-Yvette. He was awarded the Brouwer Medal in 1970, the Grand Prix Scientifique de la Ville de Paris in 1974, and became a Member of the Académie des Sciences of Paris in 1976.

After a fellowship in the United States, he went on to teach at the Universities of Grenoble (1953–1954) and Strasbourg (1954–1963), where he was appointed Professor in 1957. In 1964, he moved to the Institut des Hautes Études Scientifiques, in Bures-sur-Yvette. He was awarded the Brouwer Medal in 1970, the Grand Prix Scientifique de la Ville de Paris in 1974, and became a Member of the Académie des Sciences of Paris in 1976.

在美国获得奖学金后，他继续在格勒诺布尔大学（1953-1954）和斯特拉斯堡大学（1954-1963）任教，并于1957年被任命为教授。1964年，他转而进入伊维特河畔布尔斯Bures-sur-Yvette的高等科学研究院工作。后来于1970年获得了布劳维尔奖章，于1974年获得了巴黎科学奖，且于1976年成为巴黎科学院的成员。


While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972, in which he uses his earlier work on differential topology to develop a general theory of biological form,[1] his academic achievements concern mostly his mathematical work on topology. In the early 1950s it concerned what are now called Thom spaces, characteristic classes, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture, versions of which have been investigated using gauge theory. From the mid 1950s he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom–Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga. While Thom found general recognition among the general public for the popularized version of his work on biology (later developed by Christopher Zeeman), this work struggled to gain traction among natural scientists due to the inaccessibility of its mathematics.[1]

While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972, in which he uses his earlier work on differential topology to develop a general theory of biological form, his academic achievements concern mostly his mathematical work on topology. In the early 1950s it concerned what are now called Thom spaces, characteristic classes, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture, versions of which have been investigated using gauge theory. From the mid 1950s he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom–Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga. While Thom found general recognition among the general public for the popularized version of his work on biology (later developed by Christopher Zeeman), this work struggled to gain traction among natural scientists due to the inaccessibility of its mathematics.

勒内·托姆1968年至1972年间发展的突变理论利用了他先前在微分拓扑上的工作成果，进而发展了生物形式的通用理论。尽管后来因此而广为公众所知，但他的学术成就主要还是涉及在拓扑上的数学研究。在1950年代初，托姆就开始研究诸如 托姆空间Thom spaces， 特征类Characteristic classes， 托姆配边理论Cobordism theory和 托姆横截性定理定理Thom transversality theorem。另一个例子是 托姆猜想Thom conjecture，后已使用 规范理论Gauge theory研究了其形式。从1950年代中期开始，他开始研究奇点理论，其包含了突变理论，在1960年至1969年之间的一系列较深入（当时并不明确）的论文中，他提出了 分层集合论stratified sets和 分层映射理论stratified maps，证明了描述 惠特尼分层集合的局部圆锥结构 local conical structure of Whitney stratified sets的基本 分层同质化定理stratified isotopy theorem，现称为 托姆 - 马瑟同质化定理Thom–Mather isotopy theorem。他在分层集上所做的大部分工作都是为了理解拓扑稳定图的概念而开发的，并最终证明了两个平滑流形之间的拓扑稳定映射集是一个密集集的结果。托姆于1960年在波恩大学发表的关于微分映射图稳定性的演讲，后来由Harold Levine详细记载，并在1969-70年于利物浦大学举行的为期一年的奇点研讨会论文集中发表，该研讨会由C.T.C. Wall编辑。约翰·马瑟John Mather于1970年根据托姆在过去十年中提出的思想，完成了拓扑稳定映射密度的证明。克里斯托弗·吉布森Christopher Gibson，克劳斯·维特穆勒Klaus Wirthmüller，安德鲁·迪·普莱西斯Andrew du Plessis和爱德华·洛伊吉恩加Eduard Looijenga于1976年发表了更详尽的论述。尽管汤姆的生物学著作（后来由克里斯托弗·塞曼Christopher Zeeman继续）的大众化版本获得了公众的普遍认可，但由于其数学的不可及性，这项研究仍在努力吸引自然科学家的注意。


During the last twenty years of his life Thom's published work was mainly in philosophy and epistemology, and he undertook a reevaluation of Aristotle's writings on science. In 1992, he was one of eighteen academics who sent a letter to Cambridge University protesting against plans to award Jacques Derrida an honorary doctorate.[2]

Beyond Thom's contributions to algebraic topology, he studied differentiable mappings, through the study of generic properties. In his final years, he turned his attention to an effort to apply his ideas about structural topography to the questions of thought, language, and meaning in the form of a "semiophysics".

在托姆一生的最后二十年中，他的主要著作涉及的是哲学和认识论，他对亚里士多德的科学著作进行了重新评估。1992年，他是18位向剑桥大学致信的抗议者之一，抗议计划授予雅克·德里达名誉博士学位。


Beyond Thom's contributions to algebraic topology, he studied differentiable mappings, through the study of generic properties. In his final years, he turned his attention to an effort to apply his ideas about structural topography to the questions of thought, language, and meaning in the form of a "semiophysics".

除了对代数拓扑的贡献之外，托姆还通过通有性质来研究可微映射。在他的最后几年里，他将注意力转移到以“ Semiophysics”的形式将有关结构地形的思想应用于思维，语言和含义的问题中去。


Bibliography 参考书目

  • Thom, René (1952), "Espaces fibrés en sphères et carrés de Steenrod" (PDF), Annales Scientifiques de l'École Normale Supérieure, Série 3, 69: 109–182, doi:10.24033/asens.998, MR 0054960
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