勒内·托姆René Thom

勒内·托姆 以其对突变论的发展而闻名，突变论是对连续作用产生不连续结果的数学处理。

从 1931 年起，托姆 在他出生的城市蒙贝利亚尔上小学，他的父母在那里开店。正是在这所小学，托姆 首次展现出他的学术潜力，赢得了奖学金。他在蒙贝利亚尔的居维叶中学就读，并于 1940 年在贝桑松获得初等数学业士学位。然而，他的生活即将被第二次世界大战打乱。

托姆的父母把他和弟弟送到南方以躲避冲突，尽管他们自己仍留在蒙贝利亚尔。托姆和弟弟最终到达了瑞士。他在[6]中写道：-

我们在那里受到的热烈欢迎，所有那些在路边提供食物和水的人们，至今仍让我感动。

在罗蒙附近帮助收割后，托姆回到法国，被带到里昂，在那里他与母亲的一位朋友同住。在里昂期间，他继续学业，于1941年6月获得哲学学士学位。此后，他回到蒙贝利亚尔父母家中，但很快又到巴黎继续学业。

托姆 在巴黎的圣路易中学就读，并申请进入巴黎高等师范学校，但在1942年未能获得入学资格。他决心利用在巴黎高等师范学校接受大学教育的机会，于1943年再次申请，这一次他[6]：-

...成功了（但并非特别出色）！

在巴黎高等师范学校，日子很艰难，因为巴黎被德军占领。然而，对托姆来说，数学上这是一个激动人心的时期，他深受亨利·嘉当和尼古拉·布尔巴基数学方法的影响。第二次世界大战结束时，托姆仍在巴黎高等师范学校学习，[6]：-

...最后一年，在“胜利”之后，是一个开放的一年，带来了再次充分生活的感觉。对于这种重生，我能回忆起一种难以控制的自由感。

1946年，托姆从巴黎高等师范学校毕业，随后搬到斯特拉斯堡，接受了法国国家科学研究中心的研究职位，以便能继续与亨利·嘉当合作。在那里，他受到了包括夏尔·埃雷斯曼和让-路易·科斯居尔在内的其他人的影响。他的博士学位由亨利·嘉当指导，于1951年授予，学位论文题为Fibre spaces in spheres and Steenrod squares。论文的工作是在斯特拉斯堡进行的，但托姆将其提交给了巴黎。配边理论的基础，后来使托姆获得了菲尔兹奖，已经出现在他的博士论文中。

托姆 于 1951 年获得奖学金，得以前往美国，他在 [6] 中讲述了这如何使他得以见到 阿尔伯特·爱因斯坦、赫尔曼·外尔 和 诺曼·斯廷罗德，并参加卡拉比和 小平邦彦 的讨论班。托姆 回到法国，1953-54 年在格勒诺布尔任教，然后从 1954 年到 1963 年在斯特拉斯堡任教。他于 1957 年被任命为教授。

托姆最为人所知的是作为突变理论的发明者，但他在研究突变理论之前，早期的工作已经使他闻名。他在拓扑学方面的工作，特别是关于示性类、配边理论和托姆横截定理，使他在1958年被授予约翰·查尔斯·菲尔兹奖章。然而，托姆觉得在某种意义上他不配获得这一荣誉[6]：-

……我的印象是，稍后不久所做的工作在深度和睿智上都超过了我，其作者即使不比我更配得上这枚奖章，也完全配得上（比如与我共同获奖的克劳斯·罗特）。我还想到巴里·马聚尔对阿图尔·舍恩弗利斯猜想的证明：$\mathbb{R}^{n}$中每个具有正则边界的球面$S^{n-1}$都是一个$n$球的边界。更不用说约翰·米尔诺对异种球面的发现。

海因茨·霍普夫在爱丁堡向托姆颁发了菲尔兹奖章，他在颁奖致辞中指出了托姆理论的重要性：-

……他的基本思想——我曾谈到过其宏大的简洁性——具有非常几何和直观的性质。这些思想极大地丰富了数学，而一切似乎都表明，托姆思想的影响——无论它们是在已知著作中还是在即将问世的著作中得到表达——远未穷尽。

然而，菲尔兹奖章的颁发给了托姆选择新研究方向的自由[1]：-

托姆说，菲尔兹奖章给他带来了选择自己想做什么研究的自由，而对他来说这是至关重要的。他开始把整个科学作为自己的画布。从设计实验和预测结果的意义上说，他不是一位理论或实验科学家，而是一位科学哲学家，撰写关于科学中需要发生的长期未来发展的文章。

1964年，他搬到了比叙尔伊韦特的法国高等科学研究所。然而，正如他在[6]中所解释的，这促使了方向的改变：-

与我的同事亚历山大·格罗滕迪克的关系对我来说不那么愉快。他在技术上的优势令人难以招架。他的讨论班吸引了整个巴黎的数学界，而我却拿不出什么新东西。这使我离开了纯粹的数学世界，去研究更一般的概念，比如形态发生理论，这个课题更让我感兴趣，并把我引向一种非常一般的“哲学”生物学。

托姆的理论试图以微分学无法做到的方式来描述这样一些情形：逐渐变化的力导致所谓的突变，即突然的变化。该理论在物理科学、生物科学以及社会科学中有着广泛的应用。该理论由托姆在Structural Stability and Morphogenesis（1972）中提出，此后由许多数学家加以发展。然而，托姆在[6]中撰文解释了为什么曾是marked by enormous popular success的理论已经失宠：——

突变论已死，这是事实。但可以说，它是死于自身的成功。它被从解析（或代数）模型推广到仅仅光滑的模型所拖垮。因为一旦人们清楚该理论不允许定量预测，所有优秀的头脑……就认定它毫无价值。归根结底，这一推广源自B Malgrange对预备定理的推广。

托姆于1974年被授予巴黎市科学大奖。他于1990年成为伦敦数学会的荣誉成员。

作为讨论班演讲者，托姆可能让人难以听懂[1]：——

[他]的讨论班常常令人困惑，因为他的思路往往跳跃向前，让听众去填补空白。

然而，与他进行数学交谈可能是一种美妙的体验[1]：-

……与他的一对一交谈非常精彩：如果被要求填补一个空白，他常常会揭示出一座金矿。他表现出温和的机智、极大的怀疑精神，以及对人类境况的平静的乐趣。他对天下万事都有独到的想法。他的著作常常具有挑衅性，以激发读者看到真理。

René Thom is known for his development of catastrophe theory, a mathematical treatment of continuous action producing a discontinuous result.

From 1931 Thom attended Primary School in Montbéliard, the town of his birth in which his parents were shopkeepers. It was at this primary school that Thom first showed his academic potential winning a scholarship. He attended Collège Cuvier at Montbéliard and received his baccalaureate in elementary mathematics from Besançon in 1940. However his life was about to be disrupted by World War II.

Thom's parents sent him and his brother south to avoid the conflict although they themselves remained in Montbéliard. Thom and his brother eventually reached Switzerland. He writes in [6]:-

The surprising warmth with which we were welcomed there, all those people offering food and drink at the roadside, still fills me with emotion.

After helping with the harvest near Romont, Thom returned to France being taken to Lyon where he lived with a friend of his mother. While in Lyon he continued his education, receiving his baccalaureate in philosophy in June 1941. After this he returned to his parents home in Montbéliard but was soon in Paris again to continue his education.

Thom attended the Lycée Saint-Louis in Paris and applied to enter the École Normale Supérieure but failed to gain entrance in 1942. Determined to take advantage of a university education at the École Normale Supérieure, he applied again in 1943 and this time he was [6]:-

... successful (but not brilliantly so)!

At the École Normale Supérieure times were difficult as Paris was occupied by the German forces. However, mathematically it was an exciting time for Thom who was to be strongly influenced by Henri Cartan and the Bourbaki approach to mathematics. World War II ended while Thom was still studying at the École Normale Supérieure and [6]:-

... the last year, after the 'victory', was a year of opening, bringing with it the impression of once more living life to the full. Of this rebirth I can recall a sensation of freedom that I found hard to control.

In 1946 Thom graduated from the École Normale Supérieure and then moved to Strasbourg, taking a CNRS research post, so that he could continue to work with Henri Cartan. There he was influenced by others including Ehresmann and Koszul. His doctorate, supervised by Henri Cartan, was awarded in 1951 for a thesis entitled Fibre spaces in spheres and Steenrod squares. The work of the thesis was carried out in Strasbourg but Thom presented it to Paris. The foundations of the theory of cobordism, for which Thom later received a Fields Medal, already appear in his doctoral thesis.

Thom was awarded a fellowship to allow him to travel to the United States in 1951 and he relates in [6] how this enabled him to meet Einstein, Weyl, and Steenrod, and to attend the seminars of Calabi and Kodaira. Thom returned to France and taught at Grenoble in 1953-54, then at Strasbourg from 1954 until 1963. He was appointed a professor in 1957.

It is as the inventor of catastrophe theory that Thom is best known but his earlier work had made him well known before he worked on catastrophe theory. His work on topology, in particular on characteristic classes, cobordism theory and the Thom transversality theorem led to his being awarded a Fields medal in 1958. However, Thom feels that in some sense he did not deserve the honour [6]:-

... I have the impression that work was done just a little while later that was greater in depth and sagacity than mine and whose authors were quite as deserving, if not more so, of the medal (such as my co-medallist Klaus Roth). I am thinking too of Barry Mazur's demonstration of the Schönflies conjecture: Every sphere $S^{n-1}$ in $\mathbb{R}^{n}$ with regular boundary is the boundary of an $n$-ball. Not to mention the discovery by Milnor of exotic spheres.

Hopf, who awarded the Fields Medal to Thom in Edinburgh, pointed in his presentation address to the importance of Thom's theory:-

... his basic ideas, the grand simplicity of which I have talked of, are of a very geometric and intuitive nature. These ideas have significantly enriched mathematics, and everything seems to indicate that the impact of Thom's ideas - whether they find their expression in the already known or in forthcoming works - is not exhausted by far.

However, the award of the Fields Medal gave Thom freedom to choose a new research direction [1]:-

Thom said that the Fields Medal had brought him the freedom to choose what research he wanted to do, and for him that was essential. He began to take the whole of science as his canvas. He was not a theoretical or experimental scientist, in the sense of designing experiments and predicting results, but rather a philosopher of science, writing about the long-term future developments in the sciences that needed to occur.

In 1964 he moved to the Institut des Hautes Études Scientifiques at Bures-sur-Yvette. However this prompted a change in direction as he explains in [6]:-

Relations with my colleague Grothendieck were less agreeable for me. His technical superiority was crushing. His seminar attracted the whole of Parisian mathematics, whereas I had nothing new to offer. That made me leave the strictly mathematical world and tackle more general notions, like the theory of morphogenesis, a subject which interested me more and led me towards a very general form of 'philosophical' biology.

Thom's theory is an attempt to describe, in a way that is impossible using differential calculus, those situations in which gradually changing forces lead to so-called catastrophes, or abrupt changes. The theory has widespread application in the physical and biological sciences and in the social sciences. Presented by Thom in Structural Stability and Morphogenesis (1972), the theory has since been developed by many mathematicians. However, writing in [6], Thom explains why the theory which was marked by enormous popular success has fallen from favour:-

It is a fact that catastrophe theory is dead. But one could say that it died of its own success. It was brought down by the extension from analytical (or algebraic) models to models that were only smooth. For as soon as it became clear that the theory did not permit quantitative prediction, all good minds ... decided it was of no value. When it comes down to it, this extension resulted from B Malgrange's extension of the preparation theorem.

Thom was awarded the Grand Prix Scientifique de la Ville de Paris in 1974. He was made an honorary member of the London Mathematical Society in 1990.

As a seminar speaker, Thom could be difficult to follow [1]:-

[H]is seminars were often confusing, because his mind tended to leap ahead, leaving the audience to fill in the gaps.

However, mathematical conversations with him could be a wonderful experience [1]:-

... one-to-one conversations with him were marvellous: if challenged to fill a gap he would often reveal a goldmine. He showed a gentle wit, a great scepticism, and a quiet amusement at the human condition. He had original ideas about everything under the sun. His writings were often provocative in order to stimulate the reader into seeing the truth.