亨利·嘉当Henri Cartan

亨利·嘉当是埃利·嘉当和Marie-Louise Bianconi的儿子。亨利·嘉当五岁时，他的父亲被任命为索邦大学的讲师，全家从南锡搬到了巴黎。他在那里就读于布丰中学和凡尔赛的奥什中学。他有一个姐姐和两个弟弟让和路易，两人都悲惨地去世了。让是一位作曲家，25岁时死于肺结核；路易是一位物理学家，1942年被德国人逮捕，1943年2月被驱逐到德国，在被囚禁15个月后被处决。

亨利·嘉当成长在一个音乐非常重要的家庭中，所有孩子都演奏音乐。尽管亨利·嘉当的父亲是一位数学家，但他并没有试图影响孩子们走上数学职业道路。虽然在他成长过程中总是愿意回答亨利·嘉当关于数学的问题，但他从未比其他科目更强调数学。尽管如此，亨利·嘉当从小就知道自己会成为一名数学家。

完成中学教育后，亨利·嘉当在巴黎高等师范学校学习，并很快与比他高一级的安德烈·韦伊成为朋友。亨利·嘉当在高师的老师包括茹利亚和他的父亲埃利·嘉当。高师的学生还必须参加索邦大学的普通课程，因此亨利·嘉当也在那里学习。他的博士研究由保罗·蒙泰尔指导，后者的研究兴趣是复变函数解析理论，亨利·嘉当于1928年获得Docteur ès Sciences mathématiques。获得博士学位后，亨利·嘉当于1928年至1929年在卡昂中学任教，然后于1929年至1931年在里尔大学任教。

正是亨利·嘉当的朋友安德烈·韦伊建议他研究多复变解析函数，而安德烈·韦伊向他介绍了康斯坦丁·卡拉西奥多里的工作。亨利·嘉当于1930年发表了Les transformations analytiques des domaines cerclés les uns dans les autresⓉ（有界空间之间的解析变换），由于这篇论文包含了Heinrich Behnke所证明结果的推广，他应Behnke的邀请于1931年5月访问德国，并在Behnke任教的威斯特法伦明斯特做了一系列讲座。在那里他遇到了Behnke的助手Peter Thullen，他们开始合作，成果是1932年在Mathematische Annalen上发表的联合论文Zur Theorie der Singularitäten der Funktionen mehrerer Komplexen VeränderlichenⓉ（关于多复变函数奇点理论）。Thullen（尽管他不是犹太人）在1933年纳粹上台后离开了德国，因此合作终止。亨利·嘉当确实在第二次世界大战开始前第二次访问了威斯特法伦明斯特的Behnke，接受了1937年去那里的邀请。

亨利·嘉当和他的父亲埃利·嘉当合写的一篇论文Les transformations des domaines cerclés bornésⓉ（有界空间之间的变换）于1931年发表。大多数时候这两位数学家独立工作，但在这篇论文中，他们能够利用埃利·嘉当在李群方面的专长来解决亨利·嘉当感兴趣的问题。这篇联合论文发表的同一年，亨利·嘉当离开了里尔的职位，从1931年11月开始，他在斯特拉斯堡大学任职。他于1935年9月14日与Nicole Antoinette Weiss结婚。他们有两个儿子和三个女儿：让、弗朗索瓦丝、艾蒂安、米蕾耶和苏珊娜。

亨利·嘉当数学生涯中非常重要的一部分被尼古拉·布尔巴基占据了。这个自称“尼古拉·布尔巴基”的数学家小组的第一次会议于1935年1月14日举行。在[8]中，亨利·嘉当解释了尼古拉·布尔巴基的背景：-

[A]第一次世界大战之后，法国的科学家，我指的是优秀的科学家，不多了，因为他们大多数都阵亡了。我们是战后第一代。在我们之前是一片空白，必须把一切重新建立起来。我的一些朋友去了国外，尤其是去了德国，观察那里正在做什么。这是数学复兴的开端。这要归功于像安德烈·韦伊、谢瓦莱、de Possel这样的人……。正是这些人，响应安德烈·韦伊的倡议，聚集在一起组成了布尔巴基小组。

让·勒雷和Paul Dubreil参加了1935年1月的会议，但在1935年7月小组正式确定成员之前就退出了。除了亨利·嘉当之外，在那次7月会议上尼古拉·布尔巴基的创始成员还有安德烈·韦伊、让·迪厄多内、索勒姆·曼德尔布罗伊特、谢瓦莱、René de Possel和让·戴尔萨特。亨利·嘉当描述了小组的运作方式[8]：-

我们经常意见不合，经常激烈争论——但我们仍然是好朋友。每个主题都指定一位“rédacteur”（撰写人）。之后，他的rédaction（撰写稿）被大声朗读并彻底审查。下一位“rédacteur”得到相应的指示，如此继续。每一章最多可能有九份rédaction。但到最后，每个人都fatigué——累了。让·迪厄多内会说：“现在完成了。我来写最后一份rédaction。”他确实这么做了。最终，尽管似乎不可能达成完全一致，但还是达成了一致。但这需要时间。就团队合作而言，这也许不是最好的方式，但这就是我们采取的方式。

第二次世界大战爆发时，亨利·嘉当正在斯特拉斯堡大学任教[8]：-

但在1939年9月，斯特拉斯堡的居民不得不疏散。大学迁到了克莱蒙费朗，我在那里教了一年书，之后于1940年11月在巴黎被任命为索邦大学教授（事实上，我负责高等师范学校的数学学生）。整个战争期间，我都不被允许去我在斯特拉斯堡的公寓。有一天，Behnke提出试着取回我留在那里的一些数学论文。他真的去了斯特拉斯堡，但没有成功。他又试了一次，成功了。他设法拿到了一些文件，把它们留在了弗莱堡大学的图书馆。1945年，驻德法军的一些成员碰巧在那里发现了它们，并把它们还给了我。

必须记住，战争时期对亨利·嘉当家族来说是一个巨大的悲剧。亨利·嘉当的兄弟路易是抵抗运动的成员，在法国与占领的德国军队作战。1943年2月他被捕后，家人再没有收到任何消息，但他们担心最坏的情况。直到1945年5月，他们才得知他于1943年12月被纳粹斩首。1945年底，第二次世界大战结束后，亨利·嘉当回到斯特拉斯堡大学，在那里又教了两年。1946年11月，他访问了黑森林的奥伯沃尔法赫研究所时，与德国同事重新建立了联系[8]：-

天气非常冷；有雪和冰。我见到了威廉·聚斯教授（奥伯沃尔法赫的创始人）和威廉·聚斯夫人，还有Heinrich Behnke。我记得他们让我弹钢琴。那是一架漂亮的钢琴——那里有两架钢琴。奥伯沃尔法赫的老城堡已经不存在了。之后我又访问了奥伯沃尔法赫几次。

亨利·嘉当在1942年曾被邀请去美国，但他决定为了家人，特别是当时已年迈的父亲，必须留在法国。1948年1月，他应安德烈·韦伊邀请访问芝加哥，并收到邀请去哈佛大学访问四个月，从1948年2月到5月。他于1947年12月飞抵美国，在纽约机场由塞缪尔·艾伦伯格迎接。他与塞缪尔·艾伦伯格的主要合作是1956年首次出版的Homological Algebra一书。这是一部经典著作，在近半个世纪的时间里对该领域产生了深远的影响。

我们在上文解释过，亨利·嘉当于1940年11月在巴黎被任命为索邦大学教授。他从那时起在巴黎任教直到1969年，然后从1970年到1975年在奥赛的巴黎南大学任教。他于1975年退休。在巴黎高等师范学校，当让-皮埃尔·塞尔是他的博士生之一时，他创办了亨利·嘉当讨论班。正是让-皮埃尔·塞尔建议把这些讨论班整理出版，1948年至1964年间出版了十五种由亨利·嘉当撰写的巴黎高等师范学校讨论班记录。

亨利·嘉当研究解析函数、层论、homological theory、代数拓扑学和potential theory，在所有这些领域都做出了重要发展。他的部分工作由R O Wells Jr在评论[2]时置于背景中：-

多复变函数论从世纪之交后不久Hartogs、Levi和儒勒·昂利·庞加莱的工作中的初创阶段，发展到当前作为现代数学中心领域的地位，正如其前身单复变函数论在19世纪所经历的那样。这一发展中的核心人物是亨利·嘉当，他从1920年代开始在该领域的系列论文涉及与罗尔夫·内万林纳理论相关的基本问题、将约斯塔·米塔格-莱弗勒和卡尔·魏尔斯特拉斯定理推广到多变量函数、与biholomorphic映射和双全纯等价问题有关的课题、全纯域和全纯凸性等。1930年至1950年该理论的主要发展来自亨利·嘉当及其在法国的学派、 Behnke在明斯特的学派，以及日本的冈洁。直到那时为止的核心思想在1950年代初亨利·嘉当的讨论班中被综合起来，这些对接下来几代数学家产生了非常大的影响。亨利·嘉当的成就广泛，他以非凡的方式通过他的写作、教学、讨论班和学生影响了数学。

在[2]的序言中，Remmert和让-皮埃尔·塞尔写道：-

读者应当注意，这些卷册并未完全反映亨利·嘉当的工作，其中很大一部分还包含在他的十五次ENS讨论班（1948—1964）以及他与塞缪尔·艾伦伯格合著的《同调代数》一书中。特别是，若不研究他1950/51、1951/52和1953/54年的讨论班，就无法体会亨利·嘉当对层论、Stein流形和解析空间的贡献的重要性。尽管如此，我们相信，全世界的数学家都会欢迎这位数学家的《全集》问世，他的写作与教学对我们这一代人产生了如此大的影响。

我们还应提到亨利·嘉当工作的另一个方面，它与政治有关，尤其是支持人权。1974年出现了一个事件，当时俄罗斯当局把数学家Leonid Plyushch关进了一所特殊精神病院。Andrei Sakharov指出这是一项政治行为，亨利·嘉当于是开始了一场争取Plyushch获释的艰苦运动。1974年国际数学家大会在温哥华举行，这提供了一个机会，为Plyushch争取到广泛的国际支持，有一千人签署了要求释放他的请愿书。大会之后，亨利·嘉当在建立Comité des Mathématiciens方面发挥了主要作用，以特别支持Plyushch，并支持所有持不同政见的数学家。1976年1月，苏联当局释放了Plyushch，这是亨利·嘉当和Comité des Mathématiciens的一大成功。但该委员会在这次成功之后并未停止。它还支持了其他因政治观点而遭受迫害的数学家，例如乌拉圭数学家José Luis Massera。由于他在援助持不同政见者方面的杰出工作，亨利·嘉当获得了纽约科学院颁发的Pagels奖。

亨利·嘉当 是巴黎 Académie des Sciences 以及欧洲、美国和日本其他科学院的成员。他当选为 丹麦皇家科学院（1962 年）、Royal Society of London（1971 年）、American Academy（1950 年）、华盛顿的 国家科学院（1972 年）、比利时皇家科学院（1978 年）、Akademie der Wissenschaften Göttingen（1971 年）、马德里皇家科学院（1971 年）、巴伐利亚科学院（1974 年）、日本科学院（1979 年）、芬兰科学院（1979 年）、Royal Swedish Academy of Sciences（1981 年）、Polish Academy of Sciences（1985 年）、Russian Academy（1999 年）的成员。授予他的众多荣誉中包括 1959 年 伦敦数学会 的荣誉会员。他获得了多所大学的荣誉博士学位，包括苏黎世联邦理工学院（1955 年）、明斯特（1952 年）、奥斯陆（1961 年）、萨塞克斯（1969 年）、剑桥（1969 年）、斯德哥尔摩（1978 年）、牛津（1980 年）、萨拉戈萨（1985 年）和雅典（1992 年）。他还获得了国家科学研究中心金质奖章（1976 年）、1980 年沃尔夫数学奖，并于 1989 年被授予荣誉军团司令勋章。

Henri Cartan is the son of Élie Cartan and Marie-Louise Bianconi. When Henri was five years old, his father was appointed as a lecturer at the Sorbonne and the family moved from Nancy to Paris. There he attended the Lycée Buffon and the Lycée Hoche in Versailles. He had a sister and two younger brothers Jean and Louis who both died tragically. Jean, a composer, died of tuberculosis at the age of 25 while Louis, a physicist, was arrested by the Germans in 1942, deported to Germany in February 1943, and executed after 15 months in captivity.

Henri grew up in a home in which music was very important with all the children playing music. Despite Henri's father being a mathematician, he did not try to influence his children towards a career in mathematics. Although always prepared to answer Henri's questions about mathematics as he was growing up, he never emphasised the subject more than others. Despite this Henri always knew from a very young age that he would be a mathematician.

After completing his school education Henri studied at the École Normale Supérieure in Paris and soon became friendly with André Weil who was one year ahead. Among Henri Cartan's teachers at the École Normale were Gaston Julia and his father Élie Cartan. The students at the École Normale also had to attend general courses at the Sorbonne, so Henri studied there too. His doctoral studies were supervised by Paul Montel, whose research interests were the theory of analytic functions of a complex variable, and Cartan received his Docteur ès Sciences mathématiques in 1928. After being awarded his doctorate, Cartan taught at the Lycée Caen from 1928 to 1929, then at the University of Lille from 1929 to 1931.

It had been Cartan's friend André Weil who had suggested that he work on analytic functions of several complex variables and Weil told him about the work of Carathéodory. Cartan published Les transformations analytiques des domaines cerclés les uns dans les autres Ⓣ in 1930 and, since this paper contained generalisations of results proved by Heinrich Behnke, he was invited by Behnke to visit Germany in May 1931 and give a series of lectures at Münster in Westphalen where Behnke taught. While there he met Peter Thullen, Behnke's assistant, and they began a collaboration which resulted in the joint paper Zur Theorie der Singularitäten der Funktionen mehrerer Komplexen Veränderlichen Ⓣ published in Mathematische Annalen in 1932. Thullen (although he was not Jewish) left Germany after the Nazis came to power in 1933 so the collaboration came to an end. Cartan did visit Behnke in Münster in Westfalen for a second time before the start of World War II, accepting an invitation to go there in 1937.

A joint paper written by Henri and his father Élie Cartan, Les transformations des domaines cerclés bornés Ⓣ appeared in 1931. Most of the time the two mathematicians worked independently, but for this paper they were able to use Élie Cartan's expertise on Lie groups to tackle questions that Henri had been interested in. The year in which this joint paper appeared was the one when Cartan left his position in Lille and, starting in November 1931, he took up a post at the University of Strasbourg. He married Nicole Antoinette Weiss on 14 September 1935. They had two sons and three daughters: Jean, Francoise, Étienne, Mireille, and Suzanne.

A very important part of Cartan's mathematical life was taken up with Bourbaki. The first meeting of the group of mathematicians who called themselves "Bourbaki" took place on 14 January 1935. In [8] Cartan explains the background to Bourbaki:-

[A]fter the First World War, there were not so many scientists, I mean good scientists, in France, because most of them had been killed. We were the first generation after the war. Before us there was a vacuum, and it was necessary to make everything new. Some of my friends went abroad, notably to Germany, and observed what was being done there. This was the beginning of a mathematical renewal. It was due to such people as Weil, Chevalley, de Possel .... The same people, responding to André Weil's initiative, came together to form the Bourbaki group.

Leray and Paul Dubreil attended the January 1935 meeting, but dropped out before membership of the group was finalised in July of 1935. In addition to Henri Cartan the founding members of Bourbaki at that July meeting were André Weil, Jean Dieudonné, Szolem Mandelbrojt, Claude Chevalley, René de Possel, and Jean Delsarte. Cartan described how the group operated [8]:-

We often disagreed, we often had big arguments - but we remained good friends. For each subject, a "rédacteur" was appointed. Later, his rédaction was read aloud and thoroughly examined. The next "rédacteur" was given the appropriate instructions, and so on. For each chapter there could be up to nine rédactions. But in the end, everybody was fatigué - tired. And Dieudonné would say, "It is finished now. I shall write the last rédaction." Which he did. And eventually, although it seemed to be impossible to reach a complete agreement, there was an agreement. But it took time. It is perhaps not the best way in terms of teamwork, but that was the way we took.

Cartan was teaching at the University of Strasbourg when World War II started [8]:-

But in September 1939, the inhabitants of Strasbourg had to be evacuated. The university was displaced to Clermont-Ferrand, where I taught for a year before I was appointed professor at the Sorbonne in Paris, in November 1940 (in fact, I was to be in charge of the mathematics students at the École Normale). Throughout the war I was not permitted to go to my apartment in Strasbourg. One day, Behnke offered to try and retrieve some mathematical papers I had left there. He actually went to Strasbourg, but to no avail. He tried again and succeeded. He managed to get hold of some documents, which he left with the library of the University of Freiburg. In 1945, some members of the French Forces in Germany happened to find them there and returned them to me.

It has to be remembered that the period of the war was one of great tragedy for the Cartan family. Henri's brother Louis was a member of the Resistance fighting in France against the occupying German forces. After his arrest in February 1943 the family received no further news but they feared the worst. Only in May 1945 did they learn that he had been beheaded by the Nazis in December 1943. Late in 1945, when World War II had ended, Cartan returned to the University of Strasbourg and taught there for a further two years. He renewed contacts with his German colleagues when he visited the Research Institute in Oberwolfach, in the Black Forrest, in November 1946 [8]:-

It was very cold; there was snow and ice. I saw Professor Süss (the founder of Oberwolfach) and Frau Süss, and also Heinrich Behnke. I remember they asked me to play the piano. It was a beautiful piano - there were two pianos there. The old château at Oberwolfach doesn't exist anymore. I visited Oberwolfach several times after that.

Cartan had been invited to the United States in 1942 but he decided that he had to remain in France for the sake of his family, particularly his father who by this time was an old man. He was invited by André Weil to visit Chicago in January 1948, and he received an invitation to visit Harvard University for four months, February to May 1948. He flew into the United States in December 1947 and was met in the airport at New York by Samuel Eilenberg. His major collaboration with Eilenberg was the book Homological Algebra first published in 1956. It is a classic text which has had a profound influence on the subject over a period of nearly half a century.

We explained above that Cartan was appointed professor at the Sorbonne in Paris in November 1940. He taught in Paris from that time until 1969 and then at the Université de Paris-Sud at Orsay from 1970 to 1975. He retired in 1975. At the École Normale Supérieure he started the Séminaire Cartan at the time that Serre was one of his doctoral students. It was Serre who suggested that the seminars should be written up for publication and fifteen ENS-Seminars written by Cartan were published between 1948 and 1964.

Cartan worked on analytic functions, the theory of sheaves, homological theory, algebraic topology and potential theory, producing significant developments in all these areas. Some of his work is put in context by R O Wells Jr reviewing [2]:-

The theory of functions of several complex variables has gone from its infancy with the work of Hartogs, Levi and Poincaré shortly after the turn of the century to its current role as a central field of modern mathematics, much as its predecessor, function theory in one complex variable, did in the 19th century. A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalizations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc. The major developments in the theory from 1930 to 1950 came from Cartan and his school in France, Behnke's school in Münster, and Oka in Japan. The central ideas up to that time were synthesized in Cartan's Séminaires in the early 1950's, and these were very influential to the next several generations of mathematicians. Cartan's accomplishments were broad and he influenced mathematics through his writing, his teaching, his seminars, and his students in a remarkable manner.

In the Preface of [2] Remmert and Serre write:-

The reader should be aware that these volumes do not fully reflect H Cartan's work, a large part of which is also contained in his fifteen ENS-Seminars (1948-1964) and in his book Homological algebra with S Eilenberg. In particular one cannot appreciate the importance of Cartan's contributions to sheaf theory, Stein manifolds and analytic spaces without studying his 1950/51, 1951/52 and 1953/54 Seminars. Still, we trust that mathematicians throughout the world will welcome the availability of the 'Oeuvres' of a mathematician whose writing and teaching has had such an influence on our generation.

We should mention another aspect of Cartan's work which has been involved with politics and in particular supporting human rights. In 1974 a case arose when the Russian authorities placed the mathematician Leonid Plyushch in a special psychiatric hospital. Andrei Sakharov pointed out that this was a political act and Cartan began a strenuous campaign for Plyushch's release. The International Congress of Mathematicians was held in Vancouver in 1974 and this presented an opportunity to gain wide international support for Plyushch with a thousand signatures to a petition for his release. After the Congress Cartan played a major role in setting up the Comité des Mathématiciens to support Plyushch in particular, and all dissident mathematicians. In January 1976 the Soviet authorities released Plyushch which was a major success for Cartan and the Comité des Mathématiciens. But the Committee did not stop after this success. It has supported other mathematicians who have suffered for their political views, such as the Uruguayan mathematician José Luis Massera. For his outstanding work in assisting dissidents Cartan received the Pagels Award from the New York Academy of Sciences.

Cartan is a member of the Académie des Sciences of Paris and of other academies in Europe, the United States, and Japan. He was elected to membership of the Royal Danish Academy of Sciences (1962), The Royal Society of London (1971), the American Academy (1950), the National Academy of Sciences in Washington (1972), The Royal Academy of Belgium (1978), the Akademie der Wissenschaften Göttingen (1971), the Royal Academy of Science Madrid (1971), Bayerische Akademie der Wissenschaften (1974), the Academy of Japan (1979), the Academy of Finland (1979), the Royal Swedish Academy of Sciences (1981), the Polish Academy of Sciences (1985), the Russian Academy (1999). Among the many honours which have been awarded to him was honorary membership of the London Mathematical Society in 1959. He has received honorary doctorates from several universities including ETH Zürich (1955), Münster (1952), Oslo (1961), Sussex (1969), Cambridge (1969), Stockholm (1978), Oxford (1980), Zaragoza (1985), and Athens (1992). He also received the Gold Medal of the National Centre for Scientific research (1976), the Wolf Prize in Mathematics in 1980 and was made Commandeur de la Légion d'Honneur in 1989.