埃利·嘉当Élie Cartan

埃利·嘉当的母亲是Anne Florentine Cottaz（1841-1927），父亲是Joseph Antoine 埃利·嘉当（1837-1917），是一名铁匠。让我们再追溯这些家族一代。Anne Cottaz是François Cottaz和Françoise Mallen的女儿，而Joseph Cartan是Benoît Bordel 埃利·嘉当（一名磨坊主）和Jeanne Denard的儿子。Joseph和Anne Cartan有四个孩子：Jeanne Marie Cartan（1867-1931）；本传记的主人公埃利·嘉当 Joseph Cartan；Léon 埃利·嘉当（1872-1956），他追随父亲加入了家族铁匠生意；以及Anna Cartan（1878-1923），他成为了一名数学教师。埃利·嘉当与家人住在Dolomieu的Square Champ-de-Mars的一所房子里。他回忆起自己的童年时光（引自[3]）：-

……铁砧的敲击声，每天清晨从黎明开始。……他的母亲，在那些难得的不用照顾孩子和料理家务的几分钟里，正操作着纺车。

这个家庭非常贫穷，正如埃利·嘉当后来所说，他的父母是（引自[3]）：-

……朴实无华的农民，在他们漫长的一生中，向孩子们展示了快乐完成工作和勇敢承担重负的榜样。

在19世纪末的法国，来自贫困家庭的孩子不可能获得大学教育。正是埃利·嘉当的非凡才能，加上许多运气，才使他有可能接受高质量的教育。当他在小学时，他就展现出了非凡的才华，给他的老师M Collomb和M Dupuis留下了深刻印象。后者说：-

埃利·嘉当是个害羞的男孩，但他的眼睛闪耀着非凡的智慧之光，并且记忆力极佳。

如果不是年轻的学校督学、后来的重要政治家Antonin Dubost（1844-1921），埃利·嘉当可能永远不会成为一位杰出的数学家。Dubost当时受雇为小学督学，正是在访问法国阿尔卑斯山区多洛米厄的一所小学时，他发现了这位非凡的少年埃利·嘉当。Dubost鼓励埃利·嘉当参加国家助学金的竞争，以便让埃利·嘉当进入一所公立中学。他的老师M Dupuis为他准备在格勒诺布尔举行的竞争性考试。出色的表现使他得以进入维埃纳学院，他在那里度过了1880至1885这五年。在整个求学过程中，Dubost一直支持这个男孩，并为他争取更多的资助。从维埃纳学院毕业后，他又在格勒诺布尔的公立中学学习了1885至1887这两年，然后在巴黎的让松-德-萨伊公立中学度过一年，专攻数学，完成了中学教育。国家助学金得以延续，使他能够在巴黎高等师范学校学习。

埃利·嘉当于1888年成为巴黎高等师范学校的学生，在那里他聆听了当时顶尖数学家的课程，包括儒勒·昂利·庞加莱、夏尔·埃尔米特、于乐·达奈希、让·加斯东·达布、保尔·阿佩尔、埃米尔·皮卡和爱徳华·古尔萨。埃利·嘉当于1891年毕业，随后在军队服役一年，然后继续在巴黎高等师范学校攻读博士学位。当埃利·嘉当在军队中（他在那里晋升为中士）时，他的朋友Arthur Tresse（1868-1958）正在莱比锡跟随索菲斯·李学习。回来后，Tresse告诉埃利·嘉当关于威廉·基灵在有限连续变换群结构方面的杰出工作。埃利·嘉当着手完成威廉·基灵的分类，并且从索菲斯·李于1892年对巴黎的六个月访问中受益匪浅。在1892至1894这两年埃利·嘉当撰写博士论文期间，他获得了Peccot基金会提供的著名奖学金资助。埃利·嘉当1894年的博士论文对李代数做出了重大贡献，他完成了复域上半单代数的分类，这基本上是威廉·基灵发现的。然而，尽管威廉·基灵已经表明只有某些例外单代数可能存在，但他并未证明这些代数确实存在。埃利·嘉当在他的论文中证明了这一点，他在复域上构造了每一个例外单李代数。他于1893年发表的第一批论文是两篇短文，陈述了他关于单索菲斯·李群的结果。Robert Bryant在[12]中写道，在1893年的短文中：-

……《论单变换群》……他特别宣布，他已经找到了对应于威廉·基灵发现的每一个“例外”根系索菲斯·李群的例子。我觉得这项工作令人瞩目的一点是埃利·嘉当将例外群解释为变换群的方式。

埃利·嘉当在第三篇论文中发表了分类的完整细节，这篇论文基本上就是他的博士论文。他于1894年从索邦大学理学院获得博士学位。随后他被任命到蒙彼利埃大学，从1894年到1896年在那里授课。此后，他被任命为里昂大学的讲师，从1896年到1903年在那里任教。1903年在里昂，他与Marie-Louise Bianconi（1880-1950）结婚，她是Pierre-Louis Bianconi的女儿，后者曾是化学教授，后来成为里昂的督学。埃利·嘉当和Marie-Louise Cartan有四个孩子：亨利·嘉当；Jean Cartan；Louis Cartan；和Hélène 埃利·嘉当。长子亨利·嘉当在数学上做出了杰出的工作，本档案中有他的传记。另外两个儿子不幸去世。Jean是一位优秀的音乐作曲家，1932年因肺结核去世，年仅25岁；而他们的儿子Louis成为普瓦捷大学的物理学家。他是抵抗运动的成员，在法国与占领的德国军队作战。1943年2月他被捕后，家人再未收到任何消息，但他们担心最坏的情况。直到1945年5月，他们才得知他已于1943年12月被纳粹斩首。当他们收到Louis被德国人杀害的消息时，埃利·嘉当已经75岁，这对他是一个毁灭性的打击。他们的第四个孩子是女儿Hélène，她成为费奈隆公立中学的数学教师。

1903年，埃利·嘉当被任命为南锡大学的教授，但他也在电气工程与应用力学研究所授课。他一直留在那里，直到1909年搬到巴黎[3]：-

1909年，埃利·嘉当在家乡村庄多洛米厄建了一座房子，他经常在那里度假。在多洛米厄，埃利·嘉当继续进行科学研究，但有时会去家里的铁匠铺，帮助父亲和兄弟拉风箱。

1909年，他在巴黎被任命为索邦大学的助理讲师，但三年后，他被任命为巴黎的微分与积分学讲席。1915年至1918年第一次世界大战期间，他被征召入伍，继续保留他之前的军士军衔。他得以继续他的数学事业，同时在隶属于巴黎高等师范学校的军事医院工作。1920年，他被任命为理性力学教授，随后从1924年到1940年担任高等几何学教授。他于1940年退休，但并未停止教学，因为他继续在女子巴黎高等师范学校任教。

埃利·嘉当研究连续群、索菲斯·李代数、微分方程和几何。他的工作实现了这些领域之间的综合。他极大地扩展了由索菲斯·李开创的连续群理论。在他的关于有限连续李群的论文工作之后，他后来分类了实域上的半单索菲斯·李代数，并找到了单索菲斯·李代数的所有不可约线性表示。他转向结合代数理论，并研究了这些代数在实域和复域上的结构。约瑟夫·韦德伯恩将完成埃利·嘉当在这一领域的工作。

随后，他将注意力转向半单索菲斯·李群的表示。他的工作是索菲斯·李理论、经典几何、微分几何和拓扑学的惊人综合，这在埃利·嘉当的所有工作中都能找到。他将赫尔曼·格拉斯曼代数应用于外微分形式理论。他在1894年至1904年间发展了这一理论，并将他的外微分形式理论应用于微分几何、动力学和相对论中的各种问题。让·迪厄多内在[1]中写道：-

他讨论了大量的例子，以极其简洁的风格处理它们，这只有凭借他不可思议的代数和几何洞察力才可能实现，并且让两代数学家感到困惑。

1899年，埃利·嘉当发表了他的第一篇关于约翰·弗里德里希·普法夫问题Sur certaines expressions différentielles et le probleme de PfaffⓉ（关于某些微分表达式和约翰·弗里德里希·普法夫的一个问题）的论文。在这篇论文中，埃利·嘉当给出了微分形式的第一个正式定义。Victor Katz写道[26]：-

他的定义是“纯符号”的；即，他将“微分表达式”定义为由微分dx、dy、dz、...的有限次加法和乘法以及某些可微系数函数形成的齐次表达式。

在随后的几年里，他围绕这一主题又写了几篇重要论文，其中包括Sur l'intégration de certaines systèmes de Pfaff de caractère deux Ⓣ（《论某些特征为二的普法夫系统的积分》）（1901年）。1936—37年，他在索邦大学开设了一系列讲座，内容涵盖了他对这一主题的贡献。这些讲座于1945年出版成书Les systèmes différentiels extérieurs et leurs applications géométriques Ⓣ（《外微分系统及其几何应用》）。

埃利·嘉当关于微分方程的论文在许多方面是他最令人印象深刻的工作。他的方法再次完全创新，他表述问题的方式使其具有不变性，不依赖于特定的变量或未知函数。这使埃利·嘉当能够定义任意微分系统的一般解究竟是什么，但他不仅对一般解感兴趣，还研究了奇解。他通过从一个给定系统转向一个新的关联系统来做到这一点，新系统的一般解给出了原系统的奇解。然而，他未能证明所有奇解都由他的技术给出，直到他去世四年后才实现了这一点。

从1916年起，他主要发表微分几何方面的著作。菲利克斯·克莱因的“埃尔朗根纲领”被赫尔曼·外尔和奥斯瓦尔德·维布伦视为不足以作为几何学的一般描述，而埃利·嘉当将发挥重要作用。他研究了一个由任意索菲斯·李变换群作用的空问，发展了活动标架理论，推广了让·加斯东·达布的运动学理论。事实上，这项工作引导埃利·嘉当走向了纤维丛的概念，尽管他在工作中没有给出这一概念的明确定义。

埃利·嘉当还以其对称空间理论对几何学作出了进一步贡献，这一理论源于他1926年所写的论文。在这些论文中，他发展了最初由威廉·金顿·克利福德和阿瑟·凯莱研究的思想，并使用了赫尔曼·外尔在1925年发展的拓扑方法。这项工作于1932年完成，因此提供了[1]：

……这是少数几个例子之一：一个数学理论的创始者同时也是将其带到完成的人。

埃利·嘉当随后继续研究一个最初由儒勒·昂利·庞加莱研究的主题上的问题。到这一阶段，他的儿子亨利·嘉当正在对数学作出重大贡献，而埃利·嘉当能够在他儿子证明的定理基础上继续推进。亨利·嘉当说[24]：

[我父亲]对索菲斯·李群的了解比我多，而为了确定所有允许传递群的有界圆形域，必须使用这些知识。因此我们共同写了一篇关于这个主题的文章[《有界圆形域的变换》Ⓣ（Transformations of areas bounded by circles），C. R. Acad. Sci. Paris 192 (1931), 709-712]。但总的来说，我父亲在他的角落里工作，我在我的角落里工作。

埃利·嘉当于1913年发现了旋量理论。旋量是用于将三维旋转转换为二维表示的复向量，它们后来在量子力学中发挥了基础性作用。埃利·嘉当于1938年出版了两卷本著作Leçons sur la théorie des spineurs Ⓣ（《旋量理论讲义》）[37]：

在两卷本的序言中……M 埃利·嘉当 指出，在最一般的数学形式下，旋量是他在1913年关于单群线性表示的工作中发现的，并且他强调旋量与威廉·金顿·克利福德-鲁道夫·利普希茨超复数之间的联系。……M 埃利·嘉当的书对群论的几何与物理方面感兴趣的数学家来说将不可或缺，因为它从几何观点出发，对旋量的代数理论作了完整而权威的综述。

我们在THIS LINK给出了埃利·嘉当全部法语或英语书籍的尽可能完整的列表。

我们在THIS LINK给出了其中一些书籍的书评的简短摘录。

至于他的教学能力，陈省身和谢瓦莱写道[14]：-

埃利·嘉当是一位优秀的教师；他的讲座是令人愉悦的智识体验，给学生留下一种通常错误的印象，以为自己已经掌握了该主题的全部内容。因此更令人惊讶的是，在很长一段时间里，他的思想没有对年轻数学家产生它们本应充分享有的影响。这或许部分是由于埃利·嘉当极其谦逊。与儒勒·昂利·庞加莱不同，他并不试图避免让学生在他的指导下工作。然而，他太有幽默感，不会在自己周围组织起那种有助于形成一个数学学派的热烈狂热。

他无疑是20世纪上半叶最重要的数学家之一。让·迪厄多内在[1]中写道：-

埃利·嘉当作为一流数学家的认可直到他晚年才到来；1930年之前，儒勒·昂利·庞加莱和赫尔曼·外尔可能是唯一正确评价了他非凡能力和深度的著名数学家。这部分是由于他极其谦逊，部分是由于1900年后法国数学研究的主要趋势在函数论领域，但主要是由于他非凡的独创性。直到1930年以后，年轻一代才开始探索埋藏在他论文中的丰富思想与成果宝藏。自那时起，他的影响稳步增长，并且除了儒勒·昂利·庞加莱和大卫·希尔伯特之外，可能没有其他人对赋予我们今天的数学以现有形态和观点做出了如此多的贡献。

J H C Whitehead写道[48]：-

埃利·嘉当是当代数学的伟大建筑师之一。

[6]的作者写道：-

埃利·嘉当是他那一代的主要数学家之一，尤其因其在几何学和李代数理论方面的工作而具有影响力。在第一次世界大战后的黯淡岁月里，他是法国最杰出的数学家之一。他最终对尼古拉·布尔巴基群体产生了显著影响，他的儿子亨利·嘉当——另一位杰出的数学家——是该群体七位创始成员之一。

威廉·瓦兰斯·道格拉斯·霍奇认为埃利·嘉当是[23]：-

……一位伟大的数学天才，以广阔的视野审视全局，挑出本质，从而以大师之笔直击问题的核心。他对无数特殊情形的了解，以及对复杂论证的精通，使他能够以巨大的步伐推进他的学科，并在广阔的数学探索领域中留下持久的印记。随着他的去世，世界确实失去了一位现代数学的伟大建筑师

Robert Hermann写道[22]：-

埃利·嘉当 无疑是数学史上最伟大、最具独创性的头脑之一，他在 索菲斯·李 群、微分几何和微分方程的几何理论方面的工作，是我们今天许多工作的基础。在我看来，他在数学中的地位类似于其他知识领域中那些世纪之交的伟大宗师。正如弗洛伊德受到19世纪科学机械论世界观的影响，却利用这一背景创造出深刻影响20世纪思想的新颖而革命性的东西一样，埃利·嘉当 在19世纪90年代巴黎、柏林和哥廷根流行的数学基础上，建立了一座我们至今仍在探究其含义的数学大厦。他的工作高度依赖直觉和几何，但也建立在独创的计算与分析方法的有力结合之上，其数学专长范围从代数一直延伸到拓扑学。

由于他杰出的贡献，埃利·嘉当 获得了许多荣誉，但正如 让·迪厄多内 在上文引述中所解释的，这些荣誉直到他职业生涯的后期才到来。他于1934年获得列日大学的名誉学位，1936年获得哈佛大学的名誉学位。1947年，他被柏林自由大学、布加勒斯特大学和鲁汶天主教大学授予三个名誉学位。次年，他被比萨大学授予名誉博士学位。他于1921年当选为 波兰科学院 成员，1926年当选为 挪威科学与文学院 成员，1927年当选为 Accademia dei Lincei 成员，并于1947年5月1日当选为 伦敦皇家学会 会士。他于1931年3月9日当选为 法国科学院 成员，1945年任该科学院副院长，1946年任院长。他于1939年成为 伦敦数学会 的名誉成员。月球上的一座环形山以他的名字命名。

1939年5月18日，在索邦大学举行了一场庆祝活动，以庆祝埃利·嘉当的70岁生日。朋友和同事们纷纷致意，描述了他对数学众多不同领域的贡献。1969年，为庆祝埃利·嘉当诞辰100周年，在布加勒斯特举行了一次会议。会议论文集已出版[5]，我们的参考文献列表中包含在该会议上宣读的几篇论文，即[17]、[18]、[19]、[30]、[31]、[33]、[45]和[46]。为庆祝埃利·嘉当诞辰115周年，“埃利·嘉当的数学遗产”会议于1984年6月25日至6月29日在法国里昂举行。

Élie Cartan's mother was Anne Florentine Cottaz (1841-1927) and his father was Joseph Antoine Cartan (1837-1917) who was a blacksmith. Let us trace these families back one more generation. Anne Cottaz was the daughter of François Cottaz and Françoise Mallen while Joseph Cartan was the son of Benoît Bordel Cartan (who was a miller) and Jeanne Denard. Joseph and Anne Cartan had four children: Jeanne Marie Cartan (1867-1931); Élie Joseph Cartan, the subject of this biography; Léon Cartan (1872-1956), who followed his father and joined the family blacksmith business; and Anna Cartan (1878-1923), who became a teacher of mathematics. Élie lived with his family in a house on Square Champ-de-Mars in Dolomieu. He remembered his childhood spent with the (quoted in [3]):-

... blows of the anvil, which started every morning from dawn. ... his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning wheel.

The family were very poor and, as Élie Cartan later said, his parents were (quoted in [3]):-

... unpretentious peasants who during their long lives demonstrated to their children an example of joyful accomplished work and courageous acceptance of burdens.

In late 19th century France it was not possible for children from poor families to obtain a university education. It was Élie's exceptional abilities, together with a lot of luck, which made a high quality education possible for him. When he was in primary school he showed his remarkable talents which impressed his teachers M Collomb and M Dupuis. The latter said:-

Élie Cartan was a shy boy, but his eyes shone with an unusual light of great intelligence, and this was combined with an excellent memory.

Cartan may never have become a leading mathematician were it not for the young school inspector, later important politician, Antonin Dubost (1844-1921). Dubost was at this time employed as an inspector of primary schools and it was on a visit to the primary school in Dolomieu, in the French Alps, that he discovered the remarkable young Élie. Dubost encouraged Élie to enter the competition for state funds to allow Élie to attend a Lycée. His teacher M Dupuis prepared him to sit the competitive examinations which were held in Grenoble. An excellent performance allowed him to enter the Collège de Vienne which he attended for the five years 1880-1885. Throughout his school career Dubost continued to support the young boy and obtain further financial support for him. After the Collège de Vienne, he then studied at the Lycée in Genoble for the two years 1885-87 before completing his school education by spending one year at the Janson-de-Sailly Lycée in Paris where he specialised in mathematics. The state stipend was extended to allow him to study at the École Normale Supérieure in Paris.

Cartan became a student at the École Normale Supérieure in 1888 where he attended courses by the leading mathematicians of the day including Henri Poincaré, Charles Hermite, Jules Tannery, Gaston Darboux, Paul Appell, Émile Picard and Édouard Goursat. Cartan graduated in 1891 and then served for a year in the army before continuing his studies for his doctorate at the École Normale Supérieure. While Cartan was in the army, where he reached the rank of sergeant, his friend Arthur Tresse (1868-1958) was studying under Sophus Lie in Leipzig. On his return, Tresse told Cartan about Wilhelm Killing's remarkable work on the structure of finite continuous groups of transformations. Cartan set about completing Killing's classification and he was able to benefit greatly from a six-month visit by Sophus Lie to Paris in 1892. During the two years 1892-94 that Cartan spent working on his doctoral thesis, he was supported by a prestigious bursary from the Peccot Foundation. Cartan's doctoral thesis of 1894 contains a major contribution to Lie algebras where he completed the classification of the semisimple algebras over the complex field which Killing had essentially found. However, although Killing had shown that only certain exceptional simple algebras were possible, he had not proved that in fact these algebras exist. This was shown by Cartan in his thesis when he constructed each of the exceptional simple Lie algebras over the complex field. His first papers, published in 1893, were two notes stating his results on simple Lie groups. Robert Bryant writes in [12] that in the 1893 note:-

... Über die einfachen Transformationgruppen ... he announces, in particular, that he has found examples of Lie groups corresponding to each of the 'exceptional' root systems found by Killing. One of the things that I find remarkable about this work is the way that Cartan found interpretations of the exceptional groups as transformation groups.

Cartan published full details of the classification in a third paper which was essentially his doctoral thesis. He obtained his doctorate in 1894 from the Faculty of Science at the Sorbonne. He was then appointed to the University at Montpellier where he lectured from 1894 to 1896. Following this, he was appointed as a lecturer at the University of Lyon, where he taught from 1896 to 1903. In Lyon in 1903 he married Marie-Louise Bianconi (1880-1950), the daughter of Pierre-Louis Bianconi who had been a professor of chemistry but had become an inspector in Lyon. Élie and Marie-Louise Cartan had four children: Henri Paul Cartan; Jean Cartan; Louis Cartan; and Hélène Cartan. The eldest son, Henri Cartan, was to produce brilliant work in mathematics and has a biography in this archive. The two other sons died tragically. Jean, a composer of fine music, died of tuberculosis in 1932 at the age of 25 while their son Louis became a physicist at the University of Poitiers. He 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. By the time they received the news of Louis' murder by the Germans, Cartan was 75 years old and it was a devastating blow for him. Their fourth child was a daughter Hélène who became a teacher of mathematics at the Lycée Fénelon.

In 1903 Cartan was appointed as a professor at the University of Nancy but he also taught at the Institute of Electrical Engineering and Applied Mechanics. He remained there until 1909 when he moved to Paris [3]:-

In 1909 Cartan built a house in his home village Dolomieu, where he regularly spent his vacations. In Dolomieu Cartan continued his scientific research but sometimes went to the family smithy and helped his father and brother to blow the blacksmith's bellows.

His appointment in 1909 in Paris was as an assistant lecturer at the Sorbonne but three years later he was appointed to the Chair of Differential and Integral Calculus in Paris. From 1915 to 1918, during World War I, he was drafted into the army where he continued to hold his former rank of sergeant. He was able to continue his mathematical career and, at the same time, work in the military hospital attached to the École Normale Supérieure. He was appointed as Professor of Rational Mechanics in 1920, and then Professor of Higher Geometry from 1924 to 1940. He retired in 1940 but did not stop teaching at this point for he went on to teach at the École Normale Supérieure for girls.

Cartan worked on continuous groups, Lie algebras, differential equations and geometry. His work achieved a synthesis between these areas. He added greatly to the theory of continuous groups which had been initiated by Lie. After the work of his thesis on finite continuous Lie groups, he later classified the semisimple Lie algebras over the real field and found all the irreducible linear representations of the simple Lie algebras. He turned to the theory of associative algebras and investigated the structure for these algebras over the real and complex field. Joseph Wedderburn would complete Cartan's work in this area.

He then turned his attention to representations of semisimple Lie groups. His work is a striking synthesis of Lie theory, classical geometry, differential geometry and topology which was to be found in all Cartan's work. He applied Grassmann algebra to the theory of exterior differential forms. He developed this theory between 1894 and 1904 and applied his theory of exterior differential forms to a wide variety of problems in differential geometry, dynamics and relativity. Dieudonné writes in [1]:-

He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight and that has baffled two generations of mathematicians.

In 1899 Cartan published his first paper on the Pfaff problem Sur certaines expressions différentielles et le probleme de Pfaff Ⓣ. In this paper Cartan gave the first formal definition of a differential form. Victor Katz writes [26]:-

His definition was a "purely symbolic" one; namely, he defined "differential expressions" as homogeneous expressions formed by a finite number of additions and multiplications of the differentials dx, dy, d z , . ., and certain differentiable coefficient functions.

Over the following years he wrote several other important papers on this topic including Sur l'intégration de certaines systèmes de Pfaff de caractère deux Ⓣ (1901). In 1936-37 he delivered a series of lectures at the Sorbonne which covered his contributions to the topic. The lectures were published in 1945 in the book Les systèmes différentiels extérieurs et leurs applications géométriques Ⓣ.

Cartan's papers on differential equations are in many ways his most impressive work. Again his approach was totally innovative and he formulated problems so that they were invariant and did not depend on the particular variables or unknown functions. This enabled Cartan to define what the general solution of an arbitrary differential system really is but he was not only interested in the general solution for he also studied singular solutions. He did this by moving from a given system to a new associated system whose general solution gave the singular solutions to the original system. He failed to show that all singular solutions were given by his technique, however, and this was not achieved until four years after his death.

From 1916 onwards he published mainly on differential geometry. Klein's 'Erlanger Programme' was seen to be inadequate as a general description of geometry by Weyl and Veblen, and Cartan was to play a major role. He examined a space acted on by an arbitrary Lie group of transformations, developing a theory of moving frames which generalises the kinematical theory of Darboux. In fact this work led Cartan to the notion of a fibre bundle although he does not give an explicit definition of the concept in his work.

Cartan further contributed to geometry with his theory of symmetric spaces which have their origins in papers he wrote in 1926. In these he developed ideas first studied by Clifford and Cayley and used topological methods developed by Weyl in 1925. This work was completed by 1932 and so provides [1]:-

... one of the few instances in which the initiator of a mathematical theory was also the one who brought it to completion.

Cartan then went on to examine problems on a topic first studied by Poincaré. By this stage his son, Henri Cartan, was making major contributions to mathematics and Élie Cartan was able to build on theorems proved by his son. Henri Cartan said [24]:-

[My father] knew more than I did about Lie groups, and it was necessary to use this knowledge for the determination of all bounded circled domains which admit a transitive group. So we wrote an article on the subject together [Les transformations des domaines cerclés bornés Ⓣ, C. R. Acad. Sci. Paris 192 (1931), 709-712]. But in general my father worked in his corner, and I worked in mine.

Cartan discovered the theory of spinors in 1913. These are complex vectors that are used to transform three-dimensional rotations into two-dimensional representations and they later played a fundamental role in quantum mechanics. Cartan published the two volume work Leçons sur la théorie des spineurs Ⓣ in 1938 [37]:-

In the preface to the two volumes ... M Cartan points out that, in their most general mathematical form, spinors were discovered by him in 1913 in his work on linear representations of simple groups, and he emphasises their connection ... with Clifford-Lipschitz hypercomplex numbers. ... M Cartan's book will be indispensable to mathematicians interested in the geometrical and physical aspects of group theory, giving, as it does, a complete and authoritative survey of the algebraic theory of spinors treated from a geometrical point of view.

We have given a list, as complete as possible, of all Cartan's French or English books at THIS LINK.

We have given brief extracts from reviews of some of these books at THIS LINK.

As to his teaching abilities, Shiing-Shen Chern and Claude Chevalley write [14]:-

Cartan was an excellent teacher; his lectures were gratifying intellectual experiences, which left the student with a generally mistaken idea that he had grasped all there was on the subject. It is therefore the more surprising that for a long time his ideas did not exert the influence they so richly deserved to have on young mathematicians. This was perhaps partly due to Cartan's extreme modesty. Unlike Poincaré, he did not try to avoid having students work under his direction. However, he had too much of a sense of humour to organize around himself the kind of enthusiastic fanaticism which helps to form a mathematical school.

He is certainly one of the most important mathematicians of the first half of the 20th century. Dieudonné writes in [1]:-

Cartan's recognition as a first rate mathematician came to him only in his old age; before 1930 Poincaré and Weyl were probably the only prominent mathematicians who correctly assessed his uncommon powers and depth. This was due partly to his extreme modesty and partly to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. Since then his influence has been steadily increasing, and with the exception of Poincaré and Hilbert, probably no one else has done so much to give the mathematics of our day its present shape and viewpoints.

J H C Whitehead writes [48]:-

Élie Cartan is one of the great architects of contemporary mathematics.

The authors of [6] write:-

Cartan was one of the leading mathematicians of his generation, particularly influential for his work on geometry and the theory of Lie Algebras. In the bleak years after World War I he was one of the most prominent mathematicians in France. He eventually became a notable influence on the Bourbaki group, of which his son Henri, another distinguished mathematician, was one of the seven founder members.

William Hodge considers Cartan as [23]:-

... a great mathematical genius taking in the scene in a broad survey, and picking out the essentials, so that with a master-stroke he goes straight to the heart of a problem. His knowledge of innumerable special cases, and his mastery of intricate argument, enabled him to advance his subject by giant strides, and make a lasting mark on the vast range of mathematical endeavour. By his death, the world has indeed lost one of the great architects of modern mathematics

Robert Hermann writes [22]:-

Cartan is certainly one of the greatest and most original minds of mathematics, whose work on Lie groups, differential geometry, and the geometric theory of differential equations is at the foundation of much of what we do today. In my view, his place in mathematics is similar to that of the great turn-of-the-century masters in other areas of intellectual life. Just as Freud was influenced by the mechanistic world view of 19th century science, but used this background to create something new and revolutionary which has profoundly influenced 20th century thought, so Cartan built, on a foundation of the mathematics which was fashionable in the 1890's in Paris, Berlin and Göttingen, a mathematical edifice whose implications we are still investigating. His work was highly intuitive and geometric, but was also based on a formidable combination of original methods of calculation and analysis, ranging in mathematical expertise from algebra to topology.

For his outstanding contributions Cartan received many honours, but as Dieudonné explained in the above quote, these did not come until late in career. He received honorary degrees from the University of Liege in 1934, and from Harvard University in 1936. In 1947 he was awarded three honorary degrees from the Free University of Berlin, the University of Bucharest and the Catholic University of Louvain. In the following year he was awarded an honorary doctorate by the University of Pisa. He was elected to the Polish Academy of Sciences in 1921, the Norwegian Academy of Science and Letters in 1926, the Accademia dei Lincei in 1927 and elected a Fellow of the Royal Society of London on 1 May 1947. Elected to the French Academy of Sciences on 9 March 1931 he was vice-president of the Academy in 1945 and President in 1946. He became an honorary member of the London Mathematical Society in 1939. A crater on the moon is named for him.

A celebration was held on 18 May 1939 in the Sorbonne to celebrate Cartan's 70th birthday. Many tributes were made by friends and colleagues who described his contributions to a wide range of different areas of mathematics. In 1969, to celebrate the 100th anniversary of Cartan's birth, a conference was held in Bucharest. The proceedings was published [5] and our list of references contains several papers delivered at that conference, namely [17], [18], [19], [30], [31], [33], [45], and [46]. The conference 'The Mathematical Heritage of Élie Cartan' was held in Lyon, France from 25 June to 29 June 1984 to celebrate the 115th anniversary of Cartan's birth.