陈省身Shiing-Shen Chern

陈省身的父亲陈省身宝珍是一位受过古典训练的儒家学者，后来成为为政府工作的律师。陈省身宝珍娶了韩梅，他们有两个儿子和两个女儿。陈省身，其名字也可以写作陈国才省身，在家接受教育，并从1917年冬天到1920年偶尔就读于嘉兴小学。[当时中国的学校教育在陈省身出生前后该国革命之后才刚刚开始组织起来。]他的中文由他的姑姑教授，数学由他的父亲教授。陈省身的女儿在[32]中写道：-

……他没有一个非常严格的父亲和母亲强迫他一直学习。他自己做事，在年轻时代花了很多时间与他的祖母在一起，她可能尽可能地宠坏了他。

他于1920年进入秀水中学，但1922年，他的父亲搬到天津，陈省身在接下来的四年里在中国北方的天津扶轮中学度过。在那里，陈省身爱上了数学，并如饥似渴地解答H S Hall和S R Knight的Higher Algebra中的问题，以及George Albert Wentworth和大卫·尤金·史密斯的几何和三角学书中的问题。他于1926年高中毕业，然后于1930年9月开始在天津南开大学学习。这所小型大学总共约有300名学生，陈省身是四名学生组成的数学班中的一员。他特别受到该校唯一的数学教授姜立夫讲授的几何课程的启发，姜立夫曾在哈佛大学师从朱利安·罗威尔·柯立芝。姜立夫[65]：-

……以非常严格的方式训练他的学生，这为他们未来的职业生涯奠定了坚实的基础。

在南开大学学习四年后，陈省身于1930年获得文凭和数学学士学位。当时在中国进行数学研究的机会很少，但有一位正在从事几何研究的人，即陈省身感兴趣的主题，是Dan Sun，他在北京工作。Dan Sun于1928年在芝加哥大学获得博士学位，导师是Ernest Preston 桑德斯·麦克兰恩，他的学位论文是Projective Differential Geometry of Quadruples of Surfaces with Points in Correspondence。在参加北京清华大学的入学考试后，陈省身于1930年8月被任命为该大学数学系的助理。他担任这个职位一年，在此期间他进行了研究。1931年8月，他继续在清华大学研究生院进行研究。他是1930年进入该大学的唯一数学研究生，但在那里的四年里，他不仅广泛学习了射影微分几何，还开始发表自己关于该主题的论文。陈省身写道[25]：-

1932年春天，威海姆·布拉希开访问北京，并做了一系列关于微分几何中拓扑问题的讲座。这实际上是局部微分几何，他采用所有微分同胚的伪群，而不是经典微分几何中的索菲斯·李群，并研究局部不变量。我能够跟上他的讲座，并阅读了许多在《汉堡文集》和其他期刊上发表的同一总标题下的论文。这个主题现在被称为网几何。

这不是他唯一一次接触新思想（引自[60]）：-

1933年在北京，我参加了Sperner关于初等拓扑学的讲座。这是我第一次接触现代数学，它开阔了我的眼界……

1934年，他获得清华大学奖学金赴美留学，但他特别请求允许他去汉堡大学。他的理由是他认为他感兴趣的数学正在欧洲进行，而不是当时的美国。他在访问北京时与威海姆·布拉希开的会面使他确信，汉堡对他来说比巴黎、哥廷根或柏林等其他欧洲大数学中心更好。他写道（引自[60]）：-

正是威海姆·布拉希开教授对我的影响怎么夸大也不为过。1932年，他作为世界巡游的一部分访问了北京。我当时是他听众中的一名年轻大学生。我立刻被他的新思想和他坚持数学是一门生动易懂的学科所打动。与他的这次接触对我决定来汉堡学习起到了关键作用。

陈省身抵达汉堡时，他被告知埃里希·凯勒，汉堡的一位Privatdozent，刚刚写了一本书描述埃利·嘉当的数学，并即将就此主题举办讨论班。陈省身描述了该讨论班[47]：-

教室里坐满了人，书刚刚出版。埃里希·凯勒带着一摞书进来，给每人发了一本。但这个主题很难，所以几次之后，人们就不来了。我想我基本上是唯一一个坚持到最后的人。我想我坚持到最后是因为我跟上了这个主题。不仅如此，我当时正在写一篇论文，将这些方法应用于另一个问题，所以这个讨论班对我来说非常重要。

在威海姆·布拉希开指导下工作并与埃里希·凯勒进行了许多有益的讨论后，陈省身于1936年在汉堡获得博士学位，学习时间不到两年。他的奖学金为期三年，所以他还有一年的经济支持。在这个阶段，他被迫在两个有吸引力的选择之间做出决定，即留在汉堡在埃米尔·阿廷指导下研究代数，或去巴黎在埃利·嘉当指导下学习。尽管陈省身很了解埃米尔·阿廷，并且本想与他合作，但继续研究微分几何的愿望是决定性因素，他于1936年9月去了巴黎。在去巴黎之前，他已于8月去柏林观看那里的奥运会。他在巴黎的时光非常富有成效，他学会了以与埃利·嘉当相同的方式对待数学，见[28]：-

埃利·嘉当的著作通常被认为非常难懂，但陈省身很快适应了埃利·嘉当的思维方式。回想起来，陈省身觉得这就像学习一门新语言。数学中有一种倾向是抽象并让一切都定义明确，而埃利·嘉当则更直观地对待数学。也就是说，他从证据和特殊情况产生的现象出发对待数学，而不是从一般和抽象的观点出发。

谈到埃利·嘉当的想法，陈省身在采访中说[47]：-

没有纤维丛的记号与术语，很难令人满意地解释这些概念。

与埃利·嘉当共事对陈省身来说既有挑战性又有收获[47]：-

通常在与埃利·嘉当会面后的第二天，我就会收到他的来信。他会说：“你走后，我又多想了想你的问题……”——他有一些结果，还有一些新问题，等等。他对所有关于单索菲斯·李群、索菲斯·李代数的论文都烂熟于心。当你在街上遇到他，某个问题冒出来时，他会掏出一个旧信封，写点什么，然后把答案给你。而有时我要花上几个小时甚至几天才能得到同样的答案。我大约每两周见他一次，显然我必须非常努力地工作。

他参加了茹利亚的讨论班，那一年该讨论班专门讨论埃利·嘉当的想法。他会见了安德烈·韦伊、亨利·嘉当以及许多其他一流数学家。1937年，陈省身离开巴黎，成为清华大学数学教授。他的旅程横跨大西洋，穿越美国，然后横跨太平洋。然而，1937年7月抗日战争爆发时他正在旅途中，大学为了躲避战火两次搬迁。从1938年到1943年，他在当时名为西南联合大学（由原清华大学、北京大学和南开大学组成）工作。这所大学在中国西南的昆明市办学。在那里期间，他于1939年在昆明与Shih-ning Cheng结婚。他们有两个孩子：一个女儿Pu（人称May），后来成为物理学家，嫁给了物理学家Ching-wu Chu；一个儿子Bolong（人称Paul）。1942年他收到了普林斯顿的邀请，但他说：-

……从昆明到普林斯顿的旅程看起来令人生畏。当时中国和美国是抗日战争的盟国，美国用几乎空载的返程飞机向中国运送援助物资。所以中国政府为我安排了一个座位，乘坐美国空军飞机从印度加尔各答到美国迈阿密。这次旅程花了一周时间，途经非洲和南美洲。

他在1943-1945年间在普林斯顿度过，期间给赫尔曼·外尔和奥斯瓦尔德·维布伦留下了深刻印象，并结识了谢瓦莱和所罗门·莱夫谢茨。他与所罗门·莱夫谢茨变得友好，后者说服他成为Annals of Mathematics的编辑。他还重新联系了七年前在巴黎结识的安德烈·韦伊。安德烈·韦伊当时在宾夕法尼亚州伯利恒的理海大学工作，距离普林斯顿仅约70公里。在[67]中，安德烈·韦伊写道，当时曾与陈省身谈论埃利·嘉当的数学：-

……我们似乎对这类主题，或对数学总体上持有共同的态度；我们都努力直击每个问题的根本，同时使自己的思想摆脱关于他人可能认为处理该问题的正确或错误方式的先入之见。

安德烈·韦伊与陈省身之间的这些交谈对陈省身产生了很大影响，并促成了他在示性类方面的一些最重要工作。第二次世界大战结束时，陈省身回到中国，于1946年3月抵达上海。他被要求在当时位于南京的Academia Sinica筹建数学研究所，他非常成功地完成了这项工作。然而此时中国的内战使生活变得困难，他很高兴地接受了赫尔曼·外尔和奥斯瓦尔德·维布伦1948年的邀请，回到普林斯顿担任访问教授。此时已在芝加哥大学的安德烈·韦伊安排向陈省身提供芝加哥大学的正教授职位。陈省身于1949年1月1日回到美国，这次带着家人一同前来。

从1949年起，陈省身在美国工作，在短暂访问普林斯顿后，接受了芝加哥大学的几何学讲席。他受邀在1950年8月30日至9月6日于马萨诸塞州剑桥举行的国际数学家大会上作一小时全会报告。他作了题为Differential Geometry of Fibre Bundles的报告。Hans Samelson写道：-

本讲座的出发点是主纤维丛中联络的定义（所有空间均为可微流形，结构群是索菲斯·李群……），它推广了著名的图利奥·列维-齐维塔平行性。从几何上看，联络是丛中一个与纤维横截的接触元素场，并且在群的作用下不变。

陈省身一直留在芝加哥直到1960年，然后他去了加利福尼亚大学乔治·伯克利。正是在此时他成为了美国公民。他解释了当时的情况：-

我当选美国国家科学院院士是我获得美国公民身份的一个主要因素。1960年，我得知有可能成为院士。意识到公民身份是必要的，我便申请了。由于我与Oppenheimer的关系，这一过程被延缓了。结果，我在当选院士大约一个月前成为了美国公民。

1970年，他受邀在1970年9月1日至9月10日于法国尼斯举行的国际数学家大会上作一小时全会报告。这是一项极大的荣誉，因为很少有数学家被邀请在两次国际数学家大会上作一小时全会报告。在这第二次场合，陈省身作了题为Differential Geometry: Its Past and Its Future的报告。

他继续在伯克利工作，1979年正式退休，但在此后的六七年里，他在那里仍然保持着极高的数学活跃度。他一直住在伯克利，直到1999年，88岁的他回到中国，在天津安家，南开大学的陈省身数学研究所于1985年在那里成立。该研究所的倡议是由陈省身提出的，他提议该研究所应：-

……立足南开，面向全国，放眼世界。

陈省身的妻子于2000年1月在天津去世。在陈省身于1978年所写的论文A summary of my scientific life and works（收录于他的选集各卷中）中，陈省身写到了他妻子的贡献：-

在结束这篇叙述之前，我不能不提到我妻子在我生活和工作中的作用。历经战争与和平、顺境与逆境，我们共同生活了四十年，这生活既简单又丰富。如果我的数学工作有功劳，那功劳既属于她也属于我。

他在天津的家中因心脏病发作导致心力衰竭去世，享年93岁。

正如我们已经看到的，他的研究领域是微分几何，他在其中研究了纤维空间中的（现以他命名的）陈省身示性类。这些不仅在数学中重要，在数学物理中也很重要。他在1943-45年访问普林斯顿期间研究了示性类，同时也在此时给出了现在著名的卡尔·弗里德里希·高斯-Bonnet公式的证明。他的工作在[78]中总结如下：-

陈省身在20世纪40年代从事微分几何研究时，这一数学领域正处于低谷。整体微分几何才刚刚起步，甚至莫尔斯理论也只有极少数人理解和运用。如今，微分几何已成为数学中的主要学科，这一转变在很大程度上要归功于陈省身教授。

Richard Palais和Chuu-Lian Terng在[56]中对陈省身的数学工作给出了极好的概述：-

陈省身的数学兴趣异常广泛，远及诸多领域，他对几何学的许多领域——无论是经典的还是现代的——都做出了重要贡献。其中主要的是：几何结构及其等价问题；积分几何；欧几里得微分几何；极小曲面与极小子流形；全纯映射；网；外微分方程组与偏微分方程；卡尔·弗里德里希·高斯-Bonnet定理；以及示性类。……我们想指出贯穿这一切的一个统一主题：他对微分形式技巧的绝对精通，以及他在解决几何问题时对这些技巧的巧妙运用。这是一件魔法斗篷，由他的伟大老师埃利·嘉当传给他。这使他能够深入探索他人无法进入的新数学领域。微分形式之所以成为研究局部和整体几何性质（以及将它们相互联系起来）的理想工具，在于它们有两个互补的方面。一方面，它们允许外微分的局部运算，另一方面，它们允许在链上积分的整体运算，而这两者通过乔治·加布里埃尔·斯托克斯定理相联系。

他于1970年获得美国数学协会颁发的肖维勒奖，1975年获得国家科学奖章，1982年获得洪堡奖，1983年获得美国数学会颁发的Leroy F 凯瑟琳·斯蒂尔奖，1984年获得沃尔夫奖，2002年获得罗巴切夫斯基奖章，2004年获得香港颁发的首届邵逸夫数学奖：-

……因为他开创了整体微分几何领域，并持续引领该领域，带来了优美的进展，这些进展处于当代数学的中心，与拓扑学、代数学和分析学有着深刻的联系，简而言之，与过去六十年数学的所有主要分支都有联系。

1985年，他当选为伦敦皇家学会会士，次年成为伦敦数学会荣誉成员。他还被选为印度数学会（1950年）、New York Academy of Sciences（1987年）的荣誉成员。他当选为Academia Sinica（1948年）、United States National Academy of Sciences（1961年）、American Academy of Arts and Sciences（1963年）、Brazilian Academy of Sciences（1971年）、西西里墨西拿的佩洛里塔纳科学院（1986年）、Accademia dei Lincei（1989年）、巴黎的Académie des Sciences（1989年）、美国哲学会（1989年）、Chinese Academy of Sciences（1994年）和俄罗斯科学院（2001年）。他获得了以下大学授予的荣誉学位：芝加哥大学（1969年）、香港中文大学（1969年）、苏黎世联邦理工学院（1982年）、纽约州立大学石溪分校（1985年）、汉堡大学（1971年）、南开大学（1985年）、圣母大学（1994年）、柏林工业大学（2001年）和香港科技大学（2003年）。

我们以“陈省身之歌”结束这篇传记。1979年，为向他致敬而举行的一次陈省身研讨会以歌曲的形式向他献上了这份敬意：-

致敬陈省身！数学最伟大者！
他使卡尔·弗里德里希·高斯-博内成为家喻户晓的词，
他发现了内蕴证明，
他的真理遍布世界，
他给了我们陈省身类，
以及次级不变量，
纤维丛与层，
分布与叶状叶！
全体致敬，全体致敬陈省身。

Shiing-shen Chern's father, Baozhen Chern, was a classically trained Confucian scholar who later became a lawyer working for the government. Baozhen Chern had married Mei Han and they had two sons and two daughters. Shiing-shen Chern, whose name can also be written as Chen Xingshen, was educated at home as well as occasionally attending Chia-hsing elementary school from the winter of 1917 until 1920. [Schooling in China at this time was just beginning to get organised following the revolution in the country around the time Chern was born.] He was taught Chinese by his aunt and mathematics by his father. Chern's daughter writes in [32]:-

... he didn't have a very strict father and mother who forced him to study all the time. He did things on his own and spent a lot of time in his younger days with his grandmother who probably spoiled him as much as she could.

He entered the Xiushui Middle School in 1920 but, in 1922, his father moved to Tianjin and Chern spent the next four years at the Fulun High School in Tianjin, northern China. There Chern came to love mathematics and avidly solved the problems in Higher Algebra by H S Hall and S R Knight, and in geometry and trigonometry books by George Albert Wentworth and David Eugene Smith. He graduated from the high school in 1926 and then studied at Nankai University in Tianjin, beginning his studies in September 1930. This small university had about 300 students in total and Chern was one of a mathematics class of four students. He was particularly inspired by a geometry course given by Lifu Jiang, the only professor of mathematics at the university, who had studied at Harvard under Julian Coolidge. Lifu Jiang [65]:-

... trained his pupils in a very strict way, which built a solid foundation for their future careers.

After four years of study at Nankai University Chern was awarded a diploma and a B.Sc. in mathematics in 1930. There were few opportunities for mathematical research in China at this time but someone who was undertaking research in geometry, the topic that Chern had become interested in, was Dan Sun who worked in Peking. Dan Sun had obtained a doctorate from the University of Chicago in 1928, advised by Ernest Preston Lane, with his thesis Projective Differential Geometry of Quadruples of Surfaces with Points in Correspondence. After taking the entrance examination for the Graduate School at Tsing Hua University, Peking, Chern was appointed as an assistant in the Department of Mathematics at that university in August 1930. He held this position for a year and during this time he undertook research. In August 1931 he continued to undertake research in the Graduate School of Tsing Hua University. He was the only graduate student in mathematics to enter the university in 1930 but during his four years there he not only studied widely in projective differential geometry but he also began to publish his own papers on the topic. Chern wrote [25]:-

In the spring of 1932 Blaschke visited Peking and gave a series on topological questions in differential geometry. It was really local differential geometry where he took, instead of a Lie group as in the case of classical differential geometries, the pseudo-group of all diffeomorphisms and studied the local invariants. I was able to follow his lectures and to read many papers under the same general title published in the 'Hamburger Abhandlungen' and other journals. The subject is now known as web geometry.

This was not his only introduction to new ideas (quoted in [60]):-

In Peking in 1933 I attended Sperner's lectures on elementary topology. It was my first introduction to modern mathematics and it opened my eyes ...

He received a scholarship from Tsing Hua University in 1934 to study in the United States, but he made a special request that he be allowed to go to the University of Hamburg. His reason was that he believed the mathematics he was interested in was being done in Europe and not, at that time, in the United States. His meeting with Wilhelm Blaschke when he visited Peking had convinced him that Hamburg would be better for him than the other big European mathematics centres such as Paris, Göttingen or Berlin. He wrote (quoted in [60]):-

It was professor Blaschke whose influence on me cannot be overstated. In 1932 he visited Peking as part of his world tour. I was a young college student in his audience. I was immediately impressed by his fresh ideas and his insistence on mathematics being a lively and intelligible subject. This contact with him was instrumental in making me to decide to come to Hamburg as a student.

When Chern arrived in Hamburg he was told that Erich Kähler, a Privatdozent at Hamburg, had just written a book describing Élie Cartan's mathematics and was about to run a seminar on the topic. Chern described the seminar [47]:-

The classroom was filled, and the book had just come out. Kähler came in with a pile of the books and gave everybody a copy. But the subject was difficult, so after a number of times, people didn't come anymore. I think I was essentially the only one who stayed till the end. I think I stayed till the end because I followed the subject. Not only that, I was writing a thesis applying the methods to another problem, so the seminar was of great importance to me.

After working under Blaschke and having many useful discussions with Kähler, Chern received his doctorate from Hamburg in 1936 having studied for less than two years. His scholarship was for three years so he had still another year of financial support. At this stage he was forced to choose between two attractive options, namely to stay in Hamburg and work on algebra under Emil Artin or to go to Paris and study under Élie Cartan. Although Chern knew Artin well and would have liked to have worked with him, the desire to continue working on differential geometry was the deciding factor and he went to Paris in September 1936. Before leaving for Paris he had gone to Berlin to watch the Olympic games there in August. His time in Paris was a very productive one and he learnt to approach mathematics, in the same way that Cartan did, see [28]:-

Cartan's writings were generally regarded as very difficult, but Chern quickly accustomed himself to Cartan's way of thinking. In retrospect, Chern feels that it was like learning a new language. There is a tendency in mathematics to be abstract and have everything defined, whereas Cartan approached mathematics more intuitively. That is, he approached mathematics from evidence and the phenomena which arise from special cases rather than from a general and abstract viewpoint.

Speaking of Cartan's ideas, Chern said in the interview [47]:-

Without the notation and terminology of fibre bundles, it was difficult to explain these concepts in a satisfactory way.

Working with Élie Cartan was challenging but rewarding for Chern [47]:-

Usually the day after meeting with Cartan I would get a letter from him. He would say, "After you left, I thought more about your questions ..." - he had some results, and some more questions, and so on. He knew all these papers on simple Lie groups, Lie algebras, all by heart. When you saw him on the street, when a certain issue would come up, he would pull out some old envelope and write something and give you the answer. And sometimes it took me hours or even days to get the same answer. I saw him about once every two weeks, and clearly I had to work very hard.

He attended Gaston Julia's Seminar which, in that year, was devoted to discussing Cartan's ideas. He met André Weil, Henri Cartan and many other leading mathematicians. In 1937 Chern left Paris to become professor of mathematics at Tsing Hua University. His journey took him across the Atlantic Ocean, across the United States and then across the Pacific Ocean. However the Chinese-Japanese war began in July 1937 while he was on the journey and the university moved twice to avoid the war. He worked at what was then named Southwest Associated University (consisting of the former Tsing Hua University, Peking University and Nankai University) from 1938 until 1943. This university operated from the city of Kunming in south west China. While there he married Shih-ning Cheng in Kunming in 1939. They had two children: a daughter Pu (known as May) who became a physicist and married the physicist Ching-wu Chu, and a son Bolong (known as Paul). He received an invitation to Princeton in 1942 but, he said:-

... the trip from Kunming to Princeton looked formidable. At that time China and the US were allies in the war against Japan and the US was sending support to China with returning planes almost empty. So the Chinese government arranged for me a seat on an US Air Force plane from Calcutta, India to Miami, US. The trip took a week, through Africa and South America.

He spent 1943-1945 at Princeton where he impressed both Hermann Weyl and Oswald Veblen, and met Claude Chevalley and Solomon Lefschetz. He became friendly with Lefschetz who persuaded him to become an editor of the Annals of Mathematics. He also renewed his contacts with André Weil whom he had met in Paris seven years earlier. Weil was working at Lehigh University in Bethlehem, Pennsylvania, only about 70 km from Princeton. In [67], Weil wrote about talking about Cartan's mathematics to Chern at this time:-

... we seemed to share a common attitude towards such subjects, or towards mathematics in general; we were both striving to strike at the root of each question while freeing our minds from preconceived notions about what others might have regarded as the right or the wrong way of dealing with it.

These talks between Weil and Chern were very influential for Chern and led to some of his most important work on characteristic classes. At the end of World War II, Chern returned to China reaching Shanghai in March 1946. He was asked to set up the Institute of Mathematics of the Academia Sinica in Nanking which he did very successfully. However at this time a civil war in China began to make life difficult and he was pleased to accept an invitation in 1948 from Weyl and Veblen to return to Princeton as a visiting professor. André Weil, who by this time was at the University of Chicago, arranged for Chern to be offered a full professorship at University of Chicago. Chern returned to the United States arriving on 1 January 1949, this time bringing his family with him.

From 1949 Chern worked in the United States accepting the chair of geometry at the University of Chicago after first making a short visit to Princeton. He was an invited one-hour plenary speaker at the International Congress of Mathematicians held in Cambridge, Massachusetts, from 30 August to 6 September 1950. He gave the address Differential Geometry of Fibre Bundles. Hans Samelson writes:-

The starting point of this lecture is the definition of a connection in a principal fibre bundle (all spaces are differentiable manifolds, the structure group is a Lie group ...) generalizing the well-known Levi-Civita parallelism. Geometrically the connection is a field of contact elements in the bundle, transversal to the fibres, and invariant under the action of the group.

Chern remained at Chicago until 1960 when he went to the University of California, Berkeley. It was at this time that he became an American citizen. He explained the circumstances:-

My election to the US National Academy of Sciences was a prime factor for my US citizenship. In 1960 I was tipped about the possibility of an academy membership. Realizing that a citizenship was necessary, I applied for it. The process was slowed because of my association to Oppenheimer. As a consequence I became a US citizen about a month before my election to academy membership.

In 1970 he was an invited one-hour plenary speaker at the International Congress of Mathematicians held in Nice, France, from 1 September to 10 September 1970. This was a great honour since very few mathematicians have been asked to be one-hour plenary speaker at two International Congresses of Mathematicians. On this second occasion Chern gave the address Differential Geometry: Its Past and Its Future.

He continued working at Berkeley, retiring officially in 1979 but remaining highly mathematically active there for six of seven more years. He continued to live in Berkeley until 1999 when, at the age of 88, he returned to China where he made his home in Tianjin, where the Chern Institute of Mathematics of Nankai University had been set up in 1985. The initiative for this Institute had been by Chern who proposed that the Institute should he:-

... based at Nankai, facing the whole country, and viewing the world.

Chern's wife died in January 2000 in Tianjin. In the paper A summary of my scientific life and works which Chern wrote in 1978 (and is included in the volumes of his selected papers) Chern wrote about the contribution of his wife:-

I would not conclude this account without mentioning my wife's role in my life and work. Through war and peace and through bad and good times we have shared a life for forty years, which is both simple and rich. If there is credit for my mathematical works, it will be hers as well as mine.

He died at his home in Tianjin at the age of 93 from heart failure following a heart attack.

As we have already seen, his area of research was differential geometry where he studied the (now named) Chern characteristic classes in fibre spaces. These are important not only in mathematics but also in mathematical physics. He worked on characteristic classes during his 1943-45 visit to Princeton and, also at this time, he gave a now famous proof of the Gauss-Bonnet formula. His work is summed up in [78] as follows:-

When Chern was working on differential geometry in the 1940s, this area of mathematics was at a low point. Global differential geometry was only beginning, even Morse theory was understood and used by a very small number of people. Today, differential geometry is a major subject in mathematics and a large share of the credit for this transformation goes to Professor Chern.

Richard Palais and Chuu-Lian Terng give an excellent overview of Chern's mathematics in [56]:-

Chern's mathematical interests have been unusually wide and far-ranging and he has made significant contributions to many areas of geometry, both classical and modern. Principal among these are: Geometric structures and their equivalence problems; Integral geometry; Euclidean differential geometry; Minimal surfaces and minimal submanifolds; Holomorphic maps; Webs; Exterior Differential Systems and Partial Differential Equations; The Gauss-Bonnet Theorem; and Characteristic classes. ... we would like to point out a unifying theme that runs through all of it: his absolute mastery of the techniques of differential forms and his artful application of these techniques in solving geometric problems. This was a magic mantle, handed down to him by his great teacher, Élie Cartan. It permitted him to explore in depth new mathematical territory where others could not enter. What makes differential forms such an ideal tool for studying local and global geometric properties (and for relating them to each other) is their two complementary aspects. They admit, on the one hand, the local operation of exterior differentiation, and on the other the global operation of integration over cochains, and these are related via Stokes's Theorem.

He was awarded the Chauvenet Prize from the Mathematical Association of American 1970, the National Medal of Science in 1975, the Humboldt Prize in 1982, the Leroy F Steele Prize from the American Mathematical Society in 1983, the Wolf Prize in 1984, the Lobachevsky Medal in 2002 and the first Shaw Prize in Mathematics from Hong Kong in 2004:-

... for his initiation of the field of global differential geometry and his continued leadership of the field, resulting in beautiful developments that are at the centre of contemporary mathematics, with deep connections to topology, algebra and analysis, in short, to all major branches of mathematics of the last sixty years.

In 1985 he was elected a Fellow of the Royal Society of London and the following year he was made an honorary member of the London Mathematical Society. He has also been made an honorary member of the Indian Mathematical Society (1950), the New York Academy of Sciences (1987). He was elected to the Academia Sinica (1948), the United States National Academy of Sciences (1961), the American Academy of Arts and Sciences (1963), the Brazilian Academy of Sciences (1971), the Academia Peloritana, Messina, Sicily (1986), the Accademia dei Lincei (1989), the Académie des Sciences, Paris (1989), the American Philosophical Society (1989) the Chinese Academy of Sciences (1994), and the Russian Academy of Sciences (2001). He was awarded honorary degrees by the University of Chicago (1969), the Chinese University of Hong Kong (1969), Eidgenössische Technische Hochschule Zürich (1982), the State University of New York Stony Brook (1985), University of Hamburg (1971), Nankai University (1985), University of Notre Dame (1994), Technische Universität Berlin (2001), and Hong Kong University of Science and Technology (2003).

We end this biography with the "Chern song". In 1979 a Chern Symposium held in his honour offered him this tribute in song:-

Hail to Chern! Mathematics Greatest!
He made Gauss-Bonnet a household word,
Intrinsic proofs he found,
Throughout the World his truths abound,
Chern classes he gave us,
and Secondary Invariants,
Fibre Bundles and Sheaves,
Distributions and Foliated Leaves!
All Hail All Hail to CHERN.