拉乌尔·博特Raoul Bott

拉乌尔·博特的父亲是奥地利血统的罗马天主教徒，而他的母亲是匈牙利血统的犹太人。他的父母在他出生后不久就分开了，他由母亲作为天主教徒抚养长大。他的母亲再婚，博特与他的母亲和继父（一位讲德语的捷克人）住在斯洛伐克。他的早期教育并不十分成功，而且他确实没有展现出我们现在事后看来他必定拥有的巨大潜力。事实上，他在动手能力方面比在更学术的教育方面表现出更大的前途。甚至在他很小的时候，他就开始尝试电学并建造电气设备。

如果说他早期的教育并不成功，那肯定不是因为他缺乏机会。他住在一座豪华别墅里，由英国女家庭教师照料，因此从小就能说一口流利的英语。他还上过音乐课，而唱歌确实是他在学校成就中的亮点之一。1935年，当博特十一岁时，他的母亲去世了。这对这个年幼的男孩来说是一次毁灭性的经历，他继续与朋友Tomy Hornak一起对玩电表现出热情。在[11]中记录的一次采访中，博特说：-

当我大约十二到十四岁时，我和一个朋友一起玩电，这真的是一种合作。我们有一个实验室，在那里我们尝试制作非常原始的东西，比如麦克风。我们喜欢制造火花，并且想知道小装置是如何工作的。

博特的继父再婚，于是他有了两位继父母。1938年5月，希特勒宣布他“无条件决定”摧毁捷克斯洛伐克，政治局势开始显得严峻。9月，英国和法国建议捷克斯洛伐克接受希特勒的条件。博特的继父母对1938年事态的发展方向不抱任何幻想，他们把十五岁的男孩送到英国的一所寄宿学校，在那里他可能更安全。博特在英国待了一年后，全家移民到加拿大，他在安大略学习了一年，为大学入学做准备。

1941年秋，博特进入蒙特利尔的麦吉尔大学，在那里他学习了一门对他来说很自然的课程。他从小对电的兴趣意味着他选修了支持他主修电气工程的课程。他说，在麦吉尔，他加入了[9]：-

……工程传统中的豪饮、喧闹、吵闹、调皮和男子气概的行为。

1945年获得工程学位后，第二次世界大战仍在进行，博特加入了加拿大陆军。然而，随着8月6日原子弹投在广岛、8月9日投在长崎，战争于9月结束。正是在军队期间，他遇到了来自西印度群岛的Phyllis，她是一名英国文学专业的学生。他们两年后于1947年结婚。四个月后，博特离开军队，报名参加麦吉尔大学为期一年的硕士课程，仍然是电气工程，但此时他的兴趣正转向数学。他兴趣转变的一个原因是他在麦吉尔大学的数学教授[9]：-

我微积分的第一位老师是Williams教授。他太棒了！有些教授穿着长袍，英式风格，他就是其中之一。他的长袍旧得难以置信，沾满粉笔灰，还有点破。头发飞扬，长袍飘动，Williams教授的讲座不是清晰的典范，但我发现其中闪耀着他对这门学科的热爱，以及他对生活和人类普遍的仁慈看法。

他解释了他如何通过尝试转入医学而“皈依”数学（例如见[22]）：-

1945年，原子弹意外地结束了战争，也结束了我四个月的加拿大步兵生涯，于是我去找⟦N1⟧（麦吉尔大学医学院院长）帮忙进入那里的医学院。军队非常明智地决定尽快摆脱这些新兵，于是我们又一次意外地发现自己掌握了自己的生活。那年早些时候我已经从工程专业毕业，但已经决定不从事那个职业。院长非常热情地接待了我，并向我保证，技术训练有素的医生需求量很大。但是，他说，让我坐在他旁边，先跟我讲讲你自己吧。你对植物学，比如说，或者生物学，有过兴趣吗？嗯，其实没有，我不得不承认。化学呢——哦，我讨厌那门课。就这样继续着。过了一会儿，他说：“那么，也许是你想为人类做好事？”然后，当我尴尬地咳嗽时，他接着说：“因为他们会成为最糟糕的医生。”我谢过他，走出他的门时，我知道我将重新开始，并靠上帝的恩典努力成为一名数学家。

博特不想再花三年时间取得数学资格，然后再开始攻读数学硕士学位，如果他当初选择留在麦吉尔大学，他就必须这样做。威廉姆斯建议他申请匹兹堡的卡内基理工学院，弗瑞兹·约翰约翰·L·辛格正在那里建立一个新的数学研究生项目。这似乎是一个很好的前进方式，但博特在卡内基理工学院攻读数学硕士学位的规章制度再次要求他拥有数学学士学位。约翰·L·辛格随后提出了完美的解决方案——为什么不直接攻读博士学位呢？理查德·达芬成为博特的导师，他们解决的第一个问题是博特自己提出的。他说[17]：-

这个问题与构建滤波器有关。……[它]在我还在麦吉尔大学时就让我着迷，我把它带到了卡内基理工学院。在我与达芬的第一次面谈中，我立即向他透露了这个问题，他对此产生了兴趣。

解决这个问题导致了一篇博特-达芬的联合论文，其中包含了现在所谓的博特-达芬定理。博特于1947年因其学位论文Electrical Network Theory获得博士学位，并留在卡内基理工学院进行研究直到1949年。博特的职业生涯深受赫尔曼·外尔的影响，赫尔曼·外尔对博特所取得的成果印象深刻，并于1949年邀请他前往普林斯顿的高等爱德华·斯图迪研究所。邀请的目的是让博特能够写一本关于网络理论的书籍，但实际上这段经历将他的兴趣从网络理论转向了拓扑学[17]：-

我在那里的第一年没有写一篇论文。所以当莫尔斯在那年年底打电话给我说：“你想再待一年吗？”时，我非常高兴。我说：“当然，是的！”他说：“你的薪水够吗？”每月300美元。我说：“当然够！”因为能再待一年我太高兴了。我妻子则不那么乐观！但我们还是应付过来了。

博特在普林斯顿度过了两年，深受赫尔曼·外尔、诺曼·斯廷罗德、恩斯特·斯派克、Reidemeister和莫尔斯的影响。特别是诺曼·斯廷罗德正在讲授纤维丛课程，这是他撰写现已经典的著作The Topology of Fibre Bundles计划的一部分，该书于1951年出版。博特还参加了小平邦彦和乔治·德拉姆关于调和形式的讲座。

1951年至1959年，博特在密歇根大学任教，尽管在此期间他于1955至1957年回到高等斯图迪研究所。在密歇根大学，他[17]：-

……遇到了Hans Samelson，他是海因茨·霍普夫的学生。Samelson是几何学和Lie group theory的真正大师。在我们共事的那些年里，我从他那里学到了很多。

正是在密歇根大学，博特指导了他的第一位博士生斯蒂芬·斯梅尔。斯蒂芬·斯梅尔于1957年以学位论文Regular Curves on Riemannian Manifolds获得博士学位，并于1966年获得菲尔兹奖。很少有人能像博特那样拥有如此成功的首位博士生！

1959年，博特接受了哈佛大学的正式教授职位。他在那里度过了余下的职业生涯，于1999年从教学中退休，但成为哈佛大学的William Casper Graustein研究教授。很少有人指导过两位后来被授予约翰·查尔斯·菲尔兹奖章的博士研究，但博特拥有这一殊荣。我们在上面提到了斯蒂芬·斯梅尔，第二位是丹尼尔·奎伦，他在哈佛大学撰写了学位论文Formal Properties of Over-Determined Systems Of Linear Partial Differential Equations。丹尼尔·奎伦于1964年获得博士学位，并于1978年获得菲尔兹奖章。

让我提一下对博特在1960年代末的个人回忆。我[EFR]正在参加华威大学的一个研讨会，这时演讲室的灯熄灭了。博特跳了起来，坚定地坚持说：“我会修好它们——我是电工！”果然，大约五分钟后灯又亮了，但只持续了几秒钟，然后再次陷入黑暗。“电工博特”的第二次尝试也以类似方式失败。我们确实设法举行了研讨会——有人想出了一个聪明的主意（不是双关语！），把几辆汽车开到研讨室窗前，打开它们的前灯。

为了了解博特取得进展的数学领域范围，我们简要看一下他的论文集内容。第1卷包含关于拓扑学和索菲斯·李群的论文，其中大部分写于1949年至1962年间。Ian Hambleton在评论这一卷时写道：-

这一卷中的亮点太多，无法一一总结，但每位读者肯定会从博特优雅的风格以及对学科核心问题和时代核心问题的投入中获益。

其中包括著名的博特周期性定理（1956年）以及莫尔斯-博特函数，这是莫尔斯函数的一个重要推广，由博特在这项工作过程中引入。

第2卷收录了博特关于微分算子的论文。Serge L Tabachnikov写道：-

本书收录的论文由一个主题贯穿：分析中的拓扑约束……其中大多数涉及K理论、算子指标理论以及椭圆复形的所罗门·莱夫谢茨不动点理论。本卷中约有一半论文是与M 迈克尔·阿蒂亚合作撰写的；它们的时间跨度是指标理论的黄金十年，1964—73年。M 迈克尔·阿蒂亚关于他与博特合作及友谊的非常温暖的札记也收录在本卷中……

同一位评论者这样评述第三卷和第四卷：-

博特论文集的第三卷呈现了他在叶状结构的代数拓扑学方面以及伊斯拉埃尔·盖尔范德-Fuchs 上同调方面的工作。……

[第4卷]收录论文的主要主题是Yang-Mills方程的几何与拓扑以及向量丛的刚性现象。这些工作受物理学启发，并且极大地促成了我们今天所见证的几何学与高能物理学之间的融合。其底层主题是莫尔斯理论，特别是等变理论。本卷中的许多论文展现了R 博特作为教师和阐释者的最佳状态。我个人认为“莫尔斯及其数学工作”和“莫尔斯理论讲义，旧与新”这两篇论文确实非凡。

上面引文中关于博特作为教师的评论代表了广泛持有的观点。例如在[22]中，Tu这样描述作为讲师的博特：-

博特的讲座因其看似易于理解而堪称传奇。他的风格通常是数学演讲者所青睐的定义-定理-证明方法的对立面。通常他喜欢讨论一个简单的关键例子，该例子概括了问题的本质。常常像变魔术一样，一个具有透明意义的具体公式出现了。

博特获得了许多荣誉，例如斯隆奖学金（1956-60）、American Mathematical Society奥斯瓦尔德·维布伦奖（1964）、古根海姆奖学金（1976）、国家科学奖章（1987）、American Mathematical Society凯瑟琳·斯蒂尔终身成就奖（1990）以及沃尔夫奖（2000）。沃尔夫奖的授予理由是：-

……他在拓扑学和微分几何及其在李群、微分算子和数学物理中的应用方面做出的许多基础性贡献。

授予博特的其他荣誉包括圣母大学（1980）、麦吉尔大学（1987）、卡内基梅隆大学（1989）和莱斯特大学（1995）的荣誉博士学位。他被选为伦敦数学会（1976）和莫斯科数学会（1997）的荣誉会员。他当选为国家科学院（美国）（1964）和科学院（巴黎）（1995）的院士。

Raoul Bott's father was a Roman Catholic of Austrian descent, while his mother was Jewish of Hungarian descent. His parents split up soon after he was born and he was brought up by his mother as a Catholic. His mother remarried and Raoul lived with his mother and step father, who was a German speaking Czech, in Slovakia. His early education was not a great success, and he certainly did not show the great potential that, with hindsight, we can now say that he must have had. In fact he showed more promise at using his hands than at the more academic side of education. Even as a very young boy he began experimenting with electricity and building pieces of electrical apparatus.

If his early education was not successful it was certainly not because he lacked opportunities. He lived in a luxurious villa, was looked after by English governesses, and as a consequence spoke perfect English from a young age. He was also given music lessons, and indeed singing was one of the highlights of his achievements at school. In 1935, when Raoul was eleven years old, his mother died. This was a shattering experience for the young boy who continued to show enthusiasm for playing with electricity with his friend Tomy Hornak. In an interview recorded in [11] Bott said:-

When I was about twelve to fourteen years old, a friend and I had fun working with electricity, and it was really a collaboration. We had a lab where we tried to make very primitive things, such as a microphone. We enjoyed creating sparks, and we wanted to know how gadgets work.

Bott's step father remarried and then he had two step parents. The political situation was beginning to look serious when in May 1938 Hitler announced his 'unqualified decision' to destroy Czechoslovakia. In September, Britain and France advised Czechoslovakia to accept Hitler's terms. Bott's step parents were under no illusions about where things were heading during 1938 and they sent the fifteen year old boy to a boarding school in England where he might be safer. After Bott spent a year in England, the family emigrated to Canada where he prepared for university entrance during a year of study in Ontario.

In the autumn of 1941 Bott entered McGill University in Montreal where he took what for him was a natural course. His interest in electricity from a young age meant that he took courses supporting his main topic of electrical engineering. He said that, at McGill, he fell in with the [9]:-

... engineering tradition of hard drinking, loud, boisterous, mischievous, and macho behaviour.

After graduating with an engineering degree in 1945, with World War II still in progress, Bott joined the Canadian army. However the war ended in September following the dropping of the atomic bombs on Hiroshima on 6 August and Nagasaki on 9 August. It was while he was in the army that he met Phyllis, from the West Indies, who was a student of English literature. They married two year later in 1947. After four months Bott left the army and signed up for a one year Master's course at McGill University, still on electrical engineering but by now he was moving in interest towards mathematics. One reason for his changing interest was his mathematics professor at McGill University [9]:-

My first teacher in the calculus was Professor Williams. He was beautiful! Some professors taught in gowns, English style, and he was one of these. His gown was unbelievably old, chalk crusted and slightly torn. With his hair flying and his gown flapping, Prof Williams' lectures were not a model of clarity, but I found shining through them his love of the subject and also his general benevolent view of life and mankind.

He explained how his "conversion" to mathematics took place via an attempt to move into medicine (see for example [22]):-

I approached [the Dean of the Medical School at McGill] for help in entering the medical school there, when in 1945 the atomic bomb unexpectedly put an end to the war and to my four-month old career in the Canadian Infantry. The Army very wisely decided to get rid of such green recruits as soon as possible, and so we all again found ourselves quite unexpectedly in charge of our own lives. I had graduated in engineering earlier that year but had already decided against that career. The Dean greeted me very cordially and assured me that there was a great need for technically trained doctors. But, he said, seating me next to him, first tell me a little about yourself. Did you ever have any interest in botany, say, or biology? Well, not really, I had to admit. How about chemistry - Oh, I hated that course. And so it went. After a while he said, "Well, is it maybe that you want to do good for humanity?" And then, while I was coughing in embarrassment, he went on, "Because they make the worst doctors." I thanked him, and as I walked out of his door I knew that I would start afresh and with God's grace try and become a mathematician.

Bott did not fancy taking three more years to qualify in mathematics before starting a Master's degree in mathematics which is what he would have had to have done had he followed his initial choice to stay at McGill. Williams made the suggestion that he apply to the Carnegie Institute of Technology in Pittsburgh, where John Synge was setting up a new graduate programme in mathematics. This seemed a good way to proceed, but the regulations for Bott to take a Master's degree in mathematics at Carnegie Tech again required him to have a first degree in mathematics. Synge then came up with the perfect solution - why not move straight to a doctorate? Richard Duffin became Bott's supervisor and the first problem they solved was one which Bott suggested himself. He said [17]:-

This problem had to do with building filters. ... [It] had fascinated me when I was at McGill, and I brought it to Carnegie Tech with me. In my first interview with Duffin I immediately divulged it to him, and he became interested.

Solving this problem led to a joint Bott-Duffin paper containing what is now known as the Bott-Duffin theorem. Bott was awarded a PhD for his thesis Electrical Network Theory in 1947 and he remained at Carnegie Tech undertaking research until 1949. Bott's career was greatly influenced by Weyl who was impressed with the results that Bott was producing and invited him to the Institute for Advanced Study in Princeton in 1949. The aim of the invitation was so that Bott could write a book on network theory but in fact the experience took his interests away from network theory and towards topology [17]:-

I didn't write a single paper in my first year there. So I was very delighted when Marston Morse called me up at the end of that year and said, "Do you want to stay for another year?" And I said, "Of course, yes!" He said, "Is your salary enough?" It was $300 a month. I said, "Certainly!" because I was so delighted to be able to stay another year. My wife took a dimmer view! But we managed.

Bott spent two years at Princeton where he was greatly influenced by Weyl, Steenrod, Specker, Reidemeister, and Morse. In particular Steenrod was giving a course on fibre bundles which he was running as part of his programme of writing the now classic text The Topology of Fibre Bundles which was published in 1951. Bott also attended lectures by Kodaira and de Rham on harmonic forms.

From 1951 to 1959 Bott was on the faculty of the University of Michigan although during that time he did spend 1955-57 back at the Institute for Advanced Study. At Michigan he [17]:-

... met Hans Samelson, who was a student of Hopf. Samelson was a real master of geometry and Lie group theory. I learned a lot from him during the years we worked together.

It was at Michigan that Bott supervised his first doctoral student, Stephen Smale. Smale earned his doctorate in 1957 for the thesis Regular Curves on Riemannian Manifolds and went on to win a Fields Medal in 1966. Few people can have had a more successful first PhD student than Bott!

In 1959 Bott accepted a full professorship at Harvard University. He remained there for the rest of his career, retiring from teaching in 1999 but becoming William Casper Graustein Research Professor at Harvard. Few people have supervised the doctoral studies of two people who went on to be awarded a Fields Medal but Bott has that distinction. We mentioned Smale above, and the second was Daniel Quillen who wrote his thesis Formal Properties of Over-Determined Systems Of Linear Partial Differential Equations at Harvard. Quillen was awarded his doctorate in 1964 and a Fields Medal in 1978.

Let me mention a personal memory of Bott from the late 1960s. I [EFR] was attending a seminar at the University of Warwick when the lights in the lecture room went out. Bott leapt to his feet insisting firmly "I'll fix them - I'm an electrician!". Sure enough about five minutes later the lights came back on but only for a few seconds before again there was darkness. A second attempt by "electrician Bott" failed in a similar way. We did manage to hold the seminar - somebody had the bright (no pun intended!) idea of bringing a few cars up to the window of the seminar room and switching their headlights on.

To get an idea of the range of mathematics in which Bott made advances we look briefly at the contents of his collected papers. Volume 1 contains papers on topology and Lie groups most of which were written between 1949 and 1962. Ian Hambleton, reviewing this volume, wrote:-

There are too many high points in this volume to summarize, but each reader will certainly profit from Bott's elegant style and engagement with the central problems of the subject and time.

Included is the famous Bott periodicity theorem (1956) and the Morse-Bott functions, an important generalization of Morse functions which Bott introduced in the course of this work.

Volume 2 contains Bott's papers on differential operators. Serge L Tabachnikov writes:-

The papers included in this book are united by one theme: topological constraints on analysis .... Most of them concern K-theory, index theory of operators and Lefschetz fixed point theory for elliptic complexes. About half of the papers in this volume were written in collaboration with M Atiyah; their time span is the golden decade of index theory, 1964 - 73. M Atiyah's very warm notes on his joint work and friendship with Bott are included in the volume ...

The same reviewer writes of the third and fourth volumes:-

The third volume of collected papers by Raoul Bott represents his works on the algebraic topology aspects of foliations and Gelfand-Fuchs cohomology. ...

The main themes of the papers included in [Volume 4] are the geometry and topology of the Yang-Mills equations and the rigidity phenomena of vector bundles. These works were physics-inspired and very much contributed to the convergence between geometry and high energy physics we are witnessing today. The underlying topic is Morse theory, in particular, the equivariant one. Many papers in this volume represent R Bott, a teacher and expositor at his best. I personally find the papers "Marston Morse and his mathematical works" and "Lectures on Morse theory, old and new" truly exceptional.

The comments regarding Bott as a teacher made in the quote above represent widely held views. For example in [22] Tu writes about Bott as a lecturer:-

Bott's lectures are legendary for their seeming ease of comprehension. His style is typically the antithesis of the Definition-Theorem-Proof approach so favoured among mathematical speakers. Usually he likes to discuss a simple key example that encapsulates the essence of the problem. Often, as if by magic, a concrete formula with transparent significance appears.

Bott was awarded many honours, for example the Sloan Fellowship (1956-60), the American Mathematical Society Oswald Veblen Prize (1964), the Guggenheim Fellowship (1976), the National Medal of Science (1987), the American Mathematical Society Steele Prize for Lifetime Achievement (1990), and the Wolf Prize (2000). The Wolf Prize was awarded:-

... his many fundamental contributions in topology and differential geometry and their application to Lie groups, differential operators and mathematical physics.

Other honours which were bestowed on Bott included honorary doctorates from the University of Notre Dame (1980), McGill University (1987), Carnegie Mellon University (1989), and the University of Leicester (1995). He was made an honorary member of the London Mathematical Society (1976), the Moscow Mathematical Society (1997). He was elected to the National Academy of Sciences (United States) (1964) and the Academy of Sciences (Paris) (1995).