所罗门·莱夫谢茨Solomon Lefschetz

所罗门·莱夫谢茨是一位俄罗斯出生的犹太数学家，他是拓扑学中代数方面的主要来源。他的父亲Alexander Lefschetz和母亲Vera都是土耳其公民，但由于Alexander Lefschetz从事进口商工作，他需要大量旅行。因此，这个家庭决定在法国为自己建立一个基地，以便他们的孩子能够接受教育。莱夫谢茨出生后不久，他的家庭就在巴黎安了家。

莱夫谢茨从小在法国接受教育，法语是他的第一语言。1902年至1905年，他在巴黎的中央理工学院接受工程师培训，并在那里听了埃米尔·皮卡和保尔·阿佩尔的讲座。然而，由于不是法国公民，他会在法国获得学术职位方面遇到很大困难。他完全明白这一点，于是在1905年11月，21岁的莱夫谢茨去了美国。他在Baldwin机车厂工作了几个月，然后从1907年到1910年在匹兹堡的Westinghouse电气公司工作。1907年11月，他在一次实验室事故中不幸失去了双手，当时双手在一台变压器爆炸中被烧掉。他还失去了前臂，并在医院住了一段时间。

可以想象，这次事故除了给他造成身体残疾外，还对莱夫谢茨产生了重大的精神影响。结果是深深的抑郁，但这场悲剧最终将他推向了数学，而数学正是适合他的学科，当然，拓扑学很幸运，因为他找到了对数学的真正热爱。1910年在西屋电气公司向学徒教授数学后，他进入马萨诸塞州伍斯特的克拉克大学攻读数学博士学位，并于1910年至1911年期间担任会士。在克拉克大学攻读研究生期间，莱夫谢茨遇到了数学学生Alice Berg Hayes。他们于1913年7月3日结婚，此前一年，他于1912年6月17日成为美国公民。Alice [1]：-

……帮助他克服了残疾，鼓励他的工作，并缓和他好斗的兴致。他们没有孩子。

莱夫谢茨于1911年获得数学博士学位，其学位论文关于代数几何，题为On the existence of loci with given singularities。

同年，即1911年，莱夫谢茨被任命为林肯市内布拉斯加大学的数学讲师，两年后，他被任命到劳伦斯的堪萨斯大学。在那里，他于1916年晋升为助理教授，1919年晋升为副教授，1923年晋升为正教授。在这些年里，尽管处于数学研究主流之外，他仍撰写了一系列关于拓扑学的重要论文。莱夫谢茨在谈到这些年份对他数学发展的重要性时写道：-

我在西部完全与世隔绝的岁月，在我的成长中扮演了阿尔伯特·爱因斯坦希望每位年轻科学家都能承担的‘灯塔看守人工作’的角色，以便他能以自己的方式发展自己的思想。

他这一时期最重要的成果包含在On certain numerical invariants of algebraic varieties with application to abelian varieties中，他于1921年在Transactions of the American Mathematical Society上发表了该文，以及在他1924年的著名专著L'analysis situs et la géométrie algébrique中。莱夫谢茨用他的名言解释道：-

我的命运是将代数拓扑学的鱼叉扎入代数几何这头鲸鱼的身体。

威廉·瓦兰斯·道格拉斯·霍奇写道（见[10]或[11]）：-

我们对莱夫谢茨最大的亏欠在于他向我们表明，拓扑学的研究对所有代数几何学家都是必不可少的。

儒勒·昂利·庞加莱曾研究曲面上的曲线，但莱夫谢茨通过建立代数簇的子簇理论，将这些思想推向了更为一般的背景。由于他这一时期的杰出贡献，他于1919年被巴黎的Académie des Sciences授予博尔丹奖，并于1923年因我们上面提到的他1921年的论文而获得美国数学会颁发的马克希莫·博谢纪念奖。

在1923年发表于Proceedings of the U.S. National Academy of Sciences的Continuous transformations of 流形中，莱夫谢茨宣布他已经——

……新的、影响深远的方法……

用于研究流形的连续映射，特别是其不动点。他于1923年发表了他的紧致可定向流形不动点定理，并在1926年发表于Transactions of the American Mathematical Society的Intersections and transformations of complexes and manifolds中更完整地阐述了他著名的不动点定理。在这篇论文中，他承诺的远不止——

……流形上复形相交的深远理论。

他写道：——

在适当的限制下，所导出的公式可以推广到更广泛的流形，但这将留待日后讨论。

1927年，他兑现了自己的承诺，将其不动点定理推广到带边流形，至此勒伊岑·布劳威尔的不动点定理成为一个特例。然而，这篇论文不仅仅是一个推广，因为他在其中使用了矩阵，特别是一个矩阵的迹，极大地简化了他在1926年论文中给出的公式。他在不动点定理方面做了进一步的工作，1927年研究了任意有限复形的情形，1936年研究了任意局部连通空间的情形。

在詹姆斯·韦德尔·亚历山大的推荐下，1924年莱夫谢茨前往普林斯顿担任访问教授一年。1925年访问结束时，他获得了普林斯顿的永久职位，担任副教授，他欣然接受。1933年，他成为Henry Fine研究教授，填补了奥斯瓦尔德·维布伦从普林斯顿大学转到高等爱德华·斯图迪研究所时留下的空缺。

Sylvia Nasar写道[17]：-

当他在20世纪20年代第一次来到普林斯顿的时候，他经常说自己是一个“隐形人”。他是教职员工中最早的犹太人之一，嗓门大，粗鲁，而且穿着也很糟糕。人们在走廊里假装没看见他，在教职员工聚会上也远远避开他。但莱夫谢茨在他的一生中克服了远比一群拘谨的Wasp势利眼更为可怕的障碍。

莱夫谢茨致力于一些结果，这些结果将埃米尔·皮卡的函数论定理深刻推广到了多复变数。莱夫谢茨能够比埃米尔·皮卡走得更远，并融入了儒勒·昂利·庞加莱的思想。在这样做的时候，他发展了高维代数簇的代数拓扑理论。“拓扑学”一词来自莱夫谢茨在1930年撰写的一部专著的书名。另一部对该领域发展产生巨大影响的著作是1942年出版的Algebraic topology。在他的工作过程中，他引入了许多今天会被视为代数拓扑基本工具的东西[1]：-

他广泛使用了乘积空间；他发展了相交理论，包括流形的相交环理论；他对各种同调论做出了重要贡献，特别是相对同调、奇异同调和上同调。

莱夫谢茨有两只假手，上面总是戴着一只闪亮的黑手套。每天早上第一件事就是让一个研究生把一支粉笔塞进他手里，然后在一天结束时把它取出来。普林斯顿的学生们编了一首关于莱夫谢茨的小调：-

为莱夫谢茨，莱夫谢茨 L.干杯。
他简直桀骜不驯，
一旦他终于长眠地下，
他就会开始质问上帝。

Saunders 桑德斯·麦克兰恩补充道：-

根据我的经验，莱夫谢茨在数学研究方面既固执又热情。

[2]的作者写道：-

他的热情和威严的个性使他能够极大地影响许多晚辈的工作，以至于他们逐渐把他尊为他们事业的奠基人。

莱夫谢茨认为，独立思考和独创性才是数学研究中最重要的。与大多数数学家不同，他并不看重优雅；如果某件事在他看来显然为真，他会认为花力气去给出严格论证来验证它充其量只是浪费时间。当一个学生自豪地向他展示自己想出的一个巧妙论证，以给出莱夫谢茨某个定理的简短证明时，据说他没有称赞这个学生，而是反驳道：-

别拿你那漂亮的证明来找我。我们这儿不搞那种小儿科的东西。

即使关于莱夫谢茨流传的一个笑话中几乎没有多少真实性，即他从未写出过一个正确的证明，也从未陈述过一个错误的定理，但其中仍有一种潜在的真相，反映了他的数学风格。

Sylvia Nasar 对莱夫谢茨给普林斯顿带来的影响作了如下生动的描述[17]：-

莱夫谢茨富有创业精神且精力充沛，是一台动力超强的人类机车，……把普林斯顿数学系从温文尔雅的中庸状态一路拉到了顶峰。他招募数学家时心中只有一个标准：研究。他专横而独特的编辑方针，使《数学年刊》——普林斯顿曾经死气沉沉的月刊——成为世界上最受尊崇的数学期刊。他有时被指责为迎合反犹主义而拒绝录取许多犹太学生（他的理由是，这些人完成学位后没有人会雇用他们），但没有人否认他具有敏锐的瞬间判断力。他劝诫、发号施令、威逼胁迫，但目的是让这个系变得伟大，并把他的学生变成真正的数学家，像他一样强硬。

他从1928年到1958年担任Annals of Mathematics的编辑，把它提升到了世界一流顶级期刊之一的水准。诺曼·斯廷罗德写道：-

对美国的数学家来说，一份一流期刊的重要性在于，它为他们树立了追求的高标准。以这种多少有些间接的方式，莱夫谢茨深刻地影响了美国数学的发展。

莱夫谢茨是美国数学会的坚定支持者，于1935年至1936年担任其主席。

莱夫谢茨 从事面向战争需要的应用数学工作。与此同时，他听说苏联正在开展将现代数学工具应用于应用数学问题的工作。1944 年，这两种影响促使 莱夫谢茨 重新回到他对工程的兴趣，但此时他已经拥有深厚的数学技能，可以将其用于这些问题。他处理与耗散非线性常微分方程相关的问题，但没有采用用线性理论处理非线性微分方程的通常方法。尽管他的工作最初相当具体，但随着 莱夫谢茨 进一步发展它，这些工作多年来变得更加抽象。最终，他发展的思想成为数学一个新分支的基础，即全局分析。

莱夫谢茨 有许多学生在这一领域工作，并且在 1950 年至 1960 年间，一系列重要出版物 Contributions to the theory of nonlinear oscillations 出现在由普林斯顿大学出版社出版的 Annals of Mathematics Studies 上。从 莱夫谢茨 学派的工作中，产生了结构稳定性和通有性这两个重要概念。莱夫谢茨 甚至到八十多岁时仍做出了卓越的工作。在他的最后一本书中，他写到了自己领域之外的近期思想，即 理查德·费曼 积分的拓扑学。

在20世纪20年代和30年代，莱夫谢茨得以尽情享受他对旅行的热爱，多次前往欧洲国家。他尤其喜欢访问法国、意大利和苏联。然而，第二次世界大战的爆发使得欧洲旅行几乎不可能，因此莱夫谢茨不得不寻找新的去处。他选择了墨西哥，于1944年首次访问墨西哥国立自治大学。这是众多访问中的第一次，他最终养成了每年在那里度过夏季的习惯。由于这些访问，他对墨西哥数学的贡献具有重大意义，他帮助在那里建立了一个繁荣的学派。他的贡献得到了认可，于1964年被授予阿兹特克之鹰勋章。

这是授予莱夫谢茨的众多荣誉之一。1956年，他从Accademia dei Lincei获得了安东尼奥·费尔特里内利国际奖。他获得了巴黎、布拉格、墨西哥、克拉克、布朗和普林斯顿等大学的荣誉学位。1964年，他获得了国家科学奖章：-

……因在发展数学和培养数学家方面的不屈不挠的领导，因在代数几何和拓扑学方面的基础性出版物，以及因在非线性控制过程中激发所需的研究。

Solomon Lefschetz was a Russian born, Jewish mathematician who was the main source of the algebraic aspects of topology. His father Alexander Lefschetz and his mother Vera were both Turkish citizens but since Alexander Lefschetz worked as an importer he was required to travel a great deal. As a consequence the family decided to make a base for themselves in France where their children could be educated. Shortly after Solomon was born his family set up home in Paris.

Since Lefschetz was educated in France from a young age, French was his first language. He trained to be an engineer at the École Centrale in Paris from 1902 to 1905 and there attended lectures by Émile Picard and Paul Appell. However, not being a French citizen he would have found great difficulty obtaining an academic post in France. Fully understanding this, in November 1905 at the age of 21, Lefschetz went to the United States. For a few months he worked at the Baldwin Locomotive works, then from 1907 to 1910 he worked for Westinghouse Electric Company in Pittsburgh. He had the misfortune to lose both his hands in a laboratory accident in November 1907 when they were burnt off in a transformer explosion. He also lost his forearms and spent a while in hospital.

As one might imagine this accident had a major mental impact on Lefschetz in addition to the physical disability he suffered. The result was a deep depression, but the tragedy eventually pushed him towards mathematics, which was the right subject for him and, of course, it was fortunate for topology that he found his true love of mathematics. After teaching mathematics to apprentices at the Westinghouse Electric Company in 1910, he enrolled as a doctoral student in mathematics at Clark University in Worcester, Massachusetts, where he was a fellow from 1910 to 1911. While on the graduate program at Clark, Lefschetz met one of the mathematics students Alice Berg Hayes. They married on 3 July 1913, a year after he became an American citizen on 17 June 1912. Alice [1]:-

... helped him to overcome his handicap, encouraging him in his work and moderating his combative ebullience. They had no children.

Lefschetz received his Ph.D. in mathematics in 1911 with a thesis on algebraic geometry entitled On the existence of loci with given singularities.

That same year, 1911, Lefschetz was appointed an instructor in mathematics at the University of Nebraska in Lincoln, then, two years later, he was appointed to the University of Kansas in Lawrence. There he was promoted to assistant professor in 1916, to associate professor in 1919, and full professor in 1923. During these years he wrote a series of important papers on topology despite being out the mainstream of mathematical research. Lefschetz wrote of the importance of these years in his mathematical development:-

My years in the west with totally hermetic isolation played in my development the role of 'a job in a lighthouse' which Einstein would have every young scientist assume so that he may develop his own ideas in his own way.

His most important results from this period are contained in On certain numerical invariants of algebraic varieties with application to abelian varieties, which he had published in the Transactions of the American Mathematical Society in 1921, and in his famous monograph of 1924 L'analysis situs et la géométrie algébrique. Lefschetz explained with his famous quote:-

It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry.

Hodge wrote (see [10] or [11]):-

Our greatest debt to Lefschetz lies in the fact that he showed us that a study of topology was essential for all algebraic geometers.

Poincaré had studied curves on a surface but Lefschetz pushed the ideas into much more general settings by building a theory of subvarieties of an algebraic variety. For his remarkable contributions during this period he was awarded the Prix Bordin by the Académie des Sciences in Paris in 1919 and the Bôcher Memorial Prize from the American Mathematical Society in 1923 for his 1921 paper we mentioned above.

In Continuous transformations of manifolds published in the Proceedings of the U.S. National Academy of Sciences in 1923 Lefschetz announced that he had:-

... new and far reaching methods...

for investigating continuous maps of manifolds and, in particular, their fixed points. He published his fixed point theorem for compact orientable manifolds in 1923, giving a fuller account of his famous fixed point theorem in Intersections and transformations of complexes and manifolds published in the Transactions of the American Mathematical Society in 1926. In this paper he promised more than the:-

... far reaching theory of the intersection of complexes on a manifold.

He wrote:-

With suitable restrictions the formulas derived are susceptible of extension to a wider range of manifolds, but this will be reserved for a later occasion.

In 1927 he fulfilled his promise extending his fixed point theorems to manifolds with boundary, and by this stage Brouwer's fixed point theorem became a special case. This paper is more than an extension, however, for in it he used matrices, in particular the trace of a matrix, to greatly simplify the formulae he had presented in his 1926 paper. He did further work on fixed point theorems studying the case of any finite complex in 1927 and any locally connected space in 1936.

On Alexander's recommendation, in 1924 Lefschetz went to Princeton as a visiting professor for a year. At the end of his visit in 1925 he was offered a permanent post at Princeton as associate professor which he happily accepted. He became Henry Fine Research Professor in 1933 filling the position left vacant when Veblen moved from Princeton University to the Institute for Advanced Study.

Sylvia Nasar writes [17]:-

When he first came to Princeton in the 1920s, he often said he was an "invisible man". He was one of the first Jews on the faculty, loud, rude, and badly dressed to boot. People pretended not to see him in the hallways and gave him a wide berth at faculty parties. But Lefschetz had overcome far more formidable obstacles in his life than a bunch of prissy Wasp snobs.

Lefschetz worked on results which provided a deep generalisation of Émile Picard's theorems in function theory to several complex variables. Lefschetz was able to go further than Émile Picard and incorporate Poincaré's ideas. In doing this he developed a theory of algebraic topology of algebraic varieties of higher dimension. The word 'topology' comes from the title of a monograph written by Lefschetz in 1930. Another text which would have a huge influence on the development of the field was Algebraic topology which was published in 1942. In the course of his work he introduced many of what would be considered today the basic tools of algebraic topology [1]:-

He made extensive use of product spaces; he developed intersection theory, including the theory of the intersection ring of a manifold; and he made essential contributions to various kinds of homology theory, notably relative homology, singular homology, and cohomology.

Lefschetz had two artificial hands over which he always wore a shiny black glove. First thing every morning a graduate student had to push a piece of chalk into his hand and remove it at the end of the day. The students at Princeton made up a ditty about Lefschetz:-

Here's to Lefschetz, Solomon L.
Irrepressible as hell
When he's at last beneath the sod
He'll then begin to heckle God.

Saunders Mac Lane has added that:-

In my experience, Lefschetz was both obstreperous and enthusiastic - about research in mathematics.

The author of [2] writes:-

His enthusiasm and commanding personality enabled him to influence greatly the work of many of his juniors, so that they came to reverence him as the founder of their careers.

For Lefschetz, independent thinking and originality were what mattered in mathematical research. Unlike most mathematicians he had no respect for elegance and if something was to him clearly true, he would consider it at best a waste of time producing a rigorous argument to verify it. When a student proudly showed him a clever argument that he had produced to give a short proof of one of Lefschetz's theorems, rather than compliment the student, he is claimed to have retorted:-

Don't come to me with your pretty proofs. We don't bother with that baby stuff around here.

Even if there is little truth in a joke which circulated about Lefschetz, namely that he never wrote a correct proof or stated an incorrect theorem, there is an underlying truth in it reflecting on his style of mathematics.

Sylvia Nasar gives this vivid description of the impact Lefschetz had on Princeton [17]:-

Entrepreneurial and energetic, Lefschetz was the supercharged human locomotive that ... pulled the Princeton department out of genteel mediocrity right to the top. He recruited mathematicians with only one criterion in mind: research. His high-handed and idiosyncratic editorial policies made the Annals of Mathematics, Princeton's once-tired monthly, into the most revered mathematical journal in the world. He was sometimes accused of caving in to anti-Semitism for refusing to admit many Jewish students (his rationale being that nobody would hire them when they completed their degrees), but no one denies that he had brilliant snap judgement. He exhorted, bossed, and bullied, but with the aim of making the department great and turning his students into real mathematicians, tough like himself.

He was the editor of the Annals of Mathematics from 1928 to 1958, bringing it up to the standard of one of the very best world class journals. Steenrod wrote:-

The importance to American mathematicians of a first-class journal is that it sets high standards for them to aim at. In this somewhat indirect manner, Lefschetz profoundly affected the development of mathematics in the United States.

Lefschetz was a strong supporter of the American Mathematical Society serving as its President from 1935 to 1936.

During World War II, Lefschetz undertook applied mathematical work directed at the war effort. At the same time he heard of work being undertaken in the Soviet Union on applications of modern mathematical tools to applied mathematical problems. In 1944 these two influences inspired Lefschetz to return to his interest in engineering but now he had deep mathematical skills which he could bring to bear on the problems. He tackled problems related to dissipative nonlinear ordinary differential equations but did not take the usual approach of using linear theory to tackle nonlinear differential equations. Although initially his work was rather concrete, over the years it became more abstract as Lefschetz developed it further. In the end the ideas he developed became the foundation of a new branch of mathematics, namely global analysis.

Lefschetz had many students working in this area and, between 1950 and 1960, a series of important publications Contributions to the theory of nonlinear oscillations appeared in the Annals of Mathematics Studies, published by Princeton University Press. From this work by Lefschetz's school, came the two important concepts of structural stability and genericity. Lefschetz did remarkable work even well into his 80's. In his last book, he wrote on recent ideas outside his own area, namely the topology of Feynman integrals.

During the 1920s and 1930s Lefschetz was able to indulge his love of travel with many trips to European countries. In particular he loved to visit France, Italy and the Soviet Union. However the outbreak of World War II made European travel virtually impossible so Lefschetz had to find new places to visit. He chose Mexico, visiting the National University of Mexico for the first time in 1944. It was the first of many visits and he eventually fell into the habit of spending the summer months there every year. His contribution to mathematics in Mexico was, as a result of these visits, of major importance and he helped build a flourishing school there. His contributions were recognised when he was awarded the Order of the Aztec Eagle in 1964.

This was one of many honours which were awarded to Lefschetz. He received the Antonio Feltrinelli International Prize from the Accademia dei Lincei in 1956. He was awarded honorary degrees from the universities of Paris, Prague, Mexico, Clark, Brown, and Princeton. In 1964 he received the National Medal of Science:-

... for indomitable leadership in developing mathematics and training mathematicians, for fundamental publications in algebraic geometry and topology, and for stimulating needed research in nonlinear control processes.