塞缪尔·艾伦伯格Samuel Eilenberg

塞缪尔·艾伦伯格的父亲曾在犹太学校接受教育，但因娶入酿酒世家而成为酿酒师。萨米——人们一直这样称呼艾伦伯格——就读于华沙大学。毫不奇怪，艾伦伯格的兴趣很快转向了point set topology，而这当然是当时华沙大学蓬勃发展的领域。

艾伦伯格在华沙大学学习期间，该校数学系拥有一批杰出的数学家。例如，斯特凡·马祖尔凯维奇、卡齐米日·库拉托夫斯基、瓦茨瓦夫·谢尔宾斯基、斯坦尼斯拉夫·萨克斯和卡罗尔·博尔苏克都在那里任教。艾伦伯格于 1934 年在华沙大学获得硕士学位。随后，他在卡罗尔·博尔苏克指导下学习，于 1936 年获得博士学位。桑德斯·麦克兰恩在[1]中写道：——

他的学位论文研究平面的拓扑学，于 1936 年发表在 Fundamenta Mathematicae 上。其成果在波兰和美国都受到好评。

当时波兰的第二个数学中心是 Lwów。正是在那里，艾伦伯格遇到了领导 Lwów 数学家的斯特凡·巴拿赫。他加入了在苏格兰咖啡馆工作和饮酒的数学家群体，并为《苏格兰书》贡献了问题，这本著名的书收录了在咖啡馆工作的数学家们提出的未解决问题。

你可以在THIS LINK看到苏格兰咖啡馆的照片。

艾伦伯格这一时期的出版物大多关于point-set topology，但即使在他职业生涯的这一早期阶段，也有迹象表明他正转向更多代数的主题。桑德斯·麦克兰恩在[1]中写道：——

1938年，他在基本群上发表了一篇关于空间的高阶同伦群作用的颇具影响力的论文。代数对他的拓扑学研究来说并不陌生！

这篇论文是艾伦伯格进入他后来闻名遐迩的研究领域的一个早期迹象。这是艾伦伯格发表的一系列真正卓越论文中的一篇，因为从他本科时代起直到1939年离开波兰前往美国，他共发表了37篇论文。

1939年，艾伦伯格的父亲说服他，正确的做法是移民到美国。到达美国后，他去了普林斯顿，在那里奥斯瓦尔德·维布伦和所罗门·莱夫谢茨帮助他找到大学职位。这并没有等太久，1940年，他被任命为密歇根大学的讲师。对于艾伦伯格在美国开始教学生涯来说，这是一个极好的地方，因为在那里他可以与顶尖拓扑学家交流。Wilder是安娜堡的教职员，而诺曼·斯廷罗德早先在那里学习过，继续保持着密切联系，并于1942年回到安娜堡任职。

1940年在密歇根组织了一次重要的拓扑学会议。此时第二次世界大战正主导着国际局势，因此来自美国以外的参会人数远少于原本应有的预期。艾伦伯格在会议上就Extension and classification of continuous mappings作了报告。

艾伦伯格只做了一年讲师，随后于1941年在密歇根大学晋升为助理教授。1945年他再次晋升，这次是副教授。1945至1946年他在普林斯顿担任访问讲师，之后于1946年被任命为印第安纳大学正教授。一年后他转到纽约的哥伦比亚大学，并在那里度过了余下的职业生涯。1948年，也就是他到哥伦比亚任职后的第二年，艾伦伯格成为美国公民。1960年他与Natasa Chterenzon结婚。

也许艾伦伯格工作最显著的特点是与他人合作完成的工作量之大。一项重要的合作是他与尼古拉·布尔巴基的工作。1949年，André 安德烈·韦伊在芝加哥大学工作，他联系艾伦伯格，邀请他合作撰写关于同伦群和纤维空间的文章，作为尼古拉·布尔巴基项目的一部分。艾伦伯格成为尼古拉·布尔巴基团队的成员，1950至1951年作为访问教授在巴黎度过，并参加为期两周的夏季会议，直到1966年。他曾获得富布赖特和古根海姆奖学金，以资助他在巴黎的一年。

艾伦伯格最早的合作之一是与桑德斯·麦克兰恩。两人于1940年在安娜堡首次相遇，从那时起到大约1954年，两人合作发表了十五篇论文，涵盖了一系列主题，包括category theory、群的上同调、同调与同伦之间的关系、艾伦伯格-桑德斯·麦克兰恩空间以及一般闭链。1942年他们发表了一篇论文，首次引入了Hom和Ext。他们引入了functor和自然同构这两个术语，并在1945年增加了category和natural transformation这两个术语。

安娜堡再次提供了将艾伦伯格和诺曼·斯廷罗德聚集在一起的机会。1945年，他们提出了同调和上同调理论的公理，但在论文中并未给出证明，这些证明留待1952年他们著名的教科书Foundations of algebraic topology中发表。桑德斯·麦克兰恩在[1]中写道：-

当时，同调理论有许多不同且令人困惑的版本，有些是奇异的，有些是胞腔的。这本书使用范畴来表明，它们都可以在概念上被描述为从空间对的范畴到群或环的同调函子，满足诸如“切除”之类的适当公理。多亏了Sammy的洞察力和热情，这本教科书彻底改变了拓扑学的教学。

事实上，艾伦伯格在1944年Annals of Mathematics上的一篇论文中已经对奇异同调和上同调做出了权威性的处理。他写这篇论文是因为他发现所罗门·莱夫谢茨在其1942年的书中对该主题的处理不尽如人意。1948年，艾伦伯格与谢瓦莱合作发表了一篇论文，给出了索菲斯·李群上同调的代数方法，使用李代数作为基本对象。他们证明了在特征零的情况下，紧索菲斯·李群的上同调作为代数与相应的索菲斯·李代数的上同调同构。

另一项具有重大意义的合作是在艾伦伯格和亨利·嘉当之间进行的。两人于1947年首次相遇，并在随后的几年里开始通过信件交流思想。然而，正如我们上面提到的，艾伦伯格在1950-51年间在巴黎度过，正是在这段时间里，他们取得了显著进展。亨利·嘉当在[1]中写道：-

我们一个发现接着一个发现，萨米有一种非凡的天赋，能在每个时刻把讨论中将要得出的结论表述出来。而且总是由他随着我们的进展，用精确而简洁的英语把一切写下来。……当然，这项合作工作持续了好几年。萨米多次前往我的乡间住所（在迪耶和多洛米厄）。在工作时间之外，他参与我们的家庭生活。

这项合作的成果是Homological algebra一书，其书名是这两位数学家发明的一个术语。尽管他们在1953年就完成了手稿，但直到1956年才出版。霍赫希尔德在评论该书时写道：-

“同调代数”这个书名意在指代纯代数的一个部分，它是使代数同调理论脱离其在拓扑学中的原始栖息地，并将其构建为关于结合环上模的一般理论的结果。……同调代数的概念风味，与其说特别源自拓扑学，不如说源自数学作为一个整体的普遍“自然主义”趋势，即用对任何数学实体在与之相关联的更大数学系统的映射下的行为的分析，来补充对该实体的解剖学研究。特别是，同调代数不太关注模的内在结构，而主要关注模之间同态的复合模式，以及它们与由给定模获得新模的各种构造的相互作用。

他还指出：-

这本书的出现必定意味着同调代数的实验阶段现在已被超越。各种代数系统中最初多样的同调构造，往往具有特设和人为的性质，现已被吸收进一个一般理论，其意义远远超出其来源。同调代数的基本原则，特别是对张量积和算子同态模的运算的完全函子性控制，无疑将成为标准代数技术，甚至在初等水平上也是如此。

我们还应提到另一部重要的两卷本著作，由艾伦伯格于1974年和1976年出版。这部著作是Automata, languages, and machines，一位评论者将其描述为：-

……这是计算机科学基础数学研究和应用数学中最重要的事件之一。该著作对自动机与形式语言理论的几乎所有主要课题给出了统一的数学表述。

这是艾伦伯格从1966年起就一直感兴趣的一个课题，值得注意的是，这是艾伦伯格少数几部由他独自完成的重要著作之一。该书考察有理结构，即那些能被有限状态自动机识别的结构。

到目前为止，我们只谈了作为数学家的Sammy。然而，艾伦伯格还有另一面，因为他是艺术界的经销商，在艺术界被称为“教授”。他经营印度艺术品，并且是该领域的顶尖专家。Hyman Bass在[1]中写道：

多年来，Sammy 收集了世界上最重要的东南亚艺术收藏之一。他在某些艺术收藏家中的名声盖过了他的数学声誉。1987年，Sammy 以其一贯的慷慨与优雅，将大部分收藏捐赠给了纽约大都会艺术博物馆，后者因此受到激励，为哥伦比亚大学 艾伦伯格 数学访问教授职位提供了大量资助。

艾伦伯格因其工作获得了许多荣誉。尤其应当提到的是他于1986年与阿特勒·塞尔伯格共同获得的沃尔夫奖，以及他当选为国家科学院。

最后，让我们引用一段关于艾伦伯格性格的话。Bass在[1]中写道：

尽管他的数学思想可能显得有一种水晶般的严峻，但Sammy是一个热情、健壮且非常活跃的人。对他来说，数学是一种社会活动，因此他有许多合作。他喜欢站着做数学，经常在解释自己的想法时蹦蹦跳跳。当某件事联系起来时，可以从他顽皮的笑容和眼中的光芒中看出来。

Samuel Eilenberg's father was educated at a Jewish school but became a brewer as he married into a family of brewers. Sammy, as Eilenberg was always called, studied at the University of Warsaw. It is not surprising that Eilenberg's interests quickly turned towards point set topology which, of course, was an area which flourished at the University of Warsaw at that time.

A remarkable collection of mathematicians were on the staff at the University of Warsaw while Eilenberg studied there. For example Mazurkiewicz, Kuratowski, Sierpiński, Saks and Borsuk taught there. Eilenberg was awarded his MA from the University of Warsaw in 1934. Then in 1936 he received his doctorate after studying under Borsuk. Mac Lane writes in [1]:-

His thesis, concerned with the topology of the plane, was published in Fundamenta Mathematicae in 1936. Its results were well received both in Poland and the USA.

The second mathematical centre in Poland at this time was Lwów. It was there that Eilenberg met Banach, who led the Lwów mathematicians. He joined the community of mathematicians working and drinking in the Scottish Café and he contributed problems to the Scottish Book, the famous book in which the mathematicians working in the Café entered unsolved problems.

You can see a picture of the Scottish Café at THIS LINK.

Most of Eilenberg's publications from this period were on point-set topology but there were signs, even at this early stage of his career, that he was moving towards more algebraic topics. Mac Lane writes [1]:-

In 1938 he published in [Fundamenta Mathematicae] another influential paper on the action of the fundamental group on the higher homotopy groups of a space. Algebra was not foreign to his topology!

This paper was an early sign that Eilenberg was moving into the area for which he has become famous. It was one of a truly remarkable collection of papers published by Eilenberg for, from his days as an undergraduate up until 1939 when he left Poland for the United States, he published 37 papers.

In 1939 Eilenberg's father convinced him that the right course of action was to emigrate to the United States. Once there he went to Princeton where Veblen and Lefschetz helped him to find a university post. This was not too long in coming and, in 1940, he was appointed as an instructor at the University of Michigan. This was an excellent place for Eilenberg to begin his teaching career in the United States for there he could interact with leading topologists. Wilder was on the staff at Ann Arbor and Steenrod, who had studied there earlier, continued to have close links and returned onto the staff at Ann Arbor in 1942.

In 1940 there was an important topology conference organised at Michigan. World War II was by this time dominating the international scene so the number of participants at the conference from outside the United States was much less than one would have otherwise expected. Eilenberg lectured at the conference on Extension and classification of continuous mappings.

Eilenberg was only an instructor for one year, then in 1941 he was promoted at assistant professor at the University of Michigan. In 1945 he was promoted again, this time to associate professor. He spent the year 1945-46 as a visiting lecturer at Princeton before being appointed as a full professor at Indiana University in 1946. After one year he moved to Columbia University in New York where he remained for the rest of his career. In 1948, the year after he took up the post at Columbia, Eilenberg became a US citizen. He married Natasa Chterenzon in 1960.

Perhaps the most obvious feature of Eilenberg's work was the amount which was done in collaboration with other mathematicians. One major collaboration was his work with Bourbaki. In 1949 André Weil was working at the University of Chicago and he contacted Eilenberg to ask him to collaborate on writing about homotopy groups and fibre spaces as part of the Bourbaki project. Eilenberg became a member of the Bourbaki team spending 1950-51 as a visiting professor in Paris and participating in the two week summer meetings until 1966. He had been awarded Fulbright and Guggenheim scholarships to fund his year in Paris.

One of the first collaborations which Eilenberg entered was with Mac Lane. The two first met in 1940 in Ann Arbor and from that time until about 1954 the pair produced fifteen papers on a whole range of topics including category theory, cohomology of groups, the relation between homology and homotopy, Eilenberg-Mac Lane spaces, and generic cycles. In 1942 they published a paper in which they introduced Hom and Ext for the first time. They introduced the terms functor and natural isomorphism and, in 1945, added the terms category and natural transformation.

Ann Arbor again provided the means to bring Eilenberg and Steenrod together. In 1945 they set out the axioms for homology and cohomology theory but they did not give proofs in their paper, leaving these to appear in their famous text Foundations of algebraic topology in 1952. Mac Lane writes in [1]:-

At that time there were many different and confusing versions of homology theory, some singular some cellular. The book used categories to show that they all could be described conceptually as presenting homology functors from the category of pairs of spaces to groups or to rings, satisfying suitable axioms such as "excision". Thanks to Sammy's insight and his enthusiasm, this text drastically changed the teaching of topology.

In fact Eilenberg had written a definitive treatment of singular homology and cohomology in a paper in the Annals of Mathematics in 1944. He had written this paper since he found the treatment of the topic by Lefschetz in his 1942 book unsatisfactory. In 1948 Eilenberg, in a joint paper with Chevalley, gave an algebraic approach to the cohomology of Lie groups, using the Lie algebra as a basic object. They showed that in characteristic zero the cohomology of a compact Lie group is isomorphic as an algebra to the cohomology of the corresponding Lie algebra.

Another collaboration of major importance was between Eilenberg and Henri Cartan. The two first met in 1947 and began to exchange ideas by letter in the following years. However as we mentioned above Eilenberg spent 1950-51 in Paris and it was during this time that they made remarkable progress. Henri Cartan writes in [1]:-

We went from discovery to discovery, Sammy having an extraordinary gift for formulating at each moment the conclusions that would emerge from the discussion. And it was always he who wrote everything up as we went along in precise and concise English. ... Of course, this work together took several years. Sammy made several trips to my country houses (in Die and in Dolomieu). Outside of our work hours he participated in our family life.

The outcome of this collaboration was the book Homological algebra the title being a term which the two mathematicians invented. Although they had completed the manuscript by 1953 it was not published until 1956. Hochschild reviewing the book wrote:-

The title "Homological Algebra" is intended to designate a part of pure algebra which is the result of making algebraic homology theory independent of its original habitat in topology and building it up to a general theory of modules over associative rings. ... The conceptual flavour of homological algebra derives less specifically from topology than from the general "naturalistic" trend of mathematics as a whole to supplement the study of the anatomy of any mathematical entity with an analysis of its behaviour under the maps belonging to the larger mathematical system with which it is associated. In particular, homological algebra is concerned not so much with the intrinsic structure of modules but primarily with the pattern of compositions of homomorphisms between modules and their interplay with the various constructions by which new modules may be obtained from given ones.

He also notes:-

The appearance of this book must mean that the experimental phase of homological algebra is now surpassed. The diverse original homological constructions in various algebraic systems which were frequently of an ad hoc and artificial nature have been absorbed in a general theory whose significance goes far beyond its sources. The basic principles of homological algebra, and in particular the full functorial control over the manipulation of tensor products and modules of operator homomorphisms, will undoubtedly become standard algebraic technique already on the elementary level.

We should mention another major two volume text which Eilenberg published in 1974 and 1976. This text was Automata, languages, and machines which was described by a reviewer as:-

... one of the most important events in the mathematical study of the foundations of computer science and in applied mathematics. The work includes a unifying mathematical presentation of almost all major topics of automata and formal language theory.

It was a topic which Eilenberg had been interested in from 1966 onwards and it is worth noting that it is one of the few major works by Eilenberg which he worked on alone. The book examines rational structures, that is those that can be recognised by a finite state automaton.

So far we have only talked about Sammy the mathematician. There was another side to Eilenberg however, for he was a dealer in the art world in which he was known as "Professor". He dealt in Indian art and he was a leading expert on the subject. Hyman Bass writes in [1]:-

Over the years Sammy gathered one of the world's most important collections of Southeast Asian art. His fame among certain art collectors overshadows his mathematical reputation. In a gesture characteristically marked by its generosity and elegance, Sammy in 1987 donated much of his collection to the Metropolitan Museum of art in New York, which in turn was thus motivated to contribute substantially to the endowment of the Eilenberg Visiting Professorship in Mathematics at Columbia university.

Eilenberg received many honours for his work. In particular we should mention the Wolf Prize which he shared with Selberg in 1986 and his election to the National Academy of Sciences.

Finally let us give a quote regarding Eilenberg's personality. Bass writes in [1]:-

Though his mathematical ideas may seem to have a kind of crystalline austerity, Sammy was a warm, robust, and very animated human being. For him mathematics was a social activity, whence his many collaborations. He liked to do mathematics on his feet, often prancing while he explained his thoughts. When something connected, one could read it in his impish smile and the sparkle in his eyes.