约翰·米尔诺John Milnor

约翰·米尔诺的父母是Joseph Willard Milnor（1889-1949）和Emily Cox（1891-1973）。Joseph Milnor出生于宾夕法尼亚州威廉斯波特，1912年以数学一等荣誉毕业于理海大学。在匹兹堡的通用电气公司工作一年后，他于1913年进入西联电报公司的工程部门。九年后，他被提升为研究工程师，并于1936年成为传输工程师。他于1943年被任命为顾问工程师，次年退休。

米尔诺（朋友和同事称他为Jack）是普林斯顿大学的本科生，1951年获得学士学位。大多数杰出的数学家在学校时就对这门学科产生了热情，但米尔诺不是这样。他是在普林斯顿大学的第一年才对数学产生兴趣的[14]：-

我第一次对数学产生特别兴趣是在普林斯顿大学读大一的时候。我当时相当不适应社交，没有太多朋友，但当我来到普林斯顿，我发现自己非常适应数学公共休息室的氛围。人们在聊数学、玩游戏，你可以随时过来放松一下。我觉得讲座非常有趣。我在那里比以往任何时候都更自在，从那以后我就一直从事数学。

他在本科期间的显著成就包括在1949年和1950年的普特南数学竞赛中作为最高分获得者被命名为普特南研究员。也许更令人印象深刻的是他1950年在Annals of Mathematics上发表的第一篇论文。这篇论文，On the total curvature of knots，于1948年被接受发表，当时米尔诺只有十七岁。这篇论文是米尔诺参加阿尔伯特·W·塔克讲授的微分几何课程的结果。在讲座中提到了卡罗尔·博尔苏克关于打结曲线总曲率的问题，米尔诺“经过几天的思考”解决了这个问题。米尔诺在写论文时得到了拉尔夫·福克斯的帮助，他在论文中对此表示感谢：-

我感谢R H Fox在准备这篇论文时给予的大量帮助。

他在普林斯顿获得学士学位后开始进行研究，并于1953年在完成博士研究之前被任命为普林斯顿的教职人员。在进行研究的同时，他喜欢在公共休息室玩游戏。特别是他玩Kriegspiel（一种盲棋游戏）、围棋和约翰·福布斯·纳什（一种由约翰·福布斯·纳什发明、现在称为Hex的游戏）。事实上，约翰·福布斯·纳什在这些年里也在普林斯顿，米尔诺和约翰·福布斯·纳什经常讨论博弈论。米尔诺在进行研究时写的下一篇论文是Sums of positional games（1953年）。米尔诺在引言中写道：-

许多常见的游戏，如国际象棋和跳棋，除了定义其博弈论性质所必需的结构外，还表现出一种额外的结构。在这种游戏中的一个局面，仅仅指棋盘的物理设置，而不指定哪个玩家要走棋。在这些游戏中，对于每个局面，都为每个玩家定义了一组可能的走法，尽管在任何特定的游戏过程中，实际上只有一个玩家能够从这个局面走棋。在本文中，将为具有这种结构的游戏定义并研究一种加法运算。

这只是米尔诺在1953年发表的几篇论文之一。其他的是：The characteristics of a vector field on the two-sphere；On total curvatures of closed space curves；以及（与伊斯雷尔·内森·赫斯坦合作）An axiomatic approach to measurable utility。另一篇论文Link groups发表于1954年，但它是在1952年3月提交发表的，比刚才提到的1953年第一篇论文早了一年多。米尔诺在Link groups的引言中写道：-

链环同伦是指将一个链环变形到另一个链环，在此过程中允许链环的每个分支自交，但不允许任何两个分支相交。本文的目的是在同伦关系下研究链环。这项研究的基本工具将是链环群。链环的链环群是其补空间的基本群的一个商群，它在同伦下不变。... 我感谢R H Fox在准备本文时提供的帮助。

1954年，米尔诺凭借其在拉尔夫·福克斯指导下撰写的44页学位论文Isotopy of Links获得了博士学位。米尔诺继续留在普林斯顿任职，从1955年到1959年他是Alfred P Sloan研究员。在这些年里，他进行了关于流形的研究。他在[14]中解释了什么是流形以及它为什么重要：-

在低维中，流形是容易可视化的东西。空间中的一条曲线是一维流形的一个例子；球面和甜甜圈的表面是二维流形的例子。但对数学家来说，一维和二维只是开始；在更高维中事情变得更有趣。此外，对物理学家来说流形非常重要，他们必须研究更高维的例子。例如，假设你研究飞机的运动。仅仅描述位置就需要三个坐标，但然后你想描述它朝哪个方向飞行、机翼的角度等等。描述飞机中心在空间中的点需要三个坐标，描述其方向又需要三个坐标，所以你已经处于一个六维空间中了。当飞机移动时，你在六维空间中有一条路径，而这只是理论的开始。如果你研究气体中粒子的运动，有数量极其庞大的粒子在四处弹跳，每个粒子有三个坐标描述其位置，三个坐标描述其速度，所以一个有一千个粒子的系统将有六千个坐标。当然，会出现大得多的数字，所以数学家和物理学家习惯于在大维空间中工作。

他于1960年晋升为教授，随后在1962年，米尔诺被任命为Henry Putman讲席。

米尔诺在1962年斯德哥尔摩国际数学家大会上被授予约翰·查尔斯·菲尔兹奖章。他最杰出的成就，对获得菲尔兹奖奖起到了重要作用，是他证明了7维球面可以有几个微分结构。这项工作开辟了微分拓扑学的新领域。他证明了7维球面上存在28种不同的可微结构。他利用基于约翰·阿瑟·托德多项式的数值不变量来区分这些结构。约翰·阿瑟·托德多项式最初是在代数几何中研究的，令人惊讶的是它们在流形的分类中扮演着如此基础的角色。米尔诺能够用它们来区分流形的微分性质的原因在于它们具有算术性质，涉及伯努利数，这些性质以深刻且尚未完全理解的方式反映了这些微分性质。

参考文献[4]至[18]很好地表明了米尔诺的工作直到1992年（这些文章写作之时）的广泛影响。文章[4]是对米尔诺在代数方面工作的综述，特别是在代数$K$-理论方面，他的工作继续产生重要影响。文章[17]审视了米尔诺关于微分几何所写的九篇论文。它讨论了米尔诺定理，该定理表明纽结的总曲率至少为4。讨论的其他结果包括米尔诺的结果，表明我们不一定能“听到”16维环面的“形状”，以及另一个结果给出了基本群的有限生成子群中给定长度的不同字数的上界和下界。

在20世纪50年代，米尔诺在代数拓扑学方面做了大量工作，这在[18]中有所讨论。他构造了拓扑群的分类空间，并给出了半单纯复形的几何实现。他还研究了诺曼·斯廷罗德代数及其对偶，考察了海因茨·霍普夫代数的结构，并研究了示性类及其与数学物理的关系。

关于米尔诺数学工作的良好概述，请参见他在THIS LINK获得的各种奖项的引文。

从他的著作评论中摘录的片段以及米尔诺为这些著作所写序言的摘录包含许多有用的信息：见THIS LINK。

米尔诺目前的兴趣是动力系统，特别是全纯动力系统。Peter Makienko在评论[9]时总结了他的动力系统工作：-

现在显而易见，低维动力系统，在很大程度上由米尔诺的工作开创，是一般动力系统理论的基本部分。米尔诺在20世纪70年代中期将目光投向了动力系统理论。到那时，动力系统中的斯蒂芬·斯梅尔纲领已经完成。米尔诺的方法是从头开始，研究最简单的非平凡映射族。第一个选择，一维动力系统，成为他与威廉·瑟斯顿合著论文的主题。即使是单峰映射，即只有一个临界点的映射，也证明是极其丰富的。这项工作可以与儒勒·昂利·庞加莱关于圆微分同胚的工作相比较，后者在100年前开创了动力系统的定性理论。米尔诺的工作在该领域开辟了几个新方向，并为我们提供了许多基本概念、挑战性问题和优美的定理。

米尔诺因其极其重要的贡献而获得了许多奖项和荣誉。他于1967年获得国家科学奖章，并当选为国家科学院、American Academy of Arts and Science的成员。他是美国哲学学会的成员，并在美国数学会中发挥了重要作用。1982年8月，米尔诺获得了Leroy P 凯瑟琳·斯蒂尔奖：-

……因一篇具有根本性和持久重要性的论文《论同胚于7维球面的流形》，《数学年刊》64 (1956)，399-405。

他获得了沃尔夫奖（1989年）、Leroy P 斯蒂尔数学阐述奖（2004年）、Leroy P 斯蒂尔终身成就奖（2011年）、尼尔斯·阿贝尔奖（2011年），并于2014年成为美国数学会的会士。

参见他在THIS LINK所获各项奖项的引文摘录。

米尔诺写了八部重要著作：Morse theory（1963）；Lectures on the h-cobordism theorem（1965）；Topology from the differentiable viewpoint（1965）；Singular points of complex hypersurfaces（1968）；Introduction to algebraic K-theory（1971）；（与Dale Husemoller合著）Symmetric bilinear forms（1973）；（与James D Stasheff合著）Characteristic classes （1974）；以及Dynamics in one complex variable（1999）。

关于这些书以及已出版的七卷米尔诺论文集的评论摘录，以及米尔诺为这些书所写的序言，参见THIS LINK。

在他为数学所做的许多贡献中，有一项是编辑工作，自1962年起担任Annals of Mathematics的编辑。自1988年以来，他一直在石溪的纽约州立大学。搬到那里有几个原因，其中之一是杜沙·迈克德福，米尔诺与她已是多年朋友并最终结婚。她于1976年被任命到华威大学，但两年后辞去了那里的终身职位，接受了石溪纽约州立大学的一个非终身职位，以便能靠近米尔诺。杜沙·迈克德福写道：-

我受到Jack Milnor清晰的思想和处理数学问题的方法的影响，并得到了他的鼓励和帮助。我保留了在石溪的工作，即使这意味着要长途通勤到普林斯顿并只能周末相聚，因为对我来说，不在工作上妥协非常重要，就像我母亲曾经做的那样。几年后，我嫁给了Jack，并有了一个……孩子。

米尔诺说[14]：-

我觉得[高等爱德华·斯图迪]研究所是一个度过几年的好地方，但对我来说，也许不是一个度过一生的好地方。在某种程度上，我太孤立了。我认为与年轻人和学生的接触以及更多的连续性对我很重要，所以我很高兴在石溪找到了一个好职位。还有家庭原因：我的妻子在石溪，来回通勤，直到我们的儿子大到会说话之前都很好。然后他开始大声抱怨。

最后我们注意到米尔诺的爱好[14]：-

我喜欢通过阅读科幻小说或其他无聊的小说来放松。我当然曾经喜欢登山，尽管我从来不是专家。我也喜欢滑雪。同样我不是专家，但这是我喜欢做的事情。……我喜欢音乐，但我没有精致的音乐耳朵或音乐天赋。

John Milnor's parents were Joseph Willard Milnor (1889-1949) and Emily Cox (1891-1973). Joseph Milnor, born in Williamsport, Pennsylvania, graduated from Lehigh University in 1912 with first class honours in mathematics. After serving for a year with the General Electric Company in Pittsfield, he entered the engineering department of the Western Union Telegraph Company in 1913. Nine years later he was promoted to research engineer and, in 1936, became a transmission engineer. He was appointed consulting engineer in 1943 and retired in the following year.

Milnor (known to his friends and colleagues as Jack) was an undergraduate at Princeton University, receiving his A.B. in 1951. Most outstanding mathematicians develop a passion for the subject when at school but not so Milnor. It was in his first year at Princeton that he first became interested in mathematics [14]:-

The first time that I developed a particular interest in mathematics was as a freshman at Princeton University. I had been rather socially maladjusted and did not have too many friends, but when I came to Princeton, I found myself very much at home in the atmosphere of the mathematics common room. People were chatting about mathematics, playing games, and one could come by at any time and just relax. I found the lectures very interesting. I felt more at home there than I ever had before and I have stayed with mathematics ever since.

Among his notable achievements while an undergraduate were being named Putnam Fellow as a top scorer in the Putnam competition in mathematics in 1949 and 1950. Perhaps even more impressive was the publication of his first paper in the Annals of Mathematics in 1950. This paper, On the total curvature of knots, was accepted for publication in 1948 when Milnor was only seventeen years old. The paper came as the result of the differential geometry course Milnor attended given by Albert Tucker. In the lectures Karol Borsuk's question on the total curvature of a knotted curve was mentioned and Milnor solved the problem "after a few days thought". Milnor had been helped in writing the paper by Ralph Fox and he acknowledged this in the paper:-

I am indebted to R H Fox for substantial assistance in the preparation of this paper.

He began research at Princeton after graduating with his B.A. and, in 1953, before completing his doctoral studies, he was appointed to the faculty in Princeton. While undertaking research he enjoyed playing games in the common room. In particular he played Kriegspiel (a game of blindfold chess), Go and Nash (a game invented by John Nash and now called Hex). In fact John Nash was at Princeton during these years and Milnor and Nash often talked about game theory. Milnor's next paper, written while he was undertaking research, was Sums of positional games (1953). Milnor writes in the Introduction:-

Many common games, such as chess and checkers, exhibit a structure in addition to that which is necessary to define their game-theoretic properties. By a position in such a game will be meant merely the physical setup of the board without any specification as to which player is to move. In these games a set of possible moves is defined at each position for each of the players, even though only one of them will actually be able to move from this position in any particular play of the game. In this paper an operation of addition will be defined and studied for games having this structure.

This was only one of several papers that Milnor published in 1953. The others were: The characteristics of a vector field on the two-sphere; On total curvatures of closed space curves; and (with Israel Herstein) An axiomatic approach to measurable utility. Another paper, Link groups, was published in 1954 but it had been submitted for publication in March 1952, over a year before the first of the 1953 papers just mentioned. Milnor writes in the Introduction to Link groups:-

By a link homotopy is meant a deformation of one link onto another, during which each component of the link is allowed to cross itself, but such that no two components are allowed to intersect. The purpose of this paper is to study links under the relation of homotopy. The fundamental tool in this study will be the link group. The link group of a link is a factor group of the fundamental group of its complement, which is invariant under homotopy. ... I am indebted to R H Fox for assistance in the preparation of this paper.

In 1954 Milnor received his doctorate for his 44-page thesis Isotopy of Links written under Ralph Fox's supervision. Milnor remained on the staff at Princeton where he was an Alfred P Sloan fellow from 1955 until 1959. During these years he undertook research on manifolds. He explained in [14] what a manifold is and why it is important:-

In low dimensions manifolds are things that are easily visualized. A curve in space is an example of a one-dimensional manifold; the surfaces of a sphere and of a doughnut are examples of two-dimensional manifolds. But for mathematicians the dimensions one and two are just the beginning; things get more interesting in higher dimensions. Also, for physicists manifolds are very important, and it is essential for them to look at higher-dimensional examples. For example, suppose you study the motion of an airplane. To describe just the position takes three coordinates, but then you want to describe what direction it is going in, the angle of its wings, and so on. It takes three coordinates to describe the point in space where the plane is centred and three more coordinates to describe its orientation, so already you are in a six-dimensional space. As the plane is moving, you have a path in six-dimensional space, and this is only the beginning of the theory. If you study the motion of the particles in a gas, there are enormously many particles bouncing around, and each one has three coordinates describing its position and three coordinates describing its velocity, so a system of a thousand particles will have six thousand coordinates. Of course, much larger numbers occur, so mathematicians and physicists are used to working in large-dimensional spaces.

He was promoted to professor in 1960 then, in 1962, Milnor was appointed to the Henry Putman chair.

Milnor was awarded a Fields Medal at the 1962 International Congress of Mathematicians in Stockholm. His most remarkable achievement, which played a major role in the award of the Fields Medal, was his proof that a 7-dimensional sphere can have several differential structures. This work opened up the new field of differential topology. He showed that 28 different differentiable structures exist on the seven-dimensional sphere. He distinguished between these structures using numerical invariants based on the Todd polynomials. The Todd polynomials were first studied in algebraic geometry and it is surprising that they play this fundamental role in classification of manifolds. The reason that Milnor could use them to distinguish the differential properties of manifolds is because they have arithmetic properties, involving the Bernoulli numbers, which reflect in a deep and not fully understood way these differential properties.

The references [4] to [18] give a good indication of the wide influence of Milnor's work up to 1992 (when these articles were written). The article [4] is a survey of Milnor's work in algebra, particularly in algebraic $K$-theory, where his work continues to have important influences. The article [17] looks at nine papers which Milnor had written on differential geometry. It discusses Milnor's theorem, which shows that the total curvature of a knot is at least 4. Among other results discussed are Milnor's result showing that we cannot necessarily "hear the shape" of a 16-dimensional torus, and another result giving upper and lower bounds on the number of distinct words of a given length in a finitely generated subgroup of the fundamental group.

In the 1950s Milnor did a substantial amount of work on algebraic topology which is discussed in [18]. He constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex. He also studied the Steenrod algebra and its dual, investigated the structure of Hopf algebras, and studied characteristic classes and their relation to mathematical physics.

For a good overview of Milnor's mathematics, see the citations for the various prizes which he has won at THIS LINK.

The extracts from reviews of his books and the extracts from Milnor's Prefaces to these books contain much useful information: see THIS LINK.

Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarised by Peter Makienko in his review of [9]:-

It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.

Milnor has received many awards and honours for his extraordinarily important contributions. He received the National Medal of Science in 1967 and was elected a member of the National Academy of Sciences, the American Academy of Arts and Science. He is a member of the American Philosophy Society and has played a major role in the American Mathematical Society. In August 1982 Milnor received the Leroy P Steele Prize:-

...for a paper of fundamental and lasting importance, 'On manifolds homeomorphic to the 7-sphere', Annals of Mathematics 64 (1956), 399-405.

He received the Wolf Prize (1989), the Leroy P Steele Prize for Mathematical Exposition (2004), the Leroy P Steele Prize for Lifetime Achievement (2011), the Abel Prize (2011) and in 2014 was made a Fellow of the American Mathematical Society.

See extracts from the citations for the various prizes which he has won at THIS LINK.

Milnor has written eight important books: Morse theory (1963); Lectures on the h-cobordism theorem (1965); Topology from the differentiable viewpoint (1965); Singular points of complex hypersurfaces (1968); Introduction to algebraic K-theory (1971); (with Dale Husemoller) Symmetric bilinear forms (1973); (with James D Stasheff) Characteristic classes (1974); and Dynamics in one complex variable (1999).

For extracts from reviews of these books and of the seven volumes of Milnor's papers that have been published, and for Milnor's Prefaces to these books see THIS LINK.

Among the many services he has rendered to mathematics is editorial work, being editor of the Annals of Mathematics from 1962. Since 1988 he has been at the State University of New York at Stony Brook. There were several reasons for moving there, one being Dusa McDuff with whom Milnor had been friends for many years and eventually married. She had been appointed to the University of Warwick in 1976 but resigned her tenured post there two years later and accepted an untenured post at the State University of New York at Stony Brook so that she could be close to Milnor. Dusa McDuff wrote:-

I was influenced by the clarity of Jack Milnor's ideas and approach to mathematics, and was helped by his encouragement. I kept my job in Stony Brook, even though it meant a long commute to Princeton and a weekend relationship, since it was very important to me not to compromise on my job as my mother had done. After several years, I married Jack and had a ... child.

Milnor said [14]:-

I felt that the Institute [for Advanced Study] was a wonderful place to spend some years, but for me it was, perhaps, not a good place to spend my life. I was too isolated, in a way. I think the contact with young people and students and having more continuity was important to me, so I was happy to find a good position in Stony Brook. There were also domestic reasons: my wife was at Stony Brook and commuting back and forth, which worked very well until our son got old enough to talk. Then he started complaining loudly about it.

Finally we note Milnor's hobbies [14]:-

I like to relax by reading science fiction or other silly novels. I certainly used to love mountain climbing, although I was never an expert. I have also enjoyed skiing. Again I was not an expert, but it was something I enjoyed doing. ... I enjoy music but I don't have a refined musical ear or a talent for it.