塞尔日·兰Serge Lang

塞尔日·兰出生于靠近巴黎的圣索菲·热尔曼昂莱。他的父亲艾蒂安兰是一位商人，母亲海伦·施莱皮亚诺夫是一位音乐会钢琴家。兰有一个双胞胎兄弟，后来成为篮球教练，还有一个姐妹，后来成为演员。兰住在巴黎并在那里上学，直到第二次世界大战开始，当时他上10年级。战争于1939年9月开始，但直到1940年6月德国军队才进入巴黎。兰与他的父亲和姐妹逃往美国，定居在加利福尼亚州洛杉矶。正是在那里，兰完成了高中学业，然后进入加州理工学院。

兰于1946年毕业于加州理工学院，获得物理学学士学位。随后他在美国陆军服役约18个月，其中一段时间驻扎在意大利和德国。回到美国后，兰前往普林斯顿大学，打算攻读哲学博士学位。在哲学系待了一年后，他转到了数学系，埃米尔·阿廷成为他的学位论文导师。他于1951年因其学位论文On Quasi Algebraic Closure获得博士学位。他早期的出版物属于其学位论文领域，例如1952年他发表了三篇论文：On quasi algebraic closure; Hilbert's Nullstellensatz in infinite-dimensional space；以及（与约翰·泰特合作）On Chevalley's proof of Lüroth's theorem。也许他早期论文中最重要的是与安德烈·韦伊合作撰写的Number of points of varieties in finite fields（1954）。

兰在1951-53年间是普林斯顿的讲师，同时他也是高等爱德华·斯图迪研究所的访问学者。1953-55年间，他在芝加哥大学任讲师，随后于1955年成为哥伦比亚大学的教授。我们将在下面讨论他的一些热情所在，既有数学方面的也有非数学方面的，但在此阶段我们要说，他对政治深感兴趣，并强烈反对越南战争。1971年，他辞去了哥伦比亚大学的职位，以抗议该校对抗战抗议者的处理方式。辞职时，他并不知道自己能在哪里找到另一个职位。事实上，他在1972年获得了耶鲁大学的职位，并在那里度过了余下的职业生涯，直到2005年春天退休。退休时，耶鲁大学校长为他发表了荣誉演讲（例如见[5]）：-

你的首要热爱一直是数论，据一位同事估计，你写了50多本书和专著[实际上超过60本]，其中许多都与这个主题有关。你的几本专著是它们重要主题的唯一或几乎唯一的书籍论述。你在丢番图方程中的著名定理为你赢得了美国数学会的杰出法兰克·尼尔森·寇尔奖。你的教科书也获得了赞誉。你的《本科生微积分》在七八十年代经历了多个版本，而你的《代数》教科书是该领域的标准参考书。你作为学者如此多产，以至于你的职业中关于你有一些实际的笑话。一个笑话是：“有人打电话给耶鲁数学系，找兰。接电话的助理说：‘他现在不能说话，他正在写一本书。我会让你稍等。’”

在你的性格中，你毫不妥协地坚持你所认为的逻辑一致性和修辞诚实，你质疑了许多公认的智慧和许多权威，无论是在外部世界还是在耶鲁这里。你是一位优秀且深切关怀学生的老师，为了表彰这一点，几年前你获得了耶鲁学院教学的Dylon Hixon奖。你的学生毕业后多年仍与你保持联系，其中一位还以你的名义设立了一个捐赠基金。在你的众多专著中，有一本叫做《做数学的美》，这是你在80年代在巴黎进行的三次对话的合集。耶鲁感谢你理解、实践和教授数学艺术的热情，并祝愿你在继续终身参与其无限辉煌的过程中一切顺利。

兰的数学研究涵盖了广泛的主题，如代数几何、丢番图几何（兰发明的术语）、超越数论、丢番图逼近、解析数论及其与表示论的联系、模曲线及其在数论中的应用、$L$-级数、双曲几何、Arakelov理论以及微分几何。他的书，尽管许多是关于他正在进行研究的主题，但涵盖了更广泛的数学领域。许多是基于他讲授的研究生课程。他总是说，学习一个主题的最佳方式就是写一本关于它的书。从如此多的书中挑选出一些亮点几乎是不可能的。然而，大多数人会同意Algebra（1965年）是他研究生水平教材中最有影响力的一本[5]：-

兰单枪匹马地重新组织和振兴了这个基本且核心的主题。这本书产生了巨大的影响。

你可以在THIS LINK阅读这本书的前言。

1984年出现了第二版增订版，2002年出现了第三版。兰在2002年版的前言中写道：-

从现在起，《代数》与我的其他著作一样，由Springer-Verlag出版。随着这一变化，我考虑过出新版的可能性，但决定不出。我认为这本书非常稳定。如果从头开始，我唯一会添加的内容是SL(n)和GL(n)（在$\mathbb{R}$或$\mathbb{C}$上）的一些代数性质，超出第十三章中单性的证明。在现有情况下，我只是插入了一些关于每个人都应该知道的某些方面的习题。

兰最常用的本科教材是A first course in calculus，他于1964年首次出版。你可以在THIS LINK阅读该教材的前言。

兰的其他著作包括Introduction to algebraic geometry（1958年）、Abelian varieties（1959年）、Diophantine geometry（1962年）、Introduction to differentiable manifolds（1962年）、Algebraic numbers（1964年）、Linear algebra（1966年）、Introduction to diophantine approximations（1966年）、Introduction to transcendental numbers（1966年）、Algebraic structures（1967年）、Algebraic number theory（1970年）、Introduction to algebraic and abelian functions（1972年）、Differential manifolds（1972年）、Elliptic functions（1973年）、SL(R)（1975年）、Introduction to modular forms（1976年）、Complex analysis（1977年）、Cyclotomic fields（1978年）、Elliptic curves: Diophantine analysis（1978年）、Undergraduate analysis（1983年）、Complex multiplication（1983年）、Riemann-Roch algebra（1985年）、The beauty of doing mathematics. Three public dialogues（1985年）、Introduction to complex hyperbolic spaces（1987年）、Introduction to Arakelov theory（1988年）、Topics in Nevanlinna theory（1990年）、Basic analysis of regularized series and products（1993年）、Fundamentals of differential geometry（1999年）和Math talks for undergraduates（1999年）。

现在让我们看看兰长期反对不公正和不准确性的斗争。正如他1971年从哥伦比亚大学辞职所表明的那样，他一生都对这类事情感受强烈。然而，大约在1977年，他开始以更积极的方式投身于各种事业。让我们举两个兰所奋斗的事业的例子。首先我们提到哈佛政治学家皮埃尔·萨米埃尔 P Huntington的案例。在Huntington于1968年出版的Political order in changing societies一书中，他使用伪数学论证来证明20世纪60年代的南非是一个“满足的社会”。兰不相信这个结论，于是他查看Huntington如何为这一主张辩护，发现他使用的方法根本无效。兰怀疑他是在使用虚假的伪数学论证来为他想要证明的更大权威提供论据。兰说：-

……一种语言，它给人以科学的幻觉，却没有任何实质内容。

兰在1986年Huntington被提名后，发起了一场激烈的运动，阻止他成为国家科学院的成员。兰这次成功了，在Huntington再次被提名的第二次也成功了。兰在Academia, Journalism, and Politics: A Case Study: The Huntington Case中发表了这些事件的详细描述，该描述占据了他1998年著作Challenges的前222页。

我们作为兰运动的例子来看的第二个案例，是他试图诋毁The 1977 survey of the American professoriate的结果，该结果由Everett C Ladd和Seymour M Lipsett于1979年发表。1981年，兰发表了The File: Case Study in Correction (1977-1979)，其中包含有关该调查的信件副本。兰总是喜欢让事实自己说话，因此他对这些文件相对较少评论。然而，他清楚地阐述了他为什么进行那场运动以及他以同等精力进行的许多其他运动：-

我不喜欢我们社会中经常被当作理性讨论的胡说八道。我非常困扰于不准确、模棱两可、暗语、口号、流行语、公关手段、笼统概括和刻板印象，这些被用来（有意或无意地）影响人们。

我困扰于许多人无法认清这些事物的本质。我困扰于人们回避问题的方式，或无法澄清问题，有时是因为他们受到“同僚情谊”和其他形式的恐吓（有时微妙，有时不微妙）的抑制。大多数人忍受这些胡说八道而不采取任何行动（无论出于何种原因——惰性、缺乏精力、缺乏兴趣、缺乏时间等——无法或不愿），常常陷入愤世嫉俗和绝望。

我困扰于错误信息通过媒体不加批判地传播，以及阻碍正确信息传播的障碍。这些障碍以多种方式出现——个人的、机构的、通过自我施加的抑制、通过外部抑制、通过彻头彻尾的不诚实、通过无能——这个清单很长。

令我困扰的是，那些伪装成学术的虚假信息被用于社会、政治和教育语境中，以影响政策决定。

我困扰于错误信息被不加批判地接受，以及人们无法识别或拒绝它。

兰因其成就获得了许多荣誉。例如，他于1959年获得了美国数学会颁发的法兰克·尼尔森·寇尔奖，1967年获得了法国的卡里埃奖，1984年获得了洪堡研究与教学奖，1999年获得了美国数学会颁发的勒罗伊·P凯瑟琳·斯蒂尔数学阐述奖，以及2004年获得了迪伦·希克森'88科学教学奖。他于1985年当选为国家科学院成员。值得注意的是，尽管他在1999年接受了勒罗伊·P斯蒂尔奖，但实际上他已于1996年因一篇发表在Notices上的文章引发的争议而从美国数学会辞职。他在获奖感言中明确表示，他不希望自己的接受以任何方式减少他与该学会的争论。

Serge Lang was born in Saint-Germain-en-Laye close to Paris. His father Étienne Lang was a businessman and his mother Helene Schlepianoff was a concert pianist. Serge had a twin brother who became a basketball coach and a sister who became an actress. Serge lived in Paris and attended school there until the beginning of World War II when he was in the 10th grade. The war began in September 1939 but it was not until June 1940 that German troops entered Paris. Serge together with his father and sister fled to the United States where they settled in Los Angeles, California. It was there that Lang completed his high school education and then entered the California Institute of Technology.

Lang graduated from the California Institute of Technology in 1946 with a B.A. in physics. He then served for around eighteen months in the U.S. Army and he was stationed for part of this time in Italy and Germany. After returning to the United States, Lang went to Princeton University with the intention of studying for a doctorate in philosophy. After a year in the philosophy department, he changed to mathematics and Emil Artin became his thesis advisor. He was awarded his Ph.D. in 1951 for his dissertation On Quasi Algebraic Closure. His early publications were in the area of his thesis, for example in 1952 he published three papers: On quasi algebraic closure; Hilbert's Nullstellensatz in infinite-dimensional space; and (with John Tate) On Chevalley's proof of Lüroth's theorem. Perhaps the most significant of his early papers is Number of points of varieties in finite fields (1954) written in collaboration with André Weil.

Lang was an instructor at Princeton during 1951-53 when he was also a visitor at the Institute for Advanced Study. During 1953-55 he was an instructor at the University of Chicago, then he became a professor at Columbia University in 1955. We will discuss some of his passions below, both mathematical and non-mathematical, but let us say at this stage that he was deeply interested in politics and very strongly opposed to the Vietnam War. In 1971 he resigned his position at Columbia University in protest at their treatment of anti-war protestors. At the time he resigned he had no idea where he might find another position. In fact he was offered a position at Yale University in 1972 and he remained there for the rest of his career until he retired in the spring of 2005. On his retirement the president of Yale gave a speech in his honour (see for example [5]):-

Your primary love has always been number theory and you have written, by one colleague's estimate, over 50 books and monographs [actually over 60], many of them concerned with this topic. Several of your monographs are the only, or nearly the only, book treatments of their important subjects. Your famous theorem in Diophantine equations earned you the distinguished Cole Prize of the American Mathematical Society. Your textbooks also have garnered accolades. Your 'Calculus for undergraduates' went through many editions in the seventies and eighties, and your 'Algebra' textbook is a standard reference in the field. So prodigious are you as a scholar that there are actual jokes in your profession about you. One joke goes: "Someone calls the Yale Mathematics Department, and asks for Serge Lang. The assistant who answers says, 'He can't talk now, he is writing a book. I will put you on hold.'"

In your character, you are uncompromising in your insistence on what you perceive as logical consistency and rhetorical honesty, and you have questioned much received wisdom and many authorities in the external world as well as here at Yale. You are an excellent and deeply caring teacher, and in honour of this several years ago you received the Dylon Hixon Prize for teaching in Yale College. Your students keep in touch with you years after they graduate and one has created an endowed fund in your honour. Among your many monographs there is one called 'The Beauty of Doing Mathematics', a collection of three dialogues you gave in Paris in the '80s. Yale is grateful to you for the passion with which you understand, practice and profess the mathematical arts, and wishes you well as you continue your lifelong engagement with their illimitable splendours.

Lang's mathematical research ranged over a wide range of topics such as algebraic geometry, Diophantine geometry (a term Lang invented), transcendental number theory, Diophantine approximation, analytic number theory and its connections to representation theory, modular curves and their applications in number theory, $L$-series, hyperbolic geometry, Arakelov theory, and differential geometry. His books, although many are on the topics that he was undertaking research in, cover an even wider range of mathematics. Many are based on graduate courses that he was giving. He always said that the best way to learn a topic was to write a book on it. To single out some highlights from such a range of books is almost impossible. However most would agree that Algebra (1965) was the most influential of his graduate level texts [5]:-

Lang single-handedly reorganises and revitalises this fundamental and central subject. The book has had an enormous impact.

You can read the Foreword to this book at THIS LINK.

A second expanded edition appeared in 1984 and a third edition in 2002. Lang wrote in the Foreword to the 2002 book:-

From now on, Algebra appears with Springer-Verlag, like the rest of my books. With this change, I considered the possibility of a new edition, but decided against it. I view the book as very stable. The only addition which I would make, if starting from scratch, would be some of the algebraic properties of SL(n) and GL(n) (over $\mathbb{R}$ or $\mathbb{C}$), beyond the proof of simplicity in Chapter XIII. As things stood, I just inserted some exercises concerning some aspects which everybody should know.

Perhaps Lang's most used undergraduate text is A first course in calculus which he first published in 1964.
You can read the Foreword to the text at THIS LINK.

Other books by Lang include Introduction to algebraic geometry (1958), Abelian varieties (1959), Diophantine geometry (1962), Introduction to differentiable manifolds (1962), Algebraic numbers (1964), Linear algebra (1966), Introduction to diophantine approximations (1966), Introduction to transcendental numbers (1966), Algebraic structures (1967), Algebraic number theory (1970), Introduction to algebraic and abelian functions (1972), Differential manifolds (1972), Elliptic functions (1973), SL(R) (1975), Introduction to modular forms (1976), Complex analysis (1977), Cyclotomic fields (1978), Elliptic curves: Diophantine analysis (1978), Undergraduate analysis (1983), Complex multiplication (1983), Riemann-Roch algebra (1985), The beauty of doing mathematics. Three public dialogues (1985), Introduction to complex hyperbolic spaces (1987), Introduction to Arakelov theory (1988), Topics in Nevanlinna theory (1990), Basic analysis of regularized series and products (1993), Fundamentals of differential geometry (1999), and Math talks for undergraduates (1999).

Let us now look at Lang's long campaign against injustices and inaccuracies. He had felt strongly about such matter all his life as his resignation from Columbia University in 1971 shows. However it was around 1977 that he began to fight causes in a more aggressive way. Let us give two examples of causes that Lang fought. First we mention the case of the Harvard political scientist Samuel P Huntington. In the book Political order in changing societies that Huntington published in 1968 he used pseudo-mathematical arguments to prove that in the 1960s South Africa was a "satisfied society". Lang did not believe the conclusion so he looked how Huntington justified this claim and saw that he used methodology which was simply not valid. Lang suspected that he was using false pseudo-mathematical argument to give arguments that he wanted to justify greater authority. It was, said Lang:-

... a type of language which gives the illusion of science without any of its substance.

Lang fought a vigorous campaign to prevent Huntington becoming a member of the National Academy of Sciences in 1986 after he had been nominated. Lang was successful on this occasion and also on a second occasion when Huntington was again nominated. A detailed description of these events was published by Lang in Academia, Journalism, and Politics: A Case Study: The Huntington Case which occupies the first 222 pages of his 1998 book Challenges.

The second case we look at as an example of Lang's campaigns is his attempt to discredit the results of The 1977 survey of the American professoriate published in 1979 by Everett C Ladd and Seymour M Lipsett. In 1981 Lang published The File: Case Study in Correction (1977-1979) which consists of copies of correspondence concerning the survey. Lang always liked the facts to speak for themselves so he made relatively little commentary on the documents. He does, however, set out clearly why he fought that campaign and the many others that he fought with equal vigour:-

I don't like the nonsense that passes for rational discourse so often in our society. I am very much bothered by the inaccuracies, ambiguities, code words, slogans, catch phrases, public relation devices, sweeping generalizations, and stereotypes, which are used (consciously or otherwise) to influence people.

I am bothered by the inability of many to recognize these for what they are. I am bothered by the way people fudge issues, or are unable to clarify them, sometimes because they are inhibited by "collegiality" and other forms of intimidation (sometimes subtle, sometimes not). Most people put up with the nonsense without doing anything about it (unable or unwilling, for whatever reason - inertia, lack, of energy, lack of interest, lack of time, etc.), often falling into cynicism and despair.

I am bothered by the misinformation which gets disseminated uncritically through the media and by the obstructions which prevent correct information from being disseminated. These obstructions come about in many ways - personal, institutional, through self-imposed inhibitions, through external inhibitions, through outright dishonesty, through incompetence - the list is a long one.

I am bothered by the way misinformation, disguised as scholarship, is used in social, political, and educational contexts to affect policy decisions.

I am bothered by the way misinformation is accepted uncritically, and by the way people are unable to recognize it or reject it.

Lang received many honours for his achievements. For example he was awarded the Frank Nelson Cole Prize in 1959 from the American Mathematical Society, the Prix Carrière in France in 1967, the Humboldt Award for research and teaching in 1984, and the Leroy P Steele Prize for mathematical exposition from the American Mathematical Society in 1999, and the Dylan Hixon '88 Prize for Teaching in the Sciences in 2004. He was elected a member of the National Academy of Sciences in 1985. It is worth noting that although he accepted the Leroy P Steele Prize in 1999, he had in fact resigned from the American Mathematical Society in 1996 in a dispute about an article in the Notices. He made it very plain in his acceptance speech for the prize that he did not wish his acceptance in any way to lessen the arguments he had with the Society.