威廉·瑟斯顿William Thurston

威廉·瑟斯顿在佛罗里达州萨拉索塔的新学院学习。他于1967年在那里获得学士学位，然后转到加利福尼亚大学乔治·伯克利分校，在Morris Hirsch和斯蒂芬·斯梅尔的指导下进行研究。他于1972年获得博士学位，学位论文题为Foliations of 3-流形 which are circle bundles。这项工作证明了三维流形叶状结构中存在紧叶。

获得博士学位后，瑟斯顿于1972-73学年在普林斯顿高等爱德华·斯图迪研究所度过。然后，在1973年，他被任命为麻省理工学院数学助理教授。1974年，他被任命为普林斯顿大学数学教授。

在整个这一时期，瑟斯顿研究叶状结构。Lawson（[5]）总结这项工作：-

显然，瑟斯顿在叶状结构领域的贡献具有相当的深度。然而，使它们与众不同的是其惊人的原创性。他随后在奥斯瓦尔德·泰希米勒空间和3-流形理论方面的工作也是如此。

在[8]中，Wall描述了瑟斯顿的贡献，这些贡献使他在1982年获得了菲尔兹奖。事实上，1982年的约翰·查尔斯·菲尔兹奖章是在1982年8月初于华沙举行的国际数学联盟大会的一次会议上宣布的。它们直到在华沙举行的国际大会才颁发，该大会未能按计划于1982年举行，推迟到了次年。关于瑟斯顿因之获得奖章的工作的讲座在1983年国际大会上举行。Wall在发表该演讲时说：-

瑟斯顿具有非凡的几何洞察力和远见：他的思想彻底革新了2维和3维中拓扑学的研究，并在分析、拓扑学和几何学之间带来了新的富有成果的相互作用。

Wall [8]接着更详细地描述了瑟斯顿的工作：-

核心的新思想是：一大类闭3-流形应当承载双曲结构——即作为双曲空间对离散等距群作用的商，或等价地，承载常负曲率的度量。尽管这是2-流形情形的自然类比，在2-流形情形中这样的结果由波恩哈德·黎曼的均匀化定理给出，但在3维情形中它远不那么可信——甚至反直觉。

Kleinian群，即双曲3-空间的离散等距群，最早由儒勒·昂利·庞加莱研究，一个基本的有限性定理由拉尔斯·阿尔福斯证明。瑟斯顿关于Kleinian群的工作产生了许多新结果，并确立了一个著名的猜想。丹尼斯·苏利文在[6]中描述了这项几何工作，给出如下总结：-

瑟斯顿的结果既令人惊讶又优美。其方法是一种新层次的几何分析——一方面是在强有力的几何估计的意义上，另一方面是在空间可视化和想象的意义上，这确实是非凡的。

瑟斯顿的工作由Wall [8]总结如下：-

瑟斯顿的工作对3维拓扑学产生了巨大影响。这一领域有着“徒手”技巧的浓厚传统，与其他学科的互动相对较少。直接论证仍然必不可少，但3维拓扑学如今已牢固地重新汇入数学的主流。

瑟斯顿除了获得菲尔兹奖章外，还获得了许多荣誉。他于1974-75年获得阿尔弗雷德·P·斯隆基金会奖学金。1976年，他在叶状结构方面的工作使他获得了美国数学会的奥斯瓦尔德·维布伦几何奖。1979年，他获得了艾伦·T·沃特曼奖，是第二位获得该奖的数学家（第一位是1976年的查尔斯·费夫曼）。

1991年，瑟斯顿离开普林斯顿大学，回到加利福尼亚大学伯克利分校担任数学教授。1993年，他被任命为伯克利数学科学研究所所长。1996年，在留在加利福尼亚大学的同时，他从伯克利搬到了戴维斯。然后，在2003年，他被任命为康奈尔大学数学与计算机科学教授。

1997年，他出版了Three-dimensional geometry and topology. Vol. 1。Athanase Papadopoulos在一篇评论中解释了这本非凡著作的历史：-

1978年，W 瑟斯顿在普林斯顿大学开设了一门课程，主题是三维流形的几何与拓扑学。他为该课程写了讲义，这些讲义立即在世界各地流传开来。很可能所有从事低维拓扑学研究的人都认为，这些讲义中包含的思想是该主题有史以来最重要、最具影响力的思想。这些讲义开创了一个新的思想圈子，“瑟斯顿型几何”这一表述已变得非常普遍。1978年的普林斯顿讲义虽然风格非正式，但自成体系，拓扑学或几何学的研究生可以读懂。有些地方只给出了证明的概要，但对于大多数重要的新结果，证明中的论证是完整给出的。除了其中许多思想完全是新的之外，这些讲义以一种极具连贯性的阐述风格（尽管是特殊的）呈现了主题内容。瑟斯顿的阐述风格特殊之处在于，它要求读者积极参与其中，为心理图像留出空间，这也是为什么如果试图以线性方式阅读这些讲义，很容易陷入困境的原因之一。多年来，许多人（很可能也是他本人的意图）都要求瑟斯顿写出这些讲义更详细的版本。（某些章节的细节已由不同的个人和群体完成并发表。）本书源于作者整理和扩充这些讲义的努力，使其在形式和内容上都更易于理解。本书只包含原普林斯顿讲义一部分的发展内容，由于书名提到“第一部分”，希望会有续集。我们在这里得到的最终结果是一本书，幸运的是，它仍然以瑟斯顿的风格写成，要求读者想象力的参与，但比它所源自的1978年普林斯顿讲义各章包含了更多的细节。

他在热情洋溢的评论结尾说道：-

这里有许多美丽的数学。

2005年1月6日，在佐治亚州亚特兰大举行的联合数学会议上，瑟斯顿因Three-dimensional geometry and topology被授予美国数学会图书奖。颁奖词写道：-

这是令人兴奋且至关重要的数学。瑟斯顿的书在提供对微妙几何思想的直觉把握方面几乎是独一无二的。它对研究生和资深研究者都产生了巨大的影响。当然，从事几何化纲领研究的大批人员都将这本书视为他们工作的‘试金石’。一本在现代数学中扮演了如此重要而活跃角色的书，完全配得上AMS图书奖。

Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3-manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds.

After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University.

Throughout this period Thurston worked on foliations. Lawson ([5]) sums up this work:-

It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds.

In [8] Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall, giving that address, said:-

Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry.

Wall [8] goes on to describe Thurston's work in more detail:-

The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.

Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors. Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in [6], giving the following summary:-

Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable.

Thurston's work is summarised by Wall [8]:-

Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.

Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976).

In 1991, Thurston left Princeton University and returned to the University of California at Berkeley as Professor of Mathematics. In 1993 he was appointed Director of the Mathematical Sciences Research Institute at Berkeley. In 1996, while remaining at the University of California, he moved from Berkeley to Davis. Then, in 2003, he was appointed Professor of Mathematics and Computer Science at Cornell University.

In 1997 he published Three-dimensional geometry and topology. Vol. 1. The history of this remarkable book is explained by Athanase Papadopoulos in a review:-

In 1978, W Thurston gave a course at Princeton University, whose subject was the geometry and topology of three-dimensional manifolds. He wrote notes for that course, and the notes immediately circulated all over the world. It is probably the opinion of all the people working in low-dimensional topology that the ideas contained in these notes have been the most important and influential ideas ever written on the subject. These notes created a new circle of ideas, and the expression "Thurston type geometry" has become very common. The 1978 Princeton lecture notes, although written in an informal style, are self-contained and accessible to graduate students in topology or geometry. At some places the proofs are only sketched, but for most of the important new results, the arguments in the proofs are given completely. Besides the fact that many of these ideas are completely new, the notes present the subject matter in a great coherent expository (although special) style. Thurston's style of exposition is special in that it asks the reader to participate actively in what's going on by providing room for mental images, and this is one of the reasons why it is easy to get stuck if one tries to read these notes in a linear manner. For many years, Thurston was asked by many people (and it was probably also his own intention) to write a more detailed version of these notes. (Details of some sections have already been worked out and published by different individuals and groups of people.) The book under review grew out of the author's effort to organize and expand the notes, and to make them more accessible, both in form and in content. This book contains developments for only part of the original Princeton lecture notes, and since the title of the book refers to "Part I", there will hopefully be a sequel. The net result that we have here is a book which is, fortunately, still written in Thurston's style, demanding the participation of the reader's imagination, but with many more details than the chapters of the Princeton 1978 notes from which it grew.

He ends his enthusiastic review saying:-

There is a lot of beautiful mathematics here.

On 6 January 2005, at the Joint Mathematics Meetings in Atlanta, Georgia, Thurston was awarded the American Mathematical Society Book Prize for Three-dimensional geometry and topology. The citation for the award states:-

This is exciting and vital mathematics. Thurston's book is nearly unique in the intuitive grasp of subtle geometric ideas that it provides. It has been enormously influential, both for graduate students and seasoned researchers alike. Certainly the army of people who are working on the geometrization program regard this book as 'the touchstone' for their work. A book that has played such an important and dynamic role in modern mathematics is eminently deserving of the AMS Book Prize.