米哈伊尔·格罗莫夫Mikhail Gromov

米哈伊尔·格罗莫夫出生于Boksitogorsk，这是圣彼得堡（或他出生时所称的列宁格勒）以东约200公里的一个城镇。他的父母是Leonid Gromov和Lea Rabinovitz。格罗莫夫就读于列宁格勒大学，于1965年获得数学硕士学位。俄罗斯体系中的硕士学位基本上相当于美国的博士学位。他继续学习以成为大学教师，并于1969年获得博士学位（相当于教授资格论文（Habilitation））。他在列宁格勒的研究导师是Vladimir Abramovich Rokhlin，后者曾是安德雷·柯尔莫哥洛夫和列夫·庞特里亚金的学生。在获得博士学位之前，格罗莫夫已于1967年与Margarita 格罗莫夫结婚，并于同年被任命为列宁格勒大学的助理教授。他继续担任这一职务直到1974年。

格罗莫夫在1960年代后期发表的论文包括以下这些（全部用俄语写成）：On a geometric hypothesis of Banach（1967）；The number of simplexes in subdivisions of finite complexes（1968）；Transversal mappings of foliations（1968）；Transversal mappings of foliations（1968）；Simplexes inscribed on a hypersurface（1968）；以及Stable mappings of foliations into manifolds（1969）。
格罗莫夫从这一时期开始的数学贡献，由Hung-Hsi Wu描述如下：-

大约在1970年，微分几何界被一则消息震惊了：一位名叫格罗莫夫的年轻俄罗斯人证明了任何非紧微分流形都容许一个具有正截面曲率的黎曼度量，也容许一个具有负截面曲率的黎曼度量。我们还被告知，这是通过一种“软”的拓扑层方法实现的。此外，在同一个框架中，格罗莫夫还证明了Hirsch-斯蒂芬·斯梅尔浸入定理和A Phillips淹没定理的推广。还有更多结果被承诺。慢慢地，格罗莫夫的论文（有些是与Ya M Eliashberg和V A Rokhlin合作的）在七十年代初流传到了西方。

1970年，国际数学家大会在法国尼斯举行。格罗莫夫受邀在大会上作报告，但苏联当局不允许他离开苏联。不过，他提交了演讲稿A topological technique for the construction of solutions of differential equations and inequalities，该文于1971年发表在会议论文集中。这是吴文俊在上文引述中所提到的论文之一，同样被提及的还有格罗莫夫（与V A Rokhlin合作）的论文 Imbeddings and immersions in Riemannian geometry（1970年），以及（与Ya M Eliashberg合作）的Elimination of singularities of smooth mappings（1971年）。由于他的杰出工作，格罗莫夫于1971年被授予莫斯科数学会奖。随后，他于1973年向列宁格勒大学提交了博士后论文。

1974年，格罗莫夫离开俄罗斯前往美国，被任命为纽约州立大学石溪分校数学教授。1979年，他在巴黎第七大学讲授了一门课程Structures métriques pour les variétés riemanniennes，这门课程产生了显著的影响。这些讲义于1981年出版，格罗莫夫在序言中写道：-

这些讲义来自1979年最后一个学期在巴黎第七大学开设的一门课程。我们的目的是介绍黎曼流形V的曲率与其整体行为之间已经建立的一些联系。在这里，“整体”一词不仅适用于V的拓扑，也适用于黎曼流形及其之间映射的一族度量不变量。V最简单的度量不变量例如是其体积和直径；对于从$V_{1}$到$V_{2}$的映射，伸缩率是一个重要的不变量。事实上，这类度量不变量也出现在纯拓扑背景中，并且它们提供了V上的无穷小数据（通常由曲率条件表示）与V的拓扑之间的重要联系。例如，现在经典的Bonnet定理给出了正曲率流形V直径的上界，由此可以推出V的基本群的有限性。对黎曼流形进行更深入的拓扑研究需要比直径或体积更精细的度量不变量；我们试图对这些不变量进行系统的处理，但这项研究远未达到我们所希望的全面程度。

这些讲义的英文版于1999年以Metric structures for Riemannian and non-Riemannian spaces为书名出版。Igor Belegradek的书评开头如下：-

本书的第一版以法文出版[Structures métriques pour les variétés riemanniennes (1981)]，被认为是过去二十年中几何学领域最具影响力的著作之一。自那时起，该领域的边界急剧扩展。反映这种增长，新的英文版篇幅几乎增加了四倍。在最重要的增补中，每一部分都超过一百页，包括一章关于带测度的度量空间的收敛性，以及由Semmes撰写的关于度量空间上分析的附录。此外，全书中插入了大量的注记、例子、证明和未解决的问题。原始文本大部分得以保留，新增内容通过下标+方便地标示出来。

2005年，格罗莫夫因这本1999年的书获得了鲍耶国际数学奖。我们一直在关注格罗莫夫1979年在巴黎开设的课程所带来的显著发展。我们应该回过头来讲述他的职业生涯从那时起是如何发展的。1981年，他从纽约州立大学石溪分校转到巴黎第六大学，次年转到高等科学研究所，成为永久成员。他继续担任这一职位，但此外还在美国担任过职务。1991年至1996年，他是马里兰大学学院市分校数学教授，然后从1996年起担任理查·科朗特数学科学研究所的Jay Gould数学教授。

我们在上文已经说明，苏联当局不允许格罗莫夫参加1970年在尼斯举行的国际数学家大会，他本已受邀作为报告人。他得以接受邀请，在赫尔辛基（1978年）和华沙（1982年）的国际数学家大会上作报告。1986年，他是乔治·伯克利国际数学家大会的受邀全会报告人，在会上他就Soft and hard symplectic geometry作了报告。Hung-Hsi Wu写道：-

这是对辛几何近期工作的综述，重点在于作者本人的贡献……首先，对标题作一些解释。用作者自己的话说：“直观上，‘硬’指的是给定对象具有强而刚性的结构，而‘软’则暗示一大类对象具有某种弱的普遍性质。……本次演讲中的‘软’和‘硬’仅限于涉及光滑流形之间映射空间的几何的全局非线性分析框架。”

1985年，格罗莫夫是剑桥英国数学学术讨论会的全会报告人，当时他就Differential geometry with and without infinitesimal calculus: anatomy of curvature作了演讲。

格罗莫夫因其杰出贡献获得了一系列极为重要的数学奖项。这些奖项包括：American Mathematical Society颁发的奥斯瓦尔德·维布伦几何奖（1981年）：-

……因其在黎曼流形的拓扑性质与几何性质之间关系方面的工作；

巴黎科学院的埃利·嘉当奖（1984年）；巴黎保险联盟奖（1989年）；以及沃尔夫数学奖（1993年）。1997年，他被授予罗巴切夫斯基奖章，同年，他因对研究的开创性贡献而从美国数学会获得了Leroy P 凯瑟琳·斯蒂尔奖：-

……因他的论文⟦E1⟧，该论文彻底改变了辛几何与拓扑学这一主题，并且是当前许多研究活动的核心，包括量子上同调和镜像对称。

这篇1985年的论文为辛拓扑学的基本问题开辟了一条新的有效途径。1999年，格罗莫夫被授予巴尔赞数学奖。该奖的颁奖词如下：-

格罗莫夫毫无疑问是本世纪最伟大的几何学家之一。他的工作之所以独特，既在于他所创造的概念之丰富与有力，也在于他为解决问题而设计并应用的新技术，这些问题往往陈述和理解起来都很简单，乍看之下却似乎难以企及。其中一些问题由来已久，而它们出人意料的解答因其方法的独创性与优雅而令人惊叹和意外，这些方法由格罗莫夫构想：著名的例子包括他证明了那个古老的猜想，即一个有限生成的多项式增长群具有一个有限指数的幂零子群，或者（与I Pyatetski-Shapiro一起）在任意维数下对双曲变换的非算术离散群的精美构造。另一方面，格罗莫夫为不同目的而发展的新技术引出了全新类型的问题：人们可以想象，在所有黎曼流形（的同构类）的集合上引入一种自然的几何结构，或者发现许多新的、引人注目的流形不变量（例如K-面积、单纯体积、极小体积等），会产生多么丰富多样的问题，更不用说像双曲群这样的重要新概念，它正是微分几何中近期重大发展的源头。总而言之，格罗莫夫不仅带来了对著名而古老问题的解答，也为许多学者奠定了新研究领域的基础。上文已经强调，他倾向于从几何的角度看待所有问题：他将它们转化为专门的几何术语，并运用他非凡的几何直觉来彻底研究它们；还应当补充的是，他也能够以同样的方式处理来自数学最不同分支的问题：代数、分析、微分方程、概率论、理论物理等。由于他弟子众多，以及他的重要发现所引起的广泛反响，格罗莫夫已经并将继续对当代数学产生相当大的影响。

接下来授予格罗莫夫的重要奖项是2002年京都奖：-

2002年京都奖基础科学类得主是法国的格罗莫夫 Leonidovich 格罗莫夫。他的贡献，包括为各类几何对象的族引入度量结构，导致了几何学及数学许多其他领域的戏剧性发展。格罗莫夫在几何学和分析等多个领域开创了全新的学科，并对所有数学科学产生了实质性影响。通过应用创新思想和激进的非传统数学方法，他还解决了现代几何学中大量复杂的问题。

他于2004年获得西北大学颁发的弗雷德里克·埃瑟·内默斯数学奖：-

……因他在黎曼几何方面的工作，这一工作彻底改变了该学科；他在辛流形中伪全纯曲线方面的理论；他对多项式增长群问题的解决；以及他对双曲群理论的构建。

2008年，London Mathematical Society选举格罗莫夫为学会荣誉会员：-

……以表彰他深刻而非凡的洞见，其影响远远超出了他自己所在的几何学领域。

也许在他所获得的众多重要奖项中，最负盛名的是2009年的尼尔斯·阿贝尔奖：-

俄法数学家格罗莫夫 L 格罗莫夫是我们这个时代最杰出的数学家之一。他因在数学许多领域，尤其是几何学方面的重要贡献而闻名。几何学是数学最古老的领域之一；几个世纪以来，它一直吸引着伟大数学家的注意，但在过去50年中经历了革命性的变化。格罗莫夫引领了一些最重要的发展，产生了极具原创性的普遍思想，这些思想为几何学和数学其他领域带来了新的视角。格罗莫夫的名字永远与黎曼几何、辛几何、弦论和群论中的深刻结果和重要概念联系在一起。尼尔斯·阿贝尔委员会说：“格罗莫夫始终在追求新问题，并不断思考解决旧问题的新思路。他在整个职业生涯中做出了深刻而原创的工作，并且仍然极具创造力。格罗莫夫的工作将继续成为未来许多数学发现的灵感源泉”。

让我们引用G Elek [1]的话来结束这篇传记：-

格罗莫夫 真正地革新了几何学；奠定了全新领域的基础，引入了惊人的新观点和一种使他的论文与思想独一无二的哲学。

阿伯拉罕·伊本·埃兹拉写道：-

他的工作既极其优雅，又与应用数学中的问题直接相关，这反映了他非凡的创造力和卓越的品味。

Mikhael Leonidovich Gromov was born in Boksitogorsk, a town about 200 km east of St Petersburg (or Leningrad as it was called at the time of his birth). His parents were Leonid Gromov and Lea Rabinovitz. Gromov attended Leningrad University, graduating with a Masters degree in Mathematics in 1965. The Masters degree in the Russian system is essentially equivalent to a doctorate in the USA. He continued to study to become a university teacher being awarded a doctorate (equivalent to the habilitation) in 1969. He research supervisor at Leningrad was Vladimir Abramovich Rokhlin who had been a student of Andrei Nikolayevich Kolmogorov and Lev Semenovich Pontryagin. Before the award of his doctorate, Gromov had been married to Margarita Gromov in 1967 and appointed as an Assistant Professor at Leningrad University in the same year. He continued in this role until 1974.

The papers which Gromov published in the late 1960s include the following (all written in Russian): On a geometric hypothesis of Banach (1967); The number of simplexes in subdivisions of finite complexes (1968); Transversal mappings of foliations (1968); Transversal mappings of foliations (1968); Simplexes inscribed on a hypersurface (1968); and Stable mappings of foliations into manifolds (1969).
Gromov's mathematical contributions, beginning in this period, are described by Hung-Hsi Wu:-

Around 1970, the world of differential geometry was astounded by the news that a young Russian by the name of Mikhael Gromov had proved that any noncompact differential manifold admits a Riemannian metric of positive sectional curvature, and also one of negative sectional curvature. We were also told that this was achieved by a "soft" method of topological sheaves. Moreover, in one and the same setting, Gromov also proved generalizations of both the Hirsch-Smale immersion theorem and the A Phillips submersion theorems. Many more results were promised. Slowly, Gromov's papers (some in collaboration with Ya M Eliashberg and V A Rokhlin) filtered to the West in the early seventies.

In 1970 the International Congress of Mathematicians was held in Nice, France. Gromov had been invited to address the Congress but was not allowed to leave the USSR by the Soviet authorities. He did, however, contribute the text of his lecture A topological technique for the construction of solutions of differential equations and inequalities which was published in the Conference Proceedings in 1971. This is one of the papers referred to in the above quote by Wu, as are Gromov's papers (with V A Rokhlin) Imbeddings and immersions in Riemannian geometry (1970), and (with Ya M Eliashberg) Elimination of singularities of smooth mappings (1971). For his outstanding work Gromov was presented with the Award of the Moscow Mathematical Society in 1971. He then presented his Post-doctoral Thesis to Leningrad University in 1973.

In 1974 Gromov left Russia for the United States when he was appointed as Professor of Mathematics at the State University of New York at Stony Brook. In 1979 he gave a course of lectures Structures métriques pour les variétés riemanniennes at Paris VII which have been remarkably influential. These notes were published in 1981 and Gromov wrote in the Preface:-

These notes are from a course given at the University of Paris VII during the last trimester of 1979. Our purpose was to present some of the links that have been established between the curvature of a Riemannian manifold V and its global behaviour. Here, the word 'global' applies not only to the topology of V but also to a family of metric invariants of Riemannian manifolds and mappings between these manifolds. The simplest metric invariants of V are, for example, its volume and its diameter; the dilatation is an important invariant for a mapping from $V_{1}$ into $V_{2}$. In fact, such metric invariants also appear in a purely topological context, and they provide an important link between infinitesimal data on V (generally expressed by a condition on the curvature) and the topology of V. For example, the now classical theorem of Bonnet gives an upper bound for the diameter of a manifold V with positive curvature, from which one can deduce the finiteness of the fundamental group of V. A deeper topological study of Riemannian manifolds requires finer metric invariants than diameter or volume; we have tried to present a systematic treatment of these invariants, but this study is far from being as comprehensive as we had hoped.

An English edition of these notes was published as Metric structures for Riemannian and non-Riemannian spaces in 1999. Igor Belegradek begins a review as follows:-

The first edition of this book, published in French [Structures métriques pour les variétés riemanniennes (1981)], is considered one of the most influential books in geometry in the last twenty years. Since then the boundary of the field has dramatically exploded. Reflecting this growth, the new English edition has almost quadrupled in size. Among the most substantial additions, each taking over a hundred pages, there is a chapter on convergence of metric spaces with measures, and an appendix on analysis on metric spaces written by Semmes. In addition, numerous remarks, examples, proofs, and open problems are inserted throughout the book. The original text is mostly preserved with new items conveniently indicated by a subscript +.

In 2005 Gromov received the Janos Bolyai International Mathematical Prize for this 1999 book. We have been following the remarkable developments which have come from Gromov's 1979 course in Paris. We should return to relate how his career developed from that time. In 1981 he moved from the State University of New York at Stony Brook to the Université de Paris VI and the following year he moved to the Institut des Hautes Études Scientifiques where he was made a permanent member. He has continued to hold this position but has, in addition, had posts in the United States. He was Professor of Mathematics at University of Maryland, College Park from 1991 to 1996, and then Jay Gould Professor of Mathematics at the Courant Institute of Mathematical Sciences, New York University from 1996.

We have explained above that the Soviet authorities did not allow Gromov to attend the International Congress of Mathematicians in Nice in 1970 to which he had been invited as a speaker. He was able to accept invitations to speak at the International Congresses of Mathematicians in Helsinki (1978) and Warsaw (1982). In 1986 he was an invited plenary speaker at the International Congress of Mathematicians in Berkeley where he spoke on Soft and hard symplectic geometry. Hung-Hsi Wu writes:-

This is a survey of the recent work on symplectic geometry, with emphasis on the author's own contributions .... First, some explanations about the title. In the author's own words, "Intuitively, 'hard' refers to a strong and rigid structure of a given object, while 'soft' suggests some weak general property of a vast class of objects. ... 'Soft' and 'hard' in this talk are limited to the framework of the global nonlinear analysis concerning the geometry of spaces of maps between smooth manifolds".

In 1985 Gromov was a plenary speaker at the British Mathematical Colloquium in Cambridge when he lectured on Differential geometry with and without infinitesimal calculus: anatomy of curvature.

Gromov has received a fantastic collection of major mathematical prizes for his wonderful contributions. These include: the Oswald Veblen Prize in Geometry from the American Mathematical Society (1981):-

... for his work relating topological and geometric properties of Riemannian manifolds;

the Prix Élie Cartan of the Académie des Sciences of Paris (1984); the Prix de l'Union des Assurances de Paris (1989); and the Wolf Prize in Mathematics (1993). In 1997 he was awarded the Lobachevsky Medal and, in the same year, he received the Leroy P Steele Prize from the American Mathematical Society for Seminal Contribution to Research:-

... for his paper "Pseudo-holomorphic curves in symplectic manifolds", which revolutionized the subject of symplectic geometry and topology and is central to much current research activity, including quantum cohomology and mirror symmetry.

This 1985 paper opens a new effective approach to fundamental problems of symplectic topology. In 1999 Gromov was awarded the Balzan Prize for Mathematics. The Laudatio for this Prize reads:-

Mikhael L Gromov is, without any doubt, one of the greatest geometers of this century. His work is unique through the abundance and the force of the concepts he has created, as well as through the new techniques he has devised and applied to solve problems, often simple to state and to understand, and which seem, at first sight, inaccessible. Some of those problems were long standing, and their unexpected solutions caused wonder and surprise due to the originality and elegance of the method conceived by Gromov: famous instances are his proof of the old conjecture according to which a finitely generated group of polynomial growth has a nilpotent subgroup of finite index, or the beautiful construction (together with I Pyatetski-Shapiro) of non-arithmetic discrete groups of hyperbolic transformations in arbitrary dimension. On the other hand, new techniques developed by Gromov for different purposes led to completely new kinds of problems: one can imagine the great variety of questions arising from the introduction of a natural geometric structure on the set of all (isomorphism classes) of Riemannian manifolds, or from the discovery of many new and remarkable invariants of manifolds (e.g., the K-area, the simplicial volume, the minimal volume, etc.), not to forget important new notions, such as that of hyperbolic groups, which is at the origin of major recent developments in differential geometry. To summarise, Gromov has brought about not only solutions to famous and time-old problems, but also the bases of new fields of study for many scholars. It has been emphasised above that he tends to look at all questions from the geometric side: he translates them in ad hoc geometric terms, and uses his extraordinary geometric intuition to investigate them thoroughly; it should be added that he is also able to treat, in the same way, questions coming from the most diverse branches of mathematics: algebra, analysis, differential equations, probability theory, theoretical physics, etc. Due to the large number of his disciples and the wide repercussions aroused by his important discoveries, Mikhael Gromov has had, and will continue to have, a considerable influence on contemporary mathematics.

The next major prize to be presented to Gromov was the 2002 Kyoto Prize:-

The 2002 Kyoto Prize laureate in Basic Sciences is Mikhael Leonidovich Gromov of France. His contributions, including the introduction of a metric structure for families of various geometrical objects, have led to dramatic developments in geometry and many other fields of mathematics. Gromov has pioneered entirely new disciplines in a variety of fields, including geometry and analysis, and has had a substantial impact on all the mathematical sciences. Through the application of innovative ideas and radical nontraditional mathematical methods, he has also solved a great many complicated problems in modern geometry.

He received the 2004 Frederic Esser Nemmers Prize in Mathematics from Northwestern University:-

... for his work in Riemannian geometry, which revolutionized the subject; his theory of pseudoholomorphic curves in symplectic manifolds; his solution of the problem of groups of polynomial growth; and his construction of the theory of hyperbolic groups.

In 2008 the London Mathematical Society elected Gromov to Honorary Membership of the Society:-

... in recognition of his profound and extraordinary insights whose influence extends far beyond the boundaries of his own field of geometry.

Perhaps the most prestigious of all the major awards he has received was the Abel Prize for 2009:-

The Russian-French mathematician Mikhail L Gromov is one of the leading mathematicians of our time. He is known for important contributions in many areas of mathematics, especially geometry. Geometry is one of the oldest fields of mathematics; it has engaged the attention of great mathematicians through the centuries, but has undergone a revolutionary change in the last 50 years. Mikhail Gromov has led some of the most important developments, producing profoundly original general ideas which have resulted in new perspectives on geometry and other areas of mathematics. Gromov's name is forever attached to deep results and important concepts within Riemannian geometry, symplectic geometry, string theory and group theory. The Abel committee says: "Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions to old problems. He has produced deep and original work throughout his career and remains remarkably creative. The work of Gromov will continue to be a source of inspiration for many future mathematical discoveries".

Let us end this biography by quoting from G Elek [1]:-

[Gromov] has truly revolutionised geometry; laid the foundations of brand new fields, introduced spectacularly new viewpoints and a philosophy which makes his papers and thoughts unmistakable.

Ezra Getzler writes:-

His work is both tremendously elegant and immediately relevant to problems in applied mathematics in a way that reflects his tremendous creativity and excellent taste.