马克西姆·孔采维奇Maxim Kontsevich

马克西姆·孔采维奇出生在希姆基（有时写作Chimki或Himki）的一个有才华的家庭，该地距莫斯科市中心西北约17公里。他的父亲Lev Rafailovich 孔采维奇是朝鲜语和朝鲜历史的专家，曾在莫斯科俄罗斯科学院东方研究所担任首席研究员。他设计了朝鲜语西里尔化孔采维奇系统，这是当今用于生成朝鲜语文本俄语版本的主要系统。孔采维奇的母亲接受过工程师培训，他的哥哥Leonid（Lenny）后来从事计算机成像研究，并在旧金山工作。

孔采维奇在莫斯科上中学，从小就对数学和物理着迷。他写道：-

……多亏了我的兄弟和一些非常好的书。

在中学的最后三年，他参加了这两门科目的特别高级课程，这些课程的入学资格是通过竞赛决定的。他十六岁时在全国数学奥林匹克竞赛中获得第二名。由于他在这次竞赛中的成功，他获得了莫斯科国立大学的录取名额，无需参加入学考试。在那里，他受到许多杰出教授的教导，特别是伊斯拉埃尔·盖尔范德。1983年，当他只有十九岁时，孔采维奇的论文The growth of the Lie algebra generated by two generic vector fields on the line（俄语），与亚历山大·卡里洛夫合著，发表了。同年，他发表了Algebras of intermediate growth （俄语），与亚历山大·卡里洛夫和A I Molev合著。作者们如下描述内容：-

我们研究有限生成结合代数和李代数，其中自然过滤的第n项的维数增长快于$n$中的任何多项式但慢于任何指数$c^{n}$。由实线上的两个一般向量场生成的结合代数和索菲斯·李代数被考虑作为例子。

1985年，他离开大学，开始在该研究所从事研究，该研究所是俄罗斯科学院下属的信息处理问题研究所。他发表了进一步的论文The Virasoro algebra and Teichmüller spaces （俄语）（1988年）和Jackson networks on countable graphs（俄语）（1988年），最后一篇是合著。

也许事后看来，可以说对孔采维奇来说最重要的事件是1990年受邀在波恩的马克斯·普朗克研究所度过三个月。在他访问接近尾声时，该研究所举办了一次国际会议，会议的主要演讲者之一是迈克尔·阿蒂亚，用孔采维奇自己的话说，他是：-

……一位杰出的英国数学家，他谈到了奇妙的事情，最重要的是爱德华·威滕的猜想。

孔采维奇受到迈克尔·阿蒂亚演讲的启发，而[11]：-

……第二天，在会议参与者最后一次莱茵河乘船旅行期间，他向同事们解释了他打算如何证明爱德华·威滕的猜想。这个计划听起来如此令人印象深刻，以至于他当场被邀请回到马克斯·普朗克研究所作为访问学者整整一年。

当他第二年回到波恩时，他在波恩莱茵弗里德里希·威廉大学注册为博士生，导师是Don Zagier。他提交了博士论文Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function，并于1992年获得博士学位。在他的论文中，他实现了证明爱德华·威滕猜想的目標。他于1992年在一篇与论文同名的论文中发表了这些结果。Claude Itzykson在对该论文的评论中写道：-

本文呈现了作者就其关于爱德华·威滕一个猜想的证明所给出的最完整描述。其主题是计算由戴维·芒福德、Morita和Miller引入的稳定类的相交数，这些稳定类定义在……亏格为$g$、带有$n$个标记点的代数曲线的模空间的紧化上。爱德华·威滕猜想其生成函数是物理学家以“二维量子引力”之名研究过的矩阵积分的渐近展开。一个引人注目的推论是，它满足一个无穷可积梯队，由科特韦格-德弗里斯方程组成，并补充以一个所谓的“弦方程”。作者成功展示出第二类具有相同性质的矩阵乔治·比德尔·艾里积分。他的推导遵循威廉·瑟斯顿、戴维·芒福德、Harer和Penner的思想，优雅地归约为一个组合问题。

Clifford Henry Taubes写道[10]：-

……这一证明中的许多步骤都展现了孔采维奇在组合计算方面的独特才能。

这一非凡成就使孔采维奇收到了哈佛大学、普林斯顿高等爱德华·斯图迪研究院和波恩大学的邀请。他在1992年至1995年间访问了这三处。然而在1993年，他收到了加州大学乔治·伯克利分校的教授职位聘约，并在那里一直待到1996年，之后他移居法国[11]：-

……孔采维奇本可以在美国永久定居。他在伯克利有一个职位，离他兄弟居住的旧金山不远。事实上，他当时正要在那里购置房产，这时高等科学研究所向他提供了常任教授职位。他了解该研究所的声誉，因为1988年他在法国进行短期工作访问期间曾在那里待过几天。

孔采维奇在1992年发表杰出论文之后，次年迅速又发表了另一篇题为Vassiliev's knot invariants的论文。琼·蕾特·比尔曼在一篇评论中写道：-

V A Vassiliev[在1990年]引入了一族纽结的数值不变量。……Vassiliev证明这些不变量由一个非常复杂的组合构造所决定。它们非常强大，并且包含了沃恩·弗雷德里克·兰德尔·琼斯多项式及其所有推广。所评论文是关于Vassiliev不变量的深远结果的研究通告。……主要定理有许多推论，并且是我们撰写本评论时正在进行中的众多研究的主题。然而，也许比结果的详细陈述更重要的是，作者对Vassiliev不变量采取了非常新颖而独到的视角，并告诉我们他是如何被引向这一视角的。他还对Vassiliev的理论作了阐述，虽然缺乏细节，却使其显得既自然又清晰，而仔细阅读那篇详细论文却做不到这一点。奇怪的是，作者表示他不打算为其理论撰写完整阐述，而将这一任务留给他人，尤其是D Bar-Natan[On the Vassiliev knot invariants, Topology]。这很可惜，因为当发现某一现象的人亲自撰写它时，他能够以别人无法替代的方式让你了解他的思路。因此，这篇小论文应当反复阅读：作为该主题的入门，因为它对其背后直观思想的洞见，并且（在消化了Bar-Natan对证明的阐述之后）再回到它以获得新的洞见。

1992年在巴黎举行的第一届欧洲数学大会上，孔采维奇作了邀请报告Feynman diagrams and low dimensional topology，该报告于1994年发表在会议论文集中。他还在这次第一届欧洲数学大会上获得了欧洲数学学会奖。同样在1994年，他发表了与尤里·伊万诺维奇·马宁合著的Gromov-Witten classes, quantum cohomology, and enumerative geometry。1993年，他发表了Formal (non)commutative symplectic geometry，该文由Alexander Voronov审阅。他的评论开头如下：-

该文着重讨论了三种基本类型的代数——索菲斯·李、结合代数和交换代数——作为非交换辛几何三种假想版本的函数模型（事实上，第三种情形就是通常的交换情形）。文中概述了非交换几何中的微分形式、辛形式、哈密顿向量场和西莫恩·德尼·泊松括号的演算。作为这些思想的一个应用（以及一个动机），文中展示了如何产生代数曲线模空间的上同调类。

孔采维奇是1994年在苏黎世举行的国际数学家大会的全会报告人。他随后证明了任何西莫恩·德尼·泊松流形都容许一个形式量子化，并给出了平坦情形的显式公式。这是孔采维奇所研究的四个主要问题之一，这使他在1998年柏林国际数学家大会上获得了菲尔兹奖。另外三个问题已在上文提及：曲线模空间上的相交理论、低维拓扑学特别是通过涉及理查德·费曼图的积分研究纽结，以及有理曲线的计数。

2008年，孔采维奇和爱德华·威滕共同获得了瑞典皇家科学院颁发的克拉福德数学奖：-

……表彰他们受现代理论物理学启发而对数学作出的重要贡献。

Royal Swedish Academy of Sciences的新闻稿对孔采维奇和爱德华·威滕获奖工作的描述如下：-

数学领域的获奖者运用物理学的方法论，发展出了一门革命性的新数学，旨在研究各种类型的几何对象。他们的工作不仅在数学学科内引起了极大的兴趣，还可能在完全不同的领域找到应用。其成果对物理学及自然基本规律的研究具有相当大的价值。根据弦理论——一个旨在为所有自然力构建统一理论的雄心勃勃的尝试，构成宇宙的最小粒子是振动的弦。这一理论预言了额外维度的存在，并需要极为高深的数学。获奖者们解决了与弦理论相关的若干重要数学问题，从而为其进一步发展铺平了道路。

除上述荣誉外，孔采维奇还被授予丹尼尔·伊戈尔尼策奖，并当选为巴黎科学院的成员。

Maxim Kontsevich was born into a talented family in Khimki (sometimes written Chimki or Himki), about 17 km northwest of the centre of Moscow. His father, Lev Rafailovich Kontsevich, is an expert on the Korean language and Korean history, and he worked as Leading Researcher at the Institute of Oriental Studies of the Russian Academy of Sciences in Moscow. He devised the Kontsevich system for the Cyrillization of the Korean language, the main system in use today for producing Russian versions of Korean texts. Maxim's mother trained as an engineer, and his elder brother Leonid (Lenny) went on to undertake research into computer imaging and works in San Francisco.

Kontsevich attended secondary school in Moscow and became fascinated by mathematics and physics at an early age. This was, he wrote:-

... thanks to my brother and some very good books.

He took special advanced courses in these two subjects during his final three years at secondary school, having won the right to enter these courses which was determined by competition. When he was sixteen years old he was placed second in the national Mathematical Olympiad competition. Because of his success in this competition he was offered a place at Moscow State University without having to sit the entrance examinations. There he was taught by a number of outstanding professors, in particular by Israil Moiseevic Gelfand. In 1983, when he was only nineteen years old, Kontsevich's paper The growth of the Lie algebra generated by two generic vector fields on the line (Russian), written jointly with A A Kirillov, was published. In the same year he published Algebras of intermediate growth (Russian), written jointly with A A Kirillov and A I Molev. The authors describe the contents as follows:-

We investigate finitely generated associative and Lie algebras for which the dimension of the nth term of the natural filtration grows faster than any polynomial in $n$ but slower than any exponent $c^{n}$. The associative and Lie algebras generated by two generic vector fields on the real line are considered as examples.

In 1985, he left university to begin research at the Institute for Problems of Information Processing, an Institute attached to the Russian Academy of Sciences. He published further papers The Virasoro algebra and Teichmüller spaces (Russian) (1988) and Jackson networks on countable graphs (Russian) (1988), the last paper being a joint publication.

Perhaps in retrospect one can say that the most significant event for Kontsevich was an invitation to spend three months at the Max Planck Institute in Bonn in 1990. Towards the end of his visit an international conference was held at the Institute and one of the main speakers at the conference was Michael Atiyah who, in Kontsevich's own words, was :-

... an eminent British mathematician who spoke of wonderful things, most importantly Witten's conjecture.

Kontsevich was inspired by Atiyah's talk, and [11]:-

... the next day, during a final boat trip on the Rhine for conference participants, he explained to his colleagues how he intended to prove Witten's conjecture. The project sounded so impressive that he was invited there and then to return to the Max Planck Institute as a visitor for a full year.

When he returned to Bonn in the next year he registered as a doctoral student at the Rheinische Friedrich-Wilhelms University of Bonn with Don Zagier as his thesis supervisor. He submitted his doctoral thesis Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function and was awarded his doctorate in 1992. In his thesis he achieved his aim of proving Witten's conjecture. He published the results in a paper of the same title as his thesis in 1992. Claude Itzykson writes in a review of the paper:-

This article presents the most complete description given by its author of the proof of a conjecture by E Witten. Its subject is the computation of intersection numbers of stable classes, introduced by Mumford, Morita and Miller, on ... a compactification of the moduli space of genus-$g$ algebraic curves with $n$ marked points. Witten conjectured as generating function an asymptotic expansion of matrix integrals investigated by physicists under the name "two-dimensional quantum gravity". A striking consequence is that it satisfies an infinite integrable hierarchy of Korteweg-de Vries equations completed by a so-called "string equation". The author succeeds in exhibiting a second type of matrix Airy integral possessing the same properties. His derivation uses an elegant reduction to a combinatorial problem following ideas of Thurston, Mumford, Harer and Penner.

Clifford Henry Taubes writes [10]:-

... many of the steps in this proof exhibit Kontsevich's unique talent for combinatorial calculations.

This remarkable achievement led to Kontsevich receiving invitations to Harvard University, Princeton's Institute for Advanced Study, and the University of Bonn. He made visits to all three between 1992 and 1995. In 1993, however, he had received an offer of a Professorship at the University of California at Berkeley where he remained until 1996 when he moved to France [11]:-

... Kontsevich could have settled permanently in the United States. He had a post at Berkeley, not far from San Francisco where his brother was living. He was in fact on the point of buying a home there when the Institut des Hautes Études Scientifiques offered him the post of resident professor. He knew the institute's reputation, having spent a few days there in 1988 during a short working visit to France.

Kontsevich quickly followed his brilliant paper of 1992 with another in the following year entitled Vassiliev's knot invariants. J S Birman writes in a review:-

V A Vassiliev [in 1990] introduced a family of numerical invariants of knots. ... These invariants were shown by Vassiliev to be determined by a very complicated combinatorial construction. They are very powerful, and subsume the Jones polynomial and all of its generalizations. The paper under review is a research announcement of far-reaching results about the Vassiliev invariants. ... The main theorem has many implications, and is the subject of numerous investigations which are in progress as we write this review. However, what is perhaps even more important than the detailed statement of results is that the author has taken a very fresh and original look at the Vassiliev invariants and tells us how he was led to do it. He also gives an exposition of Vassiliev's theory which, while lacking details, makes it seem both natural and clear in a way which a careful reading of that detailed paper did not do. Curiously, the author has indicated that he does not intend to write a full exposition of his theory, leaving that task to others, notably D Bar-Natan [On the Vassiliev knot invariants, Topology]. That is a pity because when the person who has discovered a phenomenon writes about it he is able to let you in on his way of thinking in a way that another simply cannot do for him. For that reason this little paper should be read and re-read: as an introduction to the subject, for its insights into the intuitive ideas behind it, and (after digesting Bar-Natan's exposition of the proof) to return to it for fresh insights.

At the First European Congress of Mathematics in Paris in 1992 Kontsevich gave the invited address Feynman diagrams and low dimensional topology which was published in 1994 in the Proceeding of the conference. He was also awarded a European Mathematical Society Prize at this First European Congress. Also in 1994 he published Gromov-Witten classes, quantum cohomology, and enumerative geometry which was written jointly with Yuri Manin. In 1993 he published Formal (non)commutative symplectic geometry which was reviewed by Alexander Voronov. His review begins:-

The paper places emphasis on the three fundamental types of algebras - Lie, associative and commutative - as functional models of three hypothetical versions of noncommutative symplectic geometry (in fact, the usual commutative one in the third case). Calculus of differential forms, symplectic forms, Hamiltonian vector fields and Poisson brackets in noncommutative geometry are sketched. As an application of (and a motivation for) these ideas, it is shown how to produce cohomology classes of the moduli spaces of algebraic curves.

Kontsevich was a Plenary Speaker at the International Congress of Mathematicians in 1994 in Zürich. He then proved that any Poisson manifold admits a formal quantization and gave an explicit formula for the flat case. This was one of the four major problems which Kontsevich worked on, leading to him receiving a Fields Medal at the International Congress of Mathematicians in Berlin in 1998. The other three problems have been mentioned above: intersection theory on moduli spaces of curves, low-dimensional topology particularly knots via integrals related to Feynman diagrams, and the enumeration of rational curves.

In 2008 Kontsevich and Witten jointly received the Crafoord Prize in Mathematics from the Royal Swedish Academy of Sciences:-

... for their important contributions to mathematics inspired by modern theoretical physics.

The news release of the Royal Swedish Academy of Sciences described the work for which Kontsevich and Witten were awarded the prize in the following way:-

The laureates in mathematics have used the methodology of physics to develop a revolutionary new mathematics intended for the study of various types of geometrical objects. Their work is not only of great interest in the discipline of mathematics but may also find applications in totally different areas. Its results are of considerable value for physics and research into the fundamental laws of nature. According to string theory, which is an ambitious attempt to formulate a theory for all the natural forces, the smallest particles of which the universe is composed are vibrating strings. This theory predicts the existence of additional dimensions and requires very advanced mathematics. The laureates have resolved several important mathematical problems related to string theory and have in this way paved the way for its further development.

In addition to the honours mentioned above, Kontsevich was awarded the Daniel Iagolnitzer Prize and elected a member of the Academy of Sciences in Paris.