弗拉基米尔·阿诺尔德Vladimir Arnold

弗拉基米尔·阿诺尔德的父母是Igor Vladimorovich 阿诺尔德和Nina Alexandrova Isakovich。阿诺尔德的家族中连续几代人都是科学家。他对数学的兴趣始于五岁时。他解释说，这是俄罗斯数学传统[4]的结果：-

非常年幼的孩子甚至在掌握任何数字知识之前就开始思考[老商人]问题。五到六岁的孩子非常喜欢这些问题，并且能够解决它们，但对于被形式化数学训练惯坏了的大学毕业生来说，这些问题可能太难了。……许多俄罗斯家庭有给孩子出数百道[数学]题的传统，我的家庭也不例外。

他十二岁时，他的学校老师给了他一些具有挑战性的问题。他在[4]中引用了其中一个问题：-

两位老妇人日出时出发，各自以恒定速度行走。一位从A到B，另一位从B到A。她们在中午相遇，然后继续不停，分别于下午4点到达B和晚上9点到达A。这一天日出是什么时候？

阿诺尔德说：-

我花了一整天思考这个老问题，而解答……如同启示一般降临。我当时感受到的发现之感，与后来所有更为严肃的问题中的感受完全相同……

1954年，他进入莫斯科国立大学，成为力学与数学系的一名本科生。1959年，他以On mappings of a circle to itself获得第一个学位，导师是安德雷·柯尔莫哥洛夫。谈到他的本科岁月时，他说[4]：-

我在力学与数学系学习时，同一个系里聚集了如此众多伟大的数学家，这确实非同寻常，我在任何其他地方都从未见过这样的情形。安德雷·柯尔莫哥洛夫、伊斯拉埃尔·盖尔范德、伊万·彼得罗夫斯基、列夫·庞特里亚金、彼得·诺维科夫、马尔可夫、亚历山大·格尔丰德、Lusternik、亚历山大·欣钦和帕维尔·亚历山德罗夫都在教像尤里·伊万诺维奇·马宁、雅科夫·西奈、谢尔盖·彼得罗维奇·诺维科夫、V M Alexeev、德米特里·阿诺索夫、亚历山大·卡里洛夫和我这样的学生。所有这些数学家都如此不同！要听懂安德雷·柯尔莫哥洛夫的课几乎是不可能的，但他的课充满思想，确实很有收获！……我在力学与数学系做学生时，列夫·庞特里亚金已经非常虚弱，但他也许是讲课最好的人。

阿诺尔德继续在莫斯科国立大学攻读研究生，攻读副博士学位（相当于Ph.D.），导师仍是安德雷·柯尔莫哥洛夫。1961年，他因学位论文On the representation of continuous functions of 3 variables by the superpositions of continuous functions of 2 variables获得莫斯科应用数学研究所授予的该学位。该论文的评审委员会由A G Vitushkin和L V Keldysh组成，论文中包含对大卫·希尔伯特第13问题的解答。此后，他被任命为莫斯科国立大学力学与数学系的助理。他继续攻读博士学位（相当于教授资格论文（Habilitation）），并于1963年因学位论文Small denominators and stability problems in classical and celestial mechanics获得莫斯科应用数学研究所授予的该学位。他的评审人是尼古拉·博戈柳博夫、V M Volosov和G N Duboshin。获得该学位后，阿诺尔德得到晋升。

1965年，阿诺尔德成为莫斯科国立大学力学与数学系的教授，他一直担任此职位直到1986年，之后他担任了莫斯科弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所的首席研究员。除了在俄罗斯的职位外，1993年他被任命为法国巴黎第九大学的教授。他一直担任此职位直到2005年。阿诺尔德于1976年与Voronina Elionora Aleksandrova结婚；他们有一个儿子。

2001年授予阿诺尔德的沃尔夫奖的授奖词中，对他的贡献作了极好的概述：-

阿诺尔德对数量多得惊人的不同数学学科都作出了重要贡献。他的大量研究论文、著作和讲座，加上他渊博的学识和热情，对整整一代数学家产生了深远影响。阿诺尔德的Ph.D.学位论文包含对大卫·希尔伯特第13问题的解答。他在哈密顿动力学方面的工作，包括共同创立KAM（安德雷·柯尔莫哥洛夫-阿诺尔德-Moser）理论以及发现“阿诺尔德扩散”，使他在早年就闻名世界。阿诺尔德对奇点理论的贡献补充了勒内·托姆的突变论，并改变了这一领域。阿诺尔德还对微分方程理论、辛几何、实代数几何、变分法、流体动力学和磁流体动力学作出了无数根本性的贡献。他常常发现不同领域问题之间的联系。

的确，阿诺尔德工作过的不同学科数量确实令人惊叹。这些领域是动力系统、微分方程、流体动力学、磁流体动力学、经典力学与天体力学、几何学、拓扑学、代数几何、辛几何和奇点理论。

让我们也指出本引文中提到的书籍的范围。他出版了Problèmes ergodiques de la mécanique classiqueⓉ（经典力学的遍历问题）（与A Avez合著）（1967年）、Ordinary differential equations（俄文）（1971年）、Mathematical methods of classical mechanics（俄文）（1974年）、Supplementary chapters to the theory of ordinary differential equations（俄文）（1978年）、Singularity theory（1981年）、Singularities of differentiable mappings（俄文）（与A N Varchenko和S M Gusein-Zade合著）（1982年）、Catastrophe theory（1984年）、Huygens and Barrow, Newton and Hooke（俄文）（1989年）、Contact geometry and wave propagation（1989年）、Singularities of caustics and wave fronts（1990年）、The theory of singularities and its applications（1991年）、Topological invariants of plane curves and caustics（1994年）、Lectures on partial differential equations（俄文）（1997年）、Topological methods in hydrodynamics（与B A Khesin合著）（1998年）和Arnold problems（俄文）（2000年）。

Serge L Tabachnikov 在评论最后提到的那本书时，详细介绍了阿诺尔德的讨论班：-

苏联，尤其是莫斯科的数学界，以其讨论班而闻名：仅举几例，伊斯拉埃尔·盖尔范德、雅科夫·西奈、亚历山大·卡里洛夫、尤里·伊万诺维奇·马宁、彼得·诺维科夫的讨论班。这些讨论班大多每周聚会两小时，在傍晚进行。其中最著名的之一是阿诺尔德的讨论班，已存在30多年。对许多非常著名的数学家来说，这个讨论班是一次形成性的经历。每学期，讨论班的开幕会议都专门讨论未解决的问题。阿诺尔德讨论了大约十几个研究问题，并附有详细评论。其中许多问题后来被讨论班的参与者解决（或部分解决）。据阿诺尔德说，一个问题的半衰期是七年。许多讨论班参与者是阿诺尔德的研究生。他的理念是，学生应该从老师那里了解到某个问题是未解决的；具体研究问题的选择则由学生决定（引用阿诺尔德的前言：“为他选择一个问题就像为儿子选择新娘”）。

阿诺尔德在全世界受到表彰。他当选为伦敦数学会（1976年）、美国国家科学院（1983年）、Academy of Sciences of Paris（1984年）、美国科学院 of Arts and Sciences（1987年）、Royal Society of London（1988年）、罗马Accademia Nazionale dei Lincei（1988年）、Russian Academy of Sciences（1990年）、美国哲学学会（1990年）、俄罗斯自然科学院（1991年）和欧洲科学院（1991年）的成员。他获得过许多奖项，例如莫斯科数学会青年数学家奖（1958年）、列宁奖（与安德雷·柯尔莫哥洛夫共同获得）（1965年）、Swedish Academy of Sciences克拉福德奖（与路易·尼伦伯格共同获得）（1982年）、Russian Academy of Sciences的罗巴切夫斯基奖（1992年）、以色列海法理工学院的哈维奖（1994年）、俄罗斯自然科学院彼得·L·卡皮察科学发现奖章（1997年）、丹尼·海涅曼数学物理奖（2001年）、美国物理联合会奖（2001年）和沃尔夫数学奖（2001年）。

哈维奖的授予理由是：-

……表彰他对动力系统稳定性理论的基础性贡献、他在奇点理论上的开创性工作以及对分析和几何的奠基性贡献。

沃尔夫奖的授予理由是：-

……因为他在数学众多领域——包括动力系统、微分方程和奇点理论——做出了深刻而有影响的工作。

除这些荣誉外，阿诺尔德还获得了以下大学的名誉学位：巴黎皮埃尔和玛丽·居里大学（1979年）、考文垂沃里克大学（1988年）、荷兰乌得勒支大学（1991年）、意大利博洛尼亚大学（1991年）、马德里康普顿斯大学（1994年）以及加拿大多伦多大学（1997年）。

阿诺尔德公开批评许多国家的教育制度。例如，在[4]中，他嘲讽美国的教育：——

最近，甚至连国家科学院也认定，美国的科学教育应当加强。他们提议从课程中删除那些对美国儿童来说太难的不必要的科学事实，代之以真正基本的基础知识，例如所有物体都有性质，所有生物都有本性！毫无疑问，他们会在这方面走得很远！两年前，我在《今日美国》上读到，美国父母已经为每个年龄段的儿童列出了一份真正必要的知识清单。十岁时他们必须知道水有两相，十五岁时必须知道月亮有相位并绕地球旋转。在俄罗斯，我们仍在小学教儿童水有三相，但新的美国文化无疑将在不久的将来获胜。然而，自由的美国制度也有一些显著优点，比如高中生可以选修一门爵士乐史课程，而不是代数。

阿诺尔德也批评了法国的教育制度。在1997年3月7日于巴黎发现宫所作的关于数学教学的演讲中，他说：——

对于“2 + 3等于几”这个问题，一名法国小学生回答说：“3 + 2，因为加法是可交换的”。他不知道和等于多少，甚至无法理解别人问的是什么！另一名法国学生（在我看来相当理性）这样定义数学：“有一个正方形，但这还有待证明”。根据我在法国教学的经验，大学生对数学的理解（甚至包括那些在高等师范学院学习数学的学生——我最同情这些显然聪明但被扭曲的孩子）和这名学生一样贫乏。智力有问题的“抽象数学”狂热分子把所有几何（通过它，数学最常与物理和现实发生联系）都扔出了教学。爱徳华·古尔萨、夏尔·埃尔米特、埃米尔·皮卡的微积分教科书最近被巴黎第六和第七大学（朱西厄）的学生图书馆当作过时因而有害的书丢弃（只是由于我的干预才被救回）。

阿诺尔德于2002年满65岁，《莫斯科数学杂志》专门用两期发表论文来庆祝这一时刻。该杂志的编辑写了一篇引言，我们从中引用的这段话为这篇传记画上了一个恰当的句号：-

阿诺尔德是世界上最杰出的数学家之一，是莫斯科独立大学的创始人之一，是其董事会和科学委员会的主席，也是我们编辑委员会的成员。如果没有他在动力系统、经典力学与天体力学、奇点理论、拓扑学、实与复代数几何、辛几何与接触几何、流体力学、变分法、微分几何、位势论、数学物理、叠加理论等方面的工作，现代数学的面貌将无法辨认。

阿诺尔德是一位难得的教师，他的学派著名且人数众多，他有一种特殊的天赋，能发现新的优美问题来吸引和带动年轻研究者。在数学教育与研究的各个层次上，他都是一位非凡的讲师。艰深的现代理论在他的阐述中变得十分清晰而简单。很难想象没有他那些出色教科书的现代数学教育。莫斯科数学学派在很大程度上得益于他的讨论班。

他出身于一个几代科学家的家庭，将他们的科学方法和对生活各个方面的深厚兴趣融为一体，他的知识极其渊博，对周围一切事物的好奇心令人惊叹。

阿诺尔德获得了俄罗斯联邦国家奖（2007年），次年他获得了著名的邵逸夫数学科学奖。邵逸夫奖由阿诺尔德和路德维希·德米特里耶维奇·法捷耶夫平分：-

……因为他们对数学物理做出了广泛而有影响的贡献。

邵逸夫奖颁奖新闻稿的开头写道：-

阿诺尔德与Andrei Kolmogorov和Jurgen Möser一起，对动力系统稳定性研究做出了基础性贡献，其典型例子是行星绕太阳的运动。这项工作为直至今日的所有后续发展奠定了基础。阿诺尔德还提出了极具成果的思想，将经典力学与拓扑学问题联系起来。这包括著名的“阿诺尔德猜想”，该猜想直到最近才取得重要进展。在经典流体动力学中，理想流体的基本方程由莱昂哈德·欧拉于1757年导出，而理解这些方程的重大进展则由赫尔曼·冯·亥姆霍兹于1858年和开尔文于1869年取得。下一个重大突破由一个世纪后的阿诺尔德做出，这为更近期的工作提供了基础。……

Vladimir Arnold's parents were Igor Vladimorovich Arnold and Nina Alexandrova Isakovich. Several generations of Arnold's family had been scientists. His interest in mathematics began when he was as young as five years old. He explained that this was a consequence of the Russian mathematical tradition [4]:-

Very young children start thinking about [old merchant] problems even before they have any knowledge of numbers. Children five to six years old like them very much and are able to solve them, but they may be too difficult for university graduates, who are spoiled by formal mathematical training. ... Many Russian families have the tradition of giving hundreds of [mathematical] problems to their children, and mine were no exception.

When he was twelve years old he was given challenging problems by his schoolteacher. He quoted one such problem in [4]:-

Two old women started at sunrise and each walked as a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was sunrise on this day?

Arnold said:-

I spent a whole day thinking on this oldies, and the solution ... came as a revelation. The feeling of discovery I had then was exactly the same as in all the subsequent much more serious problems ...

He entered Moscow State University in 1954 as an undergraduate student in the Faculty of Mechanics and Mathematics. He was awarded his first degree in 1959 with a dissertation On mappings of a circle to itself written with Kolmogorov as advisor. Speaking of his undergraduate years he said [4]:-

The constellation of great mathematicians in the same department when I was studying at the Faculty of Mechanics and Mathematics was really exceptional, and I have never seen anything like it at any other place. Kolmogorov, Gelfand, Petrovsky, Pontryagin, P Novikov, Markov, Gelfond, Lusternik, Khinchin and P S Aleksandrov were teaching students like Manin, Sinai, Sergi Novikov, V M Alexeev, Anosov, A A Kirillov, and me. All these mathematicians were so different! It was almost impossible to understand Kolmogorov's lectures, but they were full of ideas and were really rewarding! ... Pontryagin was already very weak when I was a student at the Faculty of Mechanics and Mathematics, but he was perhaps the best of the lecturers.

Arnold continued to study as a postgraduate student at Moscow State University for his Candidate's Degree (equivalent to a Ph.D.) still with Kolmogorov as advisor. He was awarded the degree in 1961 by the Institute of Applied Mathematics in Moscow for his thesis On the representation of continuous functions of 3 variables by the superpositions of continuous functions of 2 variables. The examining committee for the thesis, which contained a solution to Hilbert's 13th problem, consisted of A G Vitushkin and L V Keldysh. Following this he was appointed as an assistant in the Faculty of Mechanics and Mathematics at Moscow State University. He continued to work towards his doctorate (equivalent to the habilitation) and this was awarded by the Institute of Applied Mathematics in Moscow in 1963 for the thesis Small denominators and stability problems in classical and celestial mechanics. He was examined by N N Bogolyubov, V M Volosov, and G N Duboshin. Following the award, Arnold was promoted.

In 1965 Arnold became a Professor in the Faculty of Mechanics and Mathematics at Moscow State University, a position he held until 1986 when he took up the position of Principal Researcher at the Steklov Institute of Mathematics in Moscow. In addition to his Russian positions, in 1993 he was appointed Professor at the University Paris-Dauphine in France. He held this position until 2005. Arnold married Voronina Elionora Aleksandrova in 1976; they had one son.

An excellent overview of Arnold's contributions is given in the citation for the Wolf Prize awarded to him in 2001:-

Vladimir I Arnold has made significant contributions to an astounding number of different mathematical disciplines. His many research papers, books, and lectures, plus his enormous erudition and enthusiasm, have had a profound influence on an entire generation of mathematicians. Arnold's Ph.D. thesis contained a solution to Hilbert's 13th problem. His work on Hamiltonian dynamics, which includes cocreation of KAM (Kolmogorov- Arnold- Moser) theory and the discovery of "Arnold diffusion", made him world famous at an early age. Arnold's contributions to the theory of singularities complement Thom's catastrophe theory and have transformed this field. Arnold has also made innumerable and fundamental contributions to the theory of differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto- hydrodynamics. He has often discovered links between problems in diverse areas.

Indeed the number of different disciplines in which Arnold has worked is truly astounding. The areas are Dynamical Systems, Differential Equations, Hydrodynamics, Magnetohydrodynamics, Classical and Celestial Mechanics, Geometry, Topology, Algebraic Geometry, Symplectic Geometry, and Singularity Theory.

Let us also indicate the range of the books referred to in this citation. He published Problèmes ergodiques de la mécanique classique Ⓣ (with A Avez) (1967), Ordinary differential equations (Russian) (1971), Mathematical methods of classical mechanics (Russian) (1974), Supplementary chapters to the theory of ordinary differential equations (Russian) (1978), Singularity theory (1981), Singularities of differentiable mappings (Russian) (with A N Varchenko and S M Gusein-Zade) (1982), Catastrophe theory (1984), Huygens and Barrow, Newton and Hooke (Russian) (1989), Contact geometry and wave propagation (1989), Singularities of caustics and wave fronts (1990), The theory of singularities and its applications (1991), Topological invariants of plane curves and caustics (1994), Lectures on partial differential equations (Russian) (1997), Topological methods in hydrodynamics (with B A Khesin) (1998), and Arnold problems (Russian) (2000).

Serge L Tabachnikov reviewing the last mentioned book gives details of Arnold's Seminar:-

Mathematical life in the Soviet Union, in particular in Moscow, was famous for its seminars: the seminars of Gelfand, Sinai, Kirillov, Manin, Novikov, to mention just a few. Most of these seminars met weekly for two hours, in late afternoon. One of the most celebrated ones is Arnold's seminar, existing for more than 30 years. For a number of very well-known mathematicians this seminar was a formative experience. Every semester, the opening meeting of the seminar was devoted to open problems. Arnold discussed about a dozen research problems with detailed comments. Many of these problems were later solved (or partially solved) by participants of the seminar. According to Arnold, the half-life of a problem is seven years. Many seminar participants are Arnold's graduate students. His philosophy is that a student should learn from his teacher that a certain problem is open; the choice of a particular research problem is then up to the student (to quote from Arnold's Preface: "To choose a problem for him is like choosing a bride for one's son").

Arnold has been honoured throughout the world. He has been elected to membership of the London Mathematical Society (1976), the National Academy of Sciences of the United States (1983), the Academy of Sciences of Paris (1984), the Academy of Arts and Sciences of the United States (1987), the Royal Society of London (1988), Accademia Nazionale dei Lincei in Rome (1988), the Russian Academy of Sciences (1990), the American Philosophical Society (1990), the Academy of Natural Sciences of Russia (1991), and the Academia Europaea (1991). He has received many prizes, for example the Young Mathematicians Prize of the Moscow Mathematical Society (1958), the Lenin Prize (with Andrei Kolmogorov) (1965), the Crafoord Prize of the Swedish Academy of Sciences (with Louis Nirenberg) (1982), the Lobachevsky Prize of Russian Academy of Sciences (1992), the Harvey Prize, Technion, Haifa, Israel (1994), the Petr L Kapitsa Medal for Scientific Discoveries, Russian Academy of Natural Sciences (1997), Dannie Heineman Prize for Mathematical Physics (2001), Prize of the American Institute of Physics (2001), and the Wolf Prize in Mathematics (2001).

The Harvey Prize was awarded:-

... In recognition of his basic contribution to the stability theory of Dynamical Systems, his pioneering work on singularity theory and seminal contributions to analysis and geometry.

The Wolf Prize was awarded:-

... for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory.

In addition to these honours, Arnold has been awarded honorary degrees from the University P and M Curie, Paris (1979), Warwick University, Coventry (1988), Utrecht University, Netherlands (1991), University of Bologna, Italy (1991), University Complutense, Madrid (1994), and the University of Toronto, Canada (1997).

Arnold is openly critical of the education system of many countries. For example in [4] he pokes fun at education in the United States:-

Recently, even the National Academy of Sciences decided that scientific education in America should be enhanced. What they propose is to eliminate from the curriculum unnecessary scientific facts too difficult for American children and replace them by really fundamental basic knowledge, such as all objects have properties and all organisms have nature! Undoubtedly they will go far with this! Two years ago, I read in USA Today that American parents have formed a list of really necessary knowledge for children of each age category. At ten they have to know that water has two phases, and at fifteen that the moon has phases and rotates round the earth. In Russia we still teach children in primary school that water has three phases, but the new American culture will undoubtedly win in the near future. There are, however, some remarkable advantages in the free American system, where a high school student may take, say, a course on the history of jazz instead of algebra.

Arnold has also been critical of the French education system. In an address on teaching of mathematics, given in the Palais de Découverte in Paris on 7 March 1997, he said:-

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about! Another French pupil (quite rational, in my opinion) defined mathematics as follows: "There is a square, but that still has to be proved". Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil. Mentally challenged zealots of "abstract mathematics" threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard were recently dumped by the student library of the Universitiés Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention).

Arnold reached 65 years of age in 2002 and the Moscow Mathematical Journal devoted two issues to papers dedicated to celebrate the occasion. The editors of the Journal wrote an introduction and the quote we give from that forms a fitting end to this biography:-

Arnold is one of the very best mathematicians of the world, one of the founders of the Independent University of Moscow, president of its Board of Trustees and its Scientific Committee, member of our Editorial Board. The face of modern mathematics would be unrecognisable without his work in dynamical systems, classical and celestial mechanics, singularity theory, topology, real and complex algebraic geometry, symplectic and contact geometry, hydrodynamics, variation calculus, differential geometry, potential theory, mathematical physics, superposition theory, etc.

Arnold is a rare teacher, his school is famous and numerous, he has a special gift for finding new beautiful problems to interest and involve young researchers. He is an extraordinary lecturer at all levels of mathematical education and research. Difficult modern theories become quite clear and simple in his exposition. One could hardly imagine modern mathematical education without his brilliant textbooks. The Moscow mathematical school owes a lot to his seminar.

Coming from a family of scientists in several generations, he brings together their scientific approach and deep interest to all sides of the life, his knowledge being extremely vast and his curiosity towards everything around him quite amazing.

Arnold was awarded the State Prize of the Russian Federation (2007), and in the following year he received the prestigious Shaw Prize in mathematical sciences. The Shaw Prize was awarded in equal shares to Vladimir Arnold and Ludwig Faddeev:-

... for their widespread and influential contributions to Mathematical Physics.

The Press Release for the award of the Shaw Prize begins:-

Vladimir Arnold, together with Andrei Kolmogorov and Jurgen Möser, made fundamental contributions to the study of stability in dynamical systems, exemplified by the motion of the planets round the sun. This work laid the foundation for all subsequent developments right up to the present time. Arnold also produced extremely fruitful ideas, relating classical mechanics to questions of topology. This includes the famous "Arnold Conjecture" which has only recently seen important progress. In classical hydrodynamics the basic equations of an ideal fluid were derived by Euler in 1757 and major steps towards understanding them were taken by Helmholtz in 1858, and Kelvin in 1869. The next significant breakthrough was made by Arnold a century later and this has provided the basis for more recent work. ...