安德雷·柯尔莫哥洛夫Andrey Kolmogorov

安德雷·柯尔莫哥洛夫的父母没有结婚，他的父亲没有参与他的成长。他的父亲Nikolai Kataev是一位牧师的儿子，是一位被流放的农学家。革命后他返回，领导农业部的一个部门，但在1919年的战斗中去世。柯尔莫哥洛夫的母亲也同样悲剧性地没有参与他的成长，因为她在柯尔莫哥洛夫出生时死于分娩。他母亲的妹妹Vera Yakovlena抚养了柯尔莫哥洛夫，他始终对她怀有最深的感情。

事实上，柯尔莫哥洛夫出生在坦波夫纯属偶然，因为家族与那个地方没有联系。柯尔莫哥洛夫的母亲当时正从克里米亚返回她在雅罗斯拉夫尔附近Tunoshna的家，而柯尔莫哥洛夫就是在Tunoshna他外祖父的家中度过了他的青年时代。柯尔莫哥洛夫的名字来自他的祖父Yakov Stepanovich 柯尔莫哥洛夫，而不是来自他自己的父亲。Yakov Stepanovich出身贵族，这在当时的俄罗斯是一个难以维持的身份，而且确实有故事说他的房子里运营着一台非法印刷机。

柯尔莫哥洛夫离开学校后，曾一度在铁路上担任列车员。在业余时间，他写了一篇关于艾萨克·牛顿力学定律的论文。然后，在1920年，柯尔莫哥洛夫进入莫斯科国立大学，但在这个阶段，他远未致力于数学。他学习了多门学科，例如除了数学，他还学习了冶金学和俄罗斯历史。也不应认为俄罗斯历史仅仅是一门填充课程的科目，事实上，他写了一篇关于15和16世纪诺夫哥罗德财产所有权的严肃科学学位论文。大卫·乔治·肯德尔在[10]中讲述了关于这篇学位论文的一则轶事，他的老师说：-

你为你的学位论文提供了一个证明，在你所学的数学中，这或许就足够了，但我们历史学家更希望至少有十个证明。

柯尔莫哥洛夫可能把这个故事当作笑话来讲，但笑话只有在其中有些真实成分时才有趣，毫无疑问，这里的情况就是如此。

在数学方面，柯尔莫哥洛夫早期受到多位杰出数学家的影响。帕维尔·亚历山德罗夫大约在柯尔莫哥洛夫开始本科生涯的时候，正在莫斯科开始他的研究（第二次）。尼古拉·卢津和德米特里·叶戈罗夫当时正在运行他们令人印象深刻的研究小组，学生们称之为“卢西塔尼亚”。除了帕维尔·亚历山德罗夫，还包括米哈伊尔·雅科夫列维奇·苏斯林和帕维尔·萨穆伊洛维奇·乌雷松。然而，当时给柯尔莫哥洛夫留下最深印象的人是维亚切斯拉夫·瓦西里耶维奇·斯捷潘诺夫，他给他讲授三角级数。

值得注意的是，柯尔莫哥洛夫虽然只是一名本科生，但在这个阶段就开始了研究并产生了具有国际重要性的成果。到1922年春天，他已经完成了一篇关于集合运算的论文，这是对米哈伊尔·雅科夫列维奇·苏斯林所得结果的重大推广。到1922年6月，他已经构造了一个几乎处处发散的可和函数。这完全出乎专家们的意料，柯尔莫哥洛夫的名字开始在世界各地为人所知。[7]和[8]的作者指出：-

几乎在同一时期，柯尔莫哥洛夫 在经典分析的若干其他领域展现了他的兴趣：在微分与积分问题、集合测度等方面。在他处理如此多样主题的每一篇论文中，他都引入了独创性的元素、方法的广度和思想的深度。

柯尔莫哥洛夫 于1925年毕业于莫斯科国立大学，并于同年开始在 尼古拉·卢津 的指导下进行研究。值得注意的是，柯尔莫哥洛夫 在1925年发表了八篇论文，全部是在他仍是本科生时写的。另一个里程碑发生在1925年，即 柯尔莫哥洛夫 关于 probability 的第一篇论文问世。这篇论文与 亚历山大·欣钦 联合发表，包含“三级数”定理以及关于随机变量部分和不等式的结果，这些结果后来成为 鞅 不等式和随机微积分的基础。

1929年，柯尔莫哥洛夫 完成了他的博士学位。此时他已有18篇出版物，大卫·乔治·肯德尔 在 [10] 中写道：-

这些包括他版本的大数定律和重对数律、微分与积分运算的一些推广，以及对 intuitional logic 的贡献。他关于最后这个主题的论文……被该领域的专家们怀着敬畏看待。柯尔莫哥洛夫 全集的俄文版包含对这些论文的回顾性评论，[柯尔莫哥洛夫] 显然认为这些论文标志着其哲学观点的一个重要发展。

对 柯尔莫哥洛夫 来说，一个重要事件是他与 帕维尔·亚历山德罗夫 的友谊，这始于1929年夏天，当时他们一起度过了三个星期。在一次从雅罗斯拉夫尔出发的旅行中，他们乘船沿伏尔加河而下，然后穿过高加索山脉到达亚美尼亚的塞万湖。在那里，帕维尔·亚历山德罗夫 致力于他与 海因茨·霍普夫 合著的 拓扑学 一书，而 柯尔莫哥洛夫 则研究具有连续状态和连续时间的 马尔可夫 过程。柯尔莫哥洛夫 在湖边工作所得的结果于1931年发表，标志着扩散理论的开始。1931年夏天，柯尔莫哥洛夫 和 帕维尔·亚历山德罗夫 又进行了一次长途旅行。他们访问了柏林、哥廷根、慕尼黑和巴黎，在那里 柯尔莫哥洛夫 与 保罗·莱维 进行了许多小时的深入讨论。此后，他们与 莫里斯·弗雷歇 在海边度过了一个月。

柯尔莫哥洛夫 于1931年被任命为莫斯科大学教授。他关于概率论的专著 Grundbegriffe der Wahrscheinlichkeitsrechnung Ⓣ（概率论的基本概念）于1933年出版，以与 欧几里得 处理几何学相媲美的方式，从基本公理出发严格地建立了概率论。这种方法的一个成功之处在于它提供了条件期望的严格定义。正如 [10] 所指出的：-

1931年可以视为柯尔莫哥洛夫一生中第二个创造性阶段的开始。他在数学各个分支中提出的广泛的一般概念是这一阶段的特征。

在提到柯尔莫哥洛夫于1938年发表的奠定马尔可夫随机过程理论基础的那篇极为重要的论文Analytic methods in probability theory之后，他们继续描述道：-

……他在集合论拓扑学、逼近论、湍流理论、泛函分析、几何基础以及数学史与方法论方面的思想。[他对]这些分支中每一个的贡献……[是]一个单一的整体，其中一个领域的重大进展会导致其他领域的实质性丰富。

帕维尔·亚历山德罗夫和柯尔莫哥洛夫于1935年在莫斯科郊外的小村庄科马罗夫卡买了一栋房子。许多著名数学家访问过科马罗夫卡：雅克·阿达马、莫里斯·弗雷歇、斯特凡·巴拿赫、海因茨·霍普夫、卡齐米日·库拉托夫斯基等人。鲍里斯·弗拉基米拉维奇·格涅坚科和其他研究生继续（[7]和[8]）：-

……数学远足[们]在科马罗夫卡结束，柯尔莫哥洛夫和帕维尔·亚历山德罗夫在那里招待全公司吃晚饭。疲惫不堪、满脑子数学思想，因意识到我们发现了书本上找不到的东西而快乐，我们会在傍晚返回莫斯科。

大约在这个时候，Malcev和伊斯拉埃尔·盖尔范德等人是柯尔莫哥洛夫的研究生，还有鲍里斯·弗拉基米拉维奇·格涅坚科，他描述了在柯尔莫哥洛夫指导下学习是什么样子（[7]和[8]）：-

对所有柯尔莫哥洛夫的学生来说，他们的研究生学习时期仍然是他们一生中难以忘怀的时期，充满了对科学和文化的高度追求、科学进步的迸发以及为科学问题的解决而奉献全部力量。无法忘记的是，[柯尔莫哥洛夫]邀请他自己所有的学生（研究生和本科生）以及其他导师的学生参加的周日美妙散步。这些在博尔舍沃、克利亚济马以及约30-35公里外的其他地方的郊游，充满了关于数学（及其应用）当前问题的讨论，以及关于文化进步问题的讨论，尤其是绘画、建筑和文学。

1938-1939年，一些来自莫斯科大学的领先数学家加入了苏联科学院的弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所，同时保留他们在大学的职位。其中包括帕维尔·亚历山德罗夫、伊斯拉埃尔·盖尔范德、柯尔莫哥洛夫、伊万·彼得罗夫斯基和亚历山大·欣钦。概率与统计系在该研究所成立，柯尔莫哥洛夫被任命为系主任。

柯尔莫哥洛夫后来将他的工作扩展到研究行星运动和喷气发动机的湍流气流。1941年，他发表了两篇关于湍流的论文，具有根本重要性。1954年，他发展了与行星运动相关的动力系统工作。他由此证明了概率论在物理学中的重要作用。

我们必须只提及柯尔莫哥洛夫在数学各个不同领域做出的众多其他重大贡献中的少数几个。在拓扑学中，柯尔莫哥洛夫几乎同时且独立于詹姆斯·韦德尔·亚历山大引入了cohomology groups的概念。1934年，柯尔莫哥洛夫研究了有限胞腔复形的链、上链、同调和上同调。在1936年发表的进一步论文中，柯尔莫哥洛夫为任意局部紧拓扑空间定义了上同调群。这一领域另一个极其重要的贡献是他1935年在莫斯科国际拓扑学会议上宣布的上同调环的定义。在这次会议上，柯尔莫哥洛夫和詹姆斯·韦德尔·亚历山大都就他们各自独立的上同调工作做了演讲。

1953年和1954年，柯尔莫哥洛夫的两篇论文各四页长，发表了。这些论文关于动力系统理论及其在哈密顿动力学中的应用。这些论文标志着KAM理论的开端，该理论以柯尔莫哥洛夫、弗拉基米尔·阿诺尔德和Moser命名。柯尔莫哥洛夫于1954年在阿姆斯特丹国际数学家大会上以他的重要演讲General theory of dynamical systems and classical mechanics讨论了这一主题。

N H Bingham [10]指出，柯尔莫哥洛夫在建立理论以回答大卫·希尔伯特第六问题中的概率部分方面发挥了重要作用，该问题要求“用公理方法处理那些数学在其中起重要作用的物理科学；首先是概率论和力学”，这体现在他1933年的专著Grundbegriffe der Wahrscheinlichkeitsrechnung Ⓣ（概率论基本概念）中。Bingham还指出：-

……保罗·莱维深刻地写道，他一看到“Grundbegriffe”就意识到自己错失的机会。马克·卡茨雄辩的著作提供了另一种视角，讲述了像胡戈·施泰因豪斯和他自己这样的波兰数学家在上世纪30年代，即使拥有“Grundbegriffe”，也难以理解随机独立性这一（表面上清晰的）概念所经历的挣扎。

柯尔莫哥洛夫对大卫·希尔伯特的第六问题做出了重大贡献，他在1957年完全解决了大卫·希尔伯特的第十三问题，当时他表明大卫·希尔伯特要求证明存在三个变量的连续函数不能由两个变量的连续函数表示是错误的。

柯尔莫哥洛夫对为天才儿童提供特殊教育的项目特别感兴趣[10]：-

多年来，他将大部分时间投入到这所学校，规划教学大纲、编写教科书、亲自给孩子们上大量课时，引导他们接触文学和音乐，参加他们的娱乐活动，带他们远足、短途旅行和探险。……柯尔莫哥洛夫力求确保这些孩子获得广泛而自然的人格发展，如果学校里的孩子没有成为数学家，他也不会担心。无论他们最终从事什么职业，只要他们的视野保持开阔，好奇心未被扼杀，他就会感到满足。的确，能属于柯尔莫哥洛夫这个大家庭一定是很美妙的。

像柯尔莫哥洛夫这样杰出的科学家自然会收到来自许多不同国家的大量荣誉。1939年，他当选为苏联科学院。他获得了1941年首批颁发的国家奖之一，1965年的列宁奖，六次列宁勋章，以及1987年的罗巴切夫斯基奖。他还当选为许多其他科学院和学会的成员，包括罗马尼亚科学院（1956年）、伦敦的皇家统计学会（1956年）、Leopoldina Academy of Germany（1959年）、American Academy of Arts and Sciences（1959年）、London Mathematical Society（1959年）、美国哲学学会（1961年）、印度统计研究所（1962年）、Royal Netherlands Academy of Sciences（1963年）、Royal Society of London（1964年）、National Academy of the United States（1967年）、French Academy of Sciences（1968年）。

除上述奖项外，柯尔莫哥洛夫还于1962年获得了巴尔赞国际奖。许多大学授予他荣誉学位，包括巴黎、斯德哥尔摩和华沙。

柯尔莫哥洛夫在数学之外有许多兴趣，特别是他对俄罗斯作家普希金诗歌的形式和结构感兴趣。

Andrei Nikolaevich Kolmogorov's parents were not married and his father took no part in his upbringing. His father Nikolai Kataev, the son of a priest, was an agriculturist who was exiled. He returned after the Revolution to head a Department in the Agricultural Ministry but died in fighting in 1919. Kolmogorov's mother also, tragically, took no part in his upbringing since she died in childbirth at Kolmogorov's birth. His mother's sister, Vera Yakovlena, brought Kolmogorov up and he always had the deepest affection for her.

In fact it was chance that had Kolmogorov born in Tambov since the family had no connections with that place. Kolmogorov's mother had been on a journey from the Crimea back to her home in Tunoshna near Yaroslavl and it was in the home of his maternal grandfather in Tunoshna that Kolmogorov spent his youth. Kolmogorov's name came from his grandfather, Yakov Stepanovich Kolmogorov, and not from his own father. Yakov Stepanovich was from the nobility, a difficult status to have in Russia at this time, and there is certainly stories told that an illegal printing press was operated from his house.

After Kolmogorov left school he worked for a while as a conductor on the railway. In his spare time he wrote a treatise on Newton's laws of mechanics. Then, in 1920, Kolmogorov entered Moscow State University but at this stage he was far from committed to mathematics. He studied a number of subjects, for example in addition to mathematics he studied metallurgy and Russian history. Nor should it be thought that Russian history was merely a topic to fill out his course, indeed he wrote a serious scientific thesis on the owning of property in Novgorod in the 15th and 16th centuries. There is an anecdote told by D G Kendall in [10] regarding this thesis, his teacher saying:-

You have supplied one proof of your thesis, and in the mathematics that you study this would perhaps suffice, but we historians prefer to have at least ten proofs.

Kolmogorov may have told this story as a joke but nevertheless jokes are only funny if there is some truth in them and undoubtedly this is the case here.

In mathematics Kolmogorov was influenced at an early stage by a number of outstanding mathematicians. P S Aleksandrov was beginning his research (for the second time) at Moscow around the time Kolmogorov began his undergraduate career. Luzin and Egorov were running their impressive research group at this time which the students called 'Luzitania'. It included M Ya Suslin and P S Urysohn, in addition to Aleksandrov. However the person who made the deepest impression on Kolmogorov at this time was Stepanov who lectured to him on trigonometric series.

It is remarkable that Kolmogorov, although only an undergraduate, began research and produced results of international importance at this stage. He had finished writing a paper on operations on sets by the spring of 1922 which was a major generalisation of results obtained by Suslin. By June of 1922 he had constructed a summable function which diverged almost everywhere. This was wholly unexpected by the experts and Kolmogorov's name began to be known around the world. The authors of [7] and [8] note that:-

Almost simultaneously [Kolmogorov] exhibited his interest in a number of other areas of classical analysis: in problems of differentiation and integration, in measures of sets etc. In every one of his papers, dealing with such a variety of topics, he introduced an element of originality, a breadth of approach, and a depth of thought.

Kolmogorov graduated from Moscow State University in 1925 and began research under Luzin's supervision in that year. It is remarkable that Kolmogorov published eight papers in 1925, all written while he was still an undergraduate. Another milestone occurred in 1925, namely Kolmogorov's first paper on probability appeared. This was published jointly with Khinchin and contains the 'three series' theorem as well as results on inequalities of partial sums of random variables which would become the basis for martingale inequalities and the stochastic calculus.

In 1929 Kolmogorov completed his doctorate. By this time he had 18 publications and Kendall writes in [10]:-

These included his versions of the strong law of large numbers and the law of the iterated logarithm, some generalisations of the operations of differentiation and integration, and a contribution to intuitional logic. His papers ... on this last topic are regarded with awe by specialists in the field. The Russian language edition of Kolmogorov's collected works contains a retrospective commentary on these papers which [Kolmogorov] evidently regarded as marking an important development in his philosophical outlook.

An important event for Kolmogorov was his friendship with Aleksandrov which began in the summer of 1929 when they spent three weeks together. On a trip starting from Yaroslavl, they went by boat down the Volga then across the Caucasus mountains to Lake Sevan in Armenia. There Aleksandrov worked on the topology book which he co-authored with Hopf, while Kolmogorov worked on Markov processes with continuous states and continuous time. Kolmogorov's results from his work by the Lake were published in 1931 and mark the beginning of diffusion theory. In the summer of 1931 Kolmogorov and Aleksandrov made another long trip. They visited Berlin, Göttingen, Munich, and Paris where Kolmogorov spent many hours in deep discussions with Paul Lévy. After this they spent a month at the seaside with Fréchet

Kolmogorov was appointed a professor at Moscow University in 1931. His monograph on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung Ⓣ published in 1933 built up probability theory in a rigorous way from fundamental axioms in a way comparable with Euclid's treatment of geometry. One success of this approach is that it provides a rigorous definition of conditional expectation. As noted in [10]:-

The year 1931 can be regarded as the beginning of the second creative stage in Kolmogorov's life. Broad general concepts advanced by him in various branched of mathematics are characteristic of this stage.

After mentioning the highly significant paper Analytic methods in probability theory which Kolmogorov published in 1938 laying the foundations of the theory of Markov random processes, they continue to describe:-

... his ideas in set-theoretic topology, approximation theory, the theory of turbulent flow, functional analysis, the foundations of geometry, and the history and methodology of mathematics. [His contributions to] each of these branches ... [is] a single whole, where a serious advance in one field leads to a substantial enrichment of the others.

Aleksandrov and Kolmogorov bought a house in Komarovka, a small village outside Moscow, in 1935. Many famous mathematicians visited Komarovka: Hadamard, Fréchet, Banach, Hopf, Kuratowski, and others. Gnedenko and other graduate students went on ([7] and [8]):-

... mathematical outings [which] ended in Komarovka, where Kolmogorov and Aleksandrov treated the whole company to dinner. Tired and full of mathematical ideas, happy from the consciousness that we had found out something which one cannot find in books, we would return in the evening to Moscow.

Around this time Malcev and Gelfand and others were graduate students of Kolmogorov along with Gnedenko who describes what it was like being supervised by Kolmogorov ([7] and [8]):-

The time of their graduate studies remains for all of Kolmogorov's students an unforgettable period in their lives, full of high scientific and cultural strivings, outbursts of scientific progress and a dedication of all one's powers to the solutions of the problems of science. It is impossible to forget the wonderful walks on Sundays to which [Kolmogorov] invited all his own students (graduates and undergraduates), as well as the students of other supervisors. These outings in the environs of Bolshevo, Klyazma, and other places about 30-35 kilometres away, were full of discussions about the current problems of mathematics (and its applications), as well as discussions about the questions of the progress of culture, especially painting, architecture and literature.

In 1938-1939 a number of leading mathematicians from the Moscow University joined the Steklov Mathematical Institute of the USSR Academy of Sciences while retaining their positions at the University. Among them were Aleksandrov, Gelfand, Kolmogorov, Petrovsky, and Khinchin. The Department of Probability and Statistics was set up at the Institute and Kolmogorov was appointed as Head of Department.

Kolmogorov later extended his work to study the motion of the planets and the turbulent flow of air from a jet engine. In 1941 he published two papers on turbulence which are of fundamental importance. In 1954 he developed his work on dynamical systems in relation to planetary motion. He thus demonstrated the vital role of probability theory in physics.

We must mention just a few of the numerous other major contributions which Kolmogorov made in a whole range of different areas of mathematics. In topology Kolmogorov introduced the notion of cohomology groups at much the same time, and independently of, Alexander. In 1934 Kolmogorov investigated chains, cochains, homology and cohomology of a finite cell complex. In further papers, published in 1936, Kolmogorov defined cohomology groups for an arbitrary locally compact topological space. Another contribution of the highest significance in this area was his definition of the cohomology ring which he announced at the International Topology Conference in Moscow in 1935. At this conference both Kolmogorov and Alexander lectured on their independent work on cohomology.

In 1953 and 1954 two papers by Kolmogorov, each of four pages in length, appeared. These are on the theory of dynamical systems with applications to Hamiltonian dynamics. These papers mark the beginning of KAM-theory, which is named after Kolmogorov, Arnold and Moser. Kolmogorov addressed the International Congress of Mathematicians in Amsterdam in 1954 on this topic with his important talk General theory of dynamical systems and classical mechanics.

N H Bingham [10] notes Kolmogorov's major part in setting up the theory to answer the probability part of Hilbert's Sixth Problem "to treat ... by means of axioms those physical sciences in which mathematics plays an important part; in the first rank are the theory of probability and mechanics" in his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung Ⓣ. Bingham also notes:-

... Paul Lévy writes poignantly of his realisation, immediately on seeing the "Grundbegriffe", of the opportunity which he himself had neglected to take. A rather different perspective is supplied by the eloquent writings of Mark Kac on the struggles that Polish mathematicians of the calibre Steinhaus and himself had in the 1930s, even armed with the "Grundbegriffe", to understand the (apparently perspicuous) notion of stochastic independence.

If Kolmogorov made a major contribution to Hilbert's sixth problem, he completely solved Hilbert's Thirteenth Problem in 1957 when he showed that Hilbert was wrong in asking for a proof that there exist continuous functions of three variables which could not be represented by continuous functions of two variables.

Kolmogorov took a special interest in a project to provide special education for gifted children [10]:-

To this school he devoted a major proportion of his time over many years, planning syllabuses, writing textbooks, spending a large number of teaching hours with the children themselves, introducing them to literature and music, joining in their recreations and taking them on hikes, excursions, and expeditions. ... [Kolmogorov] sought to ensure for these children a broad and natural development of the personality, and it did not worry him if the children in his school did not become mathematicians. Whatever profession they ultimately followed, he would be content if their outlook remained broad and their curiosity unstifled. Indeed it must have been wonderful to belong to this extended family of [Kolmogorov].

Such an outstanding scientist as Kolmogorov naturally received a whole host of honours from many different countries. In 1939 he was elected to the USSR Academy of Sciences. He received one of the first State Prizes to be awarded in 1941, the Lenin Prize in 1965, the Order of Lenin on six separate occasions, and the Lobachevsky Prize in 1987. He was also elected to the many other academies and societies including the Romanian Academy of Sciences (1956), the Royal Statistical Society of London (1956), the Leopoldina Academy of Germany (1959), the American Academy of Arts and Sciences (1959), the London Mathematical Society (1959), the American Philosophical Society (1961), The Indian Statistical Institute (1962), the Royal Netherlands Academy of Sciences (1963), the Royal Society of London (1964), the National Academy of the United States (1967), the French Academy of Sciences (1968).

In addition to the prizes mentioned above, Kolmogorov was awarded the Balzan International Prize in 1962. Many universities awarded him an honorary degree including Paris, Stockholm, and Warsaw.

Kolmogorov had many interests outside mathematics, in particular he was interested in the form and structure of the poetry of the Russian author Pushkin.