尼古拉·卢津Nikolai Luzin

尼古拉·卢津出生在伊尔库茨克，他的出生地不是像一些来源错误陈述的那样是托木斯克。卢津的父亲是一个商人，一半俄罗斯人一半布里亚特人。卢津是他父母唯一的儿子，全家在他大约十一岁时搬到托木斯克，以便他能在那里上文理中学。

人们可能会以为卢津在文理中学时会表现出特殊的数学才能，但事实远非如此（[15]和[16]）：-

这是因为……的教学体系建立在机械记忆之上：要求学生背熟定理并一字不差地复现其证明。对卢津来说，这是一种折磨。他在文理中学的数学成绩越来越差，以致他父亲不得不请一位家庭教师……

幸运的是，这位家庭教师是个有才华的年轻人，他很快发现，尽管卢津的数学成绩不佳，他却能解出难题，而且常常使用家庭教师从未见过的新方法。不久，这位家庭教师就让卢津明白，数学不是一门必须记住一长串事实的学科，而是一个创造力和想象力起主要作用的领域。

1901年，卢津离开了文理中学，此时他的父亲卖掉了生意，全家搬到了莫斯科。在那里，卢津进入莫斯科大学物理数学系，打算接受培训成为一名工程师。起初，卢津住在莫斯科的新家，但卢津的父亲开始用卖掉生意所得的钱在证券交易所赌博。由于卢津的父亲输掉了所有积蓄，全家很快陷入困境，不得不离开家。卢津和一个朋友搬进了一位医生遗孀拥有的房间。他的朋友很快卷入革命，被迫躲藏起来。卢津独自留在那个房间里，但他显然与房主相处得很好，因为后来在1908年，他娶了那位遗孀的女儿。

卢津师从尼古莱·布加耶夫，从他那里学习函数论，这极大地影响了他最终研究的方向。然而，他只是一个普通的学生，似乎对数学没有什么天赋。不过，尽管卢津似乎缺乏数学天赋，他的一位老师德米特里·叶戈罗夫发现了他的巨大才能，邀请他到家中，并开始给他出难题。

大学里有一位数学学生帕维尔·弗洛连斯基，他在毕业后经历了一场危机，转向宗教和神学研究。这对卢津产生了重大影响，他是弗洛连斯基的密友，我们将在下面描述。

1905年秋毕业后，卢津似乎不确定是否要献身于数学。事实上，卢津的危机在1905年春天就击中了他，1906年5月1日，卢津从巴黎写信给弗洛连斯基，五个月前德米特里·叶戈罗夫把他送到那里，试图让他度过危机（见[9]）：——

你在大学里发现我时，我还只是个孩子，什么都不懂。我不知道这是怎么发生的，但我再也无法满足于解析函数和布鲁克·泰勒级数了……这大约发生在一年前。……看到人们的苦难，看到生活的折磨，从数学会议回家的路上……在那里，一些妇女在寒风中瑟瑟发抖，徒劳地等待着用恐惧换来的晚餐——这是令人无法忍受的景象。看到这些之后，还平静地研究（实际上是享受）科学，是令人无法忍受的。在那之后，我无法只研究数学，我想转到医学院。……我在这里大约五个月了，但最近才开始学习。

卢津不仅因看到妓女而心烦意乱，他还在信中说到1905年革命的“可怕日子”如何影响了他。这一时期有德米特里·叶戈罗夫的信，恳求卢津不要放弃数学。回到俄罗斯后，卢津除了数学，还学习医学和神学。然而在1908年4月，他写到了他在数论中发现的快乐（见[9]）：——

这是一个神秘领域，它越来越深地包围着我。

在同一封信中，他说他刚刚结婚，并且：-

……我的妻子也非常感兴趣，并和我一样致力于探索生活的深刻真理。

卢津的危机似乎已被Florensky解决，卢津在1908年7月写信给他说：-

有两次我非常接近自杀——然后我来了……想和你谈谈，两次我都感觉好像靠在了一根柱子上，带着这种支持的感觉我回到了家……我对于生活的兴趣要归功于你……

他对数学的兴趣慢慢恢复了，但直到1909年，卢津似乎才最终完全投身于数学。在德米特里·叶戈罗夫的指导下，他从事硕士论文工作。1910年，他被任命为莫斯科大学纯粹数学的助理讲师。他与德米特里·叶戈罗夫合作了一年，他们接着发表了关于函数理论的联合论文，这标志着莫斯科函数理论学派的开端。

1910年，卢津出国旅行，访问了哥廷根，在那里他受到了埃德蒙·朗道的影响。他于1914年回到莫斯科，并完成了他的学位论文The integral and trigonometric series，于1915年提交。在口试之后，他被授予博士学位，尽管他提交的是硕士学位论文。德米特里·叶戈罗夫对这项工作印象非常深刻，并极力主张授予博士学位，但它的写作风格与当时公认的俄罗斯风格截然不同。有些结果没有严格证明，而是用诸如“在我看来”和“我确信”这样的短语来证明。当时其他数学家并没有那么印象深刻，例如弗拉基米尔·安德烈耶维奇·斯捷克洛夫在页边写下了诸如“在他看来是这样，但在我看来不是”和“哥廷根的闲聊”之类的评论。

然而，这项工作具有根本的重要性，正如[15]和[16]中所述：-

卢津的学位论文对函数论未来发展的影响怎么估计也不过分。它的基本结果、深刻的研究方法和基本问题的陈述，使它跻身于当时任何学位论文或专著都难以与之相比的著作之列。

1914年，卢津和妻子短暂分居，弗洛连斯基似乎再次帮助他们度过了难关。他写信给卢津的妻子（见[9]）：-

卢津尼古拉耶维奇是一个非常可爱和优秀的人；但在人际关系中，他一点也不成熟，尤其是在直觉地感知生活中隐藏的潮流方面。……你必须掌握这种关系，创造一种家庭氛围，简单朴素。相反，据我观察……你建立起来的是一种熟人之间的语气，而不是家庭的语气。

弗洛连斯基似乎给出了很好的建议，因为卢津和妻子恢复了成功的婚姻。

1917年，就在革命前夕，卢津被任命为莫斯科大学纯粹数学教授。革命使卢津重新思考了一些他在危机时期曾有过的同样想法，他再次与弗洛连斯基通信。然而，到这时，他的数学事业已经极其成功，第二次危机没有成为现实。

在接下来的十年里，卢津和德米特里·叶戈罗夫在莫斯科大学建立了一个令人印象深刻的研究小组，学生们称之为“卢西塔尼亚”。第一批学生包括帕维尔·亚历山德罗夫、米哈伊尔·雅科夫列维奇·苏斯林、迪米特里·缅绍夫和亚历山大·欣钦。下一批学生包括帕维尔·萨穆伊洛维奇·乌雷松、安德雷·柯尔莫哥洛夫、N K Bari、L A Lusternik和列夫·杰里科维奇·史尼尔曼。1923年，彼得·诺维科夫和L V Keldysh加入了这个小组。

当时卢西塔尼亚研究小组的另一位成员是米哈伊尔·阿列克谢耶维奇·拉夫连季耶夫。事实上，米哈伊尔·阿列克谢耶维奇·拉夫连季耶夫描绘了该小组的如下画面：-

德米特里·叶戈罗夫矜持而正式，而卢津则外向而戏剧化，在这些学生和年轻同事中激发了真正的忠诚。……存在着强烈的同志情谊……由卢津所激发。

卢津的主要贡献在数学基础与测度论领域。他还在描述集合拓扑学方面做出了重要贡献。在解析函数的边界性质理论中，他于1919年证明了一个关于边界点集在共形映射下不变性的重要结果。他还与Privalov一起研究了解析函数的边界唯一性性质。

从1917年起，卢津研究描述集合论。他提出了基本问题（[15]和[16]）：-

集合论的目标是一个极其重要的问题：我们能否将直线原子式地视为点的集合：顺便说一句，这个问题并不新鲜，而是可以追溯到希腊人。

卢津在集合论方面的大部分工作涉及有效集的研究，即无需选择公理即可构造的集合。Keldysh在[12]和[13]中描述了这项工作：-

……卢津从法国学派（埃米尔·博雷尔，昂利·勒贝格）的观点出发，该学派对他影响很大。但是，法国人分析的是借助选择公理进行的集合论构造，而卢津则走得远得多，考虑了有效集理论内部产生的困难。他所着手的有效集研究持续深入进行了二十多年，并导致集合论中许多重要问题的解决……

卢津的学派在1922年至1926年间处于鼎盛时期，但随后卢津专注于撰写他的第二部函数论专著，与学派中年轻数学家的相处时间减少了。这些数学家中的许多人转向了其他主题，如拓扑学、微分方程和复变函数。

1927年，卢津当选为苏联科学院成员。两年后，他成为正式成员，先是哲学部，然后是纯粹数学部。他从这时起一直在苏联科学院工作，直到去世。从1935年起，他领导弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所的实变量函数论研究室。

1931年，卢津本人转向了一个新领域，开始研究微分方程及其在几何学和控制理论中的应用。他在这一领域的工作使他研究曲面的弯曲，这在[15]和[16]中有描述：-

曲面在主子基上的弯曲是曲面的一种连续弯曲，在这种弯曲下，曲面上某些曲线网的共轭性保持不变。... Finikov 推导出了确定给定曲面上所有主子的微分方程，而 Byushgens 获得了确定具有给定线元并允许在主子基上弯曲的曲面的微分方程。然而，这些方程的可解性问题一般来说仍不清楚。... 直到1938年，卢津通过对这些方程的精妙分析，确定主子基的存在相当罕见，此前没有发现方程...不可解的例子。

[19]提请我们注意，1936年，卢津成为苏联当局通过报纸Pravda组织的暴力政治运动的受害者。他被指控进行反苏宣传和破坏活动，因为他将所有重要成果发表在国外，而只在苏联期刊上发表次要论文。目的显然是要除掉作为莫斯科前苏联时期旧数学学派代表的卢津：他的导师德米特里·叶戈罗夫本人在1930年也曾是此类运动的受害者（基于他的宗教同情），并于1931年在绝望和苦难中去世。关于“卢津事件”的当代记录被奇迹般地保存下来，最近由 Demidov 和 Levchin 在莫斯科编辑出版[3]，[23]。它表明卢津险些遭遇悲剧命运，因为苏联当局可能担心对一位在国外如此著名的科学家进行过于强烈的攻击会带来国际后果。卢津事件的主要可见后果是，从这一确切时刻起，苏联数学家开始几乎只在苏联期刊上用俄语发表论文。

卢津一直对数学史感兴趣，在职业生涯后期，他撰写了关于艾萨克·牛顿和莱昂哈德·欧拉的重要文章。

作为教师，他的非凡才能由 Kuznetsov 描述（[15]或[16]）：-

他的表述总是非常优雅，乍看之下似乎不必要地简单——这是他伟大教学才能的结果。他所承担的大问题的解决方案以其精妙、优雅和表述的简洁而著称。

Keldysh 和彼得·诺维科夫在[14]中写道：-

由于他非凡的直觉和深入问题核心的能力，卢津经常预测出数学事实，而这些事实的证明只有在多年以后才成为可能，并且需要创造全新的数学方法。他是我们时代杰出的数学家和思想家之一……

Nikolai Nikolaevich Luzin was born in Irkutsk, and his birthplace was not, as is incorrectly stated in a number of sources, Tomsk. Nikolai's father was a businessman, half Russian and half Buryat. Nikolai was the only son of his parents and the family moved to Tomsk when he was about eleven years old so that he could attend the Gymnasium there.

One might expect that Nikolai would have shown a special talent for mathematics at the Gymnasium, but this was far from the case ([15] and [16]):-

This was because the system of instruction ... was based on mechanical memory: it was required to learn the theorems by heart and to reproduce their proofs exactly. For Luzin this was torture. His progress in mathematics at the Gymnasium became worse and worse, so that his father was obliged to engage a tutor ...

Fortunately the tutor was a talented young man who quickly discovered that, despite Luzin's poor performance in mathematics, he could solve hard problems but often using a novel method that the tutor had never seen before. Soon the tutor had shown Luzin that mathematics was not a subject where one had to learn long lists of facts, but a topic where creativity and imagination played a major role.

In 1901 Luzin left the Gymnasium and at this time his father sold his business and the family moved to Moscow. There Luzin entered the Faculty of Physics and Mathematics at Moscow University intending to train to become an engineer. At first Luzin lived in the new family home in Moscow, but Luzin's father began to gamble on the stock exchange with the money he had made from the sale of his business. The family soon hit hard times as Luzin's father lost all their savings and the family had to leave their home. Luzin, together with a friend, moved into a room owned by the widow of a doctor. His friend soon became involved with the Revolution and was forced into hiding. Luzin stayed on by himself in the room but he clearly got on well with the owners since he later, in 1908, married the widow's daughter.

At Moscow University Luzin studied under Bugaev, learning from him the theory of functions which was to influence greatly the direction his research would eventually take. However he was only an average student who seemed to show little flair for mathematics. However, although Luzin appeared to lack talent in mathematics, one of his teachers Egorov spotted his great talent, invited him to his home, and began to set him hard problems.

There was a mathematics student at the university, Pavel Florensky, who experienced a crisis after graduating and turned to religion and the study of theology. This had a major effect on Luzin, who was a close friend of Florensky, as we shall describe below.

After graduating in the autumn of 1905 Luzin seemed unsure whether to devote himself to mathematics. In fact Luzin's crisis had hit him in the spring of 1905 and, on 1 May 1906, Luzin wrote to Florensky from Paris where Egorov had sent him five months earlier in an attempt to get him through the crisis (see [9]):-

You found me a mere child at the University, knowing nothing. I don't know how it happened, but I cannot be satisfied any more with analytic functions and Taylor series ... it happened about a year ago. ... To see the misery of people, to see the torment of life, to wend my way home from a mathematical meeting ... where, shivering in the cold, some women stand waiting in vain for dinner purchased with horror - this is an unbearable sight. It is unbearable, having seen this, to calmly study (in fact to enjoy) science. After that I could not study only mathematics, and I wanted to transfer to the medical school. ... I have been here about five months, but have only recently begun to study.

Luzin was not only upset by seeing the prostitutes, he also says in the letter how he had been affected by the 'terrible days' of the 1905 Revolution. There are letters from Egorov at this time pleading with Luzin not to give up mathematics. After returning to Russia, Luzin studied medicine and theology as well as mathematics. However in April 1908 he wrote of the joy he was finding in number theory (see [9]):-

It is a mysterious area that envelops me deeper and deeper.

In the same letter he says that he has just married and:-

... my wife is also very interested and shares my commitment to the search for the profound truths of life.

Largely Luzin's crisis seems to have been solved by Florensky to whom Luzin wrote in July 1908:-

Two times I was very close to suicide - then I came ... looking to talk with you, and both times I felt as if I had leaned on a pillar and with this feeling of support I returned home ... I owe my interest in life to you...

His interest in mathematics slowly returned but it was not until 1909 that Luzin seems to have finally committed himself completely to mathematics. Under Egorov's supervision he worked on his master's thesis. In 1910 he was appointed as assistant lecturer in Pure Mathematics at Moscow University. He worked for a year with Egorov and they went on to publish joint papers on function theory which mark the beginnings of the Moscow school of function theory.

In 1910 Luzin travelled abroad visiting Göttingen where he was influenced by Edmund Landau. He returned to Moscow in 1914 and he completed his thesis The integral and trigonometric series which he submitted in 1915. After his oral examination he was awarded a doctorate, despite having submitted his thesis for the Master's Degree. Egorov was extraordinarily impressed by the work and had pressed for the award of the doctorate, but it was written in a style quite different from the accepted Russian style of the time. Some of the results were not rigorously proved but were justified using phrases such as 'it seems to me' and 'I am convinced'. Other mathematicians were not so impressed at the time, for example Steklov wrote comments in the margin such as 'it seems to him, but it doesn't seem to me' and 'Göttingen chatter'.

However, the work was of fundamental importance as is stated in [15] and [16]:-

The influence of Luzin's dissertation on the future development of the theory of functions cannot be overestimated. Its fundamental results, deep methods of investigation and fundamental statements of problems put it into the ranks of works with which it is difficult to compare any dissertation or monograph of the time.

In 1914 Luzin and his wife separated for a short time and again Florensky seems to have helped them through the difficult time. He wrote to Luzin's wife (see [9]):-

Nikolai Nikolaevich is a very sweet and fine person; but in personal relationships he is not at all mature, especially in intuitively perceiving the hidden currents of life. ... You will have to take the relationship in hand and create a family tone, simplicity. Instead, as I perceive it ... you have established the tone of an acquaintanceship rather than a family.

Florensky seems to have given good advice since Luzin and his wife returned to a successful marriage.

In 1917 Luzin was appointed as Professor of Pure Mathematics at Moscow University just before the Revolution. The Revolution caused Luzin to rethink some of the same thoughts as he had done at the time of his crisis and again he exchanged letters with Florensky. By this stage, however, his mathematical career was extremely successful and the second crisis did not materialise.

Over the next ten years Luzin and Egorov built up an impressive research group at the University of Moscow which the students called 'Luzitania'. The first students included P S Aleksandrov, M Ya Suslin, D E Menshov and A Ya Khinchin. The next students included P S Urysohn, A N Kolmogorov, N K Bari, L A Lusternik and N G Shnirelman. In 1923 P S Novikov and L V Keldysh joined the group.

Another of the members of the Luzitania research group at this time was Lavrent'ev. In fact Lavrent'ev draws the following picture of the group:-

Whereas Egorov was reserved and formal, Luzin was extroverted and theatrical, inspiring real devotion among these students and young colleagues. ... There was intense camaraderie ... inspired by Luzin.

Luzin's main contributions are in the area of foundations of mathematics and measure theory. He also made significant contributions to descriptive set topology. In the theory of boundary properties of analytic functions he proved an important result in 1919 on the invariance of sets of boundary points under conformal mappings. He also studied, together with Privalov, boundary uniqueness properties of analytic functions.

From 1917 onwards, Luzin studied descriptive set theory. He stated the fundamental problem ([15] and [16]):-

The aim of set theory is a question of great importance: can we regard a line atomistically as a set of points: incidentally this question is not new, but goes back to the Greeks.

Much of Luzin's work on set theory involved the study of effective sets, that is sets which can be constructed without the axiom of choice. Keldysh describes this work in [12] and [13]:-

... Luzin proceeded from the point of view of the French school (Borel, Lebesgue), which greatly influenced him. But whereas the French had analysed set-theoretical constructions carried out with the help of the Axiom of Choice, Luzin went considerably further and considered difficulties arising within the theory of effective sets. The study of effective sets that he embarked upon was pursued intensively for more than two decades and led to the solution of many important problems of set theory ...

Luzin's school was at its peak during the years 1922 to 1926, but then Luzin concentrated on writing his second monograph on the theory of functions and spent less time with the young mathematicians in the school. Many of these mathematicians turned to other topics such as topology, differential equations, and functions of a complex variable.

In 1927 Luzin was elected as a member of the USSR Academy of Sciences. Two years later he became a full member of first the Department of Philosophy, then to the Department of Pure Mathematics. He worked from this time until his death in the USSR Academy of Sciences. From 1935 he headed the Department of the Theory of Functions of Real Variables at the Steklov Institute.

In 1931 Luzin himself turned to a new area when he began to study differential equations and their application to geometry and to control theory. His work in this area led him to study the bending of surfaces which is described in [15] and [16]:-

The bending of a surface on a principal base is a continuous bending of a surface under which the conjugacy of the net of certain curves on the surface is preserved. ... Finikov had derived differential equations that determine all principal on a given surface, and Byushgens had obtained differential equations that determine surfaces which have a given linear element and admit a bending on a principal base. However, the question of solubility of these equations, in general, remained unclear. ... no example was found in which the equations ... were insoluble ... up to 1938, when Luzin, by means of a subtle analysis of these equations, established that the existence of a principal base is rather rare.

It has been drawn to our attention by [19], that in 1936, Luzin was the victim of a violent political campaign organized by the Soviet authorities through the newspaper Pravda. He was accused of anti-Soviet propaganda and sabotage by publishing all his important results abroad and only minor papers in Soviet journals. The aim was obviously to get rid of Luzin as a representative of the old pre-Soviet mathematical school of Moscow: his master, Egorov, had been himself the victim of such a campaign in 1930 (based on his religious sympathies) and died shortly after in 1931 in despair and misery. A contemporary record of the "Luzin affair" has been miraculously preserved and recently edited in Moscow by Demidov and Levchin [3], [23]. It shows that Luzin had had a narrow escape from a tragic fate as the Soviet authorities may have feared the international consequences of a too strong attack on a scientist so famous abroad. The main visible consequence of the Luzin affair was that, from this precise moment, Soviet mathematicians began to publish almost exclusively in Soviet journals and in Russian.

Luzin always had an interest in the history of mathematics and late in his career he wrote important articles on Newton and on Euler.

As a teacher his remarkable talents are described by Kuznetsov ([15] or [16]):-

His presentation was always very elegant and at first sight apparently unnecessarily simple - the result of his great pedagogic talent. The solution of the large problems that he undertook is distinguished by their subtlety, elegance, and simplicity of presentation.

Keldysh and Novikov wrote in [14]:-

Thanks to his exceptional intuition and his ability to see deeply into the heart of a question, Luzin frequently predicted mathematical facts whose proof turned out to be possible only after many years and required the creation of completely new mathematical methods. He was one of the outstanding mathematicians and thinkers of our time ...