列夫·庞特里亚金Lev Pontryagin

列夫·庞特里亚金的父亲Semen Akimovich 庞特里亚金是一名公务员。庞特里亚金的母亲Tat'yana Andreevna Pontryagina在他出生时29岁，她是一位非凡的女性，在他成为数学家的道路上发挥了关键作用。也许“公务员”这一描述虽然准确，却给人错误的印象，以为家庭相当富裕。事实上，Semen Akimovich的工作使家庭没有足够的钱让儿子接受良好的教育，Tat'yana Andreevna利用她的缝纫技能工作以帮助家庭财务。

庞特里亚金就读于镇上的学校，那里的教育水平远低于较好的学校，但家庭贫困使这些好学校在经济上遥不可及。14岁时，庞特里亚金遭遇了一场事故，一次爆炸使他失明。这本来可能意味着他的教育和职业生涯的终结，但他的母亲另有想法，并致力于帮助他克服失明这一几乎不可能的困难而取得成功。她给予庞特里亚金的帮助在[3]和[4]中有所描述：-

从这一刻起，塔季扬娜·安德烈耶夫娜完全承担起了在儿子生活各个方面满足其需求的责任。尽管她不得不面对巨大的困难，但她在自己承担的任务上如此成功，以至于她确实值得全世界科学的感激……多年来，她实际上一直担任庞特里亚金的秘书，为他朗读科学著作，在他的手稿中写入公式，修改他的工作等等。为了做到这一点，她尤其必须学会阅读外语。塔季扬娜·安德烈耶夫娜在其他所有方面都帮助庞特里亚金，照顾他的需求，对他关怀备至。

我们不妨稍作停顿，想一想塔季扬娜·安德烈耶夫娜，一个没有数学训练或知识的人，如何凭借她的决心和极大努力，让庞特里亚金克服重重困难成为数学家，从而为数学做出了重大贡献。一定还有许多其他非数学家，其中许多人也许未被历史记载，他们也通过无私的行为让数学得以繁荣。正如我们在本档案中试图表明的，数学的发展取决于数学家自身才能之外的众多影响：政治影响、经济影响、社会影响，以及像塔季扬娜·安德烈耶夫娜这样的非数学家的行为。

但是，一个人如果不懂数学，又怎么能读懂数学论文呢？当然，论文里满是神秘的符号，塔季扬娜·安德烈耶夫娜不知道它们的数学含义或名称，只能根据它们的外形来描述。例如，交符号成了“尾巴朝下”，而并符号成了“尾巴朝上”。如果她读到“$A$尾巴朝右$B$”，那么庞特里亚金就知道$A$是$B$的子集！

庞特里亚金于1925年进入莫斯科大学，他的讲师们很快便发现他是一个出类拔萃的学生。当然，一个不能做笔记的盲人学生却能够记住最复杂的符号运算，这本身就非常了不起。更了不起的是，庞特里亚金能够比他的任何同学都更清楚地“看到”（请原谅这个糟糕的双关语）呈现在他面前的主题的深层含义。在他所修的高级课程中，庞特里亚金对亚历山大·欣钦的分析课程感到不太满意，但他特别喜欢帕维尔·亚历山德罗夫的课程。庞特里亚金深受帕维尔·亚历山德罗夫的影响，而帕维尔·亚历山德罗夫的研究方向决定了庞特里亚金多年工作的领域。然而，这既与帕维尔·亚历山德罗夫的数学有关，也同样与帕维尔·亚历山德罗夫本人有关（[3]和[4]）：-

亚历山德罗夫的个人魅力、他的关注和乐于助人，对庞特里亚金科学兴趣的形成产生了显著影响，事实上不亚于这位年轻学者本人的个人能力和倾向。

1927年是庞特里亚金父亲去世的一年。到1927年，尽管庞特里亚金还只有19岁，他已经开始在詹姆斯·韦德尔·亚历山大对偶定理上产生重要结果。他的主要工具是使用由勒伊岑·布劳威尔引入的环绕数，并且到1932年，他证明了欧几里得空间中有界闭集的homology groups与该空间补集中的同调群之间的对偶性，从而得出了这些对偶性结果中最重要的一个。

庞特里亚金于1929年毕业于莫斯科大学，并被聘入力学与数学系。1934年，他成为弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所的成员，1935年，他成为该研究所拓扑学与泛函分析系的主任。

庞特里亚金研究拓扑学和代数中的问题。事实上，他对自己所研究领域的描述是：-

……这些问题把数学的这两个领域结合到了一起。

庞特里亚金关于对偶性的这项工作（[3]和[4]）的意义在于：——

……不仅在于它对拓扑学的进一步发展所产生的影响；同样重要的是，他的定理使他能够为交换topological group构造一般的characters理论。这一理论在历史上是拓扑代数这个数学新分支中第一个真正非凡的成就，是本世纪整个数学中最根本的进展之一……

大卫·希尔伯特在1900年提出的23个问题之一，是证明他的猜想：任何局部欧几里得拓扑群都可以赋予解析流形的结构，从而成为一个李群。这后来被称为大卫·希尔伯特第五问题。1929年，冯·诺伊曼利用他引入的一般紧群上的积分，解决了大卫·希尔伯特第五问题中关于紧的群的情形。1934年，庞特里亚金利用他引入的局部紧阿贝尔群上的特征标理论，证明了大卫·希尔伯特第五问题中关于阿贝尔群的情形。

庞特里亚金关于上述主题的最重要著作是topological groups（1938）。[3]和[4]的作者正确地断言：——

这本书属于那种罕见的数学著作，它们真正堪称经典——历经数十年仍保持其重要性，并对整整几代数学家的科学观产生塑造性影响。

此处一段未译出，以下为英文原文 In 1934 Cartan visited Moscow and lectured in the Mechanics and Mathematics Faculty. Pontryagin attended Cartan's lecture which was in French but Pontryagin did not understand French so he listened to a whispered translation by Nina Bari who sat beside him. Cartan's lecture was based around the problem of calculating the homology groups of the classical compact Lie groups. Cartan had some ideas how this might be achieved and he explained these in the lecture but, the following year, Pontryagin was able to solve the problem completely using a totally different approach to the one suggested by Cartan. In fact Pontryagin used ideas introduced by Morse on equipotential surfaces.

庞特里亚金的名字与许多数学概念联系在一起。配边理论的基本工具是庞特里亚金-勒内·托姆构造。关于流形示性类的一个基本定理涉及称为流形的庞特里亚金示性类的特殊类。示性类的主要问题之一直到谢尔盖·彼得罗维奇·诺维科夫证明了它们的拓扑不变性才得到解决。

1952年，庞特里亚金完全改变了他的研究方向。他开始研究应用数学问题，特别是研究微分方程和控制理论。事实上，这一方向的改变并不像看起来那么突然。从20世纪30年代起，庞特里亚金就与物理学家A A Andronov交好，并经常与他讨论Andronov正在研究的振动理论和自动控制理论中的问题。1932年，他与Andronov合作发表了一篇关于动力系统的论文，但庞特里亚金在1952年工作中的重大转变发生在Andronov去世前后。

1961年，他与他的学生V G Boltyanskii、R V Gamrelidze和E F Mishchenko一起发表了The Mathematical Theory of Optimal Processes。第二年出现了英译本，同样在1962年，庞特里亚金因其著作获得了列宁奖。随后，他发表了一系列关于微分博弈的论文，扩展了他在控制理论方面的工作。庞特里亚金在控制理论方面的工作在历史综述[5]中有所讨论。

庞特里亚金的另一本书Ordinary differential equations的英译本也于1962年问世。

庞特里亚金因其工作获得了许多荣誉。他于1939年当选为科学院院士，1959年成为正式成员。1941年，他是斯大林奖（后称国家奖）的首批获得者之一。1970年，他荣幸地当选为国际数学联盟副主席。

Lev Semenovich Pontryagin's father, Semen Akimovich Pontryagin was a civil servant. Pontryagin's mother, Tat'yana Andreevna Pontryagina, was 29 years old when he was born and she was a remarkable woman who played a crucial role in his path to becoming a mathematician. Perhaps the description of 'civil servant', although accurate, gives the wrong impression that the family were reasonably well off. In fact Semen Akimovich's job left the family without enough money to allow them to give their son a good education and Tat'yana Andreevna worked using her sewing skills to help out the family finances.

Pontryagin attended the town school where the standard of education was well below that of the better schools but the family's poor circumstances put these well out of reach financially. At the age of 14 years Pontryagin suffered an accident and an explosion left him blind. This might have meant an end to his education and career but his mother had other ideas and devoted herself to help him succeed despite the almost impossible difficulties of being blind. The help that she gave Pontryagin is described in [3] and [4]:-

From this moment Tat'yana Andreevna assumed complete responsibility for ministering to the needs of her son in all aspects of his life. In spite of the great difficulties with which she had to contend, she was so successful in her self-appointed task that she truly deserves the gratitude ... of science throughout the world. For many years she worked, in effect, as Pontryagin's secretary, reading scientific works aloud to him, writing in the formulas in his manuscripts, correcting his work and so on. In order to do this she had, in particular, to learn to read foreign languages. Tat'yana Andreevna helped Pontryagin in all other respects, seeing to his needs and taking very great care of him.

It is not unreasonable to pause for a moment and think about how Tat'yana Andreevna, with no mathematical training or knowledge, made by her determination and extreme efforts a major contribution to mathematics by allowing Pontryagin to become a mathematician against all the odds. There must be many other non-mathematicians, perhaps many of whom are unrecorded by history, who have also by their unselfish acts allowed mathematics to flourish. As we try to show in this archive, the development of mathematics depends on a wide number of influences other than the talents of the mathematicians themselves: political influences, economic influences, social influences, and the acts of non-mathematicians like Tat'yana Andreevna.

But how does one read a mathematics paper without knowing any mathematics? Of course it is full of mysterious symbols and Tat'yana Andreevna, not knowing their mathematical meaning or name, could only describe them by their appearance. For example an intersection sign became a 'tails down' while a union symbol became a 'tails up'. If she read '$A$ tails right $B$' then Pontryagin knew that $A$ was a subset of $B$!

Pontryagin entered the University of Moscow in 1925 and it quickly became apparent to his lecturers that he was an exceptional student. Of course that a blind student who could not make notes yet was able to remember the most complicated manipulations with symbols was in itself truly remarkable. Even more remarkable was the fact that Pontryagin could 'see' (if you will excuse the bad pun) far more clearly than any of his fellow students the depth of meaning in the topics presented to him. Of the advanced courses he took, Pontryagin felt less happy with Khinchin's analysis course but he took a special liking to Aleksandrov's courses. Pontryagin was strongly influenced by Aleksandrov and the direction of Aleksandrov's research was to determine the area of Pontryagin's work for many years. However this was as much to do with Aleksandrov himself as with his mathematics ([3] and [4]):-

Aleksandrov's personal charm, his attention and helpfulness influenced the formation of Pontryagin's scientific interests to a remarkable extent, as much in fact as the personal abilities and inclinations of the young scholar himself.

The year 1927 was the year of the death of Pontryagin's father. By 1927, although he was still only 19 years old, Pontryagin had begun to produce important results on the Alexander duality theorem. His main tool was to use link numbers which had been introduced by Brouwer and, by 1932, he had produced the most significant of these duality results when he proved the duality between the homology groups of bounded closed sets in Euclidean space and the homology groups in the complement of the space.

Pontryagin graduated from the University of Moscow in 1929 and was appointed to the Mechanics and Mathematics Faculty. In 1934 he became a member of the Steklov Institute and in 1935 he became head of the Department of Topology and Functional Analysis at the Institute.

Pontryagin worked on problems in topology and algebra. In fact his own description of this area that he worked on was:-

... problems where these two domains of mathematics come together.

The significance of this work of Pontryagin on duality ([3] and [4]):-

... lies not merely in its effect on the further development of topology; of equal significance is the fact that his theorem enabled him to construct a general theory of characters for commutative topological groups. This theory, historically the first really exceptional achievement in a new branch of mathematics, that of topological algebra, was one of the most fundamental advances in the whole of mathematics during the present century...

One of the 23 problems posed by Hilbert in 1900 was to prove his conjecture that any locally Euclidean topological group can be given the structure of an analytic manifold so as to become a Lie group. This became known as Hilbert's Fifth Problem. In 1929 von Neumann, using integration on general compact groups which he had introduced, was able to solve Hilbert's Fifth Problem for compact groups. In 1934 Pontryagin was able to prove Hilbert's Fifth Problem for abelian groups using the theory of characters on locally compact abelian groups which he had introduced.

Among Pontryagin's most important books on the above topics is topological groups (1938). The authors of [3] and [4] rightly assert:-

This book belongs to that rare category of mathematical works that can truly be called classical - book which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians.

In 1934 Cartan visited Moscow and lectured in the Mechanics and Mathematics Faculty. Pontryagin attended Cartan's lecture which was in French but Pontryagin did not understand French so he listened to a whispered translation by Nina Bari who sat beside him. Cartan's lecture was based around the problem of calculating the homology groups of the classical compact Lie groups. Cartan had some ideas how this might be achieved and he explained these in the lecture but, the following year, Pontryagin was able to solve the problem completely using a totally different approach to the one suggested by Cartan. In fact Pontryagin used ideas introduced by Morse on equipotential surfaces.

Pontryagin's name is attached to many mathematical concepts. The essential tool of cobordism theory is the Pontryagin-Thom construction. A fundamental theorem concerning characteristic classes of a manifold deals with special classes called the Pontryagin characteristic class of the manifold. One of the main problems of characteristic classes was not solved until Sergei Novikov proved their topological invariance.

In 1952 Pontryagin changed the direction of his research completely. He began to study applied mathematics problems, in particular studying differential equations and control theory. In fact this change of direction was not quite as sudden as it appeared. From the 1930s Pontryagin had been friendly with the physicist A A Andronov and had regularly discussed with him problems in the theory of oscillations and the theory of automatic control on which Andronov was working. He published a paper with Andronov on dynamical systems in 1932 but the big shift in Pontryagin's work in 1952 occurred around the time of Andronov's death.

In 1961 he published The Mathematical Theory of Optimal Processes with his students V G Boltyanskii, R V Gamrelidze and E F Mishchenko. The following year an English translation appeared and, also in 1962, Pontryagin received the Lenin prize for his book. He then produced a series of papers on differential games which extends his work on control theory. Pontryagin's work in control theory is discussed in the historical survey [5].

Another book by Pontryagin Ordinary differential equations appeared in English translation, also in 1962.

Pontryagin received many honours for his work. He was elected to the Academy of Sciences in 1939, becoming a full member in 1959. In 1941 he was of one the first recipients of the Stalin prizes (later called the State Prizes). He was honoured in 1970 by being elected Vice-President of the International Mathematical Union.