**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.

**Article by:** *J J O'Connor* and *E F Robertson*

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