# Mikhail Mikhailovich Postnikov

### Born: 27 October 1927 in Shatura, Russia

Died: 27 May 2004 in Moscow, Russia

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**Mikhail Mikhailovich Postnikov**'s father was an electrical engineer, working at the local power station, while his mother was a school teacher. He only spent the first few years of his life in Shatura since the family moved to Perm. However, in 1937, when Mikhail Mikhailovich was ten years old, his father was arrested and sent to the Gulag. He was never seen again and presumably died in the Gulag but some time later he was exonerated. Postnikov attended school in Perm for eight years but it was not a good experience for him since he was far ahead of his fellow pupils but was not allowed to skip classes. Some of the teachers allowed him to read advanced books in their classes although they always demanded that he could answer their questions on the current lesson.

In September 1942, when still only 14 years of age, he was allowed to take the entrance examinations for Perm University despite the regulations which required him to have spent ten years at school. The reason that the regulations were relaxed was due to the difficulties encountered during World War II - the intention was to make it possible for soldiers to enter university but Postnikov took his chance and, passing the entrance examinations, entered Perm University in 1942. (In fact Perm had been renamed Molotov in 1940 so it was in fact Molotov University that he entered - however, we shall continue to was the name Perm for both the city and its university.) Although Postnikov was not at all impressed by most of his professors, who he felt were simply reading lectures from a textbook, he did find one professor, Sof'ja Aleksandrovna Janovskaja, inspiring. Janovskaja had been a professor at Moscow University but she had been evacuated to Perm when it looked as if the German advance would threaten to capture Moscow. She had with her a brilliant third year student, Eugene Borisovich Dynkin, and quickly Postnikov became convinced that he would only have top quality lecturers if he attended Moscow University.

Once the threat of the German army capturing Moscow had receded, Janovskaja and Dynkin returned to Moscow in 1943. Postnikov had wanted to travel with them but special permission was required before anyone was allowed into Moscow and Postnikov was denied permission. Somehow or other, in November 1943 some time after Janovskaja had begun teaching again in Moscow, Postnikov arrived there on a train having hidden away and, knowing Janovskaja's address, was able to arrive unannounced at her home. She made him welcome and the next day took him to the Department of Mechanics and Mathematics of Moscow State University where she taught. The chairman of the Department was Ivan Georgievich Petrovsky and he accepted Janovskaja's advice that Postnikov was capable of passing the first year examinations and entering second year courses. By December 1943 Postnikov had passed not only the first year examinations but some second year ones too and he began his studies. Once the quality of the young student was recognised, arrangements were made to allow him the right to be in Moscow.

Postnikov, still only seventeen years old, graduated from Moscow State University in the summer of 1945 with a Master's degree. He explained in [6] how he came to have Lev Semenovich Pontryagin as thesis advisor for his senior thesis:-

Continuing to work with Pontryagin, Postnikov was awarded his Candidate's Degree (equivalent to a Ph.D.) in 1949. He published results from his senior thesis and his Ph.D. thesis in a number of papers (all written in Russian):The assistant dean ... called me in to say, "Postnikov, you need to have an advisor for your senior thesis. Who do you want, Lusternik or Aleksandrov? ... At this point a very strange thing happened. Without a moment's hesitation I blurted out, "I want Lev Semenovich Pontryagin!" The reason why this was so strange is that I did not know Pontryagin, and had only taken his very boring, formalistic course of homological algebra. Yet my spontaneous choice turned out to be an unusually good one: if it weren't for Pontryagin it's hard to say what would have become of me. ... The day after my conversation with the assistant dean, seeing Pontryagin in the hall, I said to him "Lev Semenovich! My name is Postnikov, I am a fifth-year student, and I told the dean that you are to be my advisor!" Pontryagin was startled, but answered, "Fine, come to see me at home tomorrow, and we'll talk." That is how I became Pontryagin's student.

*The structure of the ring of intersections of three-dimensional manifolds*(1948),

*The classification of continuous mappings of a three-dimensional polyhedron into a simply connected polyhedron of arbitrary dimension*(1949),

*Homology invariants of continuous mappings*(1949),

*Classification of the continuous mappings of an arbitrary n-dimensional polyhedron into a connected topological space which is aspherical in dimensions greater than unity and less than n*(1949), and

*Classification of continuous mappings of an*(

*n*+1)

*-dimensional complex into a connected topological space which is aspherical in dimensions less than n*(1950).

After the award of his Ph.D., Postnikov began working in Pontryagin's Department in the Steklov Mathematical Institute in Moscow. During the following two years he completed a remarkable piece of work concerning the homotopy classification of polyhedra and their maps. He announced his results in a number of papers (all written in Russian):

*Determination of the homology groups of a space by means of the homotopy invariants*(1951),

*On the homotopy type of polyhedra*(1951), and

*On the classification of continuous mappings*(1951). Peter Hilton, who reviewed them, was impressed and circulated details to his colleagues. In 1953 J C H Whitehead described Postnikov's results as "the greatest achievement in algebraic topology in recent years". For this work, Postnikov was awarded the Lenin Prize in 1961. The authors of [5] write:-

Let us look briefly at some of the Postnikov's books referred to in this quotation. All were originally written in Russian but they were translated into English, and some into other languages.Postnikov played an enormous role in establishing and developing algebraic topology in the USSR. He worked in the Faculty of Mechanics and Mathematics at Moscow State University for more than twenty years and directed the scientific research seminar(in the last ten years, together with A V Chernavskii). Almost all the Soviet books on algebraic topology that have been published since the war have appeared with his active participation(direct authorship, translation, editing, recommendation for translation, and so on)and almost all the Moscow algebraic topologists are either students of Postnikov or students of his students, or students of students of his students.

1.

6.

We have changed the chronological sequence of Postnikov's publications slightly to list finally the six textbooks which he wrote corresponding to six series of lectures on geometry given to undergraduate and graduate students at Moscow State University.

7.

10.

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12.

We should also comment on Postnikov's quality as a lecturer which is emphasised in all the references [3]:-*Foundations of Galois theory*(1960). H Schwerdtfeger writes:-2.According to the preface, this book has been written for students in their second or third year in the university who, after a preliminary introduction to higher algebra, wish to proceed to Galois theory. Complicated technicalities have been avoided and with regard to its restricted aims the straightforward exposition is up-to-date in method and statement of results.

*The variational theory of geodesics*(1967). W Klingenberg writes:-3.The book contains a clear and detailed presentation of the basic theorems of Riemannian manifolds. A careful derivation of the variational properties of geodesics then follows. Highlights are Morse's and Bott's quadratic form and the index theorem.

*Introduction to Morse theory*(1971). Alan Weinstein writes:-4.Most of this book consists of a well-written, self-contained text on homotopy theory and differential geometry, in preparation for chapters on Morse theory on finite dimensional manifolds, the variational theory of geodesics, and the study of path spaces by finite-dimensional approximation.

*Introduction to the theory of algebraic numbers*(1978). J B Roberts writes:-5.The aim of the book is to give an exposition of the theory of algebraic numbers showing how the subject arises in a natural way from the investigation of Fermat's "last theorem''.

*Stable polynomials*(1981).6.

*Lectures in algebraic topology*(1984, 1985). This book is based on special lectures delivered by Postnikov to undergraduate and graduate students at Moscow State University. The first volume (subtitled*Elements of homotopy theory*) is devoted to general homotopy theory and to homotopy groups. The second volume (subtitled*Homotopy theory of cell complexes*) concentrates, as the title suggests, on homotopy theory of cell complexes.We have changed the chronological sequence of Postnikov's publications slightly to list finally the six textbooks which he wrote corresponding to six series of lectures on geometry given to undergraduate and graduate students at Moscow State University.

7.

*Lie groups and algebras*(1982). Rolf Sulanke writes:-8.The textbook is a profound description of lectures about Lie groups and Lie algebras given by the author at Moscow State University. ... The author's aim is to develop the Lie theory from the very beginning up to E Cartan's theorem about the equivalence of the category of simply connected Lie groups with the category of Lie algebras.

*Linear algebra and differential geometry*(1986). B Reichstein writes:-9.Written by a famous author and mathematician with a worldwide reputation, a winner of the Lenin prize, the highest honour that a Soviet mathematician can be awarded, the book gives a very clear and at the same time rigorous presentation of different topics in linear algebra and their numerous applications to modern geometry and analysis. The main emphasis is on the ideas rather than on computational techniques.

*Analytic geometry*(1986).10.

*Smooth manifolds*(1987).11.

*Differential geometry*(1988).12.

*Riemannian geometry*(1998). Robert J. Low writes:-This volume is the sixth in Postnikov's series of lecture notes in differential geometry, and provides an advanced overview of various topics in Riemannian geometry. ... I found the presentation insightful and stimulating.

He also made an impact in other ways [3]:-From1954until1960he lectured in the higher geometry and topology department at Moscow State University. His brilliant lectures, once heard, were never forgotten.

Finally we note that among his other interests were travel, chess and bridge.The interests of Postnikov are not restricted to pure mathematics. He has assembled a huge collection of puzzles and games. Postnikov's articles and television appearances concerning the reform of the education system have made a large impact on society at large.

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

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