Andrey Kolmogorov

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Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. He worked early in his career on intuitionistic logic, and Fourier series. He also worked on turbulence, and classical mechanics; and was a founder of algorithmic complexity theory. He is considered to be one of the greatest mathematicians of all time with influences not only in mathematics, but also in many engineering disciplines.

In 1920, Kolmogorov entered Moscow State University where he studied a number of subjects, including metallurgy and Russian history. 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 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. 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. 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. 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 the paper Analytic methods in probability theory which Kolmogorov published in 1938 laying the foundations of the theory of Markov random processes, his made substantial contributions to set-theoretic topology, approximation theory, the theory of turbulent flow, functional analysis, the foundations of geometry, and the history and methodology of mathematics.

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.

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:

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 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. 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].

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 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.

Kolmogorov worked at Moscow State University. He studied under Nikolai Luzin, earning his Ph.D. in 1925, in 1931 he became the professor of the university. In 1939 he received the title of academician of the USSR Academy of Sciences.

Quote:

"The theory of probability as mathematical discipline can and should be developed from axioms in exactly the same way as Geometry and Algebra."

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Bibliography

Selected works of A.N. Kolmogorov / edited by V.M. Tikhomirov; translated from the Russian by V.M. Volosov. 3 volumes. Dordrecht; Boston : Kluwer Academic Publishers, c1991-c1993 ISBN 9027727953 .

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