Kolmogorov, Andrey Nikolayevich

At the age of 17 Kolmogorov was enrolled in the Moscow State University. In the autumn of 1921 he worked on complex problems--in the theory of trigonometrical series and operations on sets--closely tied with the basic directions of the Moscow mathematical school of that time, namely, with ideas in the theory of functions of real variables. In the spring of 1922 he completed a study in the theory of operations on sets (published in 1928).

In 1925 Kolmogorov was graduated from the faculty of physics and mathematics of Moscow State University and was appointed a research associate of the faculty. At this time he became interested in probability theory. His important "General Theory of Measure and Probability Theory," published in 1929, gave the first description of an axiomatic construction of probability theory based on measure theory. In 1933 Kolmogorov expanded the paper into the monograph Grundbegriffe der Wahrscheinlichkeitsrechnung, which in 1950 was translated into English under the title Foundations of the Theory of Probability.

In 1931 Kolmogorov was elected a professor of Moscow State University; two years later he was appointed a director of the Institute of Mathematics of the university. During the period of this appointment he elaborated the principles of stochastic processes, including the Markov processes, in the monograph "Analytical Methods of Probability Theory." Other contributions concerned aspects of functional analysis--a comparatively recent branch of mathematics in which the traditional techniques of algebra and calculus are applied to entire collections of functions. He also made contributions to studies of topology and turbulent flow of fluids.

In the study of probability theory, which is considered his basic speciality, Kolmogorov formulated two systems of partial differential equations that bear his name; they describe transition probabilities controlling a Markov process. The Kolmogorov equations provide a means of dealing with problems in the theories of Brownian motion and diffusion. The work marked a new period in the development of probability theory and its application in physics, chemistry, civil engineering, and biology.

This fundamental study in probability was succeeded by a fundamental study of Markov chains that embodied many new ideas in the theory of dynamic systems and served as a starting point for numerous works of other mathematicians.

Of considerable importance were Kolmogorov's investigations in problems of random stationary processes, which he related to the analysis of locally isotropic turbulent flow. Norbert Wiener, the founder of cybernetics (the theory of control and communication as applied to animals and machines), had independently investigated many aspects of stationary processes, particularly those identified with statistical prediction, and he pointed out that the investigations by Kolmogorov were relevant to the statistical theory of information upon which the science of cybernetics in part depended.

In topology, Kolmogorov, simultaneously with U.S. mathematician James Alexander but independently of him put forward certain considerations relating to an operator called the nabla operator and applied these to complexes and later to any topological space. Related algebraic structures called nabla groups turned out to be very effective and provided suitable instrumentation in the study of numerous problems of topology, including continuous maps. On this foundation, Kolmogorov put forward a conception of a homological ring, an important idea in topology. His formulation of the duality law (on equivalent representations of certain mathematical properties) was very important for the period from 1935 to 36. In this context the duality law concerned closed sets situated in a topological space that is locally bicompact and fully regular.

During the 1930s Kolmogorov also published the articles entitled "On Topological Group Formulation of Geometry" and "On Formulation of Projective Geometry," as well as others on the areas of functional analysis and the optimum approximation of functions.

Later Kolmogorov's interests broadened into problems of mathematical logic and the foundations of mathematics. In 1938, he published a large article, "Mathematics," in the first edition of the Bolshaya Sovyetskaya Entsiklopediya ("Great Soviet Encyclopaedia"), in which he described the development of mathematics from ancient to modern times and interpreted it in terms of dialectical materialism.

In 1939 Kolmogorov was elected an academician of the Academy of Sciences of the U.S.S.R. and, a little later, an academician-secretary of the department of physical and mathematical sciences of the academy; the department had been established to unite and direct scientific work of the most prominent researchers in this field.

In the mid-1950s Kolmogorov turned his attention to information theory, dynamic-system theory, interconnections of information theory with function theory, classical mechanics, the 13th Hilbert Problem (representation of functions of a large number of variables by functions of a lesser number of arguments), and complex theory.

Throughout the latter part of his life, Kolmogorov paid great attention to problems of mathematical education of schoolchildren. He was appointed chairman of the Commission for Mathematical Education under the Presidium of the Academy of Sciences of the U.S.S.R. Under his leadership a new state program of training in mathematics in the Soviet general school was developed and introduced.

Introductory Real Analysis (rev. Eng. ed. by Richard A. Silverman, 1970; reprinted 1975), by Kolmogorov and S.V. Fomin, contains a good bibliography.