Kolmogorov, Andrei Nikolaevich

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KOLMOGOROV, ANDREI NIKOLAEVICH

(b. Tambov, Russia, 25 April 1903; d. Moscow, Russia, 20 October 1987)

mathematics.

Kolmogorov was one of the twentieth century’s greatest mathematicians. He made fundamental contributions to probability theory, algorithmic information theory, the theory of turbulent flow, cohomology, dynamical systems theory, ergodic theory, Fourier series, and intuitionistic logic. Mathematical talent at this level of creativity and versatility is rarely encountered.

Early Development . Kolmogorov was born in western Russia. His mother having died as a result of his birth, he was brought up by his aunt. Kolmogorov’s father was an agronomist who played little part in Kolmogorov’s upbringing, and the name “Kolmogorov” was his maternal grandfather’s, rather than his father’s, name. Kolmogorov matriculated at Moscow University in 1920 to study mathematics, taking classes in set theory, projective geometry, and the theory of analytic functions in addition to Russian history. He studied real functions with Nikolai N. Luzin and early in his undergraduate career began to produce creative mathematics—most notably, in 1922, the construction of a summable function, the Fourier series of which diverged almost everywhere. This result brought him wide recognition at an early age. Following graduation in 1925 and a further four years as a research student after which he received his doctorate, Kolmogorov taught at Moscow University’s Institute of Mathematics and Mechanics, being appointed professor there in 1931.

Probability Theory . Kolmogorov’s most famous contributions are to the foundations of probability theory. From the mid-seventeenth century, probability had been explored in a somewhat unsystematic fashion. By bringing to bear on the topic the apparatus of measure theory, Kolmogorov’s principal work in probability theory, Grundbegriffe der Wahrscheinlichkeitsrechnung (1933; Foundations of the Theory of Probability, 1956), established probability theory as a core area of rigorous mathematics. In so doing he transformed one-half of David Hilbert’s sixth problem: “To treat in the same manner, 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.” Besides its foundational importance, the monograph presented a framework for the theory of stochastic processes and, building on a result of Otton Nikodym, it gave a general treatment of conditional probabilities and expectations. The book was the culmination of an interest in probability that had begun as a collaboration with Aleksandr Y. Khinchin in 1924. This led in the ensuing four years to Kolmogorov’s publishing his celebrated three-series theorem (Kolmogorov, 1928), which gives necessary and sufficient conditions for the convergence of sums of independent random variables, to his discovering necessary and sufficient conditions for the strong law of large numbers (1930) and to his proving the law of the iterated logarithm for sums of independent random variables (1929). His 1931 paper, on continuous time Markov processes with continuous states, is widely regarded as having laid the foundations of modern diffusion theory. His 1949 work, Limit Distributions for Sums of Independent Random Variables, co-authored with B. V. Gnedenko, was for many years the standard source on the central limit theorem and surrounding topics.

Other Work . Among Kolmogorov’s other achievements is his introduction in 1935 of cohomology, the study of algebraic invariants on topological spaces (a field of which J. W. Alexander was an independent codeveloper). The year 1941 saw the publication of two papers on turbulent flow (Kolmogorov 1941a, 1941b). They contained the first clear quantitative predictions in the area of turbulence based on Kolmogorov’s two-thirds law and described the equilibrium processes underlying the transfer of energy at different scales of the flow. The importance of this work has persisted as modern computational methods have allowed increasingly detailed investigations of this area of applied mathematics. As a result of this work Kolmogorov was appointed head of the Turbulence Laboratory of the USSR Academy of Sciences in 1946, having been elected to the academy in 1939. In dynamical systems theory, the widely used KAM theory (named after Kolmogorov, Vladimir Arnold, and Jürgen Moser) provides a foundation for the understanding of chaotic motions in Hamiltonian systems, another area in which the later development of computational resources was required for the full importance of Kolmogorov’s work to be realized. In 1957 Kolmogorov made a major contribution to the solution of Hilbert’s thirteenth problem—to find a proof of the hypothesis that there are continuous functions of three variables that are not representable by continuous functions of two variables—by giving a disproof of it.

Although Kolmogorov published only two papers in logic, the first in 1925 had considerable influence. In it he proved the consistency of classical logic relative to intuitionistic logic by translating formulae of classical logic into formulae of intuitionistic logic, showing that if intuitionistic logic was consistent, then so was classical logic. This is a restricted version of a result later proved by Kurt Gödel. A 1932 paper by Kolmogorov provides an objective reading of negation within intuitionistic mathematics.

In 1965 Kolmogorov unveiled a definition of a random sequence based on the idea that a sequence of integers is random just in case any algorithm that will generate that sequence has a length essentially equal to the sequence itself; that is, the information contained in the sequence cannot be compressed. This approach is often called Kolmogorov complexity, although it was independently arrived at by Gregory Chaitin and somewhat earlier by Ray Solomonoff. This work interestingly inverts Kolmogorov’s earlier emphasis on probability in that it allows probabilistic concepts to be based on information theory rather than the reverse, which hitherto had been the standard approach.

Kolmogorov was actively involved for many years in teaching mathematically gifted children and served as the director for almost seventy advanced research students, many of whom became significant mathematicians in their own right. He had wide-ranging intellectual interests, including Russian history and Aleksandr Pushkin’s poetry. Kolmogorov’s fifty-three-year friendship with the topologist Pavel Sergeevich Alexandrov had an important influence on him. He maintained a deep commitment to the truth, clashing with Trofim Lysenko in 1940 over the interpretation of a geneticist’s experimental data. In 1942 he married Anna Egorova. They had no children.

BIBLIOGRAPHY

For a full bibliography of Kolmogorov’s published writings, see “Publications of A. N. Kolmogorov,” Annals of Probability 17 (1989): 945–964.

WORKS BY KOLMOGOROV

“On the ‘Tertium non Datur’ Principle.” Matemticheski Sbornik 32 (1925): 646–667.

“Über die Summen durch den Zufall bestimmter unabhängiger Grössen Mathematische Annalen 99 (1928): 309–319.

“Über das Gesetz des iterierten Logarithmus.” Mathematische Annalen 101 (1929): 126–135.

“Sur la loi fortes des grands nombres.” Comptes rendus de l’Acadmie des sciences 191 (1930): 910–912.

“Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung.” Mathematische Annalen 104 (1931): 415–458.

“Zur Deutung der intuitionistischen Logik.” Mathematische Zeitschrift 35 (1932): 58–65.

Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability). Berlin: Springer, 1933.

“The Local Structure of Turbulence in an Incompressible Fluid with Very Large Reynolds Numbers.” C omptes rendus de l’Acadmie des sciences de l’URSS 30 (1941a): 301–305.

“Dissipation of Energy under Locally Isotropic Turbulence.” C omptes rendus de l’Acadmie des sciences de l’URSS 32 (1941b):16–18.

With Boris V. Gnedenko. Limit Distributions for Sums of Independent Random Variables. Cambridge, MA: Addison-Wesley, 1954.

Selected Works of A. N. Kolmogorov. Vol. 1: Mathematics and Mechanics. Edited by Vladimir M. Tikhomirov. Berlin: Springer, 1989. Vol. 2: probability Theory and Mathematical Statistics. Edited by Albert N. Shirayayev. Berlin: Springer, 2001. Vol. 3: Information Theory and the Theory of Algorithms. Edited by Albert N. Shirayev. Berlin: Springer, 2006. Note variant spelling of Shirayev’s name used by Springer.

OTHER SOURCES

Aleksandrov, Pavel S. “Pages from an Autobiography.” Uspekhi Matematicheskikh Nauk 34 (1979): 219–249; 35 (1980): 241–278.

American Mathematical Society. Kolmogorov in Perspective. Translated by Harold H. McFaden. History of Mathematics Series, vol. 20. Providence, RI: American Mathematical Society; London: London Mathematical Society, 2000.

Kendall, D., G. K. Batchelor, N. H. Bingham, et al. “Andrei Nikolaevich Kolmogorov (1903–1987).” Bulletin of the London Mathematical Society 22, no. 1 (1990): 31–100.

Shiryaev, Albert. “A. N. Kolmogorov: Life and Creative Activities.” Annals of Probability 17 (1989): 866–944.

Paul Humphreys

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