World of Mathematics on Andrei Nikolaevich Kolmogorov
Andrei Nikolaevich Kolmogorov made major contributions to almost all areas of mathematics and many fields of science and is considered one of the 20th century's most eminent mathematicians. He was the founder of modern probability theory, having formulated its axiomatic foundations and developed many of its mathematical tools. Kolmogorov also helped make advances in many applied sciences, from physics to linguistics. A great teacher, he did much to keep the Soviet Union in the forefront of research in theoretical and applied mathematics and was responsible for reforms in mathematics education at the elementary and high-school levels.
Kolmogorov was born in the town of Tambov in central Russia on April 25, 1903. His father, Nikolai Kataev, became a professional agriculturalist and was killed during World War I. His mother, Mariya Yakovlevna Kolmogorova, was not formally married to his father and died during his birth. Her sister, Vera Yakovlevna Kolmogorova, adopted and raised the boy in the family's home village of Tunoshna. As a child, young Kolmogorov and his friends attended a school run by his two aunts. At the age of five, he made his first mathematical discovery by noticing the pattern that 1=12, 1+3=2 2, 1+3+5=32, etc.
In 1920, at the age of 17, Kolmogorov enrolled in Moscow University. To help support himself while he attended the university, he worked as a secondary school teacher. He took an active role in the school, and he is said to have been more proud of that work than of the honors he garnered for his own academic progress. Within two years, Kolmogorov had completed a study in the theory of operations on sets, which was eventually published in 1928. A second project he also completed in 1922 brought immediate recognition: He formulated the first known example of an integrable function with a Fourier series that diverged almost everywhere (he soon extended that result to everywhere). The international mathematics community took notice of the bright 19-year-old. During his years as a university student, he published 18 mathematical papers including the strong law of large numbers, generalizations of calculus operations, and discourses in intuitionistic logic. In 1925, Kolmogorov received a doctoral degree from the department of physics and mathematics and became a research associate at Moscow University. At the age of 28, he was made a full professor of mathematics; two years later, in 1933, he was appointed director of the university's Institute of Mathematics. In 1942, Kolmogorov married Anna Dmitrievna Egorova.
While he was still a research associate, Kolmogorov published a paper, "General Theory of Measure and Probability Theory," in which he gave an axiomatic representation of some aspects of probability theoryon the basis of measure theory. His work in this area, which a younger colleague once called the "New Testament" of mathematics, was fully described in a monograph that was published in 1933. The paper was translated into English and published in 1950 as Foundations of the Theory of Probability. Kolmogorov's contribution to probability theory has been compared to Euclid's role in establishing the basis of geometry. He also made major contributions to the understanding of stochastic processes (involving random variables), and he advanced the knowledge of chains of linked probabilities.
Kolmogorov developed many applications of probability theory, creating a powerful technique for using probability to make observa- tion-based predictions in the face of randomness and researching statistical inspection methods for mass production. One of the applications that Kolmogorov developed is known as reaction-diffusion theory, which deals with the manner in which an event spreads through a given population. It is now used to study the dispersion of epidemics, cultural changes, effects of advertising, and a variety of other situations in such fields as biology and chemistry. He was largely responsible for demonstrating the applicability of the emerging fields of probability and statistics on substantial problems in science and engineering. In fact, Kolmogorov contributed to the war effort in the 1940s by solving statistical problems relating to artillery fire and by applying stochastic theory to suggest the most effective placement of barrage balloons for protecting Moscow from Nazi bombing assaults.
An appointment as Chair of Theory of Probability at Moscow University in 1937 served as official recognition of Kolmogorov's achievements in probability theory. Communication with the West was sporadic, however, and it was not until the late 1950s that Western mathematicians discovered that Kolmogorov had already determined the nature of many issues in probability theory that they were still working to discover. In 1939, at the age of 36, Kolmogorov became one of the youngest full members elected to the Soviet Academy of Sciences. He was later appointed academician-secretary of the Academy's department of physical and mathematical sciences. These honors were in recognition not only of his work in probability theory but of his contributions to other areas of theoretical and applied mathematics.
Kolmogorov made significant contributions to set theory, measure theory, integration theory, topology, functional analysis, constructive logic, differential equations, and the theory of approximation of functions. Among his many accomplishments in applied fields, Kolmogorov developed important results in such areas as biological statistics, econometrics, mathematical linguistics, and the theory of fluid turbulence. His work in fluid turbulence was so profound that in 1946 he was chosen to head the Turbulence Laboratory at the Academy Institute of Theoretical Geophysics. He continued to work in this field for many years, sailing around the world in 1970-1972 to study ocean turbulence. He helped construct the Kolmogorov-Arnold-Moser (KAM) theorem, which is used to analyze stability in dynamic systems. Kolmogorov introduced the concept of entropy (a theoretical measure of unavailable energy in a thermodynamic system) as a measure of disorder. In the first detailed solution of the three-body problem, which had been proposed by Isaac Newton, Kolmogorov analyzed the interactions of two celestial bodies orbiting in the same plane, one with an elliptical orbit and the other with a circular path.
In an obituary published in Physics Today, V. I. Arnold of the Steklov Mathematical Institute wrote that "Kolmogorov considered his most difficult achievements to be his work from 1955 to 1957 on the 13th [David] Hilbert problem." The problem involves finding a way to represent a function of many variables in terms of a combination of functions having fewer variables.
In a remembrance of Kolmogorov published in Statistical Science, A. N. Shiryaev wrote that "One sensed that he had continuously intensive brain activity." His active intellect led him to investigate questions in a wide range of mathematical fields as well as a number of applied subjects including meteorology, hydrodynamics, celestial mechanics, genetics, history, and linguistics. Shiryaev wrote, "According to his own words, Kolmogorov had a lively interest in a problem only until it became clear what the answer should be. As soon as the picture became clear he tried to avoid writing down the results and proofs; he would look for someone else to take over." Indeed, by the time Kolmogorov solved a problem, he would have identified other topics to investigate. Shiryaev quoted the mathematician A. Ya. Khinchin as saying, "The most important and most fascinating feature of [Kolmogorov] as a mathematician is the wealth of his ideas. Each sentence of his about any work could become the basis for a Ph.D. dissertation." Also speaking at a 50th birthday tribute to Kolmogorov, Aleksandr Gelfondsaid, "The fact that mathematics is viewed as a unified discipline is due to a large extent to Kolmogorov."
Kolmogorov was also actively interested in mathematical education in the U.S.S.R., working as the chairman of the Academy of Sciences Commission of Mathematical Education. He played a pivotal role in overhauling the teaching of mathematics during the 1960s, and his leadership in mathematics education for secondary schools and universities helped move the U.S.S.R. to the forefront of mathematics internationally during the following decades. In fact, being of the opinion that no mathematician could possibly do meaningful research after the age of 60, Kolmogorov retired in 1963 and spent the following 20 years teaching high school. During his final years, he compiled his collected works; an English version was published in the United States in 1991. He died in Moscow on October 20, 1987, at the age of 84.
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