Andrey Nikolaevich Kolmogorov-25 April 1903 – 20 October 1987) was a 20th-century Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionisti logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.
Andrey was born on April 12 1903 in Tambov, Where his mother stayed on the way home from Crimea to Yaroslavl.
Kolmogorov's mother-Maria Yakovlevna Kolmogorov (1871-1903), the daughter of the leader of the Uglich nobility, the Trustee of public schools of the Yaroslavl province Yakov Stepanovich Kolmogorov — died in childbirth.
Father-Nikolai Matveevich Kataev, an agronomist by education (graduated from the Moscow agricultural Institute), belonged to the party of right-wing esers, was exiled (from St. Petersburg) for participation in the populist movement in the Yaroslavl province, where he met Maria Yakovlevna; died in 1919 during the Denikin offensive. His paternal grandfather was a rural priest in the Vyatka province.
Father Kolmogorov's brother Ivan Matveevich Kataev (1875-1946) — historian, Professor, doctor of historical Sciences, graduate of Moscow University, author of works on Archeography, Russian history, history of Moscow, essays on Russian history. He is known as the organizer of historical science in Perm, Magnitogorsk. Ivan Matveevich Kataev is the author of a textbook on Russian history for high school in three parts. Published in 1907, the textbook was published in four editions and had a significant impact on the historical knowledge of students of the early XX century. Son of Ivan Matveevich-Ivan Ivanovich Kataev, Russian writer, cousin of Andrei Kolmogorov.
Andrey Nikolaevich Kolmogorov was brought up in Yaroslavl (the modern address — Sovetskaya St., house 3) by sisters of mother; one of them, Vera Yakovlevna Kolmogorova, officially adopted Andrey and in 1910 moved with it to Moscow for definition to a gymnasium. Aunts Andrey in the house organized school for children of different age who lived nearby, were engaged with them, for children the hand-written magazine "spring swallows"was published. It published creative works of students-drawings, poems, stories. It also appeared and "scientific work" Andrew-invented them arithmetic problems. Here the boy published his first work in mathematics at the age of five. Together with Andrew in the house of his grandfather spent his childhood Peter Savvich Kuznetsov, later known Soviet linguist.
At the age of seven Kolmogorov was assigned to a private gymnasium Repman, one of the few where boys and girls studied together. Andrew in those years reveals remarkable mathematical abilities. According to the writer Vladimir Gumilevskogo, teachers do not have time to teach him, Andrew learned math by himself in the "Encyclopedic dictionary Brockhaus and Efron,". There was also a passion for history, sociology.
In the first student years, in addition to mathematics, Kolmogorov was interested in the history of Russia and took an active part in the seminar on the history of Professor S. V. Bakhrushin. At the age of 17-18 years he has performed a serious scientific study on land relations in the Novgorod land, based on the materials of the scribe books of the XV—XVI centuries, the Results of the study were presented at the seminar Bakhrushin, but for a long time remained unpublished. Kolmogorov's manuscript, however, survived and was published in 1994.
"Andrei Nikolaevich himself repeatedly told his students about the end of his"career as a historian". When the work was reported to them at the seminar, the head of the seminar Professor S. V. Bakhrushin, approving the results, noted, however, that the conclusions of the young man can not claim finality, since "in historical science, each conclusion must be justified by several proofs." Later, talking about it, he added: "And I decided to go to science, in which the final conclusion was enough to prove one thing." History has lost a brilliant researcher forever, and mathematics has acquired him.
Academician V. L. Yanin »
In 1920, Kolmogorov entered the mathematical Department of Moscow University and in parallel to the mathematical Department of the Institute of Chemical technology. D. I. Mendeleev.
"Having decided to engage in serious science, I, of course, sought to learn from the best mathematicians. I was lucky enough to study With p. S. Uryson, P. S. Alexandrov, V. V. Stepanov and N. N. Luzin, who, apparently, should be considered primarily my teacher in mathematics. But they "found" me only in the sense that they evaluated the work I brought. "The purpose of life" teenager or young man will, I think, to find himself. The older ones can only help.
A. N. Kolmogorov »
In the first months Andrew passed the exams for the course. And as a second-year student, he gets the right to" scholarship": "...I got the right to 16 kilograms of bread and 1 kilogram of oil per month, which, according to the ideas of that time, meant already complete material well-being."He had free time, which was given to attempts to solve mathematical problems.
In 1921, Kolmogorov made the first scientific report to the mathematical circle, which refutes one improvisational statement N. N. Luzin, which he used in a lecture to prove Cauchy's theorem. At the same time Kolmogorov made his first discovery in the field of trigonometric series, and in early 1922 — on descriptive set theory, Luzin invited him to become his disciple — so Kolmogorov joined the ranks of Lusitania.
In June 1922, he was born. Kolmogorov constructed an example of a Fourier series diverging almost everywhere, followed by an example of such a series diverging at each point. These works, which became a complete surprise for specialists, brought the nineteen-year-old student worldwide fame.
Discussed in the mid-twenties everywhere, including in Moscow, the issues of the foundations of mathematical analysis and closely related research on mathematical logic attracted the attention of Kolmogorov almost at the beginning of his work. He took part in the discussions between the two main opposing methodological schools — formal axiomatic (D. Hilbert) and intuitionist (L. E. J. Brauer and G. Weil). Thus he received absolutely unexpected first-class result, having proved that all formulas of arithmetic deduced by rules of classical formal logic at a certain interpretation turn into deducable formulas of intuitionistic logic-his well-known work "about the principle of tertium non datur" is dated 1925th year. Kolmogorov preserved his deep interest in the philosophy of mathematics forever.
In the 1920s, the n. Kolmogorov was one of the first in the USSR to address the problems of mathematical linguistics. He proposed to determine the case on the basis of the semantics of language constructs, and gave a formal definition of a case as a class of congruence (later the definition of the case according to Kolmogorov became the starting point for research I. I. Revzin and V. A. Uspensky, who offered their interpretations of the category of case).
In 1924 Kolmogorov first became involved in the theory of probabilities. The law of large numbers is of great importance both for this field of mathematics and for its applications to natural science. Questions its justification for decades been the largest mathematics, but that Kolmogorov was able in 1928 to detect and prove the necessary and sufficient conditions for the validity of the law of large numbers.
Many years of close and fruitful cooperation connected him with Khinchin, who also began the development of the theory of probability in the 1920s. It became the area of joint activity of these scientists, who in 1925 successfully applied to it the methods of the theory of functions of a real variable. Kolmogorov and hinchin managed to find necessary and sufficient conditions for convergence of series whose terms are mutually independent random variables; in 1929 Kolmogorov, generalizing earlier results of hinchin, proved the law of repeated logarithm for sums of independent random variables under very wide conditions imposed on terms.
The science of" case " since the Time of p. L. Chebyshev has been a kind of Russian national science. Its success was multiplied by many Soviet mathematicians, but the modern form of probability theory was due to the axiomatization proposed by Andrei Nikolaevich in 1929 and finally in 1933. His work "the Basic concepts of the theory of probability", the first edition of which was published in 1933 in German (Grundbegriffe der Wahrscheinlichkeitsrechnung). Kolmogorov laid the Foundation of modern probability theory based on the theory of measure. In particular, in his 1933 monograph, he first formulated and proved the main theorems on infinite-dimensional distributions, which later formed a reliable Foundation for the logically flawless construction of the theory of random functions and sequences of random variables.
In 1930 Kolmogorov made a business trip to Germany and France. In göttingen-the mathematical Mecca of the beginning of the century — he meets with many outstanding colleagues, and above all-With D. Hilbert and R. Courant.
In 1933. Kolmogorov proved one of the most important nonparametric criteria of mathematical statistics — Kolmogorov's goodness of fit test, used to test the hypothesis of belonging of the sample to a certain distribution law. In the 1930s, Kolmogorov also laid the foundations of the theory of Markov random processes with continuous time. Turning to the issues of topology, he in 1935 at the same time with George. W. Alexander introduced the upper boundary operator and the notion of cohomology — one of the key concepts of modern topology.
Andrei Nikolaevich until the end of his days considered the theory of probability his main specialty, although the areas of mathematics in which he worked, you can count two dozen. But then only began the road Kolmogorov and his friends in science. They worked hard, but they didn't lose their sense of humor. Jokingly called equations "the equations derived with the poor", a special term such as finite difference, pereinachila in the "other limb", and the theory of probability in the "theory of trouble."
Norbert Wiener, the " father "of Cybernetics, testified:" ...Khinchin and Kolmogorov, two of the most prominent Russian specialists in probability theory, worked in the same field for a long time as I did. For more than twenty years we have been stepping on each other's heels: they proved a theorem that I was about to prove, then I was able to reach the finish line a little earlier than them."
And another recognition of Wiener, which he once made to journalists: "For the past thirty years, when I read the works of academician Kolmogorov, I feel that this is my thoughts. That's what I wanted to say every time.»
In 1931 Kolmogorov became a Professor of MSU, from 1935 to 1939 he was Director of the Institute of mathematics and mechanics of MSU. The degree of doctor of physical and mathematical Sciences was awarded to Kolmogorov in 1935 without defending a thesis (the degree of doctor of science was restored in the USSR in 1934, a number of major mathematicians were awarded; so, together with Kolmogorov, the degree of doctor of physical and mathematical Sciences without defending a thesis was awarded to Markov (ml.) and in The same year to L. V. Kantorovich).
In 1935, he was born. Kolmogorov founded the Department of probability theory of Moscow state University and until 1965 was its head. In 1954-1958 he simultaneously worked as Dean of the faculty of mechanics and mathematics.
January 29, 1939 at the age of 35 years Kolmogorov elected immediately (bypassing the title of corresponding member) full member of the Academy of Sciences of the USSR in the Department of mathematical and natural Sciences (mathematics). He became a member of the Presidium of the Academy and, at the suggestion of O. Schmidt, Secretary of the Department of physical and mathematical Sciences of the USSR.
Since 1936, Andrey has devoted much effort to the creation of large And Small Soviet Encyclopedias. He heads the mathematical Department of the Great Soviet Encyclopedia and himself writes many articles for both encyclopedias, as well as edits articles by other authors.
Shortly before the great Patriotic war Kolmogorov and Khinchin were awarded the Stalin prize for their work on the theory of random processes (1941).
On June 23, 1941 the extended meeting of the Presidium of the USSR Academy of Sciences was held. The adopted decision marked the beginning of a restructuring of the activities of scientific institutions. Now the main thing — the military theme: all the forces, all the knowledge-victory. Soviet mathematicians on the instructions Of the main artillery Department of the army conduct complex work in the field of ballistics and mechanics. Kolmogorov, using his research on the theory of probability, gives the definition of the most advantageous dispersion of shells when firing. After the war, Kolmogorov returned to peaceful research.
At the end of the thirties by Kolmogorov was interested in the problems of turbulence. In the works of 1941-1942 and 1962, he developed the theory of the so-called locally isotropic turbulence, which allowed to find out the local structure of the turbulent flow. At the same time, he introduced an important concept of the scale of turbulence, the use of which makes it possible, in particular, to assess the influence of suspended particles and polymer solutions on the development of turbulence. In 1946 Kolmogorov organized a laboratory of atmospheric turbulence at the Geophysical Institute of the USSR Academy of Sciences.
In parallel with the work on this problem Kolmogorov continues to be successful in many areas of mathematics-research on random processes, algebraic topology, etc.
In the late 1940s, An Kolmogorov was the first lecturer in the course of theory of functions and functional analysis ("Analysis III") at the faculty of mechanics and mathematics of Moscow state University. Together with S. V. Fomin he wrote the textbook "Elements of the theory of functions and functional analysis", which has passed seven editions (7th ed. - Moscow: Fizmatlit, 2012), as well as translated into foreign languages: English, French, German, Spanish, Japanese, Czech, dari.
He read at the international Congress of mathematics in 1954 in Amsterdam, the report "General theory of dynamical systems and classical mechanics" became a world-class event.
In mathematical logic Kolmogorov in 1953 proposed a new definition of the concept of the algorithm, in which both the problem and its solution are represented as a one-dimensional topological complex, and each step of the algorithmic process is presented as the processing of one complex into another according to certain rules of processing. This definition is very General, and in its terms it is possible to present algorithms in the sense of other known definitions, and many common properties of algorithmic functions are quite simple to prove, if we proceed from the definition of the algorithm by Kolmogorov. At the same time V. Uspensky proved that the definition of Kolmogorov is equivalent to the definition of a computable function as partially recursive.
In the theory of dynamic systems of Kolmogorov, using the procedure of successive substitutions ascending To S. NEWCOM, developed methods of integration of perturbed Hamiltonian systems and published in 1954 the theorem on invariant variables, generalized in the future by V. I. Arnold and Y. Moser, which led to the creation of the Kolmogorov — Arnold — Moser theory (KAM-theory) — one of the first theories of chaos.
Kolmogorov and J. G. Sinai introduced a new invariant to ergodic theory (Kolmogorov — Sinai entropy).
In 1956 Kolmogorov obtained an unexpected and very important result in the theory of functions of a real variable: he proved that for n>3 any continuous function of n variables can be represented as a superposition of continuous functions of fewer variables. Somewhat later, V. I. Arnold obtained a similar result in the case of n=3.
In a discussion on " can a machine think?"Kolmogorov took a rather radical position, stating in 1964 in one of his articles that" the fundamental possibility of creating full-fledged living beings, built entirely on discrete (digital) mechanisms of information processing and management, does not contradict the principles of materialistic dialectics»
By the mid-1960s, the leadership of the Ministry of education of the USSR came to the conclusion that the system of teaching mathematics in the Soviet high school is in deep crisis and needs reforms. It was recognized that only outdated mathematics was taught in secondary schools, and its latest achievements were not covered. Modernization of the system of mathematical education was carried out by the Ministry of education of the USSR with the participation of the Academy of pedagogical Sciences and the Academy of Sciences of the USSR. The leadership Of the Department of mathematics of the USSR Academy of Sciences recommended to work on the modernization of academician. Kolmogorov, who played a leading role in these reforms. Under the guidance of A. N. Kolmogorov developed programs, created new repeated subsequently, textbooks on mathematics for high school: geometry textbook, textbook algebra and fundamentals of analysis. The results of this activity of the academician were evaluated ambiguously and continue to cause a lot of controversy. About textbooks geometry Alexandrov wrote
"There is hardly anything more harmful for spiritual — mental and moral — development than to teach a person to pronounce words, the meaning of which he does not really understand and, if necessary, is guided by other concepts. »
Also, the Dean of mehmat MSU read out the definition of the direction of The Kolmogorov textbook on geometry from the rostrum of the Supreme Soviet of the USSR.
In 1966 Kolmogorov was elected a full member of the Academy of pedagogical Sciences of the USSR. In 1963, he was born. Kolmogorov is one of the initiators of the boarding school at MSU and he begins to teach there. In 1970, together with academician I. K. Kikoin, A. N. Kolmogorov creates the magazine "Kvant".
Work in the "Quantum" was not for. Kolmogorov casual hobby. Creation of magazine for young people was an integral part of a broad program of improving math education that Andrei implemented throughout his creative life. This program also included the reform of mathematical education, and the creation of specialized physical and mathematical schools for children interested in mathematics and physics, and the holding of mathematical Olympiads, and the publication of special literature, and much, much more.
One of the innermost desires of Andrei Nikolaevich was to attract children living far away from the leading scientific centers to scientific creativity. They was founded the 18th of physico-mathematical boarding school (now school. A. N. Kolmogorov), the same purpose, according to Andrei Nikolayevich, had to chase and the magazine "Kvant". He had to give the chance to the school student where he lived, to get acquainted with fascinating physical and mathematical materials, to induce him to occupations by science
According to the testimony of V. Ouspensky, Kolmogorov belonged to the type of researchers-encyclopedists, able to make a fresh stream in any branch of human knowledge.
Kolmogorov made a significant contribution to the study of poetry: his name is associated with the revival in the 1960s.on a new basis of the use of mathematical methods in the study of verse. He wrote more than 10 works, including the rhythm of Mayakovsky's poetry, the Dolnik of modern Russian poetry, the study of meter and its rhythmic variants. Kolmogorov was the official opponent at the doctoral dissertation of the scholar Mikhail Gasparov.
In 1976, A. N. Kolmogorov founded the Department of mathematical statistics of Department of mechanics and mathematics of Moscow state University and 1980, was its head. In 1980, he became head of the Department of mathematical logic and remained in this position until his death in 1987. Kolmogorov also taught physics and mathematics school-internat 18 at Moscow state University (now — sunts Moscow state University named after A. N. Kolmogorov), Chairman of the Board of Trustees where he has been since 1963.
"I belong to those extremely desperate Cybernetics who do not see any fundamental limitations in the cybernetic approach to the problem of life and believe that it is possible to analyze life in its entirety, including human consciousness, by means of Cybernetics. Progress in understanding the mechanism of higher nervous activity, including the highest manifestations of human creativity, in my opinion, does not diminish anything in the value and beauty of human creative achievements.
Being engaged with some success, and sometimes with benefit, quite a wide range of practical applications of mathematics, I remain, basically, a pure mathematician. Admiring mathematicians who have become major representatives of our technology, fully appreciating the importance of computers and Cybernetics for the future of mankind, I still think that pure mathematics in its traditional aspect has not yet lost its place of honor among other Sciences. Disastrous for it could be only overly sharp separation of the mathematicians into two streams: one an abstract of the latest cultivated branches of mathematics, not focusing clearly in their relationship with gave birth to their real world, others are taking "applications", not going to an exhaustive analysis of their theoretical foundations. Therefore, I would like to emphasize the legitimacy and dignity of the position of mathematician, who understands the place and role of his science in the development of natural Sciences, technology, and the whole human culture, but quietly continues to develop "pure mathematics" in accordance with the internal logic of its development.
Mathematics is great. One person is not able to study all its ramifications. In this sense, specialization is inevitable. But at the same time mathematics is a single science. More and more connections arise between its sections, sometimes in unexpected ways. Some sections serve as tools for other sections. Therefore, the closure of mathematicians in too narrow a range should be disastrous for our science.
The situation is facilitated by the fact that the work in mathematics is, in principle, collective on. There must be a certain number of mathematicians who understand the mutual connections between the most different areas of mathematics. On the other hand, you can work with great success and in some very narrow branch of mathematics. But in this case it is necessary, at least in General terms, to understand the links between their special field of study and related areas, to understand that, in essence, scientific work in mathematics-collective work.