The national treasure of Russia is the mathematician Kolmogorov.

Andrei Nikolaevich Kolmogorov - (1903-87), Russian mathematician, founder of scientific schools on probability theory and function theory, academician of the USSR Academy of Sciences (1939), Hero of Socialist Labor (1963).

Fundamental works of Andrei Kolmogorov on the theory of functions, mathematical logic, topology, differential equations, functional analysis and especially on probability theory (axiomatic justification, theory of random processes) and information theory. Lenin Prize (1965), State Prize of the USSR (1941).

Andrei's aunts organized a school in their house for children of different ages who lived nearby, taught them - a dozen children - according to the recipes of the latest pedagogy. For the children, a handwritten magazine "Spring Swallows" was published. It published creative works of students - drawings, poems, stories. Andrey's "scientific works" also appeared in it - arithmetic problems invented by him. Here the boy published his first scientific work in mathematics at the age of five. True, it was only a well-known algebraic regularity, but the boy himself noticed it without outside help!

When, in 1920, Andrei Kolmogorov began to think about entering an institute, an eternal question arose before him: what should he devote himself to, what business? He is attracted to the mathematical department of the university, but there is also a doubt: here is pure science, and technology is, perhaps, a more serious matter. Here, for example, is the metallurgical faculty of the Mendeleev Institute! A real man's business, moreover, promising. Andrey decides to do both here and there. But it soon becomes clear to him that pure science is also very relevant, and he makes a choice in its favor.

In 1920 Andrei entered the mathematical department of Moscow University. “Having decided to engage in serious science, I, of course, strove to learn from the best mathematicians,” the scientist later recalled. - I was lucky to study with PS Uryson, PS Alexandrov, VV Stepanov and NN Luzin, who, apparently, should be considered par excellence my teacher in mathematics. But they "found" me only in the sense that they evaluated the works I brought. It seems to me that a teenager or a young man should find the "goal of life" for himself. Seniors can only help.

In the very first months, Andrei Kolmogorov passed the exams for the course. And as a second-year student, he gets the right to a "stipend": "... I got the right to 16 kilograms of bread and 1 kilogram of butter per month, which, according to the ideas of that time, already meant complete material well-being." Now there is free time. It is given to attempts to solve already set mathematical problems. The lectures of Nikolai Nikolaevich Luzin, a professor at Moscow University, were, according to contemporaries, an outstanding event. Luzin never had a prescribed form of presentation. And his lectures by no means could serve as a role model. He had a rare sense of audience. He, like a real actor, performing on the theater stage and perfectly feeling the reaction of the audience, had constant contact with students.

The professor knew how to bring students into contact with his own mathematical thought, revealing the mysteries of his scientific laboratory. Invited to joint spiritual activity, to co-creation. And what a holiday it was when Luzin invited students to his home for the famous “Wednesdays”! Conversations over a cup of tea about scientific problems ... However, why necessarily about scientific ones? There were plenty of topics for conversation. He knew how to ignite the youth with the desire for a scientific achievement, instill faith in their own strengths, and through this feeling another came - an understanding of the need for full dedication to their beloved work. Kolmogorov first attracted the attention of a professor at a lecture. Luzin, as always, led the classes, constantly addressing the audience with questions and assignments. And when he said: "Let's build a proof of the theorem based on the following assumption..." Andrey Kolmogorov raised his hand in the audience: "Professor, it is wrong..." The question "why" was followed by a short answer from the first-year student. Satisfied Luzin nodded: "Well, come to the circle, report to us your thoughts in more detail." “Although my achievement was rather childish, it made me famous in the Lusitania,” Andrei Nikolaevich recalled.

But a year later, the serious results obtained by the eighteen-year-old sophomore Andrei Kolmogorov attracted the real attention of the “patriarch”. With some solemnity, Nikolai Nikolaevich invites Kolmogorov to come on a certain day and hour of the week, intended for the students of his course. Such an invitation, according to the concepts of Lusitania, should be regarded as conferring the honorary title of student. As a recognition of ability. Over time, Kolmogorov's attitude towards Luzin changed. Under the influence of Pavel Sergeevich Alexandrov, also a former student of Luzin, he took part in the political persecution of their common teacher, the so-called Luzin case, which almost ended in repressions against Luzin. With Aleksandrov himself, Kolmogorov was bound by friendly ties until the end of his life.

Andrei Kolmogorov is the greatest Russian mathematician of the 20th century, the creator of modern probability theory, the author of classical results in the theory of functions, in mathematical logic, topology, the theory of differential equations, functional analysis, in the theory of turbulence, the theory of Hamiltonian systems.

The schools created by Kolmogorov in probability theory, function theory, functional analysis and the theory of Hamiltonian systems determined the development of these areas of mathematics in the 20th century. In the history of Russian science, his name stands next to the names of scientists who glorified Russia with their whole lives.

Andrei Nikolaevich was born on April 25, 1903 in Tambov. Since 1920 to 1925 he studies at Moscow University. While still a student, in 1922 he constructed a Fourier series that diverges almost everywhere, which brings him worldwide fame.

In 1931 Andrei Kolmogorov became a professor at Moscow State University. In 1933 he was appointed director of the Institute of Mathematics and Mechanics at Moscow State University. In 1935, at the Faculty of Mechanics and Mathematics of Moscow State University, he founded the Department of Probability Theory (which he headed until 1966).

In 1939 A.N. Kolmogorov was elected a full member of the USSR Academy of Sciences and he became (until 1942) Academician-Secretary of the Department of Physical and Mathematical Sciences. In the late 1930s and early 1940s Andrei Kolmogorov began to take an interest in the problems of turbulence and in 1946 organized the laboratory of atmospheric turbulence at the Institute of Theoretical Geophysics of the USSR Academy of Sciences. Since 1936, Andrey Nikolaevich has been devoting a lot of energy to the creation of the Great and Small Soviet Encyclopedias. He heads the mathematical department and writes many articles for encyclopedias.

In 1960, Kolmogorov created an interfaculty laboratory of probabilistic and statistical methods (which he headed from 1966 to 1976), one of the main tasks of which was the widespread use of modern methods of probability theory and mathematical statistics in natural science and humanitarian research. The decision to create this laboratory A.N. Kolmogorov accepted after his return from India, where he was struck by the scope of work in the field of applied statistics in various branches of knowledge. At that time in India, the institute headed by Mahalanobis employed about 2,000 people! There was nothing like that at that time (and now too!) in our country. Initially, about 20 employees worked in the laboratory, and by the time it was closed after the death of the rector of Moscow State University I.G. Petrovsky, there were already more than 130 people.

Very interesting about this period of life A. Kolmogorova writes in the book "Rope Walker" V.V. Nalimov, who worked as his deputy in this laboratory for many years. Here is one quote from this book. “The question posed above could be reformulated as follows: what should be the mathematical preparedness of a non-mathematician who wants to use probabilistic-statistical methods in his work? users.The danger of this kind of activity lies in the fact that applied mathematics still always remains a deductive science.A model cannot be obtained directly from experimental data without relying on the premises introduced by the researcher.For example, one must clearly understand that the results of cluster analysis always carry in themselves some uncertainty - they depend on the metric of the space constructed by the researcher (that is, on the choice of scales in which measurements are presented).

Or another example: you need to be clearly aware that the estimates of the regression coefficients in real problems of the so-called passive (i.e., unplanned) experiment always still turn out to be biased due to the fact that you can never include in consideration all the independent variables responsible for the phenomenon under study . It is possible to pose a broader problem: are the initial provisions of Fisher's concept of mathematical statistics always adequate to the situation under study? I have repeatedly discussed this topic with Andrey Nikolaevich (discussions on this topic flare up from time to time in scientific journals). Considering this topic, I proposed to introduce a new interdisciplinary specialization. It was about training graduates of a mixed profile at the University - say, mathematically oriented biologists, psychologists, etc. The ratio of the studied disciplines - mathematical and subject could be 1:1. A specialist of such a profile could act as a consultant, supporting the process of mathematization of such scientific disciplines, which traditionally developed without relying on mathematical knowledge, at the proper level. In many foreign countries, such a process has long begun. There, such a specialty as biometrics acquired the right to exist (In 1985, the first European Conference on Biometrics, organized by the International Biometric Society, took place in Hungary.

This Society has over 6,500 members from 70 countries. Our country is still not included in it (nothing has changed as of 2003 - VL). At the conference mentioned above, there were two representatives from us, and about thirty from the GDR). Specialists of this profile act not only as consultants, but also as organizers of large interclinical and interlaboratory studies. A few years ago, the training of specialists in biometrics began in the former GDR (University of Rostock, head of the program - Professor D. Rasch). In those years, Andrei Nikolaevich supported my proposal. His letter has been preserved, containing a detailed discussion of the mathematical component of such a program. But to realize this plan still failed. The rector, I. G. Petrovsky, did not support him. They reacted sharply negatively to him in the then Ministry of Higher Education. One of the leading employees of this Ministry irritably remarked: “What are we going to write in the diploma then?” Rigid regulation dominated everything, including the structure of university education. Now it has become clear that the training of interdisciplinary specialists can be justified from other, perhaps more serious, positions. Experience shows that the application of mathematics in such sciences as biology, psychology, linguistics and sociology should not be limited to solving only external tasks of an operator nature (data processing, experiment planning). Here the task of creating your own mathematicized language for building axiomatized theories is brewing, by analogy with how it happened in physics.

Essentially mathematized, as it seems to me, should become a language for creating a theory of meanings, just like, say, a language in which a theory of the manifestation of the living could be built. Understanding the role of field concepts in modern physics, one would like to think about the possibility of introducing axiomatized concepts of biological (morphophysiological) and semantic fields. But it is difficult to imagine in advance what branches of mathematics these ideas will be based on. We can only say one thing - here we need thinkers who know both the subject area and mathematics in a wide disclosure. But working in an interdisciplinary field is dangerous - you can always get hit by representatives of monodisciplinary knowledge: their local erudition will be higher than the erudition of a multidisciplinary researcher. My experience of more than 40 years of work in applied probabilistically oriented mathematics has shown me that both mathematicians and representatives of specific sciences try not to go far beyond their original education.

Mentally referring to past conversations with Andrei Kolmogorov, he would join in the search for ways to train general scientists, I think that in our days - the days of the formation of the new - he would join in the search for ways to train general scientists. A.N. himself said more than once that he was not only a mathematician, but also a naturalist. In 1976, the Department of Mathematical Statistics was opened at Moscow State University, which A.N. Kolmogorov was in charge until 1979. From 1980 until the end of his life Andrei Nikolaevich was in charge of the Department of Mathematical Logic.

In 1953 Andrei Kolmogorov was elected an honorary member of the Moscow Mathematical Society, and from 1964 to 1966 and from 1973 to 1985 he was its President.

In different years Andrei Kolmogorov was a member of the editorial boards journals "Mathematical Collection", "Reports of the Academy of Sciences of the USSR", "Advances in Mathematical Sciences". From 1946 to 1954 and from 1983 until the day of his death Andrei Nikolaevich was the editor-in-chief of Uspekhi Mathematicheskikh Nauk.

In 1956, Kolmogorov founded the journal "Theory of Probability and its Applications" and, from the first issue of 1956, was the editor-in-chief of this journal, being the initiator of the creation of the physical and mathematical journal for youth "Kvant", he has been leading the journal since its inception (1970). ) and until the end of his days was the first deputy chief editor and led the mathematical section of this journal.

Andrei Kolmogorov was the founder and first head of the Mathematics and Mechanics Editorial Board at the Foreign Literature Publishing House (now the Mir Publishing House). In 1931 his fundamental article "On Analytical Methods in Probability Theory" was published, and in 1933 his monograph "Basic Concepts of Probability Theory" was published. This completes the task of constructing the theory of probability as an integral mathematical theory. A.N. Kolmogorov made a significant contribution to the development of algebraic topology (here he introduced one of the central concepts of this theory - the concept of cohomology), the theory of dynamical systems (where he introduced a new invariant "entropy"), the theory of complexity of constructive objects, where he proposed the ideas of measuring the complexity of an object have found diverse applications in information theory, probability theory, and algorithm theory.

Andrei Kolmogorov was one of the most prominent representatives of modern mathematics in the broadest sense of the word, including applied mathematics. His name stands next to the names of Poincaré and Gilbert. This position of Andrei Nikolaevich in science enjoys undeniable recognition in the international scientific world, and it finds its external expression, in particular, in the fact that A.N. Kolmogorov holds the first place among all Soviet mathematicians in terms of the number of foreign academies and scientific communities that have elected him as their member, as well as universities that have made him their honorary doctorate.

Andrey Kolmogorov was a member of almost all the most authoritative scientific communities in the world:

Honorary doctorate from the University of Paris (1955)
- foreign member of the Polish Academy of Sciences (1956)
- honorary member of the Royal Statistical Society (Great Britain, 1956)
- Member of the International Statistical Institute (1957)
- honorary member of the American Academy of Arts and Sciences in Boston (1959)
- Member of the German Academy of Naturalists "Leopoldina" (1959)
- honorary doctorate from Stockholm University (1960)
- Foreign Member of the American Philosophical Society in Philadelphia (1961)
- Honorary Member of the Indian Statistical Society in Calcutta (1962)
- honorary member of the American Meteorological Society (1962)
- Honorary Member of the Indian Mathematical Society (1962)
- foreign member of the Royal Netherlands Academy of Sciences (1963)
- Foreign Fellow of the Royal Society of London (1964)
- honorary member of the Romanian Academy (1965)
- Honorary Member of the Hungarian Academy of Sciences (1965)
- Foreign Member of the US National Academy of Sciences (1967)
- foreign member of the Paris Academy of Sciences (1968)
- honorary member of the International Academy of the History of Science (1977)
- foreign member of the Academy of Sciences of the GDR (1977)
- foreign member of the Society of the Order "Pour la Merit" Germany (1977)
- Member of the Finnish Academy of Sciences (1985).

In world science, to celebrate achievements in those areas that are not covered by the Nobel Prizes, the Balzan Prizes were established. In 1963, the first award of the Baltsanov Prize in mathematics took place, and A. N. Kolmogorov became its laureate. This was the highest assessment of A. N. Kolmogorov's contribution to world science.

The International Prize named after N.I. Lobachevsky of the Academy of Sciences of the USSR was awarded in 1986. Andrei Kolmogorov was a laureate of the Lenin Prize (1965, for his work on classical mechanics), the State (Stalin) Prize (1941, for his work on the theory of stochastic processes), the Prize to them. Chebyshev Academy of Sciences of the USSR (1949). He was awarded the title of Hero of Socialist Labor (1963), he was awarded seven orders of Lenin, other orders and medals of the USSR, as well as the Hungarian Order of the Banner, a medal to them. Helmholtz of the Academy of Sciences of the GDR, the gold medal of the American Meteorological Society.

Many students Andrey Kolmogorov became prominent scientists in various fields of science, among them - V. I. Arnold, I. M. Gelfand, M. D. Millionshchikov, Yu. V. Prokhorov, A. M. Obukhov, A. S. Monin, A. N Shiryaev. A. Kolmogorov himself said: “I was lucky to have talented students. Many of them, having started working with me in some area, then moved on to a new topic and, completely independently of me, received wonderful results. I’ll say as a joke that at present one of my students controls the earth’s atmosphere (A. M. Obukhov), and the other controls the oceans (A. S. Monin).

Andrey Nikolaevich Kolmogorov - quotes

I have always believed that the truth is the main thing.

Dealing with some success, and sometimes with profit, in a fairly wide range of practical applications of mathematics, I remain, in the main, a pure mathematician. While admiring the 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. Only an excessively sharp stratification of mathematicians into two currents could be disastrous for it: some cultivate abstract new branches of mathematics, not clearly orienting themselves in their connections with the real world that gave birth to them, others are busy with "applications", not ascending to an exhaustive analysis of their theoretical foundations. Therefore, I would like to emphasize the legitimacy and dignity of the position of a mathematician who understands the place and role of his science in the development of natural sciences, technology, and indeed the entire human culture, but calmly continues to develop "pure mathematics" in accordance with the internal logic of its development.

The math 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 the most unforeseen way. Some sections serve as tools for other sections. Therefore, the closure of mathematicians in too narrow terms must be disastrous for our science. The situation is facilitated by the fact that work in the field of mathematics is, in principle, collective. There must be a number of mathematicians who understand the interrelationships between the most diverse areas of mathematics. On the other hand, one can work with great success in some very narrow branch of mathematics. But in this case, one must also, at least in general terms, understand the connections between one's special field of study and related fields, understand that, in essence, scientific work in mathematics is a collective work.

Humanity has always seemed to me in the form of many lights wandering in the fog, which only vaguely feel the radiance scattered by all others, but are connected by a network of clear fiery threads, each in one, two, three ... directions. And the emergence of such breakthroughs through the fog to another spark is quite reasonable to call "MIRACLE".

An outstanding Soviet mathematician, Doctor of Physical and Mathematical Sciences, Professor at Moscow State University (1931), Academician of the USSR Academy of Sciences (1939). Kolmogorov is one of the founders of modern probability theory, he obtained fundamental results in topology, mathematical logic, turbulence theory, algorithm complexity theory and a number of other areas of mathematics and its applications.

early years

Kolmogorov's mother, Maria Yakovlevna Kolmogorova (1871-1903), died in childbirth. Father - Nikolai Matveevich Kataev, an agronomist by education (graduated from the Petrovsky (Timiryazev) Academy), died in 1919 during the Denikin offensive. The boy was adopted and raised by his mother's sister, Vera Yakovlevna Kolmogorova. Andrei's aunts organized a school in their house for children of different ages who lived nearby, taught them - a dozen children - according to the recipes of the latest pedagogy. For the children, a handwritten magazine "Spring Swallows" was published. It published creative works of students - drawings, poems, stories. Andrey's "scientific works" also appeared in it - arithmetic problems invented by him. Here the boy published his first scientific work in mathematics at the age of five. True, it was only a well-known algebraic regularity, but the boy noticed it himself, without outside help!

At the age of seven, Kolmogorov was assigned to a private gymnasium. It was organized by a circle of Moscow progressive intelligentsia and was constantly under threat of closing.

Andrei already in those years showed remarkable mathematical abilities, but still it is too early to say that his further path has already been decided. There was also a passion for history and sociology. At one time he dreamed of becoming a forester. “In 1918-1920, life in Moscow was not easy,” Andrei Nikolayevich recalled. - In schools, only the most persistent were seriously engaged. At this time, I had to leave for the construction of the Kazan-Yekaterinburg railway. Simultaneously with work, I continued to study on my own, preparing to take an external student for high school. Upon returning to Moscow, I experienced some disappointment: they gave me a certificate of graduation from school, without even bothering to take an exam.

university

In 1920 he entered the mathematical department of Moscow University. “Having decided to engage in serious science, I, of course, strove to learn from the best mathematicians,” the scientist later recalled. - I was lucky to study with PS Uryson, PS Alexandrov, VV Stepanov and NN Luzin, who, apparently, should be considered par excellence my teacher in mathematics. But they "found" me only in the sense that they evaluated the works I brought. It seems to me that a teenager or a young man should find the "purpose of life" for himself. The elders can only help.”

In the very first months, Andrei passed the exams for the course. And as a second-year student, he gets the right to a "stipend": "... I got the right to 16 kilograms of bread and 1 kilogram of butter per month, which, according to the ideas of that time, already meant complete material well-being." Now there is free time. It is given to attempts to solve already set mathematical problems.

The lectures of Nikolai Nikolaevich Luzin, a professor at Moscow University, were, according to contemporaries, an outstanding event. Luzin never had a prescribed form of presentation. And his lectures by no means could serve as a role model. He had a rare sense of audience. He, like a real actor, performing on the theater stage and perfectly feeling the reaction of the audience, had constant contact with students. The professor knew how to bring students into contact with his own mathematical thought, revealing the mysteries of his scientific laboratory. Invited to joint spiritual activity, to co-creation. And what a holiday it was when Luzin invited students to his home for the famous “Wednesdays”! Conversations over a cup of tea about scientific problems ... However, why necessarily about scientific ones? There were plenty of topics for conversation. He knew how to ignite the youth with the desire for a scientific achievement, instill faith in their own strengths, and through this feeling another came - an understanding of the need for full dedication to their beloved work.

Kolmogorov first attracted the attention of a professor at a lecture. Luzin, as always, led the classes, constantly addressing the audience with questions and assignments. And when he said: "Let's build a proof of the theorem based on the following assumption..." Andrey Kolmogorov raised his hand in the audience: "Professor, it is wrong..." The question "why" was followed by a short answer from the first-year student. Satisfied Luzin nodded: "Well, come to the circle, report to us your thoughts in more detail." “Although my achievement was rather childish, it made me famous in the Lusitania,” Andrei Nikolaevich recalled.

Over time, Kolmogorov's attitude towards Luzin changed. Under the influence of Pavel Sergeevich Alexandrov, also a former student of Luzin, he took part in the political persecution of their common teacher, the so-called Luzin case, which almost ended in repressions against Luzin. With Aleksandrov himself, Kolmogorov was bound by friendly ties until the end of his life.

Kolmogorov's first publications were devoted to the problems of descriptive and metric theory of functions. The earliest of these appeared in 1923. Discussed in the mid-twenties everywhere, including in Moscow, questions of the foundations of mathematical analysis and closely related research in mathematical logic attracted Kolmogorov's attention almost at the very beginning of his work. He took part in discussions between the two main opposing methodological schools at that time - formal-axiomatic (D. Hilbert) and intuitionistic (L. E. Ya. Brouwer and G. Weyl). In doing so, he obtained a completely unexpected first-class result, proving in 1925 that all known sentences of classical formal logic, under a certain interpretation, turn into sentences of intuitionistic logic. Kolmogorov retained a deep interest in the philosophy of mathematics forever.

Of particular importance for the application of mathematical methods to the natural sciences and practical sciences was the law of large numbers. To find the necessary and sufficient conditions under which it takes place - that is what the desired result was. The leading mathematicians in many countries have been unsuccessfully trying to obtain it for decades. In 1926, these conditions were obtained by graduate student Kolmogorov.

Many years of close and fruitful cooperation connected him with A. Ya. Khinchin, who at that time began to develop problems in the theory of probability. It has become an area of joint activity of scientists. Since the time of Chebyshev, the science "about the case" has been, as it were, a Russian national science. Many Soviet mathematicians multiplied its successes, but the theory of probability received its modern form thanks to the axiomatization proposed by Andrei Nikolaevich in 1929 and finally in 1933.

Until the end of his days, Andrei Nikolaevich considered probability theory to be his main specialty, although the areas of mathematics in which he worked can be counted in a good two dozen. But then the path of Kolmogorov and his friends in science was just beginning. They worked hard, but did not lose their sense of humor. Partial differential equations were jokingly called "equations with unfortunate derivatives", such a special term as finite differences was changed into "different finitenesses", and probability theory - into "trouble theory".

Norbert Wiener, the “father” of cybernetics, testified: “... Khinchin and Kolmogorov, two of the most prominent Russian specialists in probability theory, worked for a long time in the same field as me. For more than twenty years, we stepped on each other's heels: either they proved the theorem that I was about to prove, or I managed to reach the finish line a little earlier than them.

And one more confession of Wiener, which he once made to journalists: “For thirty years now, when I read the works of Academician Kolmogorov, I feel that these are my thoughts. This is always what I myself wanted to say.

Professor

In 1930, Kolmogorov became a professor at Moscow State University, from 1933 to 1939 he was director of the Institute of Mathematics and Mechanics of Moscow State University, for many years he headed the Department of Probability Theory of the Faculty of Mechanics and Mathematics and the Interfaculty Laboratory of Statistical Methods. In 1935, Kolmogorov was awarded the degree of Doctor of Physical and Mathematical Sciences, in 1939 he was elected a member of the USSR Academy of Sciences. Shortly before the start of the Great Patriotic War Kolmogorov and Khinchin were awarded the Stalin Prize (1941) for their work on probability theory.

And on June 23, 1941, an expanded meeting of the Presidium of the USSR Academy of Sciences was held. The decision adopted at it marks the beginning of the restructuring of the activities of scientific institutions. Now the main thing is the military theme: all forces, all knowledge - to victory. Soviet mathematicians, on instructions from the Main Artillery Directorate of the Army, are conducting complex work in the field of ballistics and mechanics. Kolmogorov, using his research on probability theory, gives a definition of the most advantageous dispersion of projectiles during firing.

Post-war work

The war is over, and Kolmogorov is returning to peaceful research. It is difficult to even briefly highlight Kolmogorov's contribution to other areas of mathematics - the general theory of operations on sets, integral theory, information theory, hydrodynamics, celestial mechanics, etc., all the way to linguistics. In all these disciplines, many of Kolmogorov's methods and theorems are, admittedly, classical, and the influence of his work, as well as the work of his numerous students, among whom there are many outstanding mathematicians, on the general course of the development of mathematics is extremely great.

When one of Kolmogorov's young colleagues was asked how he felt about his teacher, he replied: "Panic respect ... You know, Andrei Nikolaevich gives us so many of his brilliant ideas that they would be enough for hundreds of excellent developments."

A remarkable pattern: many of Kolmogorov's students, gaining independence, began to play a leading role in the chosen direction of research. And the academician proudly emphasizes that the most dear to him are students who have surpassed teachers in scientific research. One can be surprised at Kolmogorov's asceticism, his ability to practice at the same time - and not without success! - many things at once. This is the management of the university laboratory of statistical methods of research, and the care of the physics and mathematics boarding school, the initiator of which Andrei Nikolayevich was the initiator of the creation of which Andrei Nikolaevich was, and the affairs of the Moscow Mathematical Society, and work on the editorial boards of Kvant, a magazine for schoolchildren, and Mathematics at School - methodical journal for teachers, and scientific and teaching activities, and the preparation of articles, brochures, books, textbooks. Kolmogorov never had to beg to speak at a student debate, to meet schoolchildren at a party. In fact, he was always surrounded by young people. He was very loved, his opinion was always listened to. Not only the authority of the world famous scientist played a role, but also the simplicity, attention, spiritual generosity that he radiated.

The circle of Andrei Nikolaevich's vital interests was not limited to pure mathematics, the unification of the individual sections of which into one whole he devoted his life to. He was fascinated by philosophical problems (for example, he formulated a new epistemological principle - the epistemological principle of A. N. Kolmogorov), and the history of science, and painting, and literature, and music.

Academician Kolmogorov is an honorary member of many foreign academies and scientific societies. In March 1963, the scientist was awarded the Balzan International Prize (he was awarded this prize together with the composer Hindemith, the biologist Frisch, the historian Morrison and the head of the Roman Catholic Church, Pope John XXIII). In the same year, Andrei Nikolaevich was awarded the title of Hero of Socialist Labor. In 1965 he was awarded the Lenin Prize (together with V. I. Arnold). In recent years, Kolmogorov headed the Department of Mathematical Logic.

“I belong,” the scientist said, “to those extremely desperate cyberneticists 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, using the methods 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.

As Stefan Banach aptly puts it: “A mathematician is one who can find analogies between statements. The best mathematician - who establishes analogy proofs. The stronger one can notice the analogies of theories. But there are those who see analogies between analogies.” Among these rare representatives of the latter is Andrei Nikolaevich Kolmogorov, one of the greatest mathematicians of the twentieth century.

Students

Many students of Kolmogorov became prominent scientists in various fields of science, among them - V. I. Arnold, I. M. Gelfand, M. D. Millionshchikov, Yu. V. Prokhorov, A. M. Obukhov, A. S. Monin, A.N. Shiryaev, S.M. Nikolsky. Kolmogorov himself said: “I was lucky to have talented students. Many of them, having started working with me in some area, then moved on to a new topic and, completely independently of me, received wonderful results. I’ll say as a joke that at present one of my students controls the earth’s atmosphere (A. M. Obukhov), and the other controls the oceans (A. S. Monin).

Kolmogorov and the reform of mathematics education in secondary school

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 secondary school is in deep crisis and needs to be reformed. It was recognized that only obsolete mathematics was taught in the secondary school, and its latest achievements were not covered. The 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 Academy of Sciences of the USSR recommended Academician A. N. Kolmogorov, who played a leading role in these reforms, for work on modernization.

The results of this activity of the academician were evaluated ambiguously and continue to cause a lot of controversy.

Andrei Nikolaevich Kolmogorov (1903-1987) is an outstanding scientist of the 20th century. The mathematician was born on April 25, 1903 in the provincial city of Tambov. The mother of the future academician, Maria Yakovlevna Kolmogorova, became very ill during childbirth and died, and his aunt V. Ya. Kolmogorova took the boy to live with her. All the early years of little Andryusha were spent in the grandfather's house in the village. Tunoshna, Yaroslavl region. Grandfather was a church father. The father of the mathematician, Kataev Nikolai Matveevich, had an agronomic education, but perished (1919) in the flames of the civil war, fighting on the Southern Front with the White Guard troops.

According to the scientist himself, he became interested in mathematics at the age of 6, discovering for himself "the joy of mathematical knowledge" in it. In that grandfather's village, his aunts set up a kind of school for a group of children and taught them the latest in pedagogy. The father did not raise his son.

In 1910, the aunt took the boy to Moscow, where he studied at the educational institution of E.A. Repman, who became school after the revolutionary events. No. 23. Ten years later, at the end of it, he enters the Moscow State University for physics and mathematics. In addition to exact science, Kolmogorov is also seriously interested in history, at the same time studying at the mathematical department of KhTI. DI. Mendeleev. A student of Kolmogorov L.A. Bassalygo found and published the early historical works of his teacher.

According to the recollections of a former Soviet student, he was very prosperous, having received after a successfully passed exam "an opportunity every month for a pood of bread and 1 kg of butter."

In 1922, a very young boy receives worldwide recognition for having constructed a Fourier series that diverges almost everywhere. Subsequently, the scientist successfully teaches, is a professor at Moscow State University, and is friends with science, he directs the Institute of Mathematics and Mechanics at his native university. In 1935 he founded a new department of probability theory. He will manage it until 1966.

From 1922 to 1925 he was a teacher of mathematics, educates schoolchildren at the Potylikhinsky People's Commissariat of Education. He explains his work in the middle link with a great need for money, but he recalls it with pleasure and moral satisfaction, as he managed to instill in his science the interest and love of students.

Since 1924 he has been fond of probability theory. The debut on this topic is "On the convergence of series whose members are determined by chance" (together with A.Ya. Khinchin). N.N. Luzin was my favorite mentor all these years.

Among the works of that time is also "On the principle of" tertium non datur ". By 1927, the completion of work on the law of the repeated logarithm belongs. Not all of Kolmogrov's works were approved by his senior comrade N.N. Luzin, some were published a few years after their writing.

In 1929, Kolmogorov completed his postgraduate studies without defending a dissertation. The current order was introduced only in 1934. Since 1936, Andrei Nikolaevich has been enthusiastically working on the creation of the famous Soviet encyclopedias (BSE and ITU). Being at the head of the department of mathematics, he creates a large number of articles for this edition.

1939 brought A.N. Kolmogorov membership in the Union Academy, and until 1942 he worked as an academician-secretary of the department of physical and mathematical sciences. At the turn of the 30s-40s. he is fond of turbulence and, after the end of hostilities in the country, establishes from scratch the laboratory of atmospheric turbulence of the Institute of Theoretical Geophysics of the USSR Academy of Sciences. Before the most treacherous German attack (1940), he was awarded the Order of the Red Banner of Labor, and in 1941-1945. does not stand aside and develops a series of articles on the theory of shooting.

In 1942 Kolmogorov got married. Anna Dmitrievna Egorova, a former school friend from the gymnasium, becomes his chosen one. With her, he lived more than a dozen happy years. His wife survived him by only one year, dying in 1988.

In the 60s. he created a unique laboratory of probabilistic and statistical methods. Until 1976, Kolmogorov was its head. The idea of creating visited the scientist after his Indian trip, where he was greatly impressed by the work of the statistical institute. Such a laboratory was an innovation for the USSR. The great mathematician came up with an idea for a completely new specialty at that time - biometrics.

Attention is also paid to the state of teaching mathematics in the school of the era of socialism. In collaboration with P.S. Alexandrov created a wonderful "Algebra", which taught more than one generation of the USSR algebraic wisdom. Together with SV Fomin they publish the textbook "Elements of the Theory of Functions..." (1st edition). Under him, a boarding school for physics and mathematics was founded at Moscow University; since 1989, the school has been named after Academician A.N. Kolmogorov. In addition, he edited the printed edition "Uspekhi matematicheskikh nauk" until his death, and the youthful journal "Kvant" appeared through his efforts.

He lives an active scientific life, participating in mathematical conferences and congresses around the world. He was awarded the Stalin Prize by the government, repeatedly the Order of Lenin (7) and the medal "For Valiant Labor", the Prize to them. P.L. Chebyshev of the Academy of Sciences of the USSR, is an honorary member of the IMO, Hero of Socialist Labor, winner of the International Balzan Prize, The Wolf Foundation and many others. others

Venerable contemporaries remember Kolmogorov not only because he was an outstanding scientist, but also because he was a real person. He warmed many talented mathematicians under his wing and saved them from attacks and misunderstanding on the part of their superiors. He was also a talented administrator, under him his favorite faculty reached its highest peak. When moving to the Mathematical Institute. V.A.Steklov Academy of Sciences of the USSR, head of the Department of Mathematical Statistics and Information Theory.

On October 20, 1987, the genius Kolmogorov, who worthily occupies an honorable place among world-class scientists, passed away. The academician was buried at the Novodevichy cemetery.

CONTENT:
Editorial (3).
Andrey Nikolaevich Kolmogorov (Biographical note) (4).
1. Fourier-Lebesgue series diverging almost everywhere (8).
2. On the order of magnitude of the coefficients of the Fourier-Lebesgue series (12).
3. Remarks on the study of the convergence of Fourier series (15).
4. Convergence of Fourier series (16).
5. Axiomatic definition of the integral (19).
6. On the boundaries of the generalization of the integral (21).
7. On the possibility of a general definition of the derivative, integral and summation of divergent series (39).
8. On harmonically conjugate functions and Fourier series (40).
9. On the principle of tertium non datur (45).
10. Convergence of Fourier series (69).
11. Fourier-Lebesgue series diverging everywhere (73).
12. Convergence of orthogonal series (75).
13. Operations on sets (85).
14. On the Denjoy integration process (93).
15. On the topological-group-theoretic substantiation of geometry (94).
16. Study of the concept of an integral (96).
17. On the definition of the mean (136).
18. On the compactness of sets of functions under convergence in the mean (139).
19. On the interpretation of intuitionistic logic (142).
20. On the justification of projective geometry (149).
21. On measure theory (150).
22. Points of discontinuity of functions of two variables (167).
23. Normability of general linear topological spaces! (168).
24. Continuation of the study on the points of discontinuity of a function of two variables (171).
25. Convergence of series in orthogonal polynomials (174).
26. Laplace transform in linear spaces (178).
27. On the order of the remainder of Fourier series of differentiable functions (179).
28. On the best approximation of functions of a given functional class (186).
29. Duality laws in combinatorial topology (190).
30. Homology ring of complexes and locally compact spaces (197).
31. Finite coverings of topological spaces (203).
32. Betti groups of locally compact spaces 2A7
33. Properties of Betti groups of locally compact spaces (209).
34. Betti groups of metric spaces (211).
35. Relative cycles. Alexander's duality theorem (214).
36. On open mappings (215).
37. Skew-symmetric quantities and topological invariants (218).
38. Study of the equation of diffusion associated with an increase in the amount of matter, and its application to a certain biological problem (221).
39. Simplified proof of the Birkhoff-Khinchin ergodic theorem (246).
40. On inequalities between upper bounds of successive derivatives of an arbitrary function on an infinite interval (252).
41. Rings of continuous functions on topological spaces (264).
42. Curves in a Hilbert space that are invariant with respect to a one-parameter group of motions (269).
43. Wiener spiral and some other interesting curves in Hilbert space (274).
44. Points of local topologicality of countably multiple open mappings of compact sets (278).
45. Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers (281).
46. On the degeneration of isotropic turbulence in an incompressible viscous fluid (287).
47 Energy dissipation in locally isotropic turbulence 290
48. Equations of turbulent motion of an incompressible fluid (294).
49. Remark on polynomials by P.L. Chebyshev, deviating least from a given function (296).
50. On the fragmentation of drops in a turbulent flow (302).
51. On dynamical systems with an integral invariant on the torus (307).
52. On the conservation of conditionally periodic motions with a small change in the Hamilton function (311).
53. General theory of dynamical systems and classical mechanics (316).
54. Some fundamental questions of approximate and exact representation of functions of one and several variables 333.
55. On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables (335).
56. On the representation of continuous functions of several variables as superpositions of continuous functions of one variable and addition (340).
57. On the linear dimension of topological vector spaces (344).
58. Refinement of ideas about the local structure of turbulence in an incompressible viscous fluid at high Reynolds numbers (348).
59. P.S. Aleksandrov and the theory of bs-operations (352).
60. Qualitative study of mathematical models of population dynamics (357).

“Humanity has always seemed to me in the form of many lights wandering in the fog, which only vaguely feel the radiance scattered by all others, but are connected by a network of clear fiery threads, each in one, two, three ... directions. And the emergence of such breakthroughs through the fog to another spark is quite reasonable to call "MIRACLES". - A. N. Kolmogorov

As Stefan Banach rightly noted: “A mathematician is one who knows how to find analogies between statements. The best mathematician - who establishes analogy proofs. The stronger one can notice the analogies of theories. But there are those who see analogies between analogies.” A rare genius with such a skill is Andrey Nikolaevich Kolmogorov - one of the greatest mathematicians of the twentieth century, Doctor of Physical and Mathematical Sciences, Professor of Moscow State University, Academician of the USSR Academy of Sciences, Stalin Prize Laureate, Hero of Socialist Labor. Kolmogorov stood at the origins of modern probability theory, the theory of turbulence, the theory of complexity of algorithms and a number of other areas of mathematical science and its applications, and obtained fundamental results in topology and mathematical logic.

Andrei Nikolayevich was born on April 12, 1903 in Tambov of Socialist Labor. Orphaned early, little Andrey was raised by his aunt, Vera Yakovlevna Kolmogorova. Vera Yakovlevna organized a school in her house. Using the recommendations of the latest pedagogy, Vera Yakovlevna raised more than a dozen children. For the children, a handwritten magazine "Spring Swallows" was specially published, in which interesting creative works of students were published. The first mathematical works of Andrey Kolmogorov were also published here. The arithmetic problems invented by five-year-old Andrei reflected a well-known algebraic pattern. The most interesting thing is that the boy came to this on his own, without outside help.

At the age of seven, Kolmogorov entered a private gymnasium organized by the Moscow Society of Progressive Intelligentsia. Working hard at school, Andrei shows himself to be a very talented mathematician.

In 1920, after much deliberation, Andrei Kolmogorov entered the Mathematics Department of Moscow State University. Deciding to devote himself to the service of science, Kolmogorov happened to listen to lectures by such famous mathematicians as P. S. Uryson, P. S. Aleksandrov, V. V. Stepanov, and N. N. Luzin. The latter had a significant impact on the development of Kolmogorov as a scientist, became his teacher in mathematics.

After only a few months, the talented Andrey Kolmogorov takes exams for the entire course. In the second year he receives a special scholarship. A promising student devotes most of his free time to solving complex mathematical problems.

A year later, eighteen-year-old sophomore Andrei Kolmogorov achieves the first serious results.

Interesting to know! Kolmogorov became a professor at Moscow State University at the age of 27.

Kolmogorov's scientific activity began with an in-depth study of the problems of descriptive and metric theory of functions. In 1923 Kolmogorov's first scientific publication appeared. The questions of the foundations of mathematical analysis, which were popular at that time, and closely related research in mathematical logic, interested the young student. Kolmogorov takes an active part in the discussions between the two methodological schools - the formal-axiomatic (D. Hilbert) and the intuitionistic (L. E. Ya. Brouwer and G. Weil). In 1925, he proves that all known sentences of classical formal logic, with a certain interpretation, turn into sentences of intuitionistic logic, which causes general interest in his philosophy of mathematics.

In 1926, graduate student Kolmogorov found necessary and sufficient conditions for the existence of the law of large numbers. This was an incredible discovery, because the world's largest mathematicians have been trying in vain to get the desired result for many decades.

For many years Andrei Nikolaevich collaborated with A.Ya. Khinchin. Together they developed a number of problems in probability theory. Thanks to the research of domestic and foreign scientists, the "science of the case" has developed rapidly. Andrei Nikolaevich Kolmogorov, who used axiomatization, gave a modern look to the theory of probability.

Throughout his scientific activity and until the end of his days, Kolmogorov considered the theory of probability the main business of his life. However, the range of interests of the scientist included several dozen branches of mathematical science, moreover, he was keenly interested in philosophy and literature, painting and music, history and sociology.

In 1930 Kolmogorov became a professor at Moscow State University. For six years from 1933 to 1939, A. N. Kolmogorov headed the Institute of Mathematics and Mechanics of Moscow State University, for many years he was the permanent head of the Department of Probability Theory of the Faculty of Mechanics and Mathematics and the Interfaculty Laboratory of Statistical Methods.

In 1941, Andrei Nikolayevich Kolmogorov was awarded the Stalin Prize for high achievements in mathematics and for his work on the theory of probability.

On October 20, 1987, the outstanding Soviet mathematician Andrei Nikolaevich Kolmogorov died in Moscow. He was buried at the Novodevichy cemetery.

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