Borisov Yuri

The presentation outlines the main points of the life and scientific work of the great 20th century mathematician A.N. Kolmogorov. The material was presented at mathematical readings in 2013 and can be used at extracurricular activities.

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MUNICIPAL BUDGETARY EDUCATIONAL INSTITUTION SECONDARY EDUCATIONAL SCHOOL No. 5 RESEARCH WORK Topic: “Life and scientific activity of A.N. Kolmogorov.” » Completed by: 11th grade student Yuri Borisov Supervisor: mathematics teacher Smirnova V.F. Kirzhach 2013

INTRODUCTION “In any case, I have always lived guided by the thesis that TRUTH is the main thing, that our duty is to find and defend it, regardless of whether it is pleasant or unpleasant. In any case, in my conscious life I have always proceeded from such principles." Andrei Nikolaevich Kolmogorov

Kolmogorov Andrey Nikolaevich (1903-1987) - an outstanding Russian Soviet mathematician. Kolmogorov - one of the founders modern theory probabilities, he obtained fundamental results in topology, geometry, mathematical logic, classical mechanics, turbulence theory, complexity theory of algorithms, information theory, function theory, theory of trigonometric series, measure theory, function approximation theory, set theory, theory differential equations, theory of dynamical systems, functional analysis and in a number of other areas of mathematics and its applications.

Relevance this study contributed to the desire to take part in city mathematical readings for high school students dedicated to the 110th anniversary of the birth of A.N. Kolmogorov. In addition, I came across the book “Algebra and the beginnings of analysis,” a textbook for grades 10-11. edited by A.N. Kolmogorov, in which the topics I studied in grade 11. were presented more simply than in the textbook edited by A.G. Mordkovich, and I decided to get acquainted with the life and work of A.N. Kolmogorov. The object of this study is the activities of A.N. Kolmogorov. The purpose of the study is to get acquainted with the biography and labor activity Kolmogorova A.N. To achieve the goal, the following tasks were solved: analyze and study the literature about the life and work of a scientist - mathematician; describe his contribution to the development of mathematical science; determine his relationships with other mathematicians of his time; establish the depth of achievements of A.N. Kolmogorov.

EARLY YEARS Andrei Nikolaevich Kolmogorov was born on April 12, 1903 in Tambov. Mother - Maria Yakovlevna Kolmogorova died during childbirth, father - Kataev Nikolai Matveevich, died in 1919 during Denikin's offensive Andrei was raised in Yaroslavl by his mother's sisters, Vera Yakovlevna Kolmogorova, one of the sisters, officially adopted Andrei and in 1910 moved with him to Moscow for placement in a gymnasium. At the age of seven, Kolmogorov was sent to the private Repman gymnasium; Andrei already in those years showed remarkable mathematical abilities. With aunt Vera Yakovlevna (1863-1951), who adopted Andrei Nikolaevich.

His rare and versatile talent manifested itself early: at the age of seven, he independently rediscovered the representation of the squares of integers as a sum of prime numbers. “I learned the joy of a mathematical “discovery” early, noticing at the age of five or six the pattern: 1 = 12 1 + 3 = 22 1 + 3 + 5 = 32, and so on...” So Andrei Nikolaevich himself wrote in his memoirs . At twelve I started studying higher mathematics. Somewhat later, in middle school, completely different hobbies prevailed - in particular, the history of Novgorod, where he made an important discovery. The estate in Tunoshna, where Andrei Kolmogorov spent his childhood Andryusha 7 years old Andryusha 7 years old Andryusha 7 years old Andryusha 7 years old

UNIVERSITY For the first time student years In addition to mathematics, Kolmogorov studied seriously in a seminar on ancient Russian history. In 1920, Kolmogorov entered the mathematics department of Moscow University. In the first months, Andrei passed the exams for the course. And as a second-year student, he receives the right to a “scholarship”: “... I received 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. "He now had free time, which he devoted to trying to solve problems that had already been raised. math problems. Moscow State University

THE BEGINNING OF SCIENTIFIC ACTIVITY In 1921, Kolmogorov made his first scientific report to a mathematical circle, in which he refuted one improvisational statement of N. N. Luzin. At the beginning of 1922, Luzin invited him to become his student. In the summer of 1922, A. N. Kolmogorov constructed a Fourier series, diverging almost everywhere. Of particular importance for the application of mathematical methods to natural science and practical sciences had the law of large numbers. In 1926, these conditions were obtained by graduate student Kolmogorov. Kolmogorov A.N. 1930

With his work “Basic Concepts of Probability Theory,” A. N. Kolmogorov laid the foundation for modern probability theory based on measure theory. In 1930, Kolmogorov went on a business trip to Germany and France. n meets with many outstanding colleagues, and above all with Hilbert and Courant. Andrei Nikolaevich considered probability theory his main specialty. Kolmogorov and his friends in science worked hard, but did not lose their sense of humor. Equations with partial derivatives were jokingly called “equations with unfortunate derivatives,” such a special term as finite differences was changed into “miscellaneous finiteness,” and the theory of probability into “trouble theory.” With Pavel Sergeevich Alexandrov. Germany. 1931 With Pavel Sergeevich Alexandrov. Germany. 1931 With Pavel Sergeevich Alexandrov. Germany. 1931 With Pavel Sergeevich Alexandrov. Germany. 1931

PROFESSORIES In 1931, 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, founded the Department of Probability Theory of the Faculty of Mechanics and Mathematics and the Interfaculty Laboratory statistical methods. Kolmogorov was awarded the degree of Doctor of Physical and Mathematical Sciences in 1935. In 1939, at the age of 35, Kolmogorov was immediately elected as a full member of the USSR Academy of Sciences, a member of the Presidium of the Academy and academician-secretary of the Department of Physical and Mathematical Sciences 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 the theory of random processes. Since 1936, Andrei Nikolaevich has devoted a lot of effort to creating the Great and Small Soviet Encyclopedias. Back in the late thirties, Kolmogorov became interested in the problems of turbulence. In 1946, after the war, he returned to these issues again. He organizes a laboratory of atmospheric turbulence at the Institute of Theoretical Geophysics of the USSR Academy of Sciences.

The 1950s saw another rise in Kolmogorov's mathematical creativity. Here it is necessary to note his outstanding, fundamental works in the following areas: celestial mechanics; on Hilbert's 13th problem; on dynamic systems; on the probability theory of constructive objects. Hilbert's Thirteenth Problem

PERSONAL LIFE In September 1942, Kolmogorov married his classmate at the gymnasium, Anna Dmitrievna Egorova, the daughter of the famous historian, professor, corresponding member of the Academy of Sciences Dmitry Nikolaevich Egorov. Their marriage lasted 45 years. Kolmogorov did not have his own children. Kolmogorov’s circle of vital interests was not limited to pure mathematics: he was also fascinated by philosophical problems, and the history of science, and painting, and literature, and music.

University in Komarovka Andrei Nikolaevich Kolmogorov, lived in the village of Komarovka from 1935 to 1986. In 1929, Kolmogorov decided to settle somewhere near Moscow. He, together with his friend Pavel Sergeevich Alexandrov, bought a house in the village of Komarovka. We spent the middle of the week in Moscow - from Tuesday to Friday, and from Friday evening to Tuesday morning - in Komarovka. One of the days in Komarovka was completely devoted to physical education - skiing, rowing, and large walking excursions. Scientists also invited “mathematical youth” to ski runs. Since then, little Komarovka has become as significant a mathematical center of the country as the largest university cities. The important thing is that Komarovka became, as it were, a branch of the Faculty of Mechanics and Mathematics. From old building Moscow State University on Manezhnaya Square an invisible path runs here. Students moved along it in both directions. There - anxiously awaiting a meeting with demanding managers. Back - armed with reprints of articles, pieces of paper with notes from academicians, which then had to be solved like puzzles. And, most importantly, armed with ideas. Ideas were distributed with extraordinary generosity in Komarovka. House in Komarovka, 50s On the ski track

Kolmogorov school In 1963, at Moscow University, by decree of the Council of Ministers of the USSR, a physics and mathematics school was opened - boarding school new type. The creation of a boarding school at Moscow State University is inextricably linked with the name of A.N. Kolmogorov. Kolmogorov considered directly personal work with schoolchildren, and then all the work to improve mathematics education in high school as important and the right country, as their civic responsibility for mathematical education. 1969 - supervised the teaching of mathematics at a boarding school at Moscow State University. I gave lectures there for students in the ninth and tenth grades; 1970 - at the Physics and Mathematics Boarding School at Moscow State University, he directed methodological unification mathematicians, gave a lecture course. Led summer school in Pushchino and the selection of applicants to the Physics and Music School. At a geometry lesson

REFORM OF SCHOOL MATHEMATICAL EDUCATION In 1968, the section of the Commission of the USSR Academy of Sciences and the USSR Academy of Pedagogical Sciences, which he headed, released new mathematics programs for grades 6-8 and 9-10, which served as the basis for writing textbooks. Andrei Nikolaevich himself took a direct part in the preparation teaching aids for 9th and 10th grades high school“Algebra and the beginning of analysis”, “Geometry for grades 6-8”. Based on materials from the school at Moscow State University, a mathematics textbook is being prepared for physics and mathematics schools (from members of the Academy of Pedagogical Sciences in the team of authors V.A. Gusev, A.A. Shershevsky), for which he wrote several chapters.

Kolmogorov's students Andrei Nikolaevich was happy in his students. He created an outstanding scientific school. Most of his students became leaders in their scientific fields, continuing the work of their teacher. Many times they tried to compile a complete list of his students, but this idea was impossible, if only because the task itself was informal. In 1963, on the occasion of Andrei Nikolaevich’s 60th birthday, a huge “Archimedean spiral” of his students was drawn at his department (probability theory) (A.N. Kolmogorov himself formed the “core”). No matter how many names were included in this spiral list, it always turned out that there were also Andrei Nikolaevich’s students and students’ students. A.N. Kolmogorov with students of Physics School No. 18 V. Tikhomirov, A. N. Kolmogorov, S. Sadikova. 1959

Recent years In recent years, Kolmogorov headed the department mathematical logic at Moscow State University and taught at Physics School No. 18 at Moscow State University. In 1963, the first Balzanov Prize in mathematics was awarded, and A. N. Kolmogorov became its laureate. This was the highest assessment of A. N. Kolmogorov’s contribution to world science. In the same year, Andrei Nikolaevich was awarded the title of Hero of Socialist Labor. In 1965 he was awarded the Lenin Prize. For his services, he was awarded the Order of Lenin seven times. Has many other awards.

Statements about Kolmogorov A.N. his colleagues and students “A. N. Kolmogorov is one of those mathematicians for whom every work in every field produces a complete revaluation of values. It is difficult to find a mathematician in recent decades, not just of such breadth, but with such an impact on mathematical tastes and on the development of mathematics ". P. S. Aleksandrov “Andrei Nikolaevich Kolmogorov occupies a unique place in modern mathematics, and in world science as a whole. In terms of the breadth and diversity of its scientific studies he resembles the classics of natural science of past centuries." N. N. Bogolyubov, B. V. Gnedenko, S. L. Sobole

CONCLUSION In the apt expression of Stefan Banach: “A mathematician is one who knows how to find analogies between statements. The best mathematician is the one who establishes analogies of proofs. The stronger one may notice the analogies of theories. But there are also those who see analogies between analogies.” These rare representatives of the latter include Andrei Nikolaevich Kolmogorov, one of the greatest mathematicians of the twentieth century. Kolmogorov died on October 20, 1987 in Moscow. He was buried at the Novodevichy cemetery.

LITERATURE http://ru.wikipedia.org http://to-name.ru http://www.math.msu.su http://www.pms.ru http:// www.famous-scientists.ru http://www. famous - scientists. ru http://kolmogorov. livejournal. com/64579. ht http://www. kolmogorov. info/index. html Kolmogorov A.N. Selected works. Mathematics and mechanics. M., 1985 Kolmogorov in memoirs, ed. A.N.Shiryaeva. M., 1993 Kolmogorov in memoirs / Ed. -composition A.N.Shiryaev. M., 1993. Kolmogorov A.N. How I became a mathematician // Ogonyok. 1963. No. 48. Kolmogorov A.N. Memories of P.S. Aleksandrov // Uspekhi Mat. 1986. T.41. Issue 6.

THANK YOU FOR ATTENTION

KOLMOGOROV Andrey Nikolaevich (born Kataev, April 12(25), 1903, Tambov - October 20, 1987, Moscow) - one of the leading mathematicians of the twentieth century, academician (1939). Hero of Socialist Labor (1963). He spent his childhood in Yaroslavl.

Born on April 12 (April 25, new style) 1903 in Tambov, where his mother stayed on the way from Crimea home to Yaroslavl Maria Yakovlevna Kolmogorova(1871-1903), daughter of the leader of the Uglich nobility, trustee of public schools of the Yaroslavl province Yakov Stepanovich Kolmogorov. In Tambov she died during childbirth.

Father - Nikolai Matveevich Kataev (? - 1919), a graduate of the Moscow Agricultural Institute, an agronomist, belonged to the Right Socialist Revolutionary Party, was exiled from St. Petersburg for participating in the revolutionary movement to the Yaroslavl province, where he met Maria Yakovlevna. His paternal grandfather was a village priest in the Vyatka province.

Uncle of Andrei Kolmogorov - Ivan Matveevich Kataev(1875-1946) - graduate of Moscow University, historian, professor, Doctor of Historical Sciences, author of works on archaeography, national history, history of Moscow, essays on Russian history, author of a textbook on Russian history for high school in three parts (published in 1907 ). The son of Ivan Matveevich is Ivan Ivanovich Kataev (1902 - 1937, executed), writer, cousin of Andrei Kolmogorov.

Until the age of seven, Andrei was raised in Yaroslavl by his mother’s sisters, who lived in a house on Ilyinskaya (Proboynaya) street, modern address- st. Sovetskaya, 3. One of them, Vera Yakovlevna Kolmogorova, officially adopted Andrey. Aunties organized a school for children in their house of different ages who lived nearby, worked with them, a handwritten magazine “Spring Swallows” was published for the children. It published the creative works of students - drawings, poems, stories. The magazine also featured “ scientific works» Andrey - arithmetic problems invented by him. Here the boy published his first work on mathematics at the age of five. Together with Andrei, Pyotr Savvich Kuznetsov, later a famous Soviet linguist, spent his childhood years in his grandfather’s house.

In 1910 Vera Yakovlevna Kolmogorova moved with Andrei to Moscow to attend the private Repman gymnasium, one of the few where boys and girls studied together. Andrei already discovered remarkable mathematical abilities in those years. Kolmogorov did not have time to graduate from high school - the revolution happened. As he later recalled, “in In 1918-1920, life in Moscow was not easy. Only the most persistent ones studied seriously in schools. At this time I had to leave for construction railway Kazan—Ekaterinburg. At the same time as working, I continued to study independently, preparing to take the high school exam as an external student. Upon returning to Moscow, I experienced some disappointment: I was given a school completion certificate without even bothering to take an exam».

In 1920, Kolmogorov entered the mathematics department of Moscow University, where his teachers were the best mathematicians of that time. In the first months, Andrei passed the exams for the course. During his student years, in addition to mathematics, Kolmogorov seriously studied in a seminar on ancient Russian history. Already in his second year at university, Kolmogorov made a number of mathematical discoveries that brought him world fame. And further work put him among the world's leading mathematicians.

In 1931, 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, founded and for many years headed the Department of Probability Theory of the Faculty of Mechanics and Mathematics and the Interfaculty Laboratory of Statistical Methods. The degree of Doctor of Physical and Mathematical Sciences was awarded to Kolmogorov in 1935 without defending a dissertation.

In 1939, at the age of 35, Kolmogorov was immediately elected as a full member (skipping the title of corresponding member) of the USSR Academy of Sciences, a member of the Presidium of the Academy and, at the suggestion of O. Yu. Schmidt, academician-secretary of the Department of Physical and Mathematical Sciences 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 the theory of random processes.

On June 23, 1941, an extended meeting of the Presidium of the USSR Academy of Sciences took place, at which a decision was made to restructure the activities of scientific institutions on military topics. Soviet mathematicians, on instructions from the Main Artillery Directorate of the Red Army, conducted complex work in ballistics and mechanics. Kolmogorov, using his research on probability theory, gave a definition of the most favorable dispersion of projectiles during firing.

In September 1942, Kolmogorov married his classmate at the gymnasium, Anna Dmitrievna Egorova, the daughter of the famous historian, professor, corresponding member of the Academy of Sciences Dmitry Nikolaevich Egorov. Their marriage lasted 45 years. Kolmogorov did not have his own children; his stepson, O. S. Ivashev-Musatov, was raised in the family.

Back in the late thirties, Kolmogorov began to study the problems of turbulence. In 1946, he returned to these issues, organizing a laboratory of atmospheric turbulence at the Institute of Theoretical Geophysics of the USSR Academy of Sciences. In parallel with his work on this problem, Kolmogorov continued his successful work in many areas of mathematics. Together with S.V. Fomin, he wrote the textbook “Elements of the Theory of Functions and functional analysis”, which went through seven editions (7th ed. - M.: Fizmatlit, 2012). The textbook has been translated into English, French, German, Spanish, Japanese, Dari, and Czech.

In the mid-1960s, on the instructions of the USSR Ministry of Education, under the leadership of A. N. Kolmogorov, programs were developed and new textbooks in mathematics for high school were created: a textbook on geometry, a textbook on algebra and fundamentals of analysis. In 1963, Kolmogorov was one of the initiators of the creation of a boarding school at Moscow State University and began teaching there himself.

In March 1966, he signed a letter from 13 figures of Soviet science, literature and art to the Presidium of the CPSU Central Committee against the rehabilitation of I.V. Stalin.

In 1966, Kolmogorov was elected a full member of the USSR Academy of Pedagogical Sciences. In 1970, together with academician I.K. Kikoin, he created the magazine “Quantum”.

In recent years, Kolmogorov headed the department of mathematical logic at Moscow State University and taught at Physics School No. 18 at Moscow State University.

Kolmogorov’s range of vital interests was not limited to pure mathematics: he was fascinated by philosophical problems, the history of science, painting, literature, and music.

Awards and prizes: Hero of Socialist Labor (1963), seven Orders of Lenin (1944, 1945, 1953, 1961, 1963, 1973, 1975), Order of the October Revolution (1983), Order of the Red Banner of Labor (1940), Stalin Prize (1941, together with A. Ya. Khinchin), Lenin Prize (1965, together with V. I. Arnold), other awards

A.N. Kolmogorov was a member of the US National Academy of Sciences (1967), the Royal Society of London (1964), the German Academy of Naturalists "Leopoldina" (1959), the French (Paris) Academy of Sciences (1968), the American Academy of Arts and Sciences (1959), the Hungarian Academy of Sciences (1965), Polish Academy of Sciences (1956), Royal Netherlands Academy of Sciences (1963), Academy of Sciences of the GDR (1977), Academy of Sciences of Finland (1985), Romanian Academy, London Mathematical Society (1962), Indian Mathematical Society (1962), American Philosophical Society (1961); honorary doctorate from the University of Paris (1955), Stockholm University (1960), Indian Statistical Institute in Calcutta (1962).

In 2003, in Yaroslavl, a memorial plaque was installed on the house where Andrei Kolmogorov lived from 1903 to 1910, and in 2008, a street in the Yaroslavl microdistrict “Sokol” was named after him.

In any case, I have always lived guided by the thesis that TRUTH is the main thing, that our duty is to find and defend it, regardless of whether it is pleasant or unpleasant. In any case, in my adult life I have always proceeded from such positions.

Andrey Nikolaevich Kolmogorov.

· Introduction……………………………………………………Art. 4

· Life path Andrei Nikolaevich……………………..st. 5

· "A. N. Kolmogorov – an extraordinary phenomenon in science”…….Article 11

· Success in teaching activities……………………Article 18

· A. N. Kolmogorov – a versatile personality……………….art.22

· Conclusion……………………………………………………………..Article 27

· Bibliography………………………………………………..Article 28

· Illustrations……………………………………………..Article 29

INTRODUCTION

In this essay, I am trying to talk about one of the most famous and talented scientists of the 20th century - Andrei Nikolaevich Kolmogorov. I want to highlight not only its truly grandiose scientific activity, but also as a talented organizer, public figure and extraordinary, highly developed personality.

The great Russian scientist, one of the greatest mathematicians of the 20th century, deservedly recognized by almost all authoritative world communities of scientists - a member of the US National Academy of Sciences and the American Academy of Arts and Sciences, a member of the Royal Netherlands Academy of Sciences and the Academy of Sciences of Finland, a member of the French and German Academy of Sciences Academy of Naturalists "Leopoldina", member International Academy history of sciences and national academies Romania, Hungary and Poland, honorary member of the Royal Statistical Society of Great Britain and the London Mathematical Society, honorary member of the International Statistical Institute and the Mathematical Society of India, foreign member of the American Philosophical and American Meteorological Society, laureate of the most honorable scientific prizes: prizes of P.L. Chebyshev and N.I. Lobachevsky of the USSR Academy of Sciences, the International Prize of the Balzan Foundation and the International Prize of the Wolf Foundation, as well as state and Lenin Prize, awarded 7 orders of Lenin, medal " Golden Star“Hero of Socialist Labor, Academician Andrei Nikolaevich Kolmogorov always called himself “just a professor at Moscow University.”

Explore the life and work of this truly genius man I'm trying.

LIFE WAY OF ANDREY NIKOLAEVICH

A. N. Kolmogorov was born on April 25, 1903 in Tambov. Kolmogorov was lucky: he began receiving education in early childhood. Andrei's aunts organized a school in their house for children of different ages who lived nearby, teaching them - dozens of children - according to the latest pedagogy. They loved children, the very matter of education. And the guys treated their teachers with love - it was so interesting to be with them! They found abilities in every boy and every girl.

A handwritten magazine “Spring Swallows” was published for the children. It published the creative works of students - drawings, poems, stories. Andrei’s first “scientific works” appeared in it—the arithmetic problems he invented.

At the age of seven he was sent to a private gymnasium. It was organized by a circle of Moscow progressive intelligentsia and was constantly under threat of closure.

His rare and versatile talent manifested itself early:

at the age of seven he independently rediscovered the representation of squares of integers as the sum of primes, and at twelve he began to study higher mathematics. Somewhat later, in middle school, completely different hobbies prevailed - in particular, the history of Novgorod, where he made an important discovery. The return to mathematics occurred in the very last grades of high school.

In 1918-1920, life in Moscow was not easy. Only the most persistent ones studied seriously in schools. At this time, Andrei Nikolaevich, together with his elders, had to leave for the construction of the Kazan-Ekaterinburg railway. At the same time as working, he continued to study independently, preparing to take external exams for high school. Upon returning to Moscow, he experienced some disappointment: he was given a school completion certificate without even bothering to take the exam.

When in 1920 Andrei Kolmogorov began to think about entering college, he faced a eternal question: what should I devote myself to, what business? He is drawn to the mathematical department of the university, but there is also doubt about pure science, and technology is perhaps a more serious matter. For example, the metallurgical department of the Mendeleev Institute! A real man's business, in addition, is promising. It was decided to go both here and there. And the seventeen-year-old boy tapped out two routes along the Moscow pavements with the wooden soles of his homemade shoes: to the university and to Mendeleevsky. Having entered the Faculty of Physics and Mathematics of Moscow University in 1920, he finally connected his life with mathematics. In his first student years, in addition to mathematics, Kolmogorov studied most seriously in a seminar on ancient Russian history by Professor S. B. Bakhrushin. He did not give up the idea of ​​a technical career, he was fascinated by metallurgy, and, in parallel with the university, he entered the metallurgical department of the Chemical-Technological Institute. Mendeleev and studied there for some time. But it soon becomes clear to him that pure science is also very relevant. There is no doubt - this is his life's work. All the rest is superfluous and aside! In the first months, exams for the course were passed. And as a second-year student, he receives the right to a “scholarship”, sixteen kilograms of bread and a kilogram of butter per month - this is real prosperity." Now there is free time It gives in to attempts to solve already posed mathematical problems

As is usually the case, A. N. Kolmogorov’s first works were devoted to solving individual previously posed difficult problems. He began broader activities to create a new direction of research with A. Ya. Khichkin in his main mathematical specialty - probability theory. In his second year, he completed his first independent scientific work. He began studying the theory of trigonometric series with Professor V.V. Stepanov together with his close friend, the unusually bright and talented mathematician T.A. Seliverstov (both Seliverstov brothers died during the Second World War). Already at the age of nineteen he managed to construct an example of “almost everywhere divergent trigonometric series", brought him worldwide fame. Its first leaders at the university were, in addition to V.V. Stepanov, V.K. Vlasov, P.S. Aleksandrov, P.S. Uryson. Somewhat later he became a student of N.N. Luzin.

The lectures of Moscow University professor Nikolai Nikolaevich Luzin, according to contemporaries, were an outstanding phenomenon of “Classics” and “Romanticism” - lecturers have long been divided into two such conditional groups. The first ones are restrained, even dry, always precise in their formulations, their phrases are honed, the material is thought out to the details. The second ones are, first of all, inspired improvisers. But here’s the detail: record the lectures of the “classics” on tape, then decipher them and you will get a textbook. It seems to be good - everything you need is here. But there is a textbook and there are lectures. Do students really not expect anything more from the lesson as soon as information, information, information.

Luzin never had a predetermined form of presentation. And his lectures could in no way serve as a role model. Yes, no one else could repeat them, even Nikolai Nikolaevich himself, if asked, would probably not have been able to cope with such a task. But he had a rare sense of audience. He, like a real actor performing on a theater stage and perfectly sensing the reaction of the audience, had constant contact with students. He knew how to bring students into contact with his own mathematical thought, revealing the mysteries of his scientific laboratory. He invited us to joint spiritual activity and co-creation.

And the famous “Wednesdays”. What a holiday it was when N. N. Luzin invited students to his home! Conversations over a cup of tea about scientific problems... However, why does it have to be about scientific ones? There were plenty of topics for conversation. He knew how to ignite young people with a desire for scientific achievement, instill faith in their own strengths, and through this feeling came another understanding of the need for complete dedication to one’s favorite work.

Kolmogorov first attracted the attention of the professor during one lecture. Luzin, as always, taught classes, constantly addressing students with questions and assignments. And when he said: “Let’s build a proof of the theorem based on the following assumption...” Andrei Kolmogorov’s hand rose in the audience: “Professor, it’s wrong...” The question “why” was followed by a short answer from the freshman. Satisfied, Luzin nodded: “Well, come to the circle, report to us your thoughts in more detail.”

Although my achievement was quite childish, it made me famous in “Lusitania,” recalls Andrei Nikolaevich. 1

1. Nikolai Gorbachev. What does it mean to be a mathematician? “Smena”, 1978, No. 12, art. 46

But a year later, the serious results obtained by 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 students of his course. Such an invitation, according to the concepts of “Lusitania,” should have been regarded as conferring an honorary title on a student. As recognition of abilities.

The twenties were the heyday of Luzin's extraordinary mathematical talent. Lusitania representatives work persistently and fruitfully with him.

The first significant works of A. N. Kolmogorov date back to the twenties. Many years of close and fruitful cooperation connected him with A. Ya. Khinchin, who at that time began to develop problems in probability theory. It became the region joint activities scientists.

Since the time of Chebyshev, the science of “case” has been, as it were, a Russian national science. Its successes were multiplied by Soviet mathematicians. The law of large numbers was of particular importance for the application of mathematical methods to natural science and practical sciences. Find the necessary and sufficient conditions, for which it takes place, this is what the desired result was. The leading mathematicians of many countries have been trying unsuccessfully to obtain it for decades. In 1926, these conditions were obtained by graduate student A. N. Kolmogorov.

Andrei Nikolaevich now considers the theory of probability to be his main specialty, although there are a good two dozen areas of mathematics in which he worked.

In the same years when Andrei Nikolaevich made his first discoveries, he became a school teacher and worked for several years in secondary school. Starting from the 30s, he gave numerous lectures to schoolchildren and students, and actively participated in the formation of school mathematical Olympiads, first Moscow, and then All-Russian and All-Union. In 1931, A. N. Kolmogorov became a professor at Moscow University, where he headed three departments at different times, created several scientific schools and founded a boarding school at Moscow State University. In 1933 (at the age of 30!) he was appointed director of the Institute of Mathematics and Mechanics at Moscow State University. The entire graduate school was under his leadership. Can one really imagine that he, as the director of this institute, met and, in essence, talked with all (!) graduate students of the Faculty of Mechanics and Mathematics? Subsequently, Andrei Nikolaevich headed the mathematical department of the Faculty of Mechanics and Mathematics, and again graduate school was under his jurisdiction. Most graduate students of those years remembered conversations with Andrei Nikolaevich for the rest of their lives and often opened the way to big science.
A.N. Kolmogorov founded two departments at the faculty. In 1935, he founded the Department of Probability Theory, and Andrei Nikolaevich became its first head (now the department is headed by A.N. Kolmogorov’s student, professor, corresponding member of the RAS A.N. Shiryaev). Then two laboratories were opened at the department, one of which, namely the laboratory of probability theory and statistical methods, was also headed by Andrei Nikolaevich himself for some time, and then by his student prof. Yu.K. Belyaev.
In 1976, Andrei Nikolaevich created and again headed another department - mathematical statistics and theory of random processes. Now it is also led by Andrei Nikolaevich’s student, Prof. Yu.A. Rozanov. In the very last years of his life, A.N. Kolmogorov headed the department of mathematical logic and theory of algorithms. Currently, it is headed by another student of Andrei Nikolaevich - Professor V.A. Uspensky. And finally, Kolmogorov’s student prof. V.M. Tikhomirov is the head of the department of general problems of management.
From 1954 to 1958, Andrei Nikolaevich was the dean of the Faculty of Mechanics and Mathematics. And although administrative activity is not Andrei Nikolaevich’s element, even in this post he tried to be a reformer, trying to “improve everything.” The Faculty of Mechanics and Mathematics owes a lot to Andrei Nikolaevich Kolmogorov.

On the days of his 80th birthday, the seriously ill Andrei Nikolaevich, recalling the years he had lived, said: “My life was filled with happiness!” On April 25 of this year, Andrei Nikolaevich Kolmogorov would have turned 95 years old. At the entrance of building “L” of the Moscow University building, where in apartment 10 he lived for 34 years (from the date of construction of the new building to the day of his death), on November 18, 1997, a bronze plaque appeared with the words forever inscribed on it: “In this house with 1953 to 1987 lived the great Russian scientist, mathematician, professor at Moscow University, academician Andrei Nikolaevich Kolmogorov." This is a modest tribute to the university’s gratitude to its professor.

Andrei Nikolaevich's entire life was devoted to the search for truth and the cause of Enlightenment. It is he who can rightfully be called the Enlightener - a man who illuminated the life and scientific path of many, many.

"A. N. KOLMOGOROV – AN EXTRAORDINARY PHENOMENON IN SCIENCE"

What is a great scientist? Terms “ great mathematician”, “great physiologist”, etc. do not yet mean “great scientist”. The greatness of a person as a scientist presupposes breadth with a touch of cosmicity. This quality was possessed, for example, by the learned keeper of the House of Weights and Measures (since 1893), full member Imperial Academy arts (since 1894) Dmitry Ivanovich Mendeleev, who climbed alone in a balloon, developed the economics of mining, created smokeless gunpowder and conducted a critical analysis of spiritualist experiments.

Kolmogorov's extremeness. Kolmogorov was precisely a great scientist, and not just a great mathematician. In 1835, Gogol published his “A few words about Pushkin”; among these words were the following: “none of our poets is higher than him” and “Pushkin is an extraordinary phenomenon.” If you replace the words “poet” and “Pushkin” with “scientist” and “Kolmogorov” here, you will get a fairly accurate description of Kolmogorov.

The breadth of Kolmogorov's interests and activities has few analogues in the 20th century. He carried out his first research while still a student. They were conducted from November 1920 to January 1922 and were dedicated to the history of Novgorod. The results of these studies were considered lost; however, after Kolmogorov's death, four manuscripts of his historical research were discovered among his papers; they have now been published. According to the authoritative testimony of V.L. Yanin, these studies of Kolmogorov were ahead of not only historical science twenties, but also contemporary historical science.

Pushkin once remarked that he had more influence on youth and Russian literature than the entire Ministry of Public Education, despite the complete inequality of funds. Kolmogorov's influence on mathematics was the same.

What does it mean to be a mathematician? A good mathematician? Outstanding, finally? As one scientist aptly put it, a mathematician is one who knows how to find analogies between statements. The best mathematician is the one who establishes analogies of proofs. The stronger one may notice the analogies of theories. But there are also those who see analogies between analogies. Andrei Nikolaevich Kolmogorov belongs to these rare representatives of the latter.

Andrei Nikolaevich’s works relate to the most diverse branches of mathematics and its applications, ranging from the most abstract sections to such application areas, such as hydrodynamics and control theory, although he was most famous for robots in probability theory - Kolmogorov put this science on a solid axiomatic foundation and significantly enriched many of its sections.

Andrei Nikolaevich is the head of the world's strongest scientific school in probability theory and mathematical statistics. For his mathematical works It is characteristic that he was a pioneer and discoverer in many areas of mathematics: he has made outstanding achievements in the theory of probability, the theory of functions, functional analysis, topology, the theory of dynamical systems, the theory of turbulent fluid motion, etc. - it is difficult to indicate the area

mathematical analysis, to which he would not have made a significant contribution, where he would not have solved old (sometimes two-hundred-year-old) problems.

Kolmogorov completed his first famous work - an example of a Fourier series of a summable function, diverging almost everywhere - at the age of 19. In 1941, for his works on probability theory, published in 1936 and 1938, the scientist was awarded the State Prize of the first degree. For a series of works on the problem of stability of Hamiltonian chains, Andrei Nikolaevich and his talented student Professor V.I. Arnold were awarded the Lenin Prize in 1965. The authors have developed completely new mathematical methods, allowing you to solve problems previously considered “inaccessible”. These methods turned out to be so fruitful that they were able to be used not only to study classical problems, but also a whole series of problems, the significance of which is only realized today (the problem of the movement of charged particles in “magnetic traps”).

Andrei Nikolaevich himself always highly valued “sports-mathematical” achievements and considered his work on Hilbert’s 13th problem to be his most difficult sports achievement.

On June 23, 1941, an extended meeting of the Presidium of the USSR Academy of Sciences took place. The decision taken there marks the beginning of a restructuring of the activities of scientific institutions. Now the main thing is the military theme: all strength, all knowledge for 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. That’s how important his choice of “pure science” turned out to be!

American scientist Norbert Wiener, one of the creators of cybernetics, testified:

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

During the war years, Wiener studied the problem of aircraft movement during anti-aircraft fire. Later it will result in the theory of forecasting, but the American scientist admits: “When I wrote my first work on the theory of forecasting, I did not realize that some of the main mathematical ideas of this article had already been published before me. But I soon discovered that shortly before the Second World War, the Soviet mathematician Kolmogorov published a small but very important note devoted to this topic... I have no confidence that Kolmogorov also did not find the possibilities of applying these methods known to me. .. Over the past twenty or thirty years, almost never has either of us published any work without very soon the closely related work of another on the same topic appearing.”

And one more confession from 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 what I myself wanted to say every time.”

In 1954, at the first post-war mathematical congress in Amsterdam, A.N. Kolmogorov made a report on one of the greatest problems of astronomy and classical mechanics - the problem of stability solar system. This question has worried all researchers since the very moment when Newton derived the equations of classical mechanics. In a report at the Amsterdam Congress, A.N. Kolmogorov spoke about the new method he had developed, which in many cases made it possible to solve the problem under consideration. Kolmogorov's method was improved by his student V.N. Arnold and the great German mathematician J. Moser and was called KAM theory, which is rightfully considered one of the largest achievements in mathematics of the 20th century. For almost half a century, A.N. Kolmogorov was a generally recognized leader in probability theory. Together with A.Ya. Khinchin and many of his students, he completed the construction of the classical stage of probability theory, the beginnings of which were laid by J. Bernoulli, Laplace and P.L. Chebyshev. Then he developed the axiomatic basis of probability theory (this achievement of A.N. Kolmogorov is perhaps best known), created the theory of the so-called Markov processes, the origins of which were Einstein, Smoluchowski and other outstanding physicists.

In addition to mathematics, where he had classical achievements in no less than two dozen fields, Andrei Nikolaevich achieved outstanding results in physics, mechanics, geophysics, oceanology, shooting theory; with great interest and success he studied problems of biology and poetry

September 24, 1956 on Faculty of Philology Moscow State University began working on the seminar “Some applications of mathematical research methods in linguistics” - the first seminar on mathematical linguistics in USSR. At the opening of the seminar, I offered its participants two educational tasks, the authorship of which belonged to Kolmogorov: to give a strict definition of the concept of case and to give a strict definition of the concept of iambic. Both of these tasks were the result of conversations between V. A. Uspensky and Kolmogorov, who was sympathetic to both the creation of such a seminar and the mathematization of philological research in general.

The origins of Kolmogorov’s interest in the theory of verse are as follows. First of all, these are his broad general humanitarian and, in particular, literary interests. Hence the interest in poetry. Further, his desire for a scientific analysis of the phenomenon, for the systematization of concepts. Hence his interest in poetry, which arose from his youth, in which he read the works of first Andrei Bely, and then Shengeli and Tomashevsky.

As V. A. Uspensky said: “ Highest level scientific analysis and systematization is mathematization. Mathematization is by no means reduced to expressing phenomena in numbers, tables and graphs. Numbers, tables and graphs may be completely absent. The main thing in mathematization is the creation of a description of a phenomenon that would be impeccable from a logical point of view, and mathematics acts here as an evaluator (and at the same time an ideal) of the degree of logical impeccability. The metrical aspect of versification lends itself most easily to mathematization.” 2 Hence Kolmogorov’s interest in that section of poetry called metric and rhythm. Due to the fact that of all the sections of poetry, it was metric and rhythm that were most advanced in the direction of formalization, the lack of proper order in its basic concepts could be detected quite quickly. It was discovered by Kolmogorov, although he, out of modesty, would hardly have agreed with such a formulation; rather, he would say that he was only expressing in explicit form generally known ideas.

Andrei Nikolaevich was also no stranger to numbers, tables and graphs. He only believed that they must certainly be preceded by a clear description of the phenomena being counted. Kolmogorov was one of the classics of statistics. The application of methods of mathematical statistics to the phenomena of speech - in particular, to the phenomena of poetic speech - could not but interest him.

In the late fifties, Kolmogorov's interests in poetry became intertwined with his studies in cybernetics. It became possible to consider both the composition of poetry (as a process) and versification (as a way of organizing the text that arises as a result of such a process) from the point of view of cybernetics and even as an object of study of the latter.

In the early sixties, Andrei Nikolaevich began to create the last of his mathematical masterpieces - the creation of the Kolmogorov complexity theory, now called the theory of Kolmogorov complexity. This theory makes it possible to assess the level of complexity of certain objects, primarily texts (i.e., finite chains of letters). Kolmogorov was interested, in particular, in the question of the complexity of literary texts, including what proportion of the complexity is due to the content of the text, and what part is due to certain literary devices; literary devices - such as rhyme, meter, etc. - are most easily formalized and isolated in poetry.

2. V. A. Uspensky. Preface for readers of “UFO” to the semiotic messages of A. N. Kolmogorov. “UFO”, 1997, No. 24, art. 142.

It remains to express regret that Kolmogorov’s poetry studies remained published only in magazines and collections and have not yet been published as a separate book. A. N. Shiryaev sums up these studies of Kolmogorov as follows:

“On the initiative of A. N. Kolmogorov, a big job to revise and clarify the results obtained by famous verse researchers A. Bely, B. Tomashevsky, G. Shengeli, K. Taranovsky, R. Yakobson and others. The main results obtained in this direction by A. N. Kolmogorov, his students and collaborators, can be formulated as follows: identification of metric laws, classification and statistics of rhythmic variations of meter, analysis of “residual” entropy, its evaluation. An estimate of the “residual” entropy was obtained and a calculation of the “entropy costs” for individual techniques sound expressiveness of verse." 3

A. N. Kolmogorov is the largest modern cyberneticist. His work on the application of scientific mathematical analysis to poetic works of fiction is known throughout the world. In the field of cybernetics, he expressed many interesting thoughts, guesses and hypotheses. In particular, he has the following very bold idea:

“The fundamental possibility of creating full-fledged living beings built on discrete digital mechanisms for information processing and control does not contradict the principles of materialist dialectics.” 4

Kolmogorov was an honorary member of the American Meteorological Society. We find his portrait in the gallery of portraits of the creators of classical mechanics, starting with Archimedes. Van Heijenoort’s well-known anthology “From Frege to Gödel” contains articles from 1879 to 1931 that determined the structure of mathematical logic; Of the domestic authors, only Kolmogorov is represented in the anthology: we find here English translation his article, completed by him on September 30, 1925, i.e. at the age of 22. Twice, in 1969 and 1971, Kolmogorov took part (and acted as a scientific supervisor) in multi-month

3. V. A. Uspensky. Preface for readers of “UFO” to the semiotic messages of A. N. Kolmogorov. “UFO”, 1997, No. 24, art. 156.

4. A. B. Sosinsky. Conversation with A. N. Kolmogorov. "Quantum", 1983, No. 4, art. 5.

oceanographic voyages on the research vessel “Dmitry Mendeleev”; the 1971 voyage even circumnavigated the world. And the concept of case according to Kolmogorov is well known to grammarians.

From communicating with Kolmogorov, there was an incomparable feeling of direct contact with a genius.

At the end of his creative life Andrei Nikolaevich proclaimed the beginning of a grandiose program to understand the unity of deterministic and chaotic phenomena: the world is one - most deterministic phenomena that have a certain instability begin to behave as random, and vice versa, random phenomena obey strict laws. The new understanding is based on the concept of complexity: a complexly described deterministic phenomenon behaves as random. This concept combines virtually all directions of its scientific research: and his research in the theory of functions, with which he began and where he achieved his first great success, and his works in the field of mathematical logic, information theory, automata theory, approximation theory, dynamical systems, classical mechanics, turbulence theory and, of course, probability theory . Thus, the creative biography of A.N. Kolmogorov appears to us as a community of ideas, theories and results, interconnected by a single philosophical and natural science concept.

SUCCESS IN TEACHING ACTIVITIES

Andrei Nikolaevich was happy in his students. He created an outstanding scientific school. Most of his students became leaders in their scientific fields, continuing the work of their teacher. Many times they tried to compile a complete list of his students, but this idea was impossible, if only because the task itself was informal. In 1963, on the occasion of Andrei Nikolaevich’s 60th birthday, a huge “Archimedean spiral” of his students was drawn at his department (probability theory) (A.N. Kolmogorov himself formed the “core”). No matter how many names were included in this spiral list, it always turned out that there were also Andrei Nikolaevich’s students and students’ students. On pp. 134-135 of the book “Kolmogorov in Memoirs” there is, as it seemed to the compiler, a fairly complete list of Kolmogorov’s students, but additions keep coming. Here are just the academicians and corresponding members: I.V. Arnold, A.A. Borovkov, I.M. Gelfand, A.N. Maltsev, M.D. Millionshchikov, V.S. Mikhalevich, S.M. Nikolsky, A.M. Obukhov, Yu.V. Prokhorov, Y.G.Sinai, B.V. Gnedienko, S.Kh.Sirazhdinov, V.A.Statulyavichyus, L.N.Bolshev, A.S.Monin, B.A.Sevastyanov, A.N.Shiryaev.

Andrei Nikolaevich was a wonderful dean. “I was lucky to have talented students,” he said. “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” 5. Kolmogorov said that talented people should be forgiven for their talent, and he saved more than one of the now very famous mathematicians from expulsion from the university. He always believed in the good beginnings in people. Andrei Nikolaevich supported many, and in almost all cases those whom he supported took their rightful place in science.

What distinguished Andrei Nikolaevich from other professors was his complete respect for the student’s personality. He always expected to hear something new and unexpected from a student, and he possessed to the highest degree that infectious passion for science, which is what students need most of all.

5. P. A. Livansky. Mathematical talents. "Quantum", 1985, No. 7, art. 9.

The main thing that Andrei Nikolaevich gave as a teacher was passion for the work and faith in one’s own strength. He knew how to make the student grow much higher than the ceiling he set for himself,” recalls A. M. Abramov, Kolmogorov’s student. - “It was somehow embarrassing not to complete the tasks given by Andrei Nikolaevich. Maybe that’s why it was possible to do a lot of things that previously seemed impossible. It is very important to have such an example before your eyes - for Andrei Nikolaevich, it seems, there were no problems that could not be solved, he knew everything.” 6

When Kolmogorov induced a certain result in his student, which in fact was almost suggested to him, he created such an environment as if the student had thought of it himself. Such psychological support for the junior partner was a very significant aspect of his activity. Andrei Nikolaevich, using very simple formulations, pushed people into an independent orbit, after which he believed that he had employees who were doing this and knew it better than him (although it was possible to know better than Kolmogorov only the details, but not the general ideas).

When one of Kolmogorov’s young colleagues was asked what feelings he had towards his teacher, he replied: “Panicized respect... You know, Andrei Nikolaevich gives us gifts like this the number of their brilliant ideas, that they would be enough for hundreds of wonderful developments” 7.

A remarkable pattern: many of Kolmogorov’s students, having gained independence, began play a leading role in the chosen area of ​​research. And the academician proudly emphasizes that the students who are most dear to him are those who have surpassed the teacher in scientific research.

Kolmogorov's students are those who directly worked with him in one or another other fields of science. There are many more indirect students of his. These are schoolchildren.

One can be amazed at Kolmogorov’s asceticism, his ability to simultaneously engage - and not unsuccessfully! - in many things at once: here is the leadership of the university laboratory of statistical research methods, and concerns about physics and mathematics

6. A. B. Sosinsky. A. N. Kolmogorov in the memories of his students. "Quantum", 1988, No. 11-12, art. 10.

7. Nikolai Gorbachev. What does it mean to be a mathematician? “Smena”, 1978, No. 12, art. 42.

boarding school, 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” - a methodological magazine for teachers, and scientific and teaching activities, and the preparation of articles, brochures, books, textbooks, and, finally, countless speeches before schoolchildren, students, teachers, and fellow scientists. Where does time come from?! Moreover, the circle of vital interests is by no means limited to pure mathematics, to the unification of individual sections of which he devoted his life to a single whole. There are philosophical problems here, and the history of science, and painting, and literature, and music.

There is a special sign of human immortality. Is your name interested in young people, does their problems concern him? If there is “no time” for this, there is no doubt that the person has stopped developing, that’s all, period. And another sign: how do young people treat you?

Kolmogorov never had to be asked to speak at a student debate or meet with schoolchildren at an evening. In fact, he was always surrounded by young people. And again there is mutual enrichment. They love him very much and always listen to his opinion. Not only the authority of a world-famous scientist plays a role, but also the simplicity, attention, and spiritual generosity that he radiates.

From a letter from A. N. Kolmogorov to sixteen-year-old schoolboy Andrei Fetisov .

“My dear namesake (my name is also Andrey)!

Modern youth often exaggerate their desire for independence. Therefore, I like your belief that in the older generation there may be a Teacher to whom you could “open your soul” and who can teach “the art of living.” In such a relationship, it is easier for an elder to be called not a Teacher, but an elder friend. Such friendships, where the elder to some extent plays the role of a mentor, teaching not only, say, mathematics, but also simply life, are not uncommon. Finding a “mentor friend” is a great happiness for a young man.

Since you ask how it was with me, I answer that I was taught, first of all, by my aunt Vera Yakovlevna Kolmogorova, who raised me like a son, to have a serious, responsible attitude towards life, to search for a big, exciting business that people need. Mathematics as a specialty to which one can devote one’s life came later...” 8

Andrei Nikolaevich devoted almost a third of his creative life to the education of youth and school. He organized a wonderful boarding school for gifted schoolchildren from the provinces (now this boarding school bears the name of A.N. Kolmogorov), was one of the founders of the Kvant magazine and its supplement, the Kvant Library, and worked math olympiads, and most importantly, was one of the initiators of a deep reform of the secondary school. A.N. Kolmogorov’s contribution to the cause of education is still waiting to be seen detailed study and recognition.

8. Nikolai Gorbachev. What does it mean to be a mathematician? “Smena”, 1978, No. 12, art. 44.

A. N. KOLMOGOROV – A VERSATILE PERSONALITY

At some point in his life (obviously, in his early youth), Andrei Nikolaevich decided that a person simply must be happy and for this he needs this and that. At the same time, it is necessary that all types of activities that a person chooses for himself truly captivate him. And Kolmogorov managed to build his life in this way: his creative achievements were extraordinary, he knew how to appreciate a lot - he loved human communication, nature, music, literature.

One day Kolmogorov said to his student: “You should not have an idea of ​​me as a person who knows only mathematics; I belong to those people who have own opinion more or less on every issue." 9 Andrei Nikolaevich was a bright, deep, unique personality. He had a limitless outlook in philosophy, economics, politics, geography, and issues related to art and literature. At the same time, he was very original, almost always unpredictable. Particularly in your passions. He considered the works of A. France T. Mann to be the peaks of world literature of the 20th century, which was unexpected for many.

Kolmogorov was a genius. This is what makes it interesting, as Mayakovsky would say. The views of geniuses on literature and art, their tastes - shouldn't this be one of the subjects of literary review, including new ones?

Poetry and music, architecture, painting and other types of plastic arts were integral and important part inner world Kolmogorov. It is not enough to say that he had extensive and deep knowledge in each of these artistic fields. He perceived poems and musical works, buildings, paintings and sculptures as a necessary environment for existence, as a kind of synchronizers, regulators, harmonizers of a person’s emotional status - as something that sets the rhythm inner life. He rejected cinema in this role, considering it not art, but entertainment. The reasoning he expressed was as follows: after listening piece of music or reading poetry arises

9. Sosinsky A. B. A. N. Kolmogorov in the memoirs of students. "Quantum", 1988, No. 11-12, art. 8.

desire for immediate repetition (of course, if you liked the music or poetry); After watching the film, such a desire does not arise. In the spring of 1965 (namely, on that day in early May when Kolmogorov met with V.A. Uspensky with Lotman), he made an attempt to captivate Kolmogorov with a recording of Galich, who, as it seemed to him, had achieved highest peaks in its genre. He chose one song - about how the bastards of physics spun the ball the other way around on a bet. This song was chosen because in it, through its attitude lyrical hero, a deep philosophical idea is expressed; the idea is the belief in the limitless power of science and the conviction that nothing good can come from the realization of this power. For Kolmogorov, however, Galich turned out to be contraindicated (this despite the fact that Kolmogorov recognized the possibility of experiencing catharsis under the influence of song.

Kolmogorov considered the novel to be the highest form of prose and said: “The greatest writers of the 20th century are Thomas Mann and Anatole France” 10 . And many people remember Kolmogorov’s disrespectful statements about Dickens, whose works he called “a kerosene stove for warming up the feelings of old maids” 11 .

As for Russian prose, from modern writers he praised Soloukhin. Andrei Nikolaevich loved nature very much and really loved Prishvin’s “Spring”, loved the expression “spring of light and water.”

Regarding A.I. Solzhenitsyn, he responded something like this: “I completely listened to the “GULAG Archipelago” on Western radio, I know that everything described there is true, but I categorically disagree with the author’s tough position: he writes that the communists , fighters for the revolution, who were shot or ended up in camps, deserved such a fate that “it serves them right.” 12 That is, Andrei Nikolaevich criticized Solzhenitsyn not from the “right”, but from the “left” - for his lack of humanism, which he could not forgive anyone for. At the same time, he was very fond of many of Solzhenitsyn’s works, especially “In the First Circle,” where the prototype of the “sharashka” artist was Andrei Nikolaevich’s gymnasium friend, the artist S. N. Ivashev-Musatov.

Even Pushkin was not forgiven for the lack of humanism. Kolmogorov reproached him for shooting with Dantes, wanting him

10, 11, 12. A. B. Sosinsky. Conversation with A. N. Kolmogorov. "Quantum", 1983, No. 4, art. 7.

death, shot at him, shouted “Bravo” when he collapsed after the shot... “After all, he wanted him dead,” Andrei Nikolaevich said excitedly. But Andrei Nikolaevich felt great admiration for Pushkin the poet.

He knew and loved not only Pushkin. Extensive quotations from Russian poetry (in particular from Sologub and Akhmatova) appear in Kolmogorov's letters to his closest friends. V. M. Tikhomirov writes: “Andrei Nikolaevich loved Tyutchev very deeply and intimately, felt a huge spiritual contact with Blok, and perceived Yesenin very touchingly and brightly (here we especially agreed with him). Klmogorov studied Mayakovsky a lot and often spoke about his poetry with admiration, although these two personalities - Kolmogorov and Mayakovsky - still did not have singular points contact." 13

Somehow the conversation turned to poetry, and Andrei Nikolaevich asked who V. M. Tikhomirov liked among modern poets (Akhmatova and Pasternak were alive, but he considered them as if from the last century). He named Slutsky and Martynov.

Andrei Nikolaevich grew gloomy. “This is strange, Volodya, I thought differently about you. It turns out that you are a supporter of rational poetry. But the essence of poetry is to express the inexpressible!” 14

Andrei Nikolaevich was very fond of Tyutchev and Yesenin. He said this about Yesenin’s poetry: “In terms of poetic talent, I place Yesenin above Pasternak, which angers Pasternak’s lovers” 15.

And more about poetry. Once V. M. Tikhomirov expressed surprise that he could like Mayakovsky. With irritation, he objected: “You, of course, have a point of view about which poets I should like and which ones I should not. But I just love good poems and don’t like bad ones” 16. However, if we consider Mayakovsky to be an optimist (which, as they say, is not “unequivocal”), then there were reasons for his surprise: Kolmogorov once told him that, being an optimist in life, he dislikes optimism in literature.

Kolmogorov was always somewhat distrustful of the fact that his interlocutor loved poetry, and always asked him to recite several

13, 16. A. B. Sosinsky. A. N. Kolmogorov in the memories of his students. "Quantum", 1988, No. 11-12, art. 14.

14, 15. A. B. Sosinsky. Conversation with A. N. Kolmogorov. "Quantum", 1983, No. 4, art. 9.

lines from a poet declared beloved. Not everyone survived this ordeal. He himself knew a lot by heart, even from poets he did not like.

Kolmogorov was connected with literature partly genetically. His father Nikolai Matveevich Kataev, although he served in the department of agriculture (he was, according to Kolmogorov, a “scientific agronomist”), wrote stories and published them from time to time in magazines; at a personal meeting in Yalta, Chekhov predicted to him literary destiny, which, however, did not take place. With greater certainty, the literary gene manifested itself in the lateral line passing through Ivan Matveevich, sibling Nikolai Matveevich (they were two of three sons dean from the Urals): his son was famous writer Ivan Kataev, who was thus Kolmogorov's cousin. Kolmogorov’s cousin, son of I. I. Kataev, Georgy Ivanovich Kataev, recalls:

“...Andrei Nikolaevich, in particular, cited some results of the work carried out: “E. Bagritsky advanced the furthest in the development of iambic. Analysis of pauses in his poems, for example, provides material for the psychology of cognition...” On other occasions he said that of all Russian poets, Pushkin is the most informative. A comparison of E. Yevtushenko with A. Voznesensky showed that the former is more informative1. Voznesensky didn’t like this, he wanted to meet with Kolmogorov, but he refused...” 17

A. N. Kolmogorov loved books very much. Here is one typical episode. A group of scientists was in Hungary at a congress on information theory. We received 1300 forints. Since they were fed and watered there, each one developed large sum money. And the question arose: what to buy? Kolmogorov immediately asked about book Shop. Arriving at the store, he immediately saw a book of Michelangelo’s drawings, which cost 1,200 forints, bought it, spent everything he had, and said to his students: “And you’ll buy me a ticket for the tram.”

A. N. Kolmogorov was a passionate and tireless skier, he loved long-distance ski trips. Once he asked to invite him

____________________________________________________________

17. Nikolai Gorbachev. What does it mean to be a mathematician? “Smena”, 1978, No. 12, art. 45.

one of the strong senior skiers to make a particularly long trek. His request was conveyed to a student with a junior rank, and he agreed. At first he frolicked, ran away, waited for Andrei Nikolaevich, then he walked next to the shore of power, and in the end Andrei Nikolaevich simply carried his skis.

Andrei Nikolaevich once told one of his students that humanity appears to him as wandering lights in the fog, which only vaguely sense the light scattered by others. But these words cannot be attributed to him: he was not only a great scientist, not only a great teacher, but also a great Enlightener. Andrei Nikolaevich belonged to those incomparable geniuses who illuminate life by the very fact of their existence.

CONCLUSION

The 20th century is the century of the atom, electronics and cybernetics, the century of great space exploration and discoveries. All this became possible thanks to the progress of mathematical science. Only modern mathematical methods allow people to solve important technical problems and introduce automation into production. We appreciate outstanding achievements Domestic mathematicians XX century.

The rapidly increasing time distance allows us to better understand the scale of the personality of Andrei Nikolaevich Kolmogorov, to evaluate his strategy for the fundamentalization of university education, his democracy, and the depth of pedagogical thinking.

A brilliant Scientist, a great Enlightener, a wonderful Person - the name of Andrei Nikolaevich Kolmogorov is inscribed in golden letters among the galaxy of the greatest people on the planet.

BIBLIOGRAPHY

· Great Soviet Encyclopedia. – M. 1981

· V. D. Chistyakov. Stories about Mathematicians. – M.: Education, 1964.

· T. A. Doronina. Next to Andrei Nikolaevich. – M.: Education, 1984.

· Nikolai Gorbachev. What does it mean to be a Mathematician? "Change", 1978, No. 12.

· A. B. Sosinsky. A. N. Kolmogorov in the memories of his students. "Quantum", 1988, No. 11-12.

· A. B. Sosinsky. Conversation with A. N. Kolmogorov. "Quantum", 1983, No. 4.

· P. A. Livansky. Mathematical talents. "Quantum", 1985, No. 7.

· V. A. Uspensky. Preface for readers of “UFO” to the semiotic messages of A. N. Kolmogorov. "UFO", 1997, No. 24.

ILLUSTRATIONS

• CONTENT:
  From the editor (3).
  Andrey Nikolaevich Kolmogorov ( Curriculum Vitae) (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. On the convergence of Fourier series (16).
  5. Axiomatic definition of the integral (19).
  6. On the limits of generalization of 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. On the convergence of Fourier series (69).
  11. Fourier-Lebesgue series, diverging everywhere (73).
  12. On the convergence of orthogonal series (75).
  13. On operations on sets (85).
  14. On the Denjoy integration process (93).
  15. On the topological-group-theoretic justification of geometry (94).
  16. Study of the concept of integral (96).
  17. On determining the average (136).
  18. On the compactness of sets of functions with convergence in mean (139).
  19. Towards the interpretation of intuitionistic logic (142).
  20. Towards the justification of projective geometry (149).
  21. Towards the theory of measure (150).
  22. About the breakpoints of the functions of two pepemen (167).
  23. On the normalizability of general linear topological spaces! (168).
  24. Continuation of the study on discontinuity points of a function of two variables (171).
  25. On the convergence of series in orthogonal polynomials (174).
  26. Laplace transform in linear spaces (178).
  27. On the order of the remainder term of Fourier series of differentiable functions (179).
  28. On the best approximation of functions of a given functional class (186).
  29. On the laws of duality 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. About 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 one biological problem (221).
  39. Simplified proof of the Birkhoff-Khinchin ergodic theorem (246).
  40. About inequalities between top edges successive derivatives arbitrary function on an infinite interval (252).
  41. About rings continuous functions on topological spaces (264).
  42. Curves in Hilbert space invariant under the one-parameter group of motions (269).
  43. Wiener spiral and some other interesting curves in Hilbert space (274).
  44. Points of local topology of countably multiple open maps of compact sets (278).
  45. Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers (281).
  46. ​​Towards 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 about polynomials by P.L. Chebyshev, who deviate least from the given function (296).
  50. On the fragmentation of droplets 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 issues 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 smaller number variables (335).
  56. On the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition (340).
  57. On the linear dimension of topological vector spaces (344).
  58. Clarification of ideas about the local structure of turbulence in an incompressible viscous fluid at high Reynolds numbers (348).
  59. P.S. Alexandrov and the theory of bs-operations (352).
  60. Qualitative study of mathematical models of population dynamics (357).

Andrey Nikolaevich Kolmogorov(April 25, 1903, Tambov - October 20, 1987, Moscow) - one of the most outstanding mathematicians of the 20th century, a person with the broadest mathematical horizons. He is one of the main initiators of the founding of the Moscow boarding school FMS 18 (now the SSC Moscow State University named after A.N. Kolmogorov). Andrei Nikolaevich is primarily known for his integral contribution to such areas of mathematics as topology, geometry, functional analysis, measure theory, theory of differential equations, theory of dynamical systems, information theory, classical mechanics and many others; in fact, he is the founder of the modern axiomatics of probability theory.

Andrei Nikolaevich was born on April 12 (25), 1903 in Tambov in the family of Nikolai Matveevich Kataev and Maria Yakovlevna Kolmogorova. His parents left him at an early age, so he was raised in Yaroslavl by his mother's sisters. Even then, Kolmogorov showed amazing mathematical abilities.

In 1920, Andrei Nikolaevich entered the mathematics department of Moscow University. Having passed all the exams for the course in the first months, Kolmogorov began his scientific activity, gradually solving more and more complex problems. This is how Andrei Nikolaevich was noticed by the famous theorist of real analysis, Nikolai Nikolaevich Luzin, who became his scientific supervisor. In 1922, Kolmogorov constructed the famous example of a Fourier series diverging almost everywhere, which gained worldwide fame.

In the first half of the 20th century, many theoretical necessary questions theories of measure, real analysis, functional analysis and probability theory gradually emerged. Many outstanding mathematicians, such as David Hilbert, Richard Courant, A.Ya. Khinchin, in fact, N.N. Luzin, worked in this area. Andrei Nikolaevich did not stand aside either. The young Kolmogorov first obtained the law of large numbers, and in 1933 he published for the first time famous work"Basic Concepts of Probability Theory", published in German. This work represented the exact axiomatics of the theory of probability, which leading minds had been thinking about since the beginning of the century.

In 1931, Kolmogorov became a professor at Moscow State University, from 1933 to 1939 he was director of the Institute of Mechanics of Moscow State University, founded and for many years headed the Department of Probability Theory of the Faculty of Mechanics and Mathematics and the Interfaculty Laboratory of Statistical Methods. The degree of Doctor of Physical and Mathematical Sciences was awarded to Kolmogorov in 1935 without defending a dissertation. In 1939, 35-year-old Kolmogorov was immediately elected as a full member (skipping the title of corresponding member) of the USSR Academy of Sciences, a member of the Presidium of the Academy and, at the suggestion of O.Yu. Schmidt, academician-secretary of the Department of Physical and Mathematical Sciences of the USSR Academy of Sciences.

All this time, Andrei Nikolaevich has been engaged not only theoretical problems, but also practical. So, during the war, you can see the results associated with the dispersion of shells (necessary in such a difficult period for the homeland), and after that he deals with issues of turbulence. In the 1950s and 1960s, along with the development of random processes as a separate discipline and the gradual exploration of space, Kolmogorov wrote many works related to these areas. In particular, Andrei Nikolaevich proves a number of facts from celestial mechanics; many results related to dynamic systems, the famous KAM theory. At the same time, the theory of algorithms and information theory were also developing, in connection with which Kolmogorov introduced the concept of algorithm complexity and, in accordance with this, set the task of measuring complexity.

Around the mid-1960s, a rethinking of the teaching system took place in the USSR. All over the country they are starting to create specialized schools. In particular, in 1963, in Moscow (as well as in Kiev, Novosibirsk and Leningrad), the Specialized Boarding School No. 18 of physics and mathematics was founded (now the SUSC MSU named after A.N. Kolmogorov), one of the initiators of the creation which was delivered by Andrei Nikolaevich. Teaching at Physics School 18 and Moscow University, in 1970, together with academician I.K. Kikoin Kolmogorov creates the magazine “Quantum”. At the end of his life, Andrei Nikolaevich focuses on teaching. Even at school, his first priority was the development of creative thinking: “It is important that here at the boarding school, schoolchildren come into contact with creative thought. This is our request, but for all subjects!.. The method of work is imitation scientific research, step by step, find, calculate something..., and not give something ready-made...".

Academician A.N. Kolmogorov died on October 20, 1987 in Moscow, at the age of 84. He was buried at the Novodevichy cemetery.

Selected publications

• A.N.Kolmogorov, Grundbegriffe der Wahrscheinlichkeitrechnung, in Ergebnisse der Mathematik, Berlin. 1933.
• A. N. Kolmogorov, On operations on sets, Mat. Sat., 1928, 35:3-4
• A. N. Kolmogorov, General theory of measure and probability calculus // Proceedings of the Communist Academy. Mathematics. - M.: 1929, vol. 1. S. 8 - 21.
• A. N. Kolmogorov, Ob analytical methods in probability theory, Uspekhi Mat. Nauk, 1938:5, 5-41
• A. N. Kolmogorov, Basic concepts of probability theory. Ed. 2nd, M. Nauka, 1974, 120 p.
• A. N. Kolmogorov, Information theory and theory of algorithms. - M.: Nauka, 1987. - 304 p.
• A. N. Kolmogorov, S. V. Fomin, Elements of the theory of functions and functional analysis. 4th ed. M. Science. 1976 544 p.
• A. N. Kolmogorov, Theory of probability and math statistics. M. Science 1986, 534 p.
• A. N. Kolmogorov, “On the profession of mathematician.” M., Moscow University Publishing House, 1988, 32 p.
• A. N. Kolmogorov, “Mathematics - science and profession.” M.: Nauka, 1988, 288 p.
• A. N. Kolmogorov, I. G. Zhurbenko, A. V. Prokhorov, “Introduction to probability theory.” M.: Nauka, 1982, 160 p.

On the initiative of Abramov A.M. (graduated from Physics School No. 18 at Moscow State University in 1964), Vavilova V.V. and Tikhomirov V.M. and with the support of the director of the Russian State Library Visly A.I. (graduated from Physics School No. 18 at Moscow State University in 1975) staffthislibrariescompiled a list of publications about the life and work of A.N. Kolmogorov, starting in 1941.