André Andrey Kolmogorov was born on April 25th, 1903 in Tampov province, Russia, and died on October 20th, 1987/kloc-0. His grandfather was a priest and his father Kataev was an agronomist. After the October Revolution, he was exiled to China and served as the leader of a Ministry of Agriculture. He died in 19 19. My mother was born into a noble family and died in childbirth. Andre Andrey Kolmogorov spent his childhood in his grandfather's house, and his menstruation brought him up. Although he lost his maternal love after his birth, and never got his father's love, Andre Andrey Kolmogorov grew up in love. At an early age, menstruation taught him to love learning knowledge and nature. At the age of five or six, Andre Andrey Kolmogorov discovered the relationship between odd numbers and square numbers:1=12,1+3 = 2 2,1+3+5 = 3 2,1+3+. Grandfather's family runs a family magazine, Chun Yan, and young Andre Andrey Kolmogorov is in charge of the math column. He published his findings in a magazine.
At the age of six, he went to Moscow with his period, where he was considered to be the most progressive preparatory school. While studying, André Andrey Kolmogorov had a wide range of interests. He studies biology and physics seriously. /kloc-When he was 0/4 years old, he learned advanced mathematics from an encyclopedia. He is also interested in chess, social issues and history.
1920 After graduating from high school, Andre Andrey Kolmogorov worked as a train conductor for a while; After work, he wrote a pamphlet about Newton's laws of mechanics. In the same year, Andrei Andrey Kolmogorov Love entered Moscow University to study. Besides mathematics, he also studied metallurgy and Russian history. He was particularly fascinated by history, and once wrote a paper on the property of landlords in Novgrad in the 6th century15-/kloc-0. About this paper, his teacher and famous historian S.V.Bakhrushin said:
"You provided a proof in your paper, which may be enough in the mathematics you studied, but it is not enough for a historian. He needs at least five kinds of proof. "
Perhaps the history professor's answer had an important influence on André Andrey Kolmogorov: he chose mathematics that only needed one kind of proof.
Break into the kingdom of mathematics
At Moscow University, André Andrey Kolmogorov listened to the class of the great mathematician Luzin (N.N. Luzin, 1883- 1950), and had a conversation with Luzin and Aleksandrov (P.S.Alexandrov,1896-1950). André Andrey Kolmogorov also took part in the discussion class of stepanov trigonometric series (1889-1950), which solved a problem raised by Luzin. Jin Lu admired him very much and offered to accept him as an apprentice.
Although Andre Andrey Kolmogorov is only a college student, he has made remarkable achievements: in February 1922, he published a paper on set operation, which popularized suslin's results; In June of the same year, he published a Fourier series that diverged almost everywhere (to 1926, and he further constructed a Fourier series that diverged everywhere). According to himself, he came up with this series on the train when he was a train conductor. Andre Andrey Kolmogorov became a shining star in the world of mathematics for a while. Almost at the same time, he became interested in many other fields in analysis, such as differential and integral problems, measure theory and so on.
1925, graduated from André Andrey Kolmogorov University and became a graduate student in Jin Lu. This year, Andre Andrey Kolmogorov published eight papers written during his college years! In every paper, he introduced new concepts, new ideas and new methods. His first paper on probability theory was published this year. This paper was completed in cooperation with Qin Xin (A.Y. Khinchin, 1894- 1959), which included the theorem of trigonometric series and the inequality about the partial sum of independent random variables, and later became the basis of martingale inequality and stochastic analysis. He proved a Chebyshev inequality of Hilbert transform, which later became the pillar of harmonic analysis. In 1928, he obtained the necessary and sufficient conditions for the sequence of independent random variables to satisfy the law of large numbers; In the second year, the broad conditions of the law of iterated logarithm were discovered. In addition, his work also includes differential and integral operations and some generalizations of intuitionistic logic.
1929 In the summer, André Andrey Kolmogorov and Aleksandrov sailed from yaroslavl, crossed the Caucasus mountains along the Volga River, and finally arrived at Sevan Lake in Armenia, where they settled on an island. There, while enjoying swimming and sunbathing, Aleksandrov wore sunglasses and a Panama hat and wrote a book on topology in the sun. This book cooperated with hopf (H.Hopf,1894-1971) and became a classic as soon as it came out. André Andrey Kolmogorov studied continuous state and continuous time Markov processes in the shade of trees. André Andrey Kolmogorov's achievement is published in 193 1, which is the origin of diffusion theory. The lifelong friendship between the two began with this long trip. Aleksandrov later recalled:
"1979 is the 50th anniversary of my friendship with Andre Andrey Kolmogorov. Throughout the half century, this friendship has never stopped, and there has never been a quarrel. There has never been any misunderstanding between us on any issues, no matter how important these issues are to our life and our philosophy; Even if there are differences on a certain issue, we fully understand and sympathize with each other's views. "
Andre Andrey Kolmogorov regards this friendship as the reason for his happy life!
/kloc-in the summer of 0/930, André Andrey Kolmogorov and Aleksandrov made another long trip. This time they went to Berlin, G? ttingen, Munich and Paris. Andre Andrey Kolmogorov met Hilbert (D. Hilbert, 1862- 1943), Courand (R. Courand, 1888- 1972) and Landau (E. Landau,/kloc- 1885- 1955), Karatai audrey (Karatai audrey, 1873- 1950), Fréchet (Fréchet,1878-/kloc-0). 187 1- 1956)、Levi (P.Lévy， 18486- 197 1)、Lebesgue (H.Lebesgue， 1875-66)。
The period of 1930 is the second creative peak of Andre Andrey Kolmogorov's mathematical career. During this period, he published more than 80 papers, involving probability theory, projective geometry, mathematical statistics, real variable function theory, topology, approximation theory, differential equations, mathematical logic, biological mathematics, philosophy, mathematical history and mathematical methodology. 193 1, André Andrey Kolmogorov was hired as a professor by Moscow University. 1933 published "Fundamentals of Probability Theory", a classic work of probability theory. This book bases probability theory on strict axioms for the first time, solves the probability part of Hilbert's sixth question, marks the beginning of a new stage of probability theory development, and has epoch-making significance. In the same year, André Andrey Kolmogorov published an important paper "Analytical Methods in Probability Theory", which laid the foundation of Markov random process theory. Since then, Markov process theory has become a powerful scientific tool.
In topology, Andre Andrey Kolmogorov is one of the founders of linear topological space theory; He and the famous American mathematician J.W. Alexander (1888- 197 1) independently introduced the concept of cohomology group at the same time. In 1934, Andre Andrey Kolmogorov studied the cohomology of chain, winding, homology and finite cell complex. In the paper published in 1936, Andre Andrey Kolmogorov defined the concept of cohomology group of any locally compact topological space. 1935, at the Moscow International Topology Conference, André Andrey Kolmogorov defined the cohomology ring.
1935, Andrei Andrey Kolmogorov and Alexanderov bought an old mansion in a small village outside Moscow called komarov Ka. Much of their math work was done here. Many famous mathematicians have been to Komalov Card, including J. Hadamard (1865- 1963), Fréchet, Banach (S.Banach, 1892- 1945), hopf and K. Kuratowski. Graduate students from Moscow University often go together. In the evening, although the students are a little tired, they always return to Moscow happily with their math gains. Malcef (A.I.Malcev, 1909- 1967) and Gelfand (I.M.Gelfand, 19 13-) who later became academicians of the Soviet Academy of Sciences are two of them. Andre Andrey Kolmogorov's doctoral student and famous mathematician Gnedenko (B.V.Gnedenko,1912-1995) recalled:
"For all the students of André Andrey Kolmogorov, the years they studied under the guidance of André Andrey Kolmogorov are unforgettable: their efforts in science and culture, their great progress in science, and their dedication to scientific issues. Unforgettable is the outing on Sunday. Andre Andrey Kolmogorov invited all his students (graduate students or undergraduates) and other tutors' students. In these outings from 30 to 35 kilometers away to Porche, Kryazma and other places, we have been discussing the current problems of mathematics (and its application) and cultural progress, especially painting, architecture and literature. "
At the end of 1930, Andre Andrey Kolmogorov developed the theory of stationary stochastic processes, and the American mathematician Wiener (N.Wiener, 1894- 1964) later got the same result. Andre Andrey Kolmogorov also expanded his research field to planetary motion and air theory of turbulence.
In the1940s, Andre Andrey Kolmogorov's interest turned to application. 194 1 year, he published two papers on turbulence of great significance, which became one of the most important contributions in the history of theory of turbulence. A famous result obtained by Andre Andrey Kolmogorov is the "two-thirds law": in turbulent flow, the square average of the velocity difference between two points with a distance of r is directly proportional to r2/3.
During this period, in addition to mathematics, André Andrey Kolmogorov also made important contributions to genetics, ballistics, meteorology and metal crystallography. In the paper published in 1940, Andre Andrey Kolmogorov proved that the materials collected by the followers of lysenko (1898- 1976) supported Mendel's law. At that time, Mendel's law was criticized in the Soviet Union, and André Andrey Kolmogorov's paper reflected his scientific spirit of pursuing truth.
The period from 65438 to 0950 was the third creative peak of André Andrey Kolmogorov's academic career. The research fields in this period include classical mechanics, ergodic theory, function theory, information theory, algorithm theory and so on.
1953 and 1954, Andre Andrey Kolmogorov published two papers on dynamical systems and their applications in Hamiltonian dynamics, which marked the beginning of KAM (Kolmogorov-Arnold- Moser) theory. From 65438 to 0954, André Andrey Kolmogorov was invited to give an important report on "General Theory and Classical Mechanics of Dynamical Systems" at the International Congress of Mathematicians in Amsterdam. Later research proved his profound insight.
During this period, Andre Andrey Kolmogorov also began to study automata theory and algorithm theory. He and his student V.A.Uspenskii established an important concept called "Andre Andrey Kolmogorov -Uspenskii machine". He also opposed it and supported the research of computing theory. Many Soviet computer scientists are students or students of André Andrey Kolmogorov. In the middle and late 1950s, André Andrey Kolmogorov devoted himself to the research of information theory and ergodicity theory of dynamical systems. He introduced the important concept of entropy into the dynamic system theory, which opened up a vast new field and later led to the birth of chaos theory. From 1958 to 1959, André Andrey Kolmogorov applied ergodic theory to a kind of turbulence phenomenon, which had a far-reaching impact on later work.
1957, Andre Andrey Kolmogorov and student Arnold completely solved Hilbert's 13 problem: there are continuous ternary functions, which cannot be expressed as the superposition of binary continuous functions. The answer is no: the continuous function of any number of variables can be expressed as the superposition of single-variable continuous functions.
/kloc-after the 1960s, Andre Andrey Kolmogorov founded two branches of calculus information theory (now called "Andre Andrey Kolmogorov's complexity theory") and calculus probability theory.
Andre Andrey Kolmogorov's research covers almost all fields of mathematics except number theory. 1963 At the conference on probability statistics held in Tbilisi, American statistician J. Wolfowitz (1910-1981) said, "The special purpose of my coming to the Soviet Union is to determine whether André Andrey Kolmogorov is a person or a research institution."
Unique teaching and research methods
During his academic career of more than half a century, André Andrey Kolmogorov kept raising new questions, constructing new ideas and creating new methods, and maintained lasting vitality in the world mathematics stage, which was due to his healthy body to some extent. He loves physical exercise very much and is called "outdoor mathematician". He and Aleksandrov live in Komalov card four days a week (the other three days in the school apartment in the city). Physical exercise lasts a whole day: skiing, boating and hiking (the average distance is as long as 30 kilometers). On a sunny March day, they often wear ski shoes and shorts to exercise outdoors for four hours. Exercise is uninterrupted in the morning on weekdays, and you have to run again in winter 10 km. They also like to swim in the water when the river ice melts. During the celebration of Andre Andrey Kolmogorov's 70th birthday, he organized a skiing trip. Andre Andrey Kolmogorov, wearing shorts and bare arms, is old and strong, leaving all the other participants behind!
Many of his wonderful and key ideas are often born while walking in the Woods, swimming in the lake and skiing on the hillside. During his visit to India on 1962, he even suggested that all universities and research institutes in India should be built on the coastline, so that teachers and students can swim first and then start serious discussions.
Andre Andrey Kolmogorov is also a famous mathematics educator. He is particularly interested in the plan of providing special education for students with mathematical talent. He believes that some parents and teachers try to dig out children with mathematical talent from students aged 10 ~ 12, which will do harm to children. But when 14 ~ 16 years old, the situation changed. It is usually obvious whether children of this age are interested in mathematics. About half of them think that mathematical physics is of little use to them, and these students should take special simplified courses. The math education of the other half students can be carried out more effectively. When choosing a major in mathematics, these students should also test their adaptability to mathematics-calculation ability, geometric intuition ability and logical reasoning ability.
André Andrey Kolmogorov founded the boarding school for mathematics at Moscow University. Over the years, he spent a lot of time in school, drawing up teaching syllabus, compiling teaching materials, giving lectures (up to 26 hours per week), leading students to explore on foot, teaching students music, art and literature, and seeking the natural development of children's personality. The students in his school are often among the best in the all-Soviet and international mathematical olympiads. But for those students who can't become mathematicians, he is not worried. No matter what occupation they end up in, he will be satisfied if they can keep their eyes open and curiosity. How lucky a student will be if he can enter Andre Andrey Kolmogorov's big family!
As one of the most outstanding mathematicians in the 20th century, André Andrey Kolmogorov won many honors: 194 1 the first Soviet State Award in; 1949 won the Chebyshev Prize of the Soviet Academy of Sciences; 1963 won the international bazin award; 1965 won the Lenin Prize; 1976 won the Helmholtz medal of the Democratic German Academy of Sciences; He won the Wolf Prize with 1980 and the Lobachevsky Prize with 1986. He also won the Lenin Medal seven times before and after.
1939, André Andrey Kolmogorov was elected as an academician of the Soviet Academy of Sciences. It is also the Polish Academy of Sciences (1956), the Royal Statistical Society of London (1956), the Romanian Academy of Sciences (1957), the German Academy of Sciences (1959), the American Academy of Arts (1959), and the United States. Foreign academicians or honorary members such as the Royal Society of London (1964), Hungarian Academy of Sciences (1965), American National Academy of Sciences (1967), French Academy of Sciences (1968) and Finnish Academy of Sciences (1983). He was awarded honorary doctorates by the University of Paris (1955), Stockholm University (1960), Indian Institute of Statistics (1962), Warsaw University and Budapest University.