In stochastic mathematics-probability theory, stochastic process theory and mathematical statistics.

1924 When he was in the fourth year of college, he established a third-order theorem about independent random variables with the then Soviet mathematician Qin Xin. In 1928, he obtained the necessary and sufficient conditions for the sequence of random variables to obey the law of large numbers. In 1929, the law of iterated logarithm of independent and identically distributed random variable sequences is obtained. In 1930, a very general sufficient condition of strong law of large numbers is obtained. 193 1 published the article "analysis method of probability theory", which laid the foundation of Markov process theory. Markov process is widely used in physics, chemistry, biology, engineering technology and economic management, and it is still one of the hot spots and focuses of mathematics research in the world today. In 1932, the necessary and sufficient conditions for the infinitely separable distribution law of second-order moment random variables are obtained. 1933 published the book "Fundamentals of Probability Theory", which established the axiomatic conclusion of probability theory on the basis of measure theory and integral theory for the first time in the world. This is an epoch-making masterpiece, which wrote the most brilliant page of mathematics in the former Soviet Union in the history of science. 1935 put forward the concept of reversible symmetric Markov process and its necessary and sufficient conditions, which became an important model in statistical physics, queuing network, simulated annealing, artificial neural network and protein structure. 1936— 1937 gives the state distribution of countable Markov chains. 1939 defines and obtains the statistic of maximum deviation between empirical distribution and theoretical distribution and its distribution function. In 1930s-1940s, together with Qin Xin, he developed the theory of Markov process and stationary stochastic process, which was applied to the automatic control of artillery and industrial and agricultural production and played a neutral role in the Great Patriotic War. 194 1 year, he obtained the prediction and interpolation formulas of stationary random processes. From 1955 to 1956, he and his student, the Soviet mathematician prokhorov, initiated the weak limit theory of the probability measure of the mean value in the function space. This theory and D-space theory introduced by Soviet mathematician A.B. Skorokhod are epoch-making achievements of weak limit theory.

In pure mathematics and mathematics of deterministic phenomena

192 1 In his sophomore year, he began to study many complicated problems such as trigonometric series and operators on sets, which made him famous all over the world. 1922 defines the basic operations in set theory. 1925 proves that law of excluded middle is established in transfinite induction, constructs an intuitive calculus system, and proves a Chebyshev inequality in Hilbert transform. 1932 studies geometry from the viewpoint of topology and group theory. The cohomology group constructed by 1936 and its operation. 1935 ——1936 introduced approximation measure, which initiated a new direction of approximation theory. In 1937, the open mapping from one-dimensional compact set to two-dimensional compact set is given. From 1934 to 1938, the concepts of linear topological space and its bounded set and convex set are defined, which promotes the development of functional analysis. In the mid-1950s, he established the KAM theory with the third-year university student V.I.Arnord and the German mathematician J.K. Morsel, and solved the basic problems in the dynamic system. He used information theory to study the ergodicity of systems, which became a new starting point for the development of power system theory. From 1956 to 1957, he put forward the basic idea of solving problems, and his student Arnold completely solved Hilbert's 13 problem.

In applied mathematics

In biology, he first constructed the stable solution of nonlinear diffusion traveling waves in 1937, proposed the branching process and its extinction probability in 1947, and verified Mendel's law of gene inheritance in 1939. In metallurgy, 1937 studied the probability that a given point belongs to a crystal cluster and the average number of crystals in the process of random crystallization of metals. 194 1 year, the prediction and interpolation formulas of stochastic processes are applied to natural phenomena such as radio engineering, automatic control of artillery, atmosphere and ocean. In fluid mechanics, the approximate formula of local isotropic turbulence was obtained in the 1940s. Throughout his life, André Andrey Kolmogorov has made outstanding contributions to pure mathematics or applied mathematics, mathematics of deterministic phenomena or random mathematics, mathematics research and mathematics education.

Honorary awards

Because of his outstanding achievements, he enjoys a high reputation at home and abroad. He is a foreign academician of more than 20 national academies such as the United States, France, the Democratic Republic of Germany, the Netherlands, Poland and Finland, and a foreign member of the Royal Society. He is an honorary doctor of many universities, such as Paris University in France and Warsaw University in Poland. 1963 won the International Balm Award, 1975 won the Hungarian Medal, 1976 won the American Meteorological Society Medal, Helmholtz Medal of the Democratic Republic of Germany, and 1980 won the world's most famous Wolff Prize. In China, 194 1 won the national prize, 195 1 won the Chebyshev Prize of the Soviet Academy of Sciences, 1963 won the title of Soviet hero, 1965 won the Lenin Prize, and1946 won the Lenin Prize. From 1944 to 1979, he won seven medals of Lenin, Venus and Brave Labor in the Great Patriotic War; Received the October Revolution Medal from 1983; Won the Lobachevsky Prize from 1986.