Selected works
Hard to find an epithet to describe the magnitude of A.N. Kolmogorov's contribution to science. Immense? Inestimable? It's as if these words don't even begin to capture his real impact. Toмavoid searching for epithets, we decided to compile facts in this section — key publications of Andrei Nikolayevich — and divided them into two categories: mathematical and non-mathematical. In the third subsection — 'Awards' — we listed the main state and scientific awards of A.N. Kolmogorov.
Trigonometric Series
In 1922 , Kolmogorov achieved his most notable result in the field of trigonometric series by constructing anмexample of a Fourier-Lebesgue series that diverges almost everywhere.
On the Convergence of Series
In 1924, jointly with Aleksandr Khinchin, Andrei Kolmogorov published his first paper on probability theory titled "On convergence of series whose terms are determined by random events".
Intuitionistic Logic
In 1925 , Kolmogorov published his first formalization of intuitionistic logic.
The Law of the Iterated Logarithm
"In proving the law of the iterated logarithm in 1927, Kolmogorov created a new method that entered the arsenal of fundamental tools of probability theory!" Albert Shiryaev said.
The Law of Large Numbers
Fundamental works on the conditions for the applicability of the law of large numbers and the strong law of large numbers date back to 1927−1929
General Measure Theory
In his 1929 paper "General Measure Theory and Probability Calculus", he presented the first version of an axiomatic construction of the foundations of probability theory.
Two Papers on Geometry
In the 1930-s , Kolmogorov published two papers on geometry: "On the Topological-Group- Theoretical Foundation of Geometry" and "On the Foundation of Projective Geometry". His works from this time on infinitely divisible laws provide a comprehensive answer to the problem posed by Bruno de Finetti.
Vladimir Arnold: "Andrei Nikolaevich always came up with a brilliant solution when it came to making sense of the findings, identifying new opportunities, and proposing a generalizing fundamental theory".
Vladimir Arnold: "Andrei Nikolaevich always came up with a brilliant solution when it came to making sense of the findings, identifying new opportunities, and proposing a generalizing fundamental theory".
Analytical Methods
In 1931, Kolmogorov's paper "On analytical methods in probability theory" came out.
Pavel Alexandrov and Aleksandr Khinchin on the significance of Andrei Kolmogorov's paper "On analytical methods in probability theory">>Fundamental Concepts of Probability Theory
In 1933 , one of Kolmogorov's most famous works, "Fundamental Concepts of Probability Theory", was published in German.
On Open Mappings
In the mid-1930-s , Kolmogorov actively engaged with problems in topology. In 1937 , he published the work "On Open Mappings", in which he provided an example of a continuous open mapping of a one-dimensional continuum onto a two-dimensional one.
Approximation Theory
In 1935−1936 , Kolmogorov published two works on approximation theory: "On the Order of the Remainder Term of Fourier Series of Differentiable Functions" and "On the Best Approximation of Functions of a Given Functional Class," which laid the foundation for a new direction in the theory of function approximations.
Markov Chains
In 1936−1937 , Kolmogorov focused on the asymptotic behavior of probability in the transitionfrom one state to another, with an indefinitely increasing number of steps for the Markov chains with a countable number of possible states.
On the Problem of Self-Organization
A joint paper by Andrei Kolmogorov, Nikolai Piskunov and Ivan Petrovsky titled "Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem" was published in 1937.
Theory of Turbulence
Before the war, Kolmogorov set out to explore the theory of turbulence. In 1941, he published his paper "Stationary sequences in Hilbert space".
Excerpts from Andrei Kolmogorov's Memoirs >>Theory of Shooting
In 1942 , Kolmogorov made several remarkable discoveries in the theory of shooting, which were published in a special issue of the MIAN Proceedings and which Kolmogorov nicknamed the "Shooting Book".
Laboratory of Atmospheric Turbulence
In 1946 , Kolmogorov resumed his turbulence research and established the Laboratory of Atmospheric Turbulence at the Institute of Theoretical Geophysics of the USSR Academy of Sciences.
Theory of Branching Random Processes
In 1947 , Kolmogorov's papers on the theory of branching random processes were published. According to Boris Sevastyanov, the term "branching process" coined by Kolmogorov was so apt that it was adopted worldwide as a calque translation.
Measures and Probability Distributions
In 1948 , in the report "Measures and probability distributions in functional space", Kolmogorovsuggested considering the distribution of a random process as a Borel algebra measure for a certain functional space. This research was continued by Kolmogorov's students Yury Prokhorov and Anatolii Skorokhod.
Algorithms
In 1953 , Kolmogorov presented his report "On the concept of an algorithm", where he offered a new definition of the term. Kolmogorov's student Vladimir Uspensky took up the development of this idea.
Theory of Dynamic Systems
At the International Mathematical Congress in 1954 , Kolmogorov presented a report titled "The general theory of dynamic systems and classical mechanics", proposing a new method that was later developed by his student Vladimir Arnold and the German mathematician Jürgen Moser, and known now as Kolmogorov-Arnold-Moser (KAM) theory.
Information Theory
In 1956 , Kolmogorov presented the report "Theory of Information Transmission", in which he outlined the basic ideas of information theory and explained the limits of its applicability.
Asymptotic Characteristics
In 1956, he wrote the article "On Some Asymptotic Characteristics of Completely Bounded Metric Spaces".
Hilbert's 13th Problem
In 1957 , Kolmogorov and his student Vladimir Arnold proposed a solution to the famous Hilbert's 13th problem in their paper "On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables".
Lebesgue Spaces
In 1958 , Kolmogorov published his paper "New metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces". Later, Kolmogorov's student Yakov Sinai published a paper giving a definition of entropy applicable to any dynamical system. Another student of Kolmogorov, Lev Meshalkin, put forward the first examples of non-trivial isomorphism forBernoulli automorphisms.
"Automata and Life"
On April 6, 1961, at Moscow State University, Kolmogorov gave a talk titled "Automata and Life", striking the audience with his penetrating and creative insights.
Information Entropy
In 1963 , Kolmogorov spoke at the Probability Theory section of the Moscow Mathematical Society, where he explained the origins of the algorithmic approach. Based on the concepts of algorithm and computable function, he formulated definitions of entropy and information quantity.
Information Quantity
1965 saw the publication of Kolmogorov's paper "Three approaches to the definition of the concept of information quantity".
Logical Foundations of Information Theory
In 1969, he authored the paper "Towards a logical foundation of information theory and probability theory", which laid the groundwork for algorithmic information theory (AIT).
Journeys around the World
In 1969 and 1971 , Kolmogorov went on round-the-world trips to explore micro and macro scale ocean turbulence.
Soviet-Japanese Symposium
In 1982, Kolmogorov gave a talk titled "On the logical, semantic and algorithmic foundations of probability theory" at the Soviet-Japanese symposium on probability theory and mathematical statistics.
Bernoulli Society Congress
In 1986 , at the 1st World Congress of the Bernoulli Society, Andrey Kolmogorov and VladimirUspensky presented a report "Algorithms and randomness" − the most complete presentation of the ideas and findings of Andrei Kolmogorov and his students in the field of the algorithmic approach to the definition of "randomness".
Andrei Kolmogorov's reply to his students' greetings on his 83rd birthday >>Pavel Alexandrov and Aleksandr Khinchin on the significance of Andrey Kolmogorov's paper "On analytical methods in probability theory"
"In the entire field of probability theory of the 20th century, one can hardly pinpoint another study that was so essential for further development of science and its applications as this paper by Andrey Nikolaevich. Today, a vast field of probability studies has evolved from this work, including the theory of random processes, which in terms of its scope and number of applications can compete with the classical parts of probability theory.
The differential "Kolmogorov equations" that govern the Markov processes and aremathematically justified in rigorous and broad manner, encompass all the particular cases of equations (Smoluchowski, Chapman, Fokker-Planck, etc.), which until then had been derived and applied by physicists on an ad hoc basis, without sufficient justification and clear understanding of their underlying premises.
These Kolmogorov equations have always served as the basis for a vast amount of research worldwide; they have proven to be fundamental both for further development of the theory and for the mathematical processing of a wide range of applied problems."
The differential "Kolmogorov equations" that govern the Markov processes and aremathematically justified in rigorous and broad manner, encompass all the particular cases of equations (Smoluchowski, Chapman, Fokker-Planck, etc.), which until then had been derived and applied by physicists on an ad hoc basis, without sufficient justification and clear understanding of their underlying premises.
These Kolmogorov equations have always served as the basis for a vast amount of research worldwide; they have proven to be fundamental both for further development of the theory and for the mathematical processing of a wide range of applied problems."
Excerpts from AndreI Kolmogorov's Memoirs
My interest in studying turbulent flows of liquids and gases emerged in the late 1930s. It immediately became clear to me that the main mathematical apparatus for research should be the theory of random functions of many variables (random fields), which was only just emerging at that time. Moreover, it became clear to me that it was difficult to hope for the creation of a self-contained pure theory. In the absence of such a theory, it would be necessary to rely on hypotheses derived from the analysis of experimental data. It was also crucial to recruit talented colleagues capable of working in this mixed approach, combining theoretical development with experimentation.
Andrey Kolmogorov's reply to his students' greetings on his 83rd birthday
"Someone here mentioned my seemingly endless youth. I am grateful for the compliment, but let me set some limits here. Old age is an inevitable and inescapable thing. Can old age be a happy time? Perhaps it can, if you decide to stop struggling for new achievements or else, reconcile yourself with the emptiness of old age. If you don't do either, old age can be as bright and joyfulas your youth, even though certain things, like doing sports or swimming in cold water, are now beyond your capability. Doctors are telling me that, objectively, I am in pretty good shape for my age, yet, my research "turnout" has declined, which implies some unfortunate limitations. I believe that my scientific career is complete now. This is sad but inevitable. In recent years, I have devoted myself to the school reform – a crucial task for our country.
If my age doesn't let me down, I can still do plenty of useful things, like writing school textbooks and books for young science enthusiasts. I would be happy to do both with as much energy and youthful enthusiasm as before. Yet, time goes by, and month after month, certain tasks keep getting postponed. That is why it is very important to choose the activity where you are hardly replaceable.
If I focus on textbooks for gifted children, I won't have time for school textbooks, so I am stuck at a crossroads here: if I fully devote myself to one task, I won't be able to put in as much energy in the other. This emotional stress is too much of a burden when you are old. That is why I highly value my young assistants, many of whom are present here today."
If my age doesn't let me down, I can still do plenty of useful things, like writing school textbooks and books for young science enthusiasts. I would be happy to do both with as much energy and youthful enthusiasm as before. Yet, time goes by, and month after month, certain tasks keep getting postponed. That is why it is very important to choose the activity where you are hardly replaceable.
If I focus on textbooks for gifted children, I won't have time for school textbooks, so I am stuck at a crossroads here: if I fully devote myself to one task, I won't be able to put in as much energy in the other. This emotional stress is too much of a burden when you are old. That is why I highly value my young assistants, many of whom are present here today."
Theory of Brownian Motion
In 1934 , Kolmogorov authored the paper "Random motions", offering a general description of Brownian motion with inertia. Kolmogorov's student, Nikolai Piskunov, largely enhanced mathematical theory based on this work.
Crystallization of Metals
In 1937, Kolmogorov's paper "On the statistical theory of metal crystallization" proposed a rigorous solution to the problem about the crystallization process speed.
Confirmation of Mendel's Laws
In 1940 , Kolmogorov published the paper "On a new confirmation of Mendel's laws", where he applied probabilistic methods to genetics problems.
Mechanism of tratification
In 1949, Kolmogorov published his "Solution of a problem in probability theory connected with the problem of the mechanism of stratification".
Great Soviet Encyclopedia
From the late 1940s, Kolmogorov worked at the "Great Soviet Encyclopedia" project, where he was in charge of the Mathematics Department. He not only prepared the glossary, selected authors, edited articles, but also wrote materials on a wide array of mathematical topics.
Speech Entropy
In the 1960s, Kolmogorov pursued active linguistic research, focusing on the analysis of speech statistics and the science of poetry. In 1960−1961, he developed a new method for the consistent estimation of "speech entropy".
New Educational Manuals
From 1964 to 1968 , Kolmogorov headed a joint commission of the USSR Academies of Sciences and Pedagogical Sciences tasked with developing teaching material for secondary schools. The commission developed new programs for grades 6-8 and 910. The textbooks "Algebra and introduction to Analysis for Grades 9-10" and "Geometry for Grades 6-8" were written under Kolmogorov's supervision.
Networks in Three-Dimensional Space
Andrei Kolmogorov and Yan Barzdin's paper "On the realization of networks in three-dimensional space" published in 1967 attempted to explain the arrangement of axons and neurons in the human brain.
Award
In 1941 , Andrei Kolmogorov and Aleksandr Khinchin were awarded the Stalin Prize, 2nd degree for a series of papers on the theory of random processes.
Honors
1940 Order of the Red Banner of Labor.
1944 Order of Lenin.
1945 Second Order of Lenin, Medal "For Valorous Labor in the Great Patriotic War". Kolmogorov received a total of seven Orders of Lenin.
1944 Order of Lenin.
1945 Second Order of Lenin, Medal "For Valorous Labor in the Great Patriotic War". Kolmogorov received a total of seven Orders of Lenin.
USSR Academy of Sciences Award
In 1949 , Kolmogorov received the Chebyshev Award from the USSR Academy of Sciences.
Balzan Prize
In 1963 , the Balzan Prize for Mathematics was first awarded, and its laureate was Andrei Kolmogorov. This was the highest recognition of the Russian scientist's contribution to world science.
Hero of Labor
In 1963, Kolmogorov was awarded the title of Hero of Socialist Labor.
Lenin Award
In 1965, Andrei Kolmogorov and Vladimir Arnold were honored with the Lenin Prize for their work on the theory of perturbations of Hamiltonian systems.
