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".

Analytical Methods

In **1931**, Kolmogorov's paper "On analytical methods in probability theory" came out.

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".

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 13^{th} 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

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).

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 1^{st} 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".

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 20^{th} 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 83^{rd} 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.

In **1940** , Kolmogorov published the paper "On a new confirmation of Mendel's laws", where he applied probabilistic methods to genetics problems.

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, 2^{nd} degree for a series of papers on the theory of random processes.

Honors

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.

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.