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Kolmogorov was born in Tambov, 25 April
His father Nikolai Kataev, the son of a priest, was an agronomist. He died in fighting in
1919. Kolmogorov's mother Maria
Kolmogorova tragically died during his birth.
Her sister, Vera Yakovlevna, very independent woman, brought Kolmogorov up and he always had the deepest affection for her.
Early Andrei’s childhood was passed in
Yaroslavl region, at her father's estate, his grandfather, Yakov Kolmogorov, a former big landowner.
1=1 2 , 1+3=
2 2 , 1+3 +5=3 2 etc. (A.K.)
Kolmogorov's family
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Together with his aunts, Andrei lived in village Tutoshna near Yaroslavl. Aunts organized a small school for around 10 children of different age. Education followed modern for that time pedagogical ideas.
-responsibility
-independence of opinion
-intolerance toward idleness
-intolerance to poorly performed tasks
-desire to understand and not only to remember
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Kolmogorov moved in Moscow. He studied at private Repman’s gymnasium organized by two intelligent and progressive women
Evgeniya Repman and Vera Fedorova. It was due to the family tradition and friendly atmosphere Andrei obtained a general principle to search for himself right serious deed.
Worked for while as a conductor on the railway.
1920, Graduated school.
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1920, Kolmogorov entered Moscow State
University. He also studied at the
Metallurgical Department of the Chemical-
Technological Institute. Besides mathematics and metallurgy he studied
Russian history.
Machinery were then as something more serious and necessary as pure science.
My first scholarly talk (which I gave at the age 17 at Moscow University) was at the seminar of professor Bakhrushin’s seminar on landowners in ancient Novgorod. (A.K.)
Crossroad moment.
Bakhrushin:” You offer only one proof. For a historian that is not enough. One needs at least five proofs”.
May be this moved Kolmogorov to prefer mathematics where just one proof is enough.
Moscow University, 1910
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N. Luzin
Nikolai Luzin’s school.
The seminar of Vyacheslav Stepanov on trigonometric series.
Teaching mathematics and physics in
Potyiliha secondary school.
1921, the first paper “Division into squares”.
1922, Trigonometric series.The paper on
Almost everywhere diverging Fourier series.
Worldwide atention (19 years old).
1924, The first results in probability theory.
Three series theorem for convergence of a series of random variables (joint with
A. Khinchin).
First paper on embedding classical logic into intuitional logic on the law of the excluded third.
V. Stepanov
D. Egorov
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The first page of the paper
An a lmost everywhere diverging
Fourier series (1922 )
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1925, Kolmogorov graduated from
Moscow State University and became
Luzin 's graduate student. Kolmogorov published eight papers in 1925.
1928, LLN (strong Law of Large Numbers) and LIL (Law of Iterated Logarithm) for sums of random variables. Kolmogorov inequality. Important methods and only the results.
Friendship with Pavel Sergeevich
Aleksandrov. The boat trip down the Volga then across the Caucasus mountains to lake Sevan in Armenia. Kolmogorov worked on Markov processes with continuous states and continuous time.
1928, the paper The general theory of measure and the calculus of probabilities.
The first sketch of the famous axioms.
A. Kolmogorov and P. Alexandrov
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In 1929 Kolmogorov completed his doctorate. By this time he had 18 publications
Researcher in Institute of Mathematics and
Mechanics of Moscow University.
06.1930-03.1931. The first abroad trip with P.
Aleksandrov. They visited Germany (Berlin,
Göttingen, Munich) and France (Paris). Kolmogorov met with Hilbert, Caratheodory, Kurant, Landau,
Levi, Frechet, Lebesque, Borel, Weyl, and other leading mathematicians
1931, Kolmogorov was appointed a professor at
Moscow University (28 years old). First PhD students.
1933, Grundbegriffe der Wahrscheinlichkeitsrechnung. Foundation of Probability theory. A rigorous definition of conditional expectation.
Probability Theory became a mathematical science.
The theory of probability as mathematical discipline can and should be developed from axioms in exactly the same way as Geometry and Algebra.
(A.K.)
A. Kolmogorov and P. Alexandrov
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1935, Full Doctor in Mathematics (without defence)
1937, Head of the chair of Probability theory
Analytic methods in probability theory which
Kolmogorov published in 1938 was laying the foundations of the theory of Markov random processes. Theory of stationary random processes as curves in Hilbert space.
Mathematical statistics.
Empirical distribution function test.
Topology.
Norms in linear topological spaces.
Algebraic topology. Introduced the notion of cohomology groups.
Approximation theory.
Best approximation of functions of a functional class.
Full member of the Academy of Sciences of the USSR.
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Aleksandrov and Kolmogorov bought a house in Komarovka, a small village outside Moscow, in 1935. Many famous mathematicians visited
Komarovka: Hadamard, Frechet, Banach, Hopf,
Kuratovski, and others.
Journal “Uspehi Matematicheskih Nauk” was organized. Kolmogorov was Editor in 1946-55,
1982-87 and always a member of the board.
Kolmogorov was elected Academician of
Academy of Science of USSR, the Secretary-
Academician of the Physics-Mathematical
Sciences Section of the Academy of Sciences
(1939-42), . The Department of Probability and Statistics was set up at the Institute of
Steklov and Kolmogorov was appointed as
Head of Department.
The house in Komarovka
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1941, the Second World War became in USSR.
Kolmogorov was evacuated in Kazan. He returned in
Moscow in 1942. Kolmogorov worked and essentially developed ballistics
Applied sciences.
Kolmogorov worked on biological problems, addressed problems in geology, meteorology, statistical control and crystallography.
Mathematics.
Stationary sequences: interpolation and extrapolation. The classic work (joint with B.
Gnedenko) Limiting distributions for sums of independent random variables. Foundation of theory of branching processes.
USSR State Prize (together with A. Ya. Khinchin in
1941), the Chebychev Prize of the USSR Academy of Sciences (together with B. V. Gnedenko in 1949).
1942, Kolmogorov married Anna Egorova. They studied together in Repman’ gymnasium in Moscow,
1910-1918.
Kolmogorov and
Anna Egorova, (a late photo)
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Turbulence theory. Law of two thirds: in a turbulent flow the average of the square of the difference of the speed of two points at a distance r from each other (average size) is proportional to r^{2/3}
Head of Turbulence Laboratory in the Institute of Theoretical Geophysics.
My interest to studying turbulence flows and gases arose in the end of 30 th . For me it became clear that the main mathematical research technique has to become theory of random functions of many variables (random fields) which was just born this time. (A.K.)
Joint Meeting of Academy of Science. Review of spectral theory of random processes.
All-Union Meeting of Mathematical Statistics,
1948. The main problems of Mathematical statistics.
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The rise in creative energy. Again, as in his younger days, the creation of important new concepts is combined with the solution of long standing and fundamental problems.
Classical mechanics.
The three body problem
(traced back to Newton and Laplace). This problem was solved in the general situation in
Kolmogorov's work for most initial conditions.
The methods of proof developed in this work played a great role in classical analysis, mechanics and other areas of natural science.
Later it was further developed in the work by V.
I. Arnold and J. Moser and became known under the name KAM-theory (the theory of
Kolmogorov-Arnold-Moser).
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Hilbert's 13th problem.
Any continuous function on an n -dimensional cube (any n ) can be represented as a superposition of continuous functions of three variables.
This reduces Hilbert's 13th problem to the concrete problem of representing functions on universal trees in three dimensional space. This last problem was solved by V. I. Arnold (solved under the guidance of A. N. in third year student work), thus refuting the hypothesis of Hilbert.
Developing ergodic theory Kolmogorov introduced a new metric invariant, the entropy of dynamical systems.
On the one hand, this solved the classic problem about the non-isomorphism of Bernoulli automorphisms, and on the other hand, a completely new field of research was opened, later leading to chaos theory.
Information theory, theory of automata, theory of algorithms. The first lectures on Cybernetics. Automata and life.
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Kolmogorov advanced the grandiose program of the simultaneous study of and complexity, the stochastic properties of deterministic processes regular indeterminacy of random processes and phenomena.
The experience of practically all his creative life is concentrated in this program. In his articles of 1965 and 1969 published in 'Problemi Peredachi Informatzi'
(Problems of Information Transfer) Andrei
Nikolaevich laid the foundation of a new field of knowledge, the algorithmic theory information , and gave a new approach to the logical foundations of Probability Theory.
of
Statistical linguistics and text entropy.
Kolmogorov organized competitions on recognition of sentence continuation.
Laboratory of mathematical statistics was organized.
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The 60th was the last ten year period when Andrei
Nikolaevich actively studied mathematics properly.
In this period he started to turn more often to pedagogical problems
1963, A mathematical boarding school, No 18 at the University of Moscow otherwise known as the
Kolmogorov school was organized. Kolmogorov took a special interest in the project to provide special education for gifted children from all over the country.
To this school he devoted a major proportion of his time over many years, planning syllabuses, writing textbooks, spending a large number of teaching hours with the children themselves, introducing then to literature and music, joining in their recreations and taking them on hikes, excursions, and expeditions.
Kolmogorov intensively worked on a new secondary school textbooks on mathematics.
Kolmogorov with pupils, 1963
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Andrei Nikolaevich has influenced many scholars
Nearly seventy graduate students worked under
Kolmogorov's supervision. Many of them later became leaders of various scholarly schools. The spectra of these schools is tremendous.
Probability theory: Yu. Prohorov (acad. A.Sc.,
USSR), A. Borovkov (corr.member,A.Sc.,USSR),B.
Sevastyanov (corr. member A.Sc.), B.Gnedenko
(acad. Ukranian A.Sc.,), S. Sirazdinov (acad.
Uzbekistan A.Sc.), Professors P. Dobrushin, V.
Zolotarev, L. Meshalkin, Yu.Belyaev, Ya.Rozanov, A.
Shlryaev, etc.
Mathematical Statistics: L.Bolshev (corr.mem.
A.Sc.).
Dynamical Systems, Mechanics : V. Arnold (acad.
A.Sc.) Professors V. Alexseev, Ya. Sinai.
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Approximation theory : S. Nikolski (acad. A.Sc.),
Professor V. Tihomirov, V. Erohin, etc.
Applied Physics, Mechanics: M. Millionchikov (acad.
A.Sc.).
Atmospheric Physics: A.Obukhov (acad. A.Sc.),
Professors M. Yaglom, G. Barenblatt.
Oceanology: A. C. Monin (corr. mem. A.Sc.).
Mathematical Logic: A. I. Mal’tsev (acad. A.Sc.),
Professor V. Uspenski.
Cybernetics: V, Mihalevich (acad. A.Sc.).
Information Theory: Professors M. Pinsker, L.
Bassaligo.
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 Abramov, Alekseev, Arato, Arnold, Asarin, Bavli, Barenblatt, Bassalygo,
Belyaev,
Bityutskov,Bolshev,Borovkov,Bulinskii,Verchenko,Vovk,Galperin,
Gelfand,Gnedenko,Gordeev,Dvoichenkov,Dmitriev,Dobrushin,
Dynkin,Erokhin,Zhurbenko,Zasukhin,Zolotarev,Ivashev-Musatov,
Kozlov V.V.,Kozlov M.V.,Levin,Leonov,Mal’tsev,Matveev,Martynov,
Martin-Löf,Medvedev,Meshalkin,Millionshchikov,Mikhalevich,Monin,
Nikol’skii,Obukhov,Ofman,Ochan,Penkov,Petrov,Pinsker,
ProkhorovA.V.,Prokhorov Yu.V.,Rozanov,Rozenblat-Rot,Sevast’yanov,
Sinai,Sirazhdinov,Skorokhod,Statulyavichus,Tikhomirov,Tulaikov,
Tyurin,Uspenskii,Fage,Fomin,Shen,Shilov,Shiryaev,Shmidov,Yunovich,
Yaglom
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His usual working week:
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Three days in week were in Moscow and the rest in Komarovka.
2 hours , a compulsory undergraduate course (usually appr. 200 students).
He developed courses Functional
Analysis (Analysis III), Mathematical
Logic etc.
2 hours, a free-choice course
(measure theory, information theory, random processes etc.)
2-4 hours a research seminar for lastyears and graduate students, and researchers.
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In the arts Kolmogorov was especially knowledgeable on ancient Russian architecture, all of Russian poetry, worldwide sculpture and painting. He spent a lot of time systematically listening music especially last years.
He liked to travel In the span of fifty years he travelled annually, with friends and later with many of his students. He was especially drawn to the Russian North, the Volga, the Crimea and the
Caucasus. Nature endowed him with excellent health, which gave him a chance to lead an active sports life, drawing the young along with him.
He was an expert cross-country skier , he could stand atop his skis for twelve hours while skiing large distances. He also engaged in swimming and downhill skiing and hiking.
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National Honours.
In 1939 he was elected to the
Academy of Sciences of the USSR. He received one of the first State Prizes in 1941, the Lenin Prize in 1965, the Order of Lenin on seven separate occasions, and the Lobachevsky Prize in 1987. He had the title of Hero of Socialist Labour.
International Honours.
He was also elected to the many other academies and societies including the Romanian
Academy of Sciences (1956), the Royal Statistical
Society of London (1956), the Leopoldina Academy of
Germany (1959), the American Academy of Arts and
Sciences (1959), the London Mathematical Society
(1959), the American Philosophical Society (1961), The
Indian Statistical Institute (1962), the Netherlands
Academy of Sciences (1963), the Royal Society of
London (1964), the National Academy of the United
States (1967), the French Academy of Sciences (1968).
Kolmogorov received the Balzan International Prize in
1962 and Wolf prize in 1981. Many universities awarded him an honorary degree including Paris, Stockholm, and
Warsaw.
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Belyaev Yuri 1956
Leonov Viktor 1957
Matveev Rostislav 1956
Meshalkin Lev 1956
Rozanov Yuri 1957
Shiryaev Albert 1957
Sinai Yakov 1957
Tikhomirov Vladimir 1957
Two types of research problems:
Submitted for personal research and all have been informed,
Problems open to be solved in competition.
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Appointed times at Moscow
Kolmogorov’s flat after seminars and in his house in Komarovka.
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Usually in the middle of a week
Kolmogorov invited several of his under-/postgraduate students in
Komarovka.
In a summer time, the long trip was planned. During the trip Kolmogorov discussed in turn with all students progress and difficulties in their research problems. By Academy’s car with sandwiches from Komarovka all drove out near 30 km from the house and returned back by foot. In winter time it was more short ski trip.
After such trip it was a joint dinner with all student and P. Alexandrov.
The evening ended up by listening classical music (Bach, Vivaldi, Mozart etc.) performed by the best musicians of the world.
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1980-1987, Head of the chair of Mathematical logic.
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(A.K.)
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