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1 Andrei Nikolaevich Kolmogorov, 1903-1987

Transcript of 1 Andrei Nikolaevich Kolmogorov, 1903-1987. 2 Andrei Nikolaevich Kolmogorov, 1903-1910 Kolmogorov...


Andrei Nikolaevich Kolmogorov, 1903-1987


Andrei Nikolaevich Kolmogorov, 1903-1910

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.

I discovered the pleasure of mathematical discoveries early, at the age five or six years. I noticed the relations 1=12, 1+3= 22 , 1+3 +5=32 etc. (A.K.)

Kolmogorov's family


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.

Aunt’s pedagogical principles of raising him with high social ideals were:-responsibility-independence of opinion-intolerance toward idleness-intolerance to poorly performed tasks-desire to understand and not only to remember

Andrei Nikolaevich Kolmogorov, 1903-1910


Andrei Nikolaevich Kolmogorov, 1910-1920

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.

Moscow, 1910


Andrei Nikolaevich Kolmogorov, 1920-1925

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


Andrei Nikolaevich Kolmogorov, 1920-1925

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.

N. Luzin

V. Stepanov D. Egorov


The first page of the paper An almost everywhere diverging Fourier series (1922 )


Andrei Nikolaevich Kolmogorov, 1926-1929

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


Andrei Nikolaevich Kolmogorov, 1929-1933

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 Wahrscheinlichkeits-rechnung. 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


Andrei Nikolaevich Kolmogorov, 1934-1939

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.


Andrei Nikolaevich Kolmogorov, 1934-1939

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


Andrei Nikolaevich Kolmogorov, 1940-1945

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)


Andrei Nikolaevich Kolmogorov, 1946-1949

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 30th. 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.



Andrei Nikolaevich Kolmogorov, 1950-1959

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).


Andrei Nikolaevich Kolmogorov, 1950-1959

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.





Andrei Nikolaevich Kolmogorov, 1960-1970

Kolmogorov advanced the grandiose program of the simultaneous study of complexity, the stochastic properties of deterministic processes and 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 of information, and gave a new approach to the logical foundations of Probability Theory.

Statistical linguistics and text entropy. Kolmogorov organized competitions on recognition of sentence continuation. Laboratory of mathematical statistics was organized.


Andrei Nikolaevich Kolmogorov, Teacher

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


Andrei Nikolaevich Kolmogorov, Teacher

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.


Andrei Nikolaevich Kolmogorov 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.



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

A.N.Kolmogorov’s PhD students



Andrei Nikolaevich Kolmogorov

His usual working week: 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 last-years and graduate students, and researchers.


Andrei Nikolaevich Kolmogorov

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.


Andrei Nikolaevich Kolmogorov

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.


A.N.Kolmogorov as supervisor ( 1956-1960)

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. Appointed times at Moscow

Kolmogorov’s flat after seminars and in his house in Komarovka.


Andrei Nikolaevich Kolmogorov, 1946-1949

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.

Komarovka in winter


Andrei Nikolaevich Kolmogorov

1980-1987, Head of the chair of Mathematical logic.


Andrei Nikolaevich Kolmogorov, 25 April, 1903- 20 October,1987

Men are cruel but Man is kind (A.K.)