Andrey Kolmogorov
Encyclopedia
Andrey Nikolaevich Kolmogorov (25 April 1903 – 20 October 1987) was a Soviet
mathematician
, preeminent in the 20th century, who advanced various scientific fields, among them probability theory
, topology
, intuitionistic logic
, turbulence
, classical mechanics
and computational complexity
.
in 1903. His unwed mother died in childbirth and he was raised by his aunts in Tunoshna near Yaroslavl
at the estate of his grandfather, a wealthy nobleman. His father, an agronomist
by trade, was deported from Saint-Petersburg for participation in the revolutionary movement. He disappeared and was presumed to have been killed in the Russian Civil War
.
Kolmogorov was educated in his aunt's village school, and his earliest literary efforts and mathematical papers were printed in the school newspaper. As an adolescent he designed perpetual motion
machines, concealing their (necessary) defects so cleverly that his secondary-school teachers could not discover them. In 1910, his aunt adopted him and then they moved to Moscow, where he went to a gymnasium
, graduating from it in 1920.
In 1920, Kolmogorov began to study at the Moscow State University
and the Chemistry Technological Institute.
Kolmogorov gained a reputation for his wide-ranging erudition. As an undergraduate, he participated in the seminars of the Russian historian S.V. Bachrushin, and he published his first research paper on the landholding practices in the Novgorod Republic in the fifteenth and sixteenth centuries. At the same time (1921–1922), Kolmogorov derived and proved several results in set theory
and in the theory of Fourier series
(trigonometrical series).
almost everywhere
, gaining international recognition. Around this time he decided to devote his life to mathematics
. In 1925 Kolmogorov graduated from Moscow State University
, and began to study under the supervision of Nikolai Luzin
. He made lifelong friends with Pavel Alexandrov who involved Kolmogorov in 1936 in an ugly political persecution of their common teacher, the so-called Luzin case or Luzin affair. Kolmogorov (together with A. Khinchin) became interested in probability theory
. Also in 1925, he published his famous work in intuitionistic logic
— On the principle of the excluded middle, where he proved that under a certain interpretation all statements of classical formal logic can be formulated as those of intuitionistic logic. In 1929 Kolmogorov earned his Doctor of Philosophy degree, Ph.D.
, at the Moscow State University
.
In 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen
and Munich
, Germany, and then to Paris, France. His pioneering work About the Analytical Methods of Probability Theory was published (in German) in 1931. Also in 1931, he became a professor at Moscow University. In 1933, Kolmogorov published the book, Foundations of the Theory of Probability, laying the modern axiomatic foundations of probability theory
and establishing his reputation as the world's leading living expert in this field. In 1935, Kolmogorov became the first chairman of probability theory at the Moscow State University
. In 1939, he was elected a full member (academician) of the USSR Academy of Sciences
. In a 1938 paper, Kolmogorov "established the basic theorems for smoothing and predicting stationary stochastic processes" — a paper that would have major military applications during the Cold War
to come. Around the same years (1936) Kolmogorov contributed to the field of ecology and generalized the Lotka-Volterra model of predator-prey systems.
In his study of stochastic processes (random processes), especially Markov process
es, Kolmogorov and the British
mathematician Sydney Chapman independently developed the pivotal set of equations in the field, the Chapman–Kolmogorov equations.
Later on, Kolmogorov changed his research interests to the area of turbulence
, where his publications beginning in 1941 had a significant influence on the field. In classical mechanics
, he is best known for the Kolmogorov–Arnold–Moser theorem
(first presented in 1954 at the International Congress of Mathematicians
). In 1957 he solved a particular interpretation of Hilbert's thirteenth problem
(a joint work with his student V. I. Arnold). He was a founder of algorithmic complexity theory, often referred to as Kolmogorov complexity theory
, which he began to develop around this time.
Kolmogorov married Anna Dmitrievna Egorova in 1942. He pursued a vigorous teaching routine throughout his life, not only at the university level but also with younger children, as he was actively involved in developing a pedagogy for gifted children, in literature, and in music, as well as in mathematics. At the Moscow State University, Kolmogorov occupied different positions, including the heads of several departments: probability
, statistics
, and random processes; mathematical logic
; and he also served as the Dean of the Moscow State University Faculty of Mechanics and Mathematics.
In 1971, Kolmogorov joined an oceanographic
expedition aboard the research vessel Dmitri Mendeleev. He wrote a number of articles for the Great Soviet Encyclopedia
. In his later years he devoted much of his effort to the mathematical and philosophical relationship between probability theory
in abstract and applied areas.
Kolmogorov died in Moscow in 1987. A quotation, "Every mathematician believes he is ahead over all others. The reason why they don't say this in public, is because they are intelligent people" is attributed to him.
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, preeminent in the 20th century, who advanced various scientific fields, among them probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
, turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
, classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
and computational complexity
Computational Complexity
Computational Complexity may refer to:*Computational complexity theory*Computational Complexity...
.
Early life
Kolmogorov was born at TambovTambov
Tambov is a city and the administrative center of Tambov Oblast, Russia, located at the confluence of the Tsna and Studenets Rivers southeast of Moscow...
in 1903. His unwed mother died in childbirth and he was raised by his aunts in Tunoshna near Yaroslavl
Yaroslavl
Yaroslavl is a city and the administrative center of Yaroslavl Oblast, Russia, located northeast of Moscow. The historical part of the city, a World Heritage Site, is located at the confluence of the Volga and the Kotorosl Rivers. It is one of the Golden Ring cities, a group of historic cities...
at the estate of his grandfather, a wealthy nobleman. His father, an agronomist
Agronomist
An agronomist is a scientist who specializes in agronomy, which is the science of utilizing plants for food, fuel, feed, and fiber. An agronomist is an expert in agricultural and allied sciences, with the exception veterinary sciences.Agronomists deal with interactions between plants, soils, and...
by trade, was deported from Saint-Petersburg for participation in the revolutionary movement. He disappeared and was presumed to have been killed in the Russian Civil War
Russian Civil War
The Russian Civil War was a multi-party war that occurred within the former Russian Empire after the Russian provisional government collapsed to the Soviets, under the domination of the Bolshevik party. Soviet forces first assumed power in Petrograd The Russian Civil War (1917–1923) was a...
.
Kolmogorov was educated in his aunt's village school, and his earliest literary efforts and mathematical papers were printed in the school newspaper. As an adolescent he designed perpetual motion
Perpetual motion
Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not....
machines, concealing their (necessary) defects so cleverly that his secondary-school teachers could not discover them. In 1910, his aunt adopted him and then they moved to Moscow, where he went to a gymnasium
Gymnasium (school)
A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...
, graduating from it in 1920.
In 1920, Kolmogorov began to study at the Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
and the Chemistry Technological Institute.
Kolmogorov gained a reputation for his wide-ranging erudition. As an undergraduate, he participated in the seminars of the Russian historian S.V. Bachrushin, and he published his first research paper on the landholding practices in the Novgorod Republic in the fifteenth and sixteenth centuries. At the same time (1921–1922), Kolmogorov derived and proved several results in set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and in the theory of Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
(trigonometrical series).
Maturity
In 1922 Kolmogorov constructed a Fourier series that divergesConvergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics...
almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
, gaining international recognition. Around this time he decided to devote his life to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. In 1925 Kolmogorov graduated from Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
, and began to study under the supervision of Nikolai Luzin
Nikolai Luzin
Nikolai Nikolaevich Luzin, , was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the...
. He made lifelong friends with Pavel Alexandrov who involved Kolmogorov in 1936 in an ugly political persecution of their common teacher, the so-called Luzin case or Luzin affair. Kolmogorov (together with A. Khinchin) became interested in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
. Also in 1925, he published his famous work in intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
— On the principle of the excluded middle, where he proved that under a certain interpretation all statements of classical formal logic can be formulated as those of intuitionistic logic. In 1929 Kolmogorov earned his Doctor of Philosophy degree, Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
, at the Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
.
In 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
and Munich
Munich
Munich The city's motto is "" . Before 2006, it was "Weltstadt mit Herz" . Its native name, , is derived from the Old High German Munichen, meaning "by the monks' place". The city's name derives from the monks of the Benedictine order who founded the city; hence the monk depicted on the city's coat...
, Germany, and then to Paris, France. His pioneering work About the Analytical Methods of Probability Theory was published (in German) in 1931. Also in 1931, he became a professor at Moscow University. In 1933, Kolmogorov published the book, Foundations of the Theory of Probability, laying the modern axiomatic foundations of probability theory
Probability axioms
In probability theory, the probability P of some event E, denoted P, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below....
and establishing his reputation as the world's leading living expert in this field. In 1935, Kolmogorov became the first chairman of probability theory at the Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
. In 1939, he was elected a full member (academician) of the USSR Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....
. In a 1938 paper, Kolmogorov "established the basic theorems for smoothing and predicting stationary stochastic processes" — a paper that would have major military applications during the Cold War
Cold War
The Cold War was the continuing state from roughly 1946 to 1991 of political conflict, military tension, proxy wars, and economic competition between the Communist World—primarily the Soviet Union and its satellite states and allies—and the powers of the Western world, primarily the United States...
to come. Around the same years (1936) Kolmogorov contributed to the field of ecology and generalized the Lotka-Volterra model of predator-prey systems.
In his study of stochastic processes (random processes), especially Markov process
Markov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
es, Kolmogorov and the British
British people
The British are citizens of the United Kingdom, of the Isle of Man, any of the Channel Islands, or of any of the British overseas territories, and their descendants...
mathematician Sydney Chapman independently developed the pivotal set of equations in the field, the Chapman–Kolmogorov equations.
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
, where his publications beginning in 1941 had a significant influence on the field. In classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, he is best known for the Kolmogorov–Arnold–Moser theorem
Kolmogorov–Arnold–Moser theorem
The Kolmogorov–Arnold–Moser theorem is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics....
(first presented in 1954 at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
). In 1957 he solved a particular interpretation of Hilbert's thirteenth problem
Hilbert's thirteenth problem
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether or not a solution exists for all 7th-degree equations using functions of two arguments...
(a joint work with his student V. I. Arnold). He was a founder of algorithmic complexity theory, often referred to as Kolmogorov complexity theory
Kolmogorov complexity
In algorithmic information theory , the Kolmogorov complexity of an object, such as a piece of text, is a measure of the computational resources needed to specify the object...
, which he began to develop around this time.
Kolmogorov married Anna Dmitrievna Egorova in 1942. He pursued a vigorous teaching routine throughout his life, not only at the university level but also with younger children, as he was actively involved in developing a pedagogy for gifted children, in literature, and in music, as well as in mathematics. At the Moscow State University, Kolmogorov occupied different positions, including the heads of several departments: probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, and random processes; mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
; and he also served as the Dean of the Moscow State University Faculty of Mechanics and Mathematics.
In 1971, Kolmogorov joined an oceanographic
Oceanography
Oceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...
expedition aboard the research vessel Dmitri Mendeleev. He wrote a number of articles for the Great Soviet Encyclopedia
Great Soviet Encyclopedia
The Great Soviet Encyclopedia is one of the largest and most comprehensive encyclopedias in Russian and in the world, issued by the Soviet state from 1926 to 1990, and again since 2002 .-Editions:There were three editions...
. In his later years he devoted much of his effort to the mathematical and philosophical relationship between probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
in abstract and applied areas.
Kolmogorov died in Moscow in 1987. A quotation, "Every mathematician believes he is ahead over all others. The reason why they don't say this in public, is because they are intelligent people" is attributed to him.
See also
- Kolmogorov axioms
- Kolmogorov equationsKolmogorov equationsKolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, characterize random dynamic processes.-Diffusion Processes vs...
(also known as the Fokker–Planck equations) - Kolmogorov dimensionMinkowski-Bouligand dimensionthumb|450px|Estimating the box-counting dimension of the coast of Great BritainIn fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, or more generally in...
(upper box dimension) - Kolmogorov continuity theoremKolmogorov continuity theoremIn mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous...
- Kolmogorov’s criterionKolmogorov’s criterionIn probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem in Markov processes concerning stationary Markov chains...
- Kolmogorov extension theoremKolmogorov extension theoremIn mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process...
- Kolmogorov's inequalityKolmogorov's inequalityIn probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound...
- Landau–Kolmogorov inequality
- Kolmogorov integralKolmogorov integralIn mathematics, the Kolmogorov integral is a general integral introduced by including the Lebesgue–Stieltjes integral, the Burkill integral, and the Hellinger integral as special cases....
- Brouwer–Heyting–Kolmogorov interpretation
- Kolmogorov microscalesKolmogorov microscalesKolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....
- Kolmogorov spaceKolmogorov spaceIn topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms...
- Kolmogorov complexityKolmogorov complexityIn algorithmic information theory , the Kolmogorov complexity of an object, such as a piece of text, is a measure of the computational resources needed to specify the object...
- Kolmogorov–Smirnov test
- Kolmogorov's zero-one lawKolmogorov's zero-one lawIn probability theory, Kolmogorov's zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.Tail...
- Kolmogorov's characterization of reversible diffusions
- Borel–Kolmogorov paradox
- Chapman–Kolmogorov equation
- Sydney Chapman (mathematician)
- Chaitin–Kolmogorov randomness
- Hahn–Kolmogorov theorem
- Astronomical seeingAstronomical seeingAstronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...
described by Kolmogorov's turbulence law - Kolmogorov structure functionKolmogorov structure functionIn 1974 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let each data be a finite binary string and models be finite sets of binary strings...
External links
- Gessen, M., "Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century". Houghton Mifflin Harcourt, 2010.
- Biography at Scholarpedia
- The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A. N. Kolmogorov. A. N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A. N. Kolmogorov.
- The origins and legacy of Kolmogorov's Grundbegriffe
- Vitanyi, P.M.B., Andrey Nikolaevich Kolmogorov. Scholarpedia, 2(2):2798; 2007
- Collection of links to Kolmogorov resources
- Interview with Kolmogorov (1958, Moscow), Eugene DynkinEugene DynkinEugene Borisovich Dynkin is a Soviet and American mathematician. He has made contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes...
Collection of Mathematics Interviews, Cornell University LibraryCornell University LibraryThe Cornell University Library is the library system of Cornell University. In 2010 it held 8 million printed volumes in open stacks, 8.5 million microfilms and microfiches, more than of manuscripts, and close to 500,000 other materials, including motion pictures, DVDs, sound recordings, and...
. - Andrei Nikolaevich Kolmogorov (in Russian)
- Kolmogorov School at Moscow University
- Annual Kolmogorov Lecture at the Computer Learning Research Centre at Royal Holloway, University of London
- Lorentz G. G., Mathematics and Politics in the Soviet Union from 1928 to 1953
- Kutateladze S. S., The Tragedy of Mathematics in Russia