Editing Andrey Kolmogorov

{{Short description|Soviet mathematician (1903–1987)}}
{{Use dmy dates|date=June 2023}} 
{{family name hatnote|Nikolaevich|Kolmogorov|lang=Eastern Slavic}}
{{Infobox scientist
| name              = Andrey Kolmogorov
| native_name       = {{nobold|Андрей Колмогоров}}
| native_name_lang  = ru
| image             = Andrej Nikolajewitsch Kolmogorov.jpg
| birth_name        = Andrey Nikolaevich Kolmogorov
| birth_date        = {{birth date|1903|04|25|df=y}}
| birth_place       = [[Tambov]], [[Russian Empire]]
| death_date        = {{death date and age|1987|10|20|1903|04|25|df=y}}
| death_place       = [[Moscow]], [[Russian Soviet Federative Socialist Republic|Russian SFSR]], [[Soviet Union]]
| alma_mater        = [[Moscow State University]] ([[Doctor of Philosophy|PhD]])
| doctoral_advisor  = [[Nikolai Luzin]]<ref name=mathgene/>
| doctoral_students = {{Plainlist |
* [[Vladimir Mikhailovich Alekseev|Vladimir Alekseev]]
* [[Vladimir Arnold]]
* [[Sergei N. Artemov]]
* [[Grigory Barenblatt]]
* [[Roland Dobrushin]]
* [[Eugene Dynkin]]
* [[Israil Gelfand]]
* [[Boris Vladimirovich Gnedenko|Boris Gnedenko]]
* [[Leonid Levin]]
* [[Valery Vasilevich Kozlov|Valerii Kozlov]]
* [[Per Martin-Löf]]
* [[Robert Minlos]]
* [[Andrei Monin]]
* [[Sergey Nikolsky]]
* [[Alexander Obukhov]]
* [[Yuri Vasilevich Prokhorov|Yuri Prokhorov]]
* [[Yakov Sinai]]
* [[Albert Shiryaev]]
* [[Anatoli Georgievich Vitushkin|Anatoli Vitushkin]]
* [[Vladimir Andreyevich Uspensky|Vladimir Uspensky]]
* [[Akiva Yaglom]]
* [[Vladimir Vovk]]<ref name=mathgene/>}}
| known_for         = {{ublist|class=nowrap|[[Probability theory]]|[[Probability space]]|[[Topology]]|[[Intuitionistic logic]]|[[Turbulence|Turbulence studies]]|[[Classical mechanics]]|[[Mathematical analysis]]|[[Kolmogorov complexity]]|[[Kolmogorov–Arnold–Moser theorem|KAM theorem]]|[[Kolmogorov–Petrovsky–Piskounov equation|KPP equation]]}}
| spouse            = {{marriage|Anna Dmitrievna Egorova|1942|1987}}
| field             = [[Mathematics]]
| work_institution  = [[Moscow State University]]
| prizes            = {{Plainlist|
* Member of the [[Russian Academy of Sciences]]<ref name=ussr>{{citation|last=Youschkevitch|first=A. P.|date=1983|
title=A. N. Kolmogorov: Historian and philosopher of mathematics on the occasion of his 80th birfhday|
journal=[[Historia Mathematica]]|volume=10|issue=4|pages=383–395|doi=10.1016/0315-0860(83)90001-0|doi-access=free}}</ref>
* [[USSR State Prize|Stalin Prize]] {{smaller|(1941)}}
* [[Balzan Prize]] {{smaller|(1962)}}
* [[Fellow of the Royal Society|ForMemRS]] {{small|(1964)}}<ref name="frs">{{Cite journal | last1 = Kendall | first1 = D. G. | author-link = David George Kendall| doi = 10.1098/rsbm.1991.0015 | title = Andrei Nikolaevich Kolmogorov. 25 April 1903-20 October 1987 | journal = [[Biographical Memoirs of Fellows of the Royal Society]] | volume = 37 | pages = 300–326 | year = 1991 | s2cid = 58080873 | doi-access = free }}</ref>
* [[Lenin Prize]] {{smaller|(1965)}}
*  [[Wolf Prize in Mathematics|Wolf Prize]] {{smaller|(1980)}}
*  [[Lobachevsky Prize]] {{smaller|(1986)}}
}}
}}

'''Andrey Nikolaevich Kolmogorov''' ({{lang-rus|Андре́й Никола́евич Колмого́ров|p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf|a=Ru-Andrey Nikolaevich Kolmogorov.ogg}}, 25 April 1903 – 20 October 1987)<ref>{{Cite journal | doi = 10.1070/RM1988v043n01ABEH001555 | title = Academician Andrei Nikolaevich Kolmogorov (obituary) | journal = Russian Mathematical Surveys | volume = 43 | issue = 1 | pages = 1–9 | year = 1988 |bibcode = 1988RuMaS..43....1. | s2cid = 250857950 }}</ref><ref>{{Cite journal | last1 = Parthasarathy | first1 = K. R. | title = Obituary: Andrei Nikolaevich Kolmogorov | journal = Journal of Applied Probability | volume = 25 | issue = 2 | pages = 445–450 | year = 1988 | jstor = 3214455 | doi = 10.1017/S0021900200041115 | doi-access = free }}</ref> was a Soviet mathematician who played a central role in the creation of modern [[probability theory]]. He also contributed to the mathematics of [[topology]], [[intuitionistic logic]], [[turbulence]], [[classical mechanics]], [[algorithmic information theory]] and [[Analysis of algorithms|computational complexity]].<ref name="Yaglom">{{cite journal |last1=Yaglom |first1=A M |title=A. N. Kolmogorov as a Fluid Mechanician and Founder of a School in Turbulence Research |journal=Annual Review of Fluid Mechanics |date=January 1994 |volume=26 |issue=1 |pages=1–23 |doi=10.1146/annurev.fl.26.010194.000245 |language=en |issn=0066-4189|doi-access=free }}</ref><ref name="frs"/><ref name=mactutor>{{MacTutor Biography|id=Kolmogorov}}</ref>

==Biography==

===Early life===
Andrey Kolmogorov was born in [[Tambov]], about 500 kilometers southeast of [[Moscow]], in 1903. His unmarried mother, Maria Yakovlevna Kolmogorova, died giving birth to him.<ref>Encyclopædia Britannica Online, s. v. "[http://www.britannica.com/EBchecked/topic/321441/Andrey-Nikolayevich-Kolmogorov Andrey Nikolayevich Kolmogorov]", accessed February 22, 2013.</ref> Andrey was raised by two of his aunts in [[Tunoshna]] (near [[Yaroslavl]]) at the estate of his grandfather, a well-to-do [[Russian nobility|nobleman]].

Little is known about Andrey's father. He was supposedly named Nikolai Matveyevich Katayev and had been an [[Agronomy|agronomist]]. Katayev had been exiled from [[Saint Petersburg]] to the Yaroslavl province after his participation in the revolutionary movement against the [[tsar]]s. He disappeared in 1919 and was presumed to have been killed in the [[Russian Civil War]].

Andrey Kolmogorov was educated in his aunt Vera's village school, and his earliest literary efforts and mathematical papers were printed in the school journal "The Swallow of Spring". Andrey (at the age of five) was the "editor" of the mathematical section of this journal. Kolmogorov's first mathematical discovery was published in this journal: at the age of five he noticed the regularity in the sum of the series of odd numbers: <math> 1 = 1^2; 1 + 3 = 2^2; 1 + 3 + 5 = 3^2, </math> etc.<ref>{{cite book|title=Wolf Prize in Mathematics, v.2|date=2001|publisher=World Scientific|isbn=9789812811769|pages=119–141|chapter-url=https://books.google.com/books?id=GHFtMc9NTkYC&q=Kolmogorov+discovered+first+mathematical+regularity+at+the+age+five+year&pg=PA119|chapter=Andrei N Kolmogorov prepared by V M Tikhomirov}}</ref>

In 1910, his aunt adopted him, and they moved to Moscow, where he graduated from [[gymnasium (school)|high school]] in 1920. Later that same year, Kolmogorov began to study at [[Moscow State University]] and at the same time [[D. Mendeleev University of Chemical Technology of Russia|Mendeleev Moscow Institute of Chemistry and Technology]].<ref>{{Cite web|url=http://new.math.msu.su/department/probab/Kolmogorov/ank_cv.html|title=Андрей Николаевич КОЛМОГОРОВ. Curriculum Vitae|accessdate=19 June 2023}}</ref> Kolmogorov writes about this time: "I arrived at Moscow University with a fair knowledge of mathematics. I knew in particular the beginning of [[set theory]]. I studied many questions in articles in the ''[[Brockhaus and Efron Encyclopedic Dictionary|Encyclopedia of Brockhaus and Efron]]'', filling out for myself what was presented too concisely in these articles."<ref>{{cite book|title=Kolmogorov in Perspective (History of Mathematics)|date=2000|isbn=978-0821829189|page=6|url=https://books.google.com/books?id=skult9z3IAUC&q=kolmogorov+in+perspective+A.+N.+Shiryaev+The+years+as+a+student&pg=PA6|last1=Society|first1=American Mathematical|publisher=American Mathematical Soc. }}</ref>

Kolmogorov gained a reputation for his wide-ranging erudition. While an undergraduate student in college, he attended the seminars of the Russian historian [[Sergey Bakhrushin|S. V. Bakhrushin]], and he published his first research paper on the fifteenth and sixteenth centuries' [[Land tenure|landholding]] practices in the [[Novgorod Republic]].<ref>{{cite book |first=David |last=Salsburg |title=The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century |url=https://archive.org/details/ladytastingteaho0000sals |url-access=registration |location=New York |publisher=W. H. Freeman |year=2001 |pages=[https://archive.org/details/ladytastingteaho0000sals/page/137 137–50] |isbn=978-0-7167-4106-0 }}</ref> During the same period (1921–22), Kolmogorov worked out and proved several results in [[set theory]] and in the theory of [[Fourier series]].

===Adulthood===
In 1922, Kolmogorov gained international recognition for constructing a [[Fourier series]] that [[Convergence of Fourier series|diverges]] [[almost everywhere]].<ref>{{cite journal | last = Kolmogorov | first = A. | author-link = Andrey Kolmogorov | journal = Fundamenta Mathematicae | title = Une série de Fourier–Lebesgue divergente presque partout | trans-title = A Fourier–Lebesgue series that diverges almost everywhere |language = fr | year = 1923 | volume = 4 | issue = 1 | pages = 324–328| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm4/fm4127.pdf | doi = 10.4064/fm-4-1-324-328 | doi-access = free }}</ref><ref>{{cite web|url=http://www.physicstoday.org/resource/1/phtoad/v42/i10/p148_s2?bypassSSO=1|title=In Brief|author=V. I. Arnold-Max Dresden|url-status=dead|archive-url=https://web.archive.org/web/20131005020429/http://www.physicstoday.org/resource/1/phtoad/v42/i10/p148_s2?bypassSSO=1|archive-date=2013-10-05}}</ref> 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]].<ref name=mathgene>{{MathGenealogy|id=10480}}</ref> He formed a lifelong close friendship with [[Pavel Alexandrov]], a fellow student of Luzin; indeed, several researchers have concluded that the two friends were sexually involved,<ref name="GrahamKantor2009">{{cite book | title = Naming infinity: a true story of religious mysticism and mathematical creativity | last1 = Graham | first1 = Loren R. | last2 = Kantor | first2 = Jean-Michel | page = 185 | publisher = [[Harvard University Press]] | year = 2009 | isbn = 978-0-674-03293-4 | quote = The police soon learned of Kolmogorov and Alexandrov's homosexual bond, and they used that knowledge to obtain the behavior that they wished. }}</ref><ref>{{cite book |first=Masha |last=Gessen|title=Perfect Rigour: A Genius and the Mathematical Breakthrough of a Lifetime|publisher=Icon Books Ltd |year=2011 |page=17 |isbn=|quote=Kolmogorov alone among the top Soviet mathematicians avoided being drafted into the postwar military effort. His students always wondered why-and the only likely explanation seems to be Kolmogorov's homosexuality. His lifelong partner, with whom he shared a home starting in 1929, was the topologist Pavel Alexandrov.}}</ref><ref>{{citation|title=Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity|first1=Loren|last1=Graham|first2=Jean-Michel|last2=Kantor|publisher=Harvard University Press|year=2009|isbn=9780674032934|page=185|url=https://books.google.com/books?id=j2IUI4pj6e8C&pg=PA185}}</ref><ref>{{citation|title=Pricing the Future: Finance, Physics, and the 300-year Journey to the Black-Scholes Equation|first=George|last=Szpiro | authorlink= George Szpiro |publisher=Basic Books|year=2011|isbn=9780465022489|page=152|url=https://books.google.com/books?id=PFOAKB4HtF0C&pg=PA152|quote=It was generally known that they had a homosexual relationship, although they never acknowledged their liaison}}</ref> although neither acknowledged this openly. Kolmogorov (together with [[Aleksandr Khinchin]]) became interested in [[probability theory]]. Also in 1925, he published his work in [[intuitionistic logic]], "On the principle of the excluded middle," in which 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 from Moscow State University. In 1929, Kolmogorov and Alexandrov during a long travel stayed about a month in an island in lake [[Lake Sevan|Sevan]] in Armenia.<ref>{{cite journal | last = Gurzadyan | first = Vahe | author-link = Vahe Gurzadyan | journal = Mathematical Intelligencer | title = Kolmogorov and Aleksandrov in Sevan monastery  | year = 2004 | volume = 26 | pages = 40–43| doi = 10.1007/BF02985651 | arxiv = math/0410397 |url = https://doi.org/10.1007/BF02985651  }}</ref>

In 1930, Kolmogorov went on his first long trip abroad, traveling to [[Göttingen, Germany|Göttingen]] and [[Munich]] and then to [[Paris]]. He had various scientific contacts in Göttingen, first with [[Richard Courant]] and his students working on limit theorems, where [[diffusion process]]es proved to be the limits of discrete random processes, then with [[Hermann Weyl]] in intuitionistic logic, and lastly with [[Edmund Landau]] in function theory. 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 State University]].

In 1933, Kolmogorov published his book ''Foundations of the Theory of Probability'', laying the modern axiomatic [[probability axioms|foundations of probability theory]] and establishing his reputation as the world's leading expert in this field. In 1935, Kolmogorov became the first chairman of the department of probability theory at Moscow State University. Around the same years (1936) Kolmogorov contributed to the field of ecology and generalized the [[Lotka–Volterra]] model of [[predator–prey]] systems.

During the [[Great Purge]] in 1936, Kolmogorov's doctoral advisor [[Nikolai Luzin]] became a high-profile target of Stalin's regime in what is now called the "Luzin Affair." Kolmogorov and several other students of Luzin testified against Luzin, accusing him of plagiarism, nepotism, and other forms of misconduct; the hearings eventually concluded that he was a servant to "fascistoid science" and thus an enemy of the Soviet people. Luzin lost his academic positions, but curiously he was neither arrested nor expelled from the [[Academy of Sciences of the Soviet Union]].<ref>{{Cite journal | doi = 10.1007/BF03024600| title = Who discovered analytic sets?| year = 2001| last1 = Lorentz | first1 = G. G.| journal = The Mathematical Intelligencer| volume = 23| issue = 4| pages = 28–32| s2cid = 121273798}}</ref><ref>{{MacTutor Biography|class=Extras|id=Luzin|title=The 1936 Luzin affair}}</ref> The question of whether Kolmogorov and others were coerced into testifying against their teacher remains a topic of considerable speculation among historians; all parties involved refused to publicly discuss the case for the rest of their lives. Soviet-Russian mathematician [[Semën Samsonovich Kutateladze]] concluded in 2013, after reviewing archival documents made available during the 1990s and other surviving testimonies, that the students of Luzin had initiated the accusations against Luzin out of personal acrimony; there was no definitive evidence that the students were coerced by the state, nor was there any definitive evidence to support their allegations of academic misconduct.<ref>{{cite web|url=http://semr.math.nsc.ru/v10/a1-6.pdf|title=СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИ|language=ru |website=semr.math.nsc.ru|access-date=19 June 2023}}</ref> Soviet historian of mathematics [[A.P. Yushkevich]] surmised that, unlike many of the other high-profile persecutions of the era, Stalin did not personally initiate the persecution of Luzin and instead eventually concluded that he was not a threat to the regime, which would explain the unusually mild punishment relative to other contemporaries.<ref>[[A.P. Yushkevich]], [http://www.ras.ru/FStorage/download.aspx?Id=e2ae0e8c-7e64-474f-9672-8fb91cd89876 The Lusin Affair] (in Russian).</ref>

In a 1938 paper, Kolmogorov "established the basic theorems for smoothing and predicting stationary [[stochastic processes]]"—a paper that had major military applications during the [[Cold War]].<ref>Salsburg, p. 139.</ref> In 1939, he was elected a full member (academician) of the [[Russian Academy of Sciences|USSR Academy of Sciences]].

During [[World War II]] Kolmogorov contributed to the Soviet war effort by applying statistical theory to artillery fire, developing a scheme of stochastic distribution of [[barrage balloon]]s intended to help protect Moscow from German bombers during the [[Battle of Moscow]].<ref>{{cite book|last=Gleick|first=James|title=The Information: a history, a theory, a flood|date=2012|publisher=Vintage Books|location=New York|isbn=978-1-4000-9623-7|page=334}}</ref>

In his study of [[stochastic processes]], especially [[Markov process]]es, Kolmogorov and the British mathematician [[Sydney Chapman (mathematician)|Sydney Chapman]] independently developed a pivotal set of equations in the field that have been given the name of the [[Chapman–Kolmogorov equation]]s.

[[File:Kolm complexity lect.jpg|thumb|Kolmogorov (left) delivers a talk at a Soviet information theory symposium. ([[Tallinn]], 1973).]]
[[File:Kolm lect prep.jpg|thumb|Kolmogorov works on his talk ([[Tallinn]], 1973).]]

Later, Kolmogorov focused his research on [[turbulence]], beginning his publications in 1941. In [[classical mechanics]], he is best known for the [[Kolmogorov–Arnold–Moser theorem]], first presented in 1954 at the [[International Congress of Mathematicians]].<ref name="Yaglom"/> In 1957, working jointly with his student [[Vladimir Arnold]], he solved a particular interpretation of [[Hilbert's thirteenth problem]]. Around this time he also began to develop, and has since been considered a founder of, [[algorithmic complexity theory]] – often referred to as [[Kolmogorov complexity|Kolmogorov complexity theory]].

Kolmogorov married Anna Dmitrievna Egorova in 1942. He pursued a vigorous teaching routine throughout his life both at the university level and also with younger children, as he was actively involved in developing a [[pedagogy]] for gifted children in literature, music, and mathematics. At Moscow State University, Kolmogorov occupied different positions including the heads of several departments: [[probability]], [[statistics]], and [[random process]]es; [[mathematical logic]]. He also served as the Dean of the Moscow State University Department 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.<ref>Salsburg, pp. 145–7.</ref>

Kolmogorov died in Moscow in 1987 and his remains were buried in the [[Novodevichy cemetery]].

A quotation attributed to Kolmogorov is [translated into English]: "Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people."

[[Vladimir Arnold]] once said: "Kolmogorov – [[Henri Poincaré|Poincaré]] – [[Carl Friedrich Gauss|Gauss]] – [[Leonhard Euler|Euler]] – [[Isaac Newton|Newton]], are only five lives separating us from the source of our science."

==Awards and honours==
Kolmogorov received numerous awards and honours both during and after his lifetime:

* Member of the [[Russian Academy of Sciences]]<ref name=ussr/>
* Awarded the [[USSR State Prize|Stalin Prize]] in 1941
* Elected an Honorary Member of the [[American Academy of Arts and Sciences]] in 1959<ref>{{Cite web |title=Andrei Nikolayevich Kolmogorov |url=https://www.amacad.org/person/andrei-nikolayevich-kolmogorov |access-date=2022-11-21 |website=American Academy of Arts & Sciences |language=en}}</ref>
* Elected member of the [[American Philosophical Society]] in 1961<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Hans+Morgenthau&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2022-11-21 |website=search.amphilsoc.org}}</ref>
* Awarded the [[Balzan Prize]] in 1962
* Elected a Foreign Member of the [[Royal Netherlands Academy of Arts and Sciences]] in 1963<ref>{{cite web|url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00001334 |title=A.N. Kolmogorov (1903–1987) |publisher=Royal Netherlands Academy of Arts and Sciences |access-date=22 July 2015}}</ref>
* Elected a [[List of Fellows of the Royal Society elected in 1964|Foreign Member of the Royal Society (ForMemRS) in 1964]].<ref name="frs"/>
* Awarded the [[Lenin Prize]] in 1965
* Elected member of the United States [[National Academy of Sciences]] in 1967<ref>{{Cite web |title=A. Kolmogorov |url=http://www.nasonline.org/member-directory/deceased-members/24674.html |access-date=2022-11-21 |website=www.nasonline.org}}</ref>
* Awarded the [[Wolf Prize]] in 1980
* Awarded the [[Lobachevsky Prize]] in 1986

The following are named in Kolmogorov's honour:
{{Div col|colwidth=30em}}
* [[Fisher–Kolmogorov equation]]
* [[Avrami equation|Johnson–Mehl–Avrami–Kolmogorov equation]]
* [[Kolmogorov axioms]]
* [[Kolmogorov equations]] (also known as the Fokker–Planck equations in the context of diffusion and in the forward case)
* [[Minkowski–Bouligand dimension|Kolmogorov dimension]] ([[upper box dimension]])
* [[Kolmogorov–Arnold representation theorem|Kolmogorov–Arnold theorem]]
* [[Kolmogorov–Arnold–Moser theorem]]
* [[Kolmogorov continuity theorem]]
* [[Kolmogorov's criterion]]
* [[Kolmogorov extension theorem]]
* [[Kolmogorov's three-series theorem]]
* [[Convergence of Fourier series#Convergence almost everywhere|Convergence of Fourier series]]
* [[Beta wavelet#Gnedenko-Kolmogorov central limit theorem revisited|Gnedenko-Kolmogorov central limit theorem]]
* [[Quasi-arithmetic mean]] (it is also called Kolmogorov mean)
* [[Kolmogorov homology]]
* [[Kolmogorov's inequality]]
* [[Landau–Kolmogorov inequality]]
* [[Kolmogorov integral]]
* [[Brouwer–Heyting–Kolmogorov interpretation]]
* [[Kolmogorov microscales]]
* [[Kolmogorov's normability criterion]]
* [[Fréchet–Kolmogorov theorem]]
* [[Kolmogorov space]]
* [[Kolmogorov complexity]]
* [[Kolmogorov–Smirnov test]]
* [[Wiener filter]] (also known as Wiener–Kolmogorov filtering theory)
* [[Kriging|Wiener–Kolmogorov prediction]]
* [[Kolmogorov automorphism]]
* [[Kolmogorov's characterization of reversible diffusions]]
* [[Borel–Kolmogorov paradox]]
* [[Chapman–Kolmogorov equation]]
* [[Hahn–Kolmogorov theorem]]
* [[Avrami equation|Johnson–Mehl–Avrami–Kolmogorov equation]]
* [[Measure-preserving dynamical system|Kolmogorov–Sinai entropy]]
* [[Astronomical seeing]] described by Kolmogorov's turbulence law
* [[Kolmogorov structure function]]
* [[Pointer machine#Kolmogorov–Uspenskii machine (KU-machine) model|Kolmogorov–Uspenskii machine model]]
* [[Kolmogorov's zero–one law]]
* [[Kolmogorov–Zurbenko filter]]
* [[Kolmogorov's two-series theorem]]
* [[Rao–Blackwell theorem|Rao–Blackwell–Kolmogorov theorem]]
* [[Wiener–Khinchin theorem|Khinchin–Kolmogorov theorem]]
* [[Kolmogorov population model]]
* Kolmogorov's Strong [[Law of Large Numbers]]
{{div col end}}

==Bibliography==
A bibliography of his works appeared in {{cite journal|date=July 1989|title=Publications of A. N. Kolmogorov|journal=Annals of Probability|volume=17|issue=3|pages=945–964|doi=10.1214/aop/1176991252|doi-access=free}}<!--|access-date=2008-03-14-->

* {{cite book |title= Grundbegriffe der Wahrscheinlichkeitsrechnung |last= Kolmogorov |first= Andrey |author-link= Andrey Kolmogorov |year= 1933 |publisher= Julius Springer |location= Berlin |language= de |ref= kol1933}}<ref>{{cite journal|author=Rietz, H. L.|author-link=Henry Louis Rietz|title=Review: ''Grundbegriffe der Wahrscheinlichkeitsrechnung'' by A. Kolmogoroff|journal=Bull. Amer. Math. Soc.|year=1934|volume=40|issue=7|pages=522–523|url=https://www.ams.org/journals/bull/1934-40-07/S0002-9904-1934-05895-6/S0002-9904-1934-05895-6.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1934-40-07/S0002-9904-1934-05895-6/S0002-9904-1934-05895-6.pdf |archive-date=2022-10-09 |url-status=live|doi=10.1090/s0002-9904-1934-05895-6|doi-access=free}}</ref>
** Translation: {{cite book |title= Foundations of the Theory of Probability |edition= 2nd |last= Kolmogorov |first= Andrey |author-link= Andrey Kolmogorov |year= 1956 |publisher= Chelsea |location= New York |isbn= 978-0-8284-0023-7 |url= http://www.mathematik.com/Kolmogorov/index.html |ref= kol1956 |access-date= 2016-02-17 |archive-url= https://web.archive.org/web/20180914120850/http://www.mathematik.com/Kolmogorov/index.html |archive-date= 2018-09-14 |url-status= dead }}<ref>{{cite web|author=Gouvêa, Fernando Q.|author-link=Fernando Q. Gouvêa|date=|url=https://old.maa.org/press/maa-reviews/foundations-of-the-theory-of-probability |title=Review of ''Foundations of the Theory of Probability'' by A. N. Kolmogorov|website=MAA Reviews, Mathematical Association of America }}</ref>
* 1991–93. ''Selected works of A.N. Kolmogorov'', 3 vols. Tikhomirov, V. M., ed., Volosov, V. M., trans. [[Dordrecht]]:[[Wolters Kluwer|Kluwer Academic Publishers]]. {{isbn|90-277-2796-1}}
* 1925. "On the principle of the excluded middle" in [[Jean van Heijenoort]], 1967. ''A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press: 414–37.
* {{cite journal|author-link=Andrei N. Kolmogorov|first=Andrei N.|last=Kolmogorov|year=1963|title=On Tables of Random Numbers| journal=Sankhyā Ser. A|volume=25|pages=369–375 | mr = 178484}}
* {{cite journal |last=Kolmogorov |first=Andrei N. |author-link=Andrei N. Kolmogorov |year=1998 |orig-year=1963 |title=On Tables of Random Numbers |journal=Theoretical Computer Science |volume=207 |issue=2 |pages=387–395 |doi=10.1016/S0304-3975(98)00075-9 |mr=1643414|doi-access= }}
* Kolmogorov, Andrei N. (2005) ''Selected works''. In 6 volumes. Moscow (in Russian)
Textbooks:
* A. N. Kolmogorov and [[Boris Vladimirovich Gnedenko|B. V. Gnedenko]]. ''"Limit distributions for sums of independent random variables"'', 1954.
* A. N. Kolmogorov and [[Sergei Fomin|S. V. Fomin]]. ''"Elements of the Theory of Functions and Functional Analysis"'', [https://www.amazon.com/Elements-Functions-Functional-Analysis-Mathematics/dp/0486406830/ Publication 1999], [https://www.amazon.com/Elements-Functions-Functional-Analysis-Volumes/dp/1614273049/ Publication 2012]
* Kolmogorov, Andrey Nikolaevich; Fomin, Sergei Vasilyevich (1975) [1970]. Introductory real analysis. New York: Dover Publications. {{ISBN|978-0-486-61226-3}}..

==References==
{{Reflist|30em}}

==External links==
{{commons category|Andrey Kolmogorov}}
* [https://web.archive.org/web/20061118061150/http://www.kolmogorov.info/ Portal dedicated to AN Kolmogorov] (his scientific and popular publications, articles about him).{{in lang|ru}}
* [https://web.archive.org/web/20180117185437/http://kolmogorov.org/ The Legacy of Andrei Nikolaevich Kolmogorov]
* [http://www.scholarpedia.org/article/Andrey_Nikolaevich_Kolmogorov Biography at Scholar''pedia'']
* [http://eng.tsutmb.ru/institsutes-and-departments/61-institute-of-mathematics,-physics-and-information-technology.html Derzhavin Tambov State University - Institute of Mathematics, Physics and Information Technology] {{Webarchive|url=https://web.archive.org/web/20191005040054/http://eng.tsutmb.ru/institsutes-and-departments/61-institute-of-mathematics,-physics-and-information-technology.html |date=2019-10-05 }}
* [http://www.probabilityandfinance.com/articles/04.pdf The origins and legacy of Kolmogorov's Grundbegriffe]
* [http://www.scholarpedia.org/article/Andrey_Nikolaevich_Kolmogorov  Vitanyi, P.M.B., Andrey Nikolaevich Kolmogorov. Scholarpedia, 2(2):2798; 2007]
* [http://www.geometry.net/scientists/kolmogorov_andrey.php Collection of links to Kolmogorov resources]
* [http://dynkincollection.library.cornell.edu/sites/default/files/Yaglom%20%28ENG%29,%20Ithaca,%20N.%20Y.,%20Dec.%202,%201988.pdf Interview with Professor A. M. Yaglom about Kolmogorov, Gelfand and other (1988, Ithaca, New York])
* [http://www.pms.ru/ Kolmogorov School] at Moscow University
* [http://www.clrc.rhul.ac.uk/events/eventsoverview.htm Annual Kolmogorov Lecture] at the Computer Learning Research Centre at Royal Holloway, University of London
* [https://web.archive.org/web/20110525230252/http://www.math.ohio-state.edu/AT/LORENTZ/JAT02-0001_final.ps Lorentz G. G., Mathematics and Politics in the Soviet Union from 1928 to 1953]
* [http://www.math.nsc.ru/LBRT/g2/english/ssk/gloria_e.html Kutateladze S. S., Sic Transit... or Heroes, Villains, and Rights of Memory].
* [http://www.math.nsc.ru/LBRT/g2/english/ssk/case_e.html Kutateladze S. S., The Tragedy of Mathematics in Russia]
* [https://www.youtube.com/watch?v=EGVOPqq3HpM Video recording of the G. Falkovich's lecture: "Andrey Nikolaevich Kolmogorov (1903–1987) and the Russian school"]
* {{MathGenealogy |id=10480}}

{{Wolf Prize in Mathematics}}

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