View source for Vladimir Arnold

{{short description|Russian mathematician (1937–2010)}}
{{family name hatnote|Igorevich|Arnold|lang=Eastern Slavic}}
{{Use dmy dates|date=March 2020}}
{{Use British English|date=June 2025}}
{{Infobox scientist
| name              = Vladimir Arnold
| native_name       = {{nobold|Владимир Арнольд}}
| native_name_lang  = ru
| image             = Vladimir Arnold-1.jpg
| image_size        = 200px
| caption           = Arnold in 2008
| birth_date        = {{birth date|df=y|1937|06|12}}
| birth_place       = [[Odessa]], [[Ukrainian SSR]], Soviet Union
| death_date        = {{death date and age|df=y|2010|06|03|1937|06|12}}
| death_place       = Paris, France
| citizenship       = {{hlist|Soviet Union|Russia}}
| fields            = Mathematics
| workplaces        = [[Paris Dauphine University]]<br />[[Steklov Institute of Mathematics]]<br />[[Independent University of Moscow]]<br />[[Moscow State University]]
| alma_mater        = [[Moscow State University]]
| doctoral_advisor  = [[Andrey Kolmogorov]]
| doctoral_students = {{Plainlist|
* [[Rifkat Bogdanov]]
* [[Alexander Givental]]
* [[Victor Goryunov]]
* [[Sabir Gusein-Zade]]
* [[Emil Horozov]]
* [[Yulij Ilyashenko]]
* [[Boris Khesin]]
* [[Askold Khovanskii]]
* [[Nikolay Nekhoroshev]]
* [[Boris Shapiro]]
* [[Alexander Varchenko]]
* [[Victor Anatolyevich Vassiliev|Victor Vassiliev]]
* [[Vladimir Zakalyukin]]}}
| known_for         = {{show|List|[[ADE classification]]<br />[[Arnold's cat map]]<br />[[Arnold conjecture]]<br />[[Arnold diffusion]]<br />[[Arnold invariants]]<br />''[[Arnold's Problems]]''<br />[[Arnold's rouble problem]]<br />[[Arnold's spectral sequence]]<br />[[Arnold tongue]]<br />[[Arnold–Beltrami–Childress flow|ABC flow]]<br />[[Arnold–Givental conjecture]]<br />[[Euler–Arnold equation]]<br />[[Gömböc]]<br />[[Gudkov's conjecture]]<br />[[Hilbert–Arnold problem]]<br />[[Hilbert's thirteenth problem]]<br />[[Kolmogorov–Arnold–Moser theorem|KAM theorem]]<br />[[Kolmogorov–Arnold representation theorem|Kolmogorov–Arnold theorem]]<br />[[Liouville–Arnold theorem]]<br />[[Topological Galois theory]]<br />''[[Mathematical Methods of Classical Mechanics]]''<br />[[Nearby Lagrangian conjecture]]<br />''[[Ordinary Differential Equations (textbook)|Ordinary Differential Equations]]''}}
| awards            = [[Shaw Prize]] (2008) <br /> [[State Prize of the Russian Federation]] (2007) <br /> [[Wolf Prize in Mathematics]] (2001) <br /> [[Dannie Heineman Prize for Mathematical Physics]] (2001) <br /> [[Harvey Prize]] (1994) <br /> [[Lobachevsky Prize#Russian Academy of Sciences|RAS Lobachevsky Prize]] (1992) <br /> [[Crafoord Prize]] (1982) <br /> [[Lenin Prize]] (1965)
| children          = 2
}}
'''Vladimir Igorevich Arnold''' (or '''Arnol'd'''; {{langx|ru|link=no|Влади́мир И́горевич Арно́льд}}, {{IPA|ru|vlɐˈdʲimʲɪr ˈiɡərʲɪvʲɪtɕ ɐrˈnolʲt|IPA}}; 12 June 1937 – 3 June 2010)<ref name=rsbm>{{cite journal|last1=Khesin|first1=Boris|author-link1=Boris Khesin|last2=Tabachnikov|first2=Sergei|author-link2=Sergei Tabachnikov|title=Vladimir Igorevich Arnold. 12 June 1937 – 3 June 2010|journal=[[Biographical Memoirs of Fellows of the Royal Society]]|volume=64|pages=7–26|year=2018 |doi=10.1098/rsbm.2017.0016|doi-access=free}}</ref><ref>[http://www.lefigaro.fr/flash-actu/2010/06/03/97001-20100603FILWWW00719-mort-d-un-grand-mathematicien-russe.php Mort d'un grand mathématicien russe], AFP (''Le Figaro'')</ref><ref name=obituary/> was a Soviet and Russian mathematician. He is best known for the [[Kolmogorov–Arnold–Moser theorem]] regarding the [[stability theory|stability]] of [[integrable system]]s, and contributed to several areas, including geometrical theory of [[Dynamical systems theory|dynamical systems]], [[algebra]], [[catastrophe theory]], [[topology]], [[real algebraic geometry]], [[symplectic geometry]], [[differential equation]]s, [[classical mechanics]], [[differential geometry|differential-geometric]] approach to [[hydrodynamics]], [[geometric analysis]] and [[singularity theory]], including posing the [[ADE classification]] problem. In his later years he shifted his research interests, investigating [[discrete mathematics]].

His first main result was the solution of [[Hilbert's thirteenth problem]] in 1957 when he was 19. He co-founded three new [[Areas of mathematics|branches of mathematics]]: [[topological Galois theory]] (with his student [[Askold Khovanskii]]), [[Kolmogorov–Arnold–Moser theorem#KAM theory|KAM theory]] (with [[Andrey Kolmogorov]] and [[Jürgen Moser]]) and [[symplectic topology]].

Arnold was also a populariser of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as ''[[Mathematical Methods of Classical Mechanics]]'' and ''[[Ordinary Differential Equations (textbook)|Ordinary Differential Equations]]'') and popular mathematics books, he influenced many mathematicians and physicists.<ref name="MacTutor">{{MacTutor Biography|id=Arnold}}</ref><ref>{{Cite book|title = Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles|url = https://books.google.com/books?id=J4nk1U3qjh0C&pg=PA211|publisher = Springer|date = 2010|isbn = 9783642136061|language = en|first1 = Claudio|last1 = Bartocci|first2 = Renato|last2 = Betti|first3 = Angelo|last3 = Guerraggio|first4 = Roberto|last4 = Lucchetti|first5 = Kim|last5 = Williams|page = 211}}</ref> Many of his books were translated into English. His views on education were opposed to those of [[Nicolas Bourbaki|Bourbaki]].

A controversial and often quoted dictum of his is "[[Relationship between mathematics and physics#Mathematics is physics|Mathematics is the part of physics where experiments are cheap]]".<ref name=obituary /><ref>{{cite journal |author1=Euler M. |title=The role of experiments in the teaching and learning of physics |journal=Proceedings of the International School of Physics "Enrico Fermi" |date=2004 |volume=156 |issue=Research on Physics Education |doi=10.3254/978-1-61499-012-3-175 }}</ref><ref>Note: This phrase generated controversy and even parodies. He wrote his defense at {{cite journal |last1=Arnol'd |first1=Vladimir I |title=Mathematics and physics: mother and daughter or sisters? |journal=Physics-Uspekhi |date=31 December 1999 |volume=42 |issue=12 |pages=1205–1217 |doi=10.1070/pu1999v042n12abeh000673 |bibcode=1999PhyU...42.1205A }}</ref> The [[Stigler's law of eponymy|Arnold principle]]: "Discoveries are rarely attributed to the correct person" is also named after him.<ref name="rsbm" />

Arnold worked at the [[Moscow State University]] from 1961 to 1986, at the [[Steklov Mathematical Institute]] since 1986, and at the [[Paris Dauphine University]] since 1993.<ref>http://www.pdmi.ras.ru/~arnsem/Arnold/arnold-cv.html</ref> He was one of the founders of the [[Independent University of Moscow]].<ref>[[Sabir Gusein-Zade|Gusein-Zade, S. M.]], [[Yulij Ilyashenko|Ilyashenko, Y. S.]], [[Askold Khovanskii|Khovanskii, A. G.]], Tsfasman, M. A. E., & [[Victor Vasiliev|Vassiliev, V. A]]. (2003). "[http://www.mathjournals.org/mmj/vol3-2-2003/dedication.html Vladimir Igorevich Arnold]". ''[[Moscow Mathematical Journal]]'', Volume 3, Number 2, pp. 261-261.</ref>

Arnold received many major prizes, including the inaugural [[Crafoord Prize]] in 1982 (with [[Louis Nirenberg]]), the [[Wolf Prize in Mathematics]] in 2001 and the [[Shaw Prize#Mathematical sciences|Shaw Prize]] in 2008 (with [[Ludwig Faddeev]]).

==Early life==
[[File:Владимир Арнольд в 1963 г.jpg|right|thumb|Arnold in 1963]]
Vladimir Igorevich Arnold was born on 12 June 1937 in [[Odessa]], [[Ukrainian SSR]], [[Soviet Union]] (now Odesa, Ukraine). His father was {{ill|Igor Vladimirovich Arnold|lt=Igor Vladimirovich Arnold|ru|Арнольд, Игорь Владимирович}} (1900–1948), a mathematician known for his work in [[mathematical education]] and who learned [[algebra]] from [[Emmy Noether]] in the late 1920s.<ref name=rsbm /> His mother was Nina Alexandrovna Arnold (1909–1986, {{nee}} Isakovich), a Jewish art historian.<ref name=obituary>{{citation|first1=Sabir M.|last1=Gusein-Zade|author1-link=Sabir Gusein-Zade|first2=Alexander N|last2= Varchenko|author2-link=Alexander Varchenko| url=http://www.ems-ph.org/journals/newsletter/pdf/2010-12-78.pdf|title= Obituary: Vladimir Arnold (12 June 1937 – 3 June 2010)|journal=Newsletter of the European Mathematical Society|volume= 78 |date=December 2010|pages= 28–29}}</ref> While a school student, Arnold once asked his father why the multiplication of two negative numbers yielded a positive number, and his father provided an answer involving the [[Field (mathematics)#Classic definition|field properties]] of [[real number]]s and the preservation of the [[distributive property]]. Arnold was deeply disappointed with this answer, and developed an aversion to the [[axiomatic method]] that lasted his whole life.<ref name="Arnold2007">{{cite book | author = Vladimir I. Arnold | date = 2007 | title = Yesterday and Long Ago | publisher = Springer | pages = 19–26 | isbn = 978-3-540-28734-6 | url = https://books.google.com/books?id=7c4oAQAAIAAJ}}</ref><ref>{{cite book |title=Axiomatic Method and Category Theory |series=Synthese Library |date=2014 |volume=364 |doi=10.1007/978-3-319-00404-4 |isbn=978-3-319-00403-7 }}{{page needed|date=October 2025}}</ref>  When Arnold was thirteen, his uncle Nikolai B. Zhitkov,<ref name="earlylife">{{cite book|editor1=Boris A. Khesin|editor2=Serge L. Tabachnikov|title=Arnold: Swimming Against the Tide|url=https://books.google.com/books?id=aBWHBAAAQBAJ|date=2014|publisher=American Mathematical Society|isbn=978-1-4704-1699-7|page=3}}</ref> who was an engineer, told him about [[calculus]] and how it could be used to understand some physical phenomena. This contributed to sparking his interest in mathematics, and he started to study the mathematics books his father had left him, which included some works by [[Leonhard Euler]] and [[Charles Hermite]].<ref>[[Sergei Tabachnikov|Табачников, С. Л.]] . "Интервью с В.И.Арнольдом", ''[[Kvant (magazine)|Квант]]'', 1990, Nº 7, pp. 2–7. (''in Russian'')</ref>

Arnold entered [[Moscow State University]] in 1954.<ref>{{cite journal |last1=Sevryuk |first1=Mikhail B. |title=Translation of the V. I. Arnold paper 'From Superpositions to KAM Theory' (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740) |journal=Regular and Chaotic Dynamics |date=November 2014 |volume=19 |issue=6 |pages=734–744 |doi=10.1134/S1560354714060100 }}</ref> Among his teachers there were [[Andrey Kolmogorov|A. N. Kolmogorov]], [[Israel Gelfand|I. M. Gelfand]], [[Lev Pontryagin|L. S. Pontriagin]] and [[Pavel Alexandrov]].<ref>{{citation|url=https://www.ams.org/journals/notices/199704/arnold.pdf|journal=[[Notices of the American Mathematical Society|Notices of the AMS]]|date=April 1997|volume=44|issue=4|title=An Interview with Vladimir Arnol'd|pages=432–438}}</ref> While a student of Andrey Kolmogorov at Moscow State University and still a teenager, Arnold showed in 1957 that any [[continuous function]] of several variables can be constructed with a finite number of two-variable functions, thereby solving [[Hilbert's thirteenth problem]].<ref>{{cite book|author=Daniel Robertz|title=Formal Algorithmic Elimination for PDEs|url=https://books.google.com/books?id=0kvPBAAAQBAJ&pg=PA192|date=13 October 2014|publisher=Springer|isbn=978-3-319-11445-3|page=192}}</ref> This is the [[Kolmogorov–Arnold representation theorem]].

==Mathematical work==
{{see also|Stability of the Solar System|ADE classification}}
Arnold obtained his PhD in 1961, with [[Andrey Kolmogorov]] as his advisor (thesis: ''On The Representation of Continuous Functions of 3 Variables By The Superpositions of Continuous Functions of 2 Variables'').<ref>{{MathGenealogy|id=17493|title=Vladimir Igorevich Arnold}}</ref>

He became an academician of the [[Academy of Sciences of the Soviet Union]] ([[Russian Academy of Science]] since 1991) in 1990.<ref name="GRE">[[Great Russian Encyclopedia]] (2005), Moscow: Bol'shaya Rossiyskaya Enciklopediya Publisher, vol. 2.</ref> Arnold can be considered to have initiated the theory of [[symplectic topology]] as a distinct discipline. The [[Arnold conjecture]] on the number of fixed points of [[symplectomorphism|Hamiltonian symplectomorphisms]] and [[Lagrangian intersection]]s was also a motivation in the development of [[Floer homology]].<ref>[[Kenji Fukaya]], [[Yong-Geun Oh]], Hiroshi Ohta (2009). ''Lagrangian Intersection Floer Theory Anomaly and Obstruction'' {{page needed|date=June 2025}}</ref>

Arnold worked at the Steklov Mathematical Institute in Moscow and at [[Paris Dauphine University]] until his death. He supervised 46 PhD students, including [[Rifkat Bogdanov]], [[Alexander Givental]], [[Victor Goryunov]], [[Sabir Gusein-Zade]], [[Emil Horozov]], [[Yulij Ilyashenko]], [[Boris Khesin]], [[Askold Khovanskii]], [[Nikolay Nekhoroshev]], [[Boris Shapiro (mathematician)|Boris Shapiro]], [[Alexander Varchenko]], [[Victor Anatolyevich Vassiliev|Victor Vassiliev]] and [[Vladimir Zakalyukin]].<ref name="mathgene">{{MathGenealogy|id=17493}}</ref>

Arnold worked on [[dynamical systems theory]], [[catastrophe theory]], [[topology]], [[algebraic geometry]], [[symplectic geometry]], [[differential equation]]s, [[classical mechanics]], [[hydrodynamics]] and [[singularity theory]].<ref name="MacTutor" /> [[Michèle Audin]] described him as "a geometer in the widest possible sense of the word" and said that "he was very fast to make connections between different fields".<ref>"Vladimir Igorevich Arnold and the Invention of Symplectic Topology", chapter I in the book ''Contact and Symplectic Topology'' (editors: Frédéric Bourgeois, Vincent Colin, András Stipsicz)</ref>

=== Hilbert's thirteenth problem ===
{{see also|Kolmogorov–Arnold representation theorem}}
[[Hilbert's thirteenth problem]] asks whether every [[continuous function]] of three variables can be expressed as a [[function composition|composition]] of finitely many continuous functions of two variables. The affirmative answer to this question was given in 1957 by Arnold, then nineteen years old and a student of [[Andrey Kolmogorov]]. Kolmogorov had shown the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were required, thus answering Hilbert's question for the class of continuous functions.<ref>{{Cite web|first=Stephen|last=Ornes|date=14 January 2021|title=Mathematicians Resurrect Hilbert's 13th Problem|url=https://www.quantamagazine.org/mathematicians-resurrect-hilberts-13th-problem-20210114/|website=[[Quanta Magazine]]}}</ref>

=== Dynamical systems ===
{{see also|Arnold diffusion|Arnold's cat map}}
[[Jürgen Moser]] and Arnold expanded the ideas of Kolmogorov (who was inspired by questions of [[Henri Poincaré]]) and gave rise to what is now known as [[Kolmogorov–Arnold–Moser theorem]] (or "KAM theory"), which concerns the persistence of some quasi-periodic motions (nearly integrable [[Hamiltonian system]]s) when they are perturbed. KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period, and specifies what the conditions for this are.<ref>{{Cite book|title = Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles|url = https://books.google.com/books?id=iEBOce-4S2EC&pg=PT39|publisher = Penguin|date = 29 July 2008|isbn = 9781440634284|first = George G.|last = Szpiro | author-link= George Szpiro}}</ref>

In 1961, he introduced [[Arnold tongue]]s; they are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes.<ref>{{cite journal|author1=Gérard, C.|author2=Goldbeter, A.|year=2012|title=The cell cycle is a limit cycle|journal=Mathematical Modelling of Natural Phenomena|volume=7|issue=6|pages=126–166|doi=10.1051/mmnp/20127607|doi-access=free}}</ref>

In 1964, Arnold introduced the [[Arnold web]], the first example of a stochastic web.<ref>Phase Space Crystals, by Lingzhen Guo https://iopscience.iop.org/book/978-0-7503-3563-8.pdf</ref><ref>''[[Scholarpedia]]'': "Zaslavsky web map", by [[George Zaslavsky]] http://www.scholarpedia.org/article/Zaslavsky_web_map</ref>

In 1974, Arnold proved the [[Liouville–Arnold theorem]], now a classic result deeply geometric in character.<ref name=rsbm />

In the 1980s, Arnold reformulated [[Hilbert's sixteenth problem]], proposing its infinitesimal version (the [[Hilbert–Arnold problem]]) that inspired many deep works in dynamical systems theory by mathematicians seeking its solution.<ref name=rsbm />

===Singularity theory===
{{see also|Arnold's spectral sequence}}
In 1965, Arnold attended [[René Thom]]'s seminar on [[catastrophe theory]]. He later said of it: "I am deeply indebted to Thom, whose singularity seminar at the [[Institut des Hautes Études Scientifiques|Institut des Hautes Etudes Scientifiques]], which I frequented throughout the year 1965, profoundly changed my mathematical universe."<ref>{{cite web |url=http://www.math.upenn.edu/Arnold/Arnold-interview1997.pdf |title=Archived copy |access-date=22 February 2015 |url-status=dead |archive-url=https://web.archive.org/web/20150714123033/https://www.math.upenn.edu/Arnold/Arnold-interview1997.pdf |archive-date=14 July 2015 }}</ref> After this event, [[singularity theory]] became one of the major interests of Arnold and his students.<ref>{{Cite web | url=http://www.ias.ac.in/resonance/Volumes/19/09/0787-0796.pdf |title = Resonance – Journal of Science Education &#124; Indian Academy of Sciences}}</ref> Among his most famous results in this area is his classification of simple singularities, contained in his paper "Normal forms of functions near degenerate critical points, the Weyl groups of A<sub>k</sub>,D<sub>k</sub>,E<sub>k</sub> and Lagrangian singularities".<ref>Note: It also appears in another article by him, but in English: ''Local Normal Forms of Functions'', http://www.maths.ed.ac.uk/~aar/papers/arnold15.pdf</ref><ref>{{cite book|author1=Dirk Siersma|author2=Charles Wall|author3=V. Zakalyukin|title=New Developments in Singularity Theory|url=https://books.google.com/books?id=rK77kWeRNqYC&pg=PA29|date=30 June 2001|publisher=Springer Science & Business Media|isbn=978-0-7923-6996-7|page=29}}</ref><ref>{{cite book |last1=Landsberg |first1=Joseph M. |last2=Manivel |first2=Laurent |title=Algebraic Transformation Groups and Algebraic Varieties |chapter=Representation Theory and Projective Geometry |series=Encyclopaedia of Mathematical Sciences |date=2004 |volume=132 |pages=71–122 |doi=10.1007/978-3-662-05652-3_6 |isbn=978-3-642-05875-2 }}</ref>

===Fluid dynamics===
{{see also|Arnold–Beltrami–Childress flow|Beltrami vector field#Beltrami fields and complexity in fluid mechanics|Euler–Arnold equation}}
In 1966, Arnold published the paper "{{lang|fr|Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits}}" ('On the [[differential geometry]] of [[infinite-dimensional Lie group]]s and its applications to the [[hydrodynamics]] of [[perfect fluid]]s'), in which he presented a common geometric interpretation for both the [[Euler's equations (rigid body dynamics)|Euler's equations for rotating rigid bodies]] and the [[Euler equations (fluid dynamics)|Euler's equations of fluid dynamics]]; this linked topics previously thought to be unrelated, and enabled mathematical solutions to many questions related to flows and [[turbulence]].<ref>{{cite book|author=Terence Tao|title=Compactness and Contradiction|url=https://books.google.com/books?id=BawxAAAAQBAJ&pg=PA205|date=22 March 2013|publisher=American Mathematical Soc.|isbn=978-0-8218-9492-7|pages=205–206|author-link=Terence Tao}}</ref><ref>{{Cite news | url=https://www.theguardian.com/science/2010/aug/19/v-i-arnold-obituary |title = VI Arnold obituary|newspaper = The Guardian|date = 19 August 2010|last1 = MacKay|first1 = Robert Sinclair|last2 = Stewart|first2 = Ian}}</ref><ref>[http://www.iamp.org/bulletins/old-bulletins/201007.pdf IAMP News Bulletin, July 2010, pp. 25–26]</ref>

===Real algebraic geometry===
In 1971, Arnold published "On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms",<ref>Note: The paper also appears with other names, as in https://web.archive.org/web/20230719150350/https://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf</ref> which gave new life to [[real algebraic geometry]]. In it, he made major advances in towards a solution to [[Gudkov's conjecture]], by finding a connection between it and [[Low-dimensional topology#Four dimensions|four-dimensional topology]].<ref>{{cite book|author1=A. G. Khovanskii|author2=Aleksandr Nikolaevich Varchenko|author3=V. A. Vasiliev|title=Topics in Singularity Theory: V. I. Arnold's 60th Anniversary Collection (preface)|url=https://books.google.com/books?id=n6MlbFDp5UwC&pg=PR10|year=1997|publisher=American Mathematical Soc.|isbn=978-0-8218-0807-8|page=10}}</ref> The [[conjecture]] was later fully solved by [[Vladimir Abramovich Rokhlin|V. A. Rokhlin]] building on Arnold's work.<ref>{{cite book|title=Arnold: Swimming Against the Tide|url=https://books.google.com/books?id=aBWHBAAAQBAJ&pg=PA159|page=159|isbn=9781470416997|last1=Khesin|first1=Boris A.|last2=Tabachnikov|first2=Serge L.|date=10 September 2014| publisher=American Mathematical Society }}</ref><ref>{{cite journal |last1=Degtyarev |first1=A I |last2=Kharlamov |first2=V M |title=Topological properties of real algebraic varieties: du coté de chez Rokhlin |journal=Russian Mathematical Surveys |date=31 August 2000 |volume=55 |issue=4 |pages=735–814 |doi=10.1070/RM2000v055n04ABEH000315 |arxiv=math/0004134 |bibcode=2000RuMaS..55..735D }}</ref>

=== Symplectic geometry ===
The [[Arnold conjecture]], linking the number of fixed points of Hamiltonian [[symplectomorphism]]s and the topology of the subjacent [[manifold]]s, was the motivating source of many of the pioneer studies in [[symplectic topology]].<ref>"Arnold and Symplectic Geometry", by [[Helmut Hofer]] (in the book ''Arnold: Swimming Against the Tide'')</ref><ref>"[http://www-irma.u-strasbg.fr/~maudin/Arnold.pdf Vladimir Igorevich Arnold and the invention of symplectic topology]", by [[Michèle Audin]] https://web.archive.org/web/20160303175152/http://www-irma.u-strasbg.fr/~maudin/Arnold.pdf</ref> He also proposed the [[nearby Lagrangian conjecture]], a still open problem in mathematics.<ref>Lisa Traynor (2024), "[https://celebratio.org/Eliashberg_Y/article/1188/ Eliashberg’s contributions towards the theory of generating functions]"</ref>

According to [[Michèle Audin]], the birth-date of symplectic topology was 27 October 1965, which is the day Arnold's paper "Sur une propriété topologique des applications globalement canoniques de la mécanique classique" was presented to the [[Paris Academy of Sciences]].<ref>András Stipsicz, Frédéric Bourgeois, Vincent Colin (Editors) 2014.''Contact and Symplectic Topology'', p. 2, Springer</ref>

===Topology===
According to [[Victor Vassiliev]], Arnold "worked comparatively little on topology for topology's sake," being motivated by problems on other areas of mathematics where topology could be of use. His contributions include the invention of a topological form of the [[Abel–Ruffini theorem]] and the initial development of some of the consequent ideas, a work which resulted in the creation of the field of [[topological Galois theory]] in the 1960s.<ref>"Topology in Arnold's work", by [[Victor Vassiliev]]</ref><ref>https://www.ams.org/journals/bull/2008-45-02/S0273-0979-07-01165-2/S0273-0979-07-01165-2.pdf Bulletin (New Series) of The American Mathematical Society Volume 45, Number 2, April 2008, pp. 329–334</ref>

=== Theory of plane curves ===
According to [[Marcel Berger]], Arnold revolutionised [[plane curve]]s theory.<ref>{{cite book |last=Berger |first=Marcel |authorlink=Marcel Berger |year= |title=A Panoramic View of Riemannian Geometry |url= |location= |publisher= |pages=24–25 |isbn= }}</ref> He developed the theory of smooth closed plane curves in the 1990s.<ref>[https://hosted.math.rochester.edu/ojac/vol9/90.pdf "On computational complexity of plane curve invariants", by Duzhin and Biaoshuai]</ref> Among his contributions are the introduction of the three [[Arnold invariants]] of plane curves: ''J''<sup>+</sup>, ''J''<sup>−</sup> and ''St''.<ref>Extrema of Arnold's invariants of curves on surfaces, by Vladimir Chernov https://math.dartmouth.edu/~chernov-china/</ref><ref>V. I. Arnold, "Plane curves, their invariants, perestroikas and classifications" (May 1993)</ref>

=== Discrete mathematics ===
In the last years of his life, Arnold's interests shifted to [[discrete mathematics]]. He investigated [[number theory]] and [[combinatorics]], producing around twenty papers on these topics, according to [[Anatoly Vershik]]. Arnold's conjecture of a [[Matrix (mathematics)|matrix]] generalization of [[Fermat's little theorem]] dates from this period.<ref>''Arnold: Swimming against the Tide'', p. 194</ref><ref>See, for example, V.I. Arnold: "On the matricial version of Fermat–Euler congruences", ''Japan. J. Math''. 1, 1–24 (2006), DOI: 10.1007/s11537-006-0501-6</ref>

=== Other ===
In 1995, Arnold conjectured the existence of the [[gömböc]], a body with one stable and one unstable [[Mechanical equilibrium|point of equilibrium]] when resting on a flat surface. This conjecture was proved by [[Gábor Domokos]] in 2006.<ref name=wolfram>{{cite web |last1=Weisstein |first1=Eric W. |title=Gömböc |url=https://mathworld.wolfram.com/Gomboc.html |website=[[MathWorld]] |access-date=29 April 2024 |language=en}}</ref><ref>{{Cite book|title = What's Happening in the Mathematical Sciences|url = https://books.google.com/books?id=la0xAAAAQBAJ&pg=PA104|publisher = American Mathematical Soc.|date = 29 December 2010|isbn = 9780821849996|language = en|first = Dana|last = Mackenzie|page = 104}}</ref>

Arnold generalised the results of [[Isaac Newton]], [[Pierre-Simon Laplace]], and [[James Ivory (mathematician)|James Ivory]] on the [[shell theorem]], showing it to be applicable to algebraic hypersurfaces.<ref>{{cite journal |last1=Izmestiev |first1=Ivan |last2=Tabachnikov |first2=Serge |title=Ivory's theorem revisited |journal=Journal of Integrable Systems |date=2017 |volume=2 |issue=1 |doi=10.1093/integr/xyx006 }}</ref>

[[Arnold's strange duality]] was one of the first examples of [[Mirror symmetry (string theory)|mirror symmetry]] (for [[K3 surface]]s).<ref>Dolgachev, I. V. "Mirror symmetry for lattice polarized K3 surfaces". ''J Math Sci'' 81, 2599–2630 (1996). https://arxiv.org/abs/alg-geom/9502005 https://doi.org/10.1007/BF02362332</ref>

In [[magnetohydrodynamics]], Arnold and E. I. Korkina investigated in 1983 the dynamo property of the [[Arnold–Beltrami–Childress flow|ABC flow]].<ref>Ismaël Bouya and Emmanuel Dormy; "Revisiting the ABC flow dynamo". ''Physics of Fluids'', 1 March 2013; 25 (3): 037103. https://arxiv.org/pdf/1206.5186 https://doi.org/10.1063/1.4795546</ref>

==Popular mathematical writings==
Arnold is well known for his lucid writing style, combining mathematical rigour with physical intuition, and an easy conversational style of teaching and education. His writings present a fresh, often [[geometric]] approach to traditional mathematical topics like [[ordinary differential equation]]s, and his many textbooks have been influential in the development of new areas of mathematics. The standard criticism about Arnold's pedagogy is that his books "are beautiful treatments of their subjects that are appreciated by experts, but too many details are omitted for students to learn the mathematics required to prove the statements that he so effortlessly justifies." His defence was that his books are meant to teach the subject to "those who truly wish to understand it".<ref>{{cite journal |last1=Chicone |first1=Carmen |title=Review of Ordinary Differential Equations |journal=SIAM Review |date=2007 |volume=49 |issue=2 |pages=335–336 |jstor=20453964 }} ''(Chicone mentions the criticism but does not agree with it.)''</ref>

Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the 20th century. He strongly believed that this approach—popularly implemented by the [[Nicolas Bourbaki|Bourbaki]] school in France—initially had a negative impact on French [[mathematical education]] and later in other countries.<ref>{{cite journal |last1=Arnol'd |first1=Vladimir I |title=On teaching mathematics |journal=Russian Mathematical Surveys |date=28 February 1998 |volume=53 |issue=1 |pages=229–236 |doi=10.1070/RM1998v053n01ABEH000005 |bibcode=1998RuMaS..53..229A }}</ref><ref name="interview1">[https://www.ams.org/notices/199704/arnold.pdf An Interview with Vladimir Arnol'd], by S. H. Lui, ''[[Notices of the AMS]]'', 1997.</ref> He was very concerned about what he saw as the divorce of mathematics from the [[natural science]]s in the 20th century.<ref>{{cite journal | last1=Ezra | first1=Gregory S. | last2=Wiggins | first2=Stephen | title=Vladimir Igorevich Arnold | journal=[[Physics Today]] | volume=63 | issue=12 | date=1 December 2010 | doi=10.1063/1.3529010 | pages=74–76| bibcode=2010PhT....63l..74E }}</ref> Arnold was very interested in the [[history of mathematics]],<ref name="Karpenkov Vladimir Igorevich Arnold">{{cite report |type=Preprint |last1=Karpenkov |first1=Oleg |title=Vladimir Igorevich Arnold |date=2010 |arxiv=1007.0688 }} (this was published at the ''Internationale Mathematische Nachrichten'' volume 214 (2010) pp. 49–57: https://www.oemg.ac.at/IMN/imn214.pdf )</ref> and in an interview,<ref name="interview1" /> remarked that he had learned much of what he knew about mathematics through the study of [[Felix Klein]]'s book ''[[Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert|Development of Mathematics in the 19th Century]]'' {{Ndash}}a book he often recommended to his students.<ref>[[Boris Khesin|B. Khesin]] and [[Sergei Tabachnikov|S. Tabachnikov]], Tribute to Vladimir Arnold, ''Notices of the AMS'', '''59''':3 (2012) 378–399.</ref> He studied the works of [[Christiaan Huygens|Huygens]], [[Isaac Newton|Newton]] and [[Henri Poincaré|Poincaré]],<ref>{{citation|last1=Goryunov|first1=V.|last2=Zakalyukin|first2=V.|title=Vladimir I. Arnold|journal=[[Moscow Mathematical Journal]]|volume=11|issue=3|url=https://www.ams.org/distribution/mmj/vol11-3-2011/vladimir-arnold.html|year=2011}}.</ref> and reported finding ideas that had yet to be explored in the works of Newton and Poincaré.<ref>See for example: Arnold, V. I.; Vasilev, V. A. (1989), "[https://www.ams.org/journals/notices/198911/198911FullIssue.pdf Newton's Principia read 300 years later]" and {{cite journal |last1=Arnol'd |first1=Vladimir I |title=Forgotten and neglected theories of Poincaré |journal=Russian Mathematical Surveys |date=28 February 2006 |volume=61 |issue=1 |pages=1–18 |doi=10.1070/RM2006v061n01ABEH004297 |bibcode=2006RuMaS..61....1A }}</ref>

==Later life and death==
In 1999 Arnold suffered a serious bicycle accident in Paris, resulting in a [[traumatic brain injury]]. He regained consciousness after a few weeks but had [[amnesia]] and for some time could not even recognise his own wife at the hospital.<ref>{{cite book | last=Arnold | first=Vladimir I. | title=Yesterday and Long Ago | publisher=Springer; Phasis | publication-place=Berlin; New York : Moscow | date=2007 | page=V | isbn=978-3-540-28734-6 | oclc=76794406}}</ref> He went on to make a good recovery.<ref>{{cite journal |author1= Leonid Polterovich|author2= Inna Scherbak|date=7 September 2011|title=V.I. Arnold (1937–2010) |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=113 |issue=4 |pages=185–219 |doi=10.1365/s13291-011-0027-6 |author1-link= Leonid Polterovich |url=https://www.mathematik.de/images/DMV/Jahresberichte/Jahresberichte_Archiv/2011.113Heft4.pdf}}</ref>

To his students and colleagues Arnold was known also for his sense of humour. For example, once at his seminar in Moscow, at the beginning of the school year, when he usually was formulating new problems, he said:

{{blockquote|There is a general principle that a stupid man can ask such questions to which one hundred wise men would not be able to answer. In accordance with this principle I shall formulate some problems.<ref>{{cite news| url=https://www.telegraph.co.uk/news/obituaries/science-obituaries/7886200/Vladimir-Arnold.html | location=London | work=The Daily Telegraph | title=Vladimir Arnold | date=12 July 2010}}</ref>}}

Arnold died of [[acute pancreatitis]]<ref>{{cite news |title=Vladimir Arnold Dies at 72; Pioneering Mathematician |author=Kenneth Chang |url=https://www.nytimes.com/2010/06/11/science/11arnold.html |newspaper=[[The New York Times]] |date=11 June 2010 |access-date=12 June 2013}}</ref> on 3 June 2010 in Paris, nine days before his 73rd birthday.<ref>{{cite news|title=Number's up as top mathematician Vladimir Arnold dies|newspaper=[[Herald Sun]]|date=4 June 2010|url=http://www.heraldsun.com.au/news/breaking-news/numbers-up-as-top-mathematician-vladimir-arnold-dies/story-e6frf7jx-1225875367896|access-date=6 June 2010|archive-url=https://web.archive.org/web/20110614172804/http://www.heraldsun.com.au/news/breaking-news/numbers-up-as-top-mathematician-vladimir-arnold-dies/story-e6frf7jx-1225875367896|archive-date=2011-06-14}}</ref> He was buried on 15 June in Moscow, at the [[Novodevichy Convent|Novodevichy Monastery]].<ref>
{{cite web |url=http://www.pdmi.ras.ru/~arnsem/Arnold/ |title=From V. I. Arnold's web page |access-date=12 June 2013}}</ref>

In a telegram to Arnold's family, [[Russian president]] [[Dmitry Medvedev]] stated:

{{blockquote|The death of Vladimir Arnold, one of the greatest mathematicians of our time, is an irretrievable loss for world science. It is difficult to overestimate the contribution made by academician Arnold to modern mathematics and the prestige of Russian science.

Teaching had a special place in Vladimir Arnold's life and he had great influence as an enlightened mentor who taught several generations of talented scientists.

The memory of Vladimir Arnold will forever remain in the hearts of his colleagues, friends and students, as well as everyone who knew and admired this brilliant man.<ref>{{cite news|url=http://eng.kremlin.ru/news/437#sel=3:1,5:29|title=Condolences to the family of Vladimir Arnold|date=15 June 2010|publisher=[[Presidential Press and Information Office]]|access-date=1 September 2011}} ([https://web.archive.org/web/20140726092511/http://eng.kremlin.ru/news/437#sel=3:1,5:29 archived version])</ref>}}

==Honours and awards==
* [[Moscow Mathematical Society#Annual Prize|Moscow Mathematical Society Prize for Young Mathematicians]] (1958).<ref name="MacTutor" />
* [[Lenin Prize]] (1965, with [[Andrey Kolmogorov]]),<ref name="Karpenkov Vladimir Igorevich Arnold"/> "for work on [[celestial mechanics]]."
* [[Crafoord Prize]] (1982, with [[Louis Nirenberg]]),<ref>{{cite news |title=American and Russian Share Prize in Mathematics |author=Harold M. Schmeck Jr. |date=27 June 1982 |newspaper=[[The New York Times]] |url=https://www.nytimes.com/1982/06/27/us/american-and-russian-share-prize-in-mathematics.html}}</ref> "for their outstanding achievements in the theory of non-linear differential equations."<ref>{{cite web|archive-url=https://web.archive.org/web/20160126153013/http://www.kva.se/globalassets/priser/crafoord/2014/rattigheter/crafoordprize1982_2014.pdf|url=http://www.kva.se/globalassets/priser/crafoord/2014/rattigheter/crafoordprize1982_2014.pdf|archive-date=2016-01-26|url-status=dead|title=The Crafoord Prize 1982–2014|publisher=The Anna-Greta and Holger Crafoord Fund}}</ref>
* Elected member of the United States [[National Academy of Sciences]] in 1983.<ref>{{Cite web |title=Vladimir I. Arnold |url=http://www.nasonline.org/member-directory/deceased-members/47101.html |access-date=2022-04-14 |website=www.nasonline.org}}</ref>
* Foreign Honorary Member of the [[American Academy of Arts and Sciences]] (1987)<ref name=AAAS>{{cite web|title=Book of Members, 1780–2010: Chapter A|url=http://www.amacad.org/publications/BookofMembers/ChapterA.pdf|publisher=American Academy of Arts and Sciences|access-date=25 April 2011}}</ref>
* Elected a [[Foreign Member of the Royal Society]] (ForMemRS) of London in 1988.<ref name=rsbm/>
* Elected member of the [[American Philosophical Society]] in 1990.<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Vladimir+Arnold&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2022-04-14 |website=search.amphilsoc.org}}</ref>
* [[Lobachevsky Prize#Russian Academy of Sciences|Lobachevsky Prize of the Russian Academy of Sciences]] (1992)<ref>D. B. Anosov, A. A. Bolibrukh, [[Ludvig Faddeev|Lyudvig D. Faddeev]], A. A. Gonchar, [[Mikhail Leonidovich Gromov|M. L. Gromov]], [[Sabir Gusein-Zade|S. M. Gusein-Zade]], Yu. S. Il'yashenko, [[Boris Khesin|B. A. Khesin]], [[Askold Khovanskii|A. G. Khovanskii]], [[Maxim Kontsevich|M. L. Kontsevich]], V. V. Kozlov, [[Yuri I. Manin|Yu. I. Manin]], A. I. Neishtadt, [[Sergei Novikov (mathematician)|S. P. Novikov]], Yu. S. Osipov, M. B. Sevryuk, [[Yakov Sinai|Yakov G. Sinai]], A. N. Tyurin, A. N. Varchenko, [[Victor Anatolyevich Vassiliev|V. A. Vasil'ev]], V. M. Vershik and V. M. Zakalyukin (1997) . [http://iopscience.iop.org/0036-0279/52/5/M10 "Vladimir Igorevich Arnol'd (on his sixtieth birthday)"]. ''[[Russian Mathematical Surveys]]'', Volume 52, Number 5. (translated from the Russian by R. F. Wheeler)</ref>
* [[Harvey Prize]] (1994), "In recognition of his basic contribution to the stability theory of Dynamical Systems, his pioneering work on singularity theory and seminal contributions to analysis and geometry."<ref>{{cite web|url=https://harveypz.net.technion.ac.il/harvey-prize-laureates/|publisher=Technion|title=Prize Winners – Harvey Prize|access-date=2024-08-24}}</ref>
* [[Dannie Heineman Prize for Mathematical Physics]] (2001), "for his fundamental contributions to our understanding of dynamics and of singularities of maps with profound consequences for [[mechanics]], [[astrophysics]], [[statistical mechanics]], [[hydrodynamics]] and [[optics]]."<ref>[http://www.aps.org/programs/honors/prizes/prizerecipient.cfm?last_nm=Arnol%27d&first_nm=Vladimir&year=2001 American Physical Society – 2001 Dannie Heineman Prize for Mathematical Physics Recipient]</ref>
* [[Wolf Prize in Mathematics]] (2001), "for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory."<ref>[http://www.wolffund.org.il/index.php?dir=site&page=winners&cs=163&language=eng The Wolf Foundation – Vladimir I. Arnold Winner of Wolf Prize in Mathematics]</ref>
* [[State Prize of the Russian Federation]] (2007),<ref name=Kommersant>{{cite news |title=Названы лауреаты Государственной премии РФ |url=https://www.kommersant.ru/doc/894018 |access-date=27 February 2025 |work=www.kommersant.ru |date=20 May 2008 |language=ru}}</ref> "for outstanding contribution to development of mathematics."
* [[Shaw Prize]] in mathematical sciences (2008, with [[Ludvig Faddeev|Ludwig Faddeev]]), "for their widespread and influential contributions to Mathematical Physics."<ref>{{cite web |title=The 2008 Prize in Mathematical Sciences |url=https://www.shawprize.org/laureates/mathematical-sciences/2008 |publisher=Shaw Prize Foundation |access-date=7 October 2022 |archive-url=https://web.archive.org/web/20221007154628/https://www.shawprize.org/laureates/mathematical-sciences/2008 |archive-date=7 October 2022}}</ref><ref>{{cite journal |title=Arnold and Faddeev Receive 2008 Shaw Prize |journal=Notices of the American Mathematical Society |date=2008 |volume=55 |issue=8 |pages=966 |url=https://www.ams.org/notices/200808/tx080800966p.pdf |access-date=8 October 2022 |archive-url=https://web.archive.org/web/20221007154713/https://www.ams.org/notices/200808/tx080800966p.pdf |archive-date=7 October 2022}}</ref>

The [[minor planet]] [[10031 Vladarnolda]] was named after him in 1981 by [[Lyudmila Georgievna Karachkina]].<ref>{{cite book|author=Lutz D. Schmadel|title=Dictionary of Minor Planet Names|url=https://books.google.com/books?id=aeAg1X7afOoC&pg=PA717|publisher=Springer Science & Business Media|isbn=978-3-642-29718-2|pages=717|date=10 June 2012}}</ref>

The ''[[Arnold Mathematical Journal]]'', published for the first time in 2015, is named after him.<ref>{{citation|journal=Arnold Mathematical Journal|volume=1|issue=1|year=2015|pages=1–3|title=Journal Description Arnold Mathematical Journal|author=Editorial|doi=10.1007/s40598-015-0006-6|doi-access=free|bibcode=2015ArnMJ...1....1.}}.</ref>

The Arnold Fellowships, of the [[London Institute]] are named after him.<ref>{{cite web | url=https://lims.ac.uk/arnold-fellowships/ | title=Arnold Fellowships }}</ref><ref>{{cite news | url=https://www.telegraph.co.uk/opinion/2022/07/01/britain-rescuing-academics-vladimir-putins-clutches/ | title=Britain is rescuing academics from Vladimir Putin's clutches | newspaper=The Telegraph | date=July 2022 | last1=Fink | first1=Thomas }}</ref>

He was a plenary speaker at both the 1974 and 1983 [[International Congress of Mathematicians]] in Vancouver and [[Warsaw]], respectively.<ref>{{Cite web |url=http://www.mathunion.org/db/ICM/Speakers/SortedByLastname.php |title=International Mathematical Union (IMU) |access-date=22 May 2015 |archive-date=24 November 2017 |archive-url=https://web.archive.org/web/20171124141541/http://www.mathunion.org/db/ICM/Speakers/SortedByLastname.php |url-status=dead }}</ref>

===Fields Medal omission===
Arnold was nominated for the 1974 [[Fields Medal]], one of the highest honours a mathematician could receive, but interference from the Soviet government led to it being withdrawn. Arnold's public opposition to the persecution of [[Dissident#Eastern bloc|dissident]]s had led him into direct conflict with influential Soviet officials, and he suffered persecution himself, including not being allowed to leave the Soviet Union during most of the 1970s and 1980s.<ref>{{cite encyclopedia |title=Vladimir Igorevich Arnold |author=Martin L. White |encyclopedia=[[Encyclopædia Britannica]] |year=2015 |url=http://www.britannica.com/EBchecked/topic/1742993/Vladimir-Igorevich-Arnold |access-date=18 March 2015 |archive-date=2 April 2015 |archive-url=https://web.archive.org/web/20150402102751/http://www.britannica.com/EBchecked/topic/1742993/Vladimir-Igorevich-Arnold |url-status=dead }}</ref><ref>{{cite news |author=Thomas H. Maugh II |date=23 June 2010 |title=Vladimir Arnold, noted Russian mathematician, dies at 72 |url=https://www.washingtonpost.com/wp-dyn/content/article/2010/06/22/AR2010062205069.html |newspaper=[[The Washington Post]]|access-date=18 March 2015}}</ref>

==Personal life==
Arnold was married to Eleonora Arnold. They had two children, Igor Arnold and Dmitry Arnold.<ref>https://www.washingtonpost.com/archive/local/2010/06/23/russian-mathematician-expanded-understanding-of-the-solar-system/596cc277-c6c0-4546-933a-a0e48a2199e7/</ref>

==Selected bibliography==

Arnold wrote around 700 research papers and many books (including ten university textbooks).<ref>preface of A. B. Givental; B. A. Khesin; J. E. Marsden; A. N. Varchenko; V. A. Vassilev; O. Ya. Viro; V. M. Zakalyukin (editors), 2010. ''Collected Works, Volume I: Representations of Functions, Celestial Mechanics, and KAM Theory (1957–1965)''. [[Springer Science+Business Media|Springer]]</ref>

* 1966: {{cite journal
 | first=Vladimir
 | last=Arnold
 | title=Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits
 | journal=[[Annales de l'Institut Fourier]]
 | volume=16
 | number=1
 | pages=319–361
 | year=1966
 | doi=10.5802/aif.233
 | url=https://aif.centre-mersenne.org/article/AIF_1966__16_1_319_0.pdf
| doi-access=free
 }}
* 1978: ''[[Ordinary Differential Equations (textbook)|Ordinary Differential Equations]]'', The MIT Press {{ISBN|0-262-51018-9}}.<ref>{{cite journal |last1=Sacker |first1=Robert J. |title=Ordinary Differential Equations |journal=Technometrics |date=August 1975 |volume=17 |issue=3 |pages=388–389 |doi=10.1080/00401706.1975.10489355 }}</ref><ref>{{cite journal |last1=Kapadia |first1=Devendra A. |title=Ordinary differential equations, by V. I. Arnold. Pp 334. DM 78. 1992. ISBN 3-540-54813-0 (Springer) |journal=The Mathematical Gazette |date=March 1995 |volume=79 |issue=484 |pages=228–229 |doi=10.2307/3620107 |jstor=3620107 }}</ref><ref>{{Cite journal |last=Chicone |first=Carmen |date=2007 |title=Review of Ordinary Differential Equations |journal=SIAM Review |volume=49 |issue=2 |pages=335–336 |jstor=20453964 }}</ref>
* 1985: {{cite book
 | first1=V. I.
 | last1=Arnold
 | first2=S. M.
 | last2=Gusein-Zade
 | first3=A. N.
 | last3=Varchenko
 | title=Singularities of Differentiable Maps, Volume I: The Classification of Critical Points Caustics and Wave Fronts
 | publisher=[[Birkhäuser]]
 | series=Monographs in Mathematics
 | year=1985
 | volume=82
 | isbn=978-1-4612-9589-1
 | doi=10.1007/978-1-4612-5154-5
}}
* 1988: {{cite book
 | first1=V. I.
 | last1=Arnold
 | first2=S. M.
 | last2=Gusein-Zade
 | first3=A. N.
 | last3=Varchenko
 | editor3-first=A. N
 | editor3-last=Varchenko
 | editor2-first=S. M
 | editor2-last=Gusein-Zade
 | editor1-first=V. I
 | editor1-last=Arnold
 | title=Singularities of Differentiable Maps, Volume II: Monodromy and Asymptotics of Integrals
 | publisher=[[Birkhäuser]]
 | series=Monographs in Mathematics
 | year=1988
 | volume=83
 | isbn=978-1-4612-8408-6
 | doi=10.1007/978-1-4612-3940-6
 | url=http://www.numdam.org/item/PMIHES_1967__33__21_0/
 }}
* 1988: {{cite book
 | first=V.I.
 | last=Arnold
 | title=Geometrical Methods in the Theory of Ordinary Differential Equations
 | publisher=[[Springer-Verlag|Springer]]
 | year=1988
 | edition=2nd
 | series=Grundlehren der mathematischen Wissenschaften
 | volume=250
 | isbn=978-1-4612-6994-6
 | doi=10.1007/978-1-4612-1037-5
}}
* 1989: {{cite book
 | first=V.I.
 | last=Arnold
 | title=[[Mathematical Methods of Classical Mechanics]]
 | publisher=[[Springer-Verlag|Springer]]
 | year=1989
 | edition=2nd
 | series= Graduate Texts in Mathematics
 | volume=60
 | isbn=978-1-4419-3087-3
 | doi=10.1007/978-1-4757-2063-1
}}<ref>Review by Ian N. Sneddon (''Bulletin of the American Mathematical Society'', Vol. 2): https://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14755-2/S0273-0979-1980-14755-2.pdf</ref><ref>Review by [[Roger A. Broucke|R. Broucke]] (''Celestial Mechanics'', Vol. 28): {{bibcode|1982CeMec..28..345A}}.</ref>
* 1989 {{cite book |last=Арнольд |first=В. И. |author-link=Vladimir Arnold |year=1989 |title=Гюйгенс и Барроу, Ньютон и Гук - Первые шаги математического анализа и теории катастроф |location=М. |publisher=[[Nauka (publisher)|Наука]] |pages=98 |isbn=5-02-013935-1}}
* 1989: (with A. Avez) ''Ergodic Problems of Classical Mechanics'', Addison-Wesley {{ISBN|0-201-09406-1}}.
* 1990: ''Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals'', Eric J.F. Primrose translator, [[Birkhäuser Verlag]] (1990) {{ISBN|3-7643-2383-3}}.<ref>{{Cite journal|title = Huygens and Barrow, Newton and Hooke: Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals (V. I. Arnol'd)|journal = SIAM Review|date = 1 September 1991 |pages = 493–495|volume = 33|issue = 3|doi = 10.1137/1033119|first = N.|last = Kazarinoff}}</ref><ref>{{Cite journal|title = Arnol'd, V. I., Huygens and Barrow, Newton and Hooke. Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals. Basel etc., Birkhäuser Verlag 1990. 118 pp., sfr 24.00. ISBN 3-7643-2383-3|journal =  Journal of Applied Mathematics and Mechanics|date = 1993 |pages = 34|volume = 73|issue = 1|doi = 10.1002/zamm.19930730109|first = R.|last = Thiele|bibcode = 1993ZaMM...73S..34T}}</ref><ref>{{Cite journal|title = V. I. Arnol'd, Huygens and Barrow, Newton and Hooke, translated by E. J. F. Primrose (Birkhäuser Verlag, Basel 1990), 118 pp., 3 7643 2383 3, sFr 24.|journal = Proceedings of the Edinburgh Mathematical Society |series=Series 2|date =June 1991 |pages = 335–336|volume = 34|issue = 2|doi = 10.1017/S0013091500007240|first = Douglas C.|last = Heggie|doi-access = free}}</ref>
* 1991: {{Cite book|url=https://books.google.com/books?id=5aCqyTCElNkC|title=The Theory of Singularities and Its Applications|last=Arnolʹd|first=Vladimir Igorevich|year=1991|publisher=Cambridge University Press|isbn=9780521422802|language=en}}
* 1995:''Topological Invariants of Plane Curves and Caustics'',<ref>{{Cite journal|last=Goryunov|first=V. V.|date=1 October 1996|title=V. I. Arnold Topological invariants of plane curves and caustics (University Lecture Series, Vol. 5, American Mathematical Society, Providence, RI, 1995), 60pp., paperback, 0 8218 0308 5, £17.50.|journal=Proceedings of the Edinburgh Mathematical Society |series=Series 2|volume=39|issue=3|pages=590–591|doi=10.1017/S0013091500023348 |doi-access=free}}</ref> American Mathematical Society (1994) {{ISBN|978-0-8218-0308-0}}
* 1998: "On the teaching of mathematics" (Russian) Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; translation in ''[[Russian Mathematical Surveys|Russian Math. Surveys]]'' 53(1): 229–236.
* 1999: (with [[Valentin Afraimovich]]) ''Bifurcation Theory And Catastrophe Theory'' Springer {{isbn|3-540-65379-1 }}
* 2001: "Tsepniye Drobi" (Continued Fractions, in Russian), Moscow (2001).
* 2002: "Что такое математика?" (What is mathematics?, in Russian) ISBN 978-5-94057-426-2.
* 2004: ''Teoriya Katastrof'' (Catastrophe Theory,<ref>{{Cite journal|last=Bernfeld|first=Stephen R.|date=1 January 1985|title=Review of Catastrophe Theory|jstor=2031497|journal=SIAM Review|volume=27|issue=1|pages=90–91|doi=10.1137/1027019}}</ref> in Russian), 4th ed. Moscow, [[Editorial-URSS]] (2004), {{ISBN|5-354-00674-0}}.
* 2004: {{cite book |title=[[Arnold's Problems]] |editor=Vladimir I. Arnold |isbn=978-3-540-20748-1 |publisher=Springer-Verlag |edition=2nd|date=15 November 2004 }}<ref>{{cite journal |last1=Sevryuk |first1=Mikhail B. |title=Book Review: Arnold's problems |journal=Bulletin of the American Mathematical Society |date=June 2005 |volume=43 |issue=1 |pages=101–110 |doi=10.1090/s0273-0979-05-01069-4 |doi-access=free }}</ref>
* 2004: {{cite book
 | first=Vladimir I.
 | last=Arnold
 | title=Lectures on Partial Differential Equations
 | publisher=[[Springer-Verlag|Springer]]
 | year=2004
 | series=Universitext
 | isbn=978-3-540-40448-4
 | doi=10.1007/978-3-662-05441-3
| url=https://cds.cern.ch/record/730993
 }}<ref>{{cite journal |last1=Guenther |first1=Ronald B. |last2=Thomann |first2=Enrique A. |title=Featured Review: Two New Books on Partial Differential Equations |journal=SIAM Review |date=2005 |volume=47 |issue=1 |pages=165–168 |jstor=20453608 }}</ref><ref>{{cite journal |last1=Groves |first1=M. |title=Book Review: Vladimir I. Arnold, Lectures on Partial Differential Equations. Universitext |journal=ZAMM |date=13 April 2005 |volume=85 |issue=4 |pages=304 |doi=10.1002/zamm.200590023 |bibcode=2005ZaMM...85..304G }}</ref>
* 2007: ''Yesterday and Long Ago'', Springer (2007), {{ISBN|978-3-540-28734-6}}.
* 2013: {{cite book
 | first=Vladimir I.
 | last=Arnold
 | editor1-first=Ilia
 | editor1-last=Itenberg
 | editor2-first=Viatcheslav
 | editor2-last=Kharlamov
 | editor3-first=Eugenii I.
 | editor3-last=Shustin
 | title=Real Algebraic Geometry
 | publisher=[[Springer-Verlag|Springer]]
 | year=2013
 | series=Unitext
 | volume=66
 | isbn=978-3-642-36242-2
 | doi=10.1007/978-3-642-36243-9
}}<ref>{{cite web|first=Fernando Q. |last=Gouvêa |author-link=Fernando Q. Gouvêa |title=Review of ''Real Algebraic Geometry'' by Arnold |url=https://old.maa.org/press/maa-reviews/real-algebraic-geometry |archive-url=https://web.archive.org/web/20230323231917/https://www.maa.org/press/maa-reviews/real-algebraic-geometry |archive-date=23 March 2023 |website=[[MAA Reviews]] |date=2013-08-15}}</ref>
* 2014: {{cite book|author=V. I. Arnold|title=Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians|url=https://books.google.com/books?id=9INzBAAAQBAJ|date=2014|publisher=American Mathematical Society|isbn=978-1-4704-1701-7}}
* 2015: ''Experimental Mathematics''. American Mathematical Society (translated from Russian, 2015) {{ISBN|978-0821894163}}
* 2015: ''Lectures and Problems: A Gift to Young Mathematicians'', American Math Society, (translated from Russian, 2015) {{ISBN|978-1470422592}}
* 1998: ''Topological Methods in Hydrodynamics'' (with [[Boris Khesin]])<ref>[https://ems.press/content/serial-article-files/28174 Review, by Daniel Peralta-Salas, of the book "Topological Methods in Hydrodynamics", by Vladimir I. Arnold and Boris A. Khesin]</ref> {{ISBN|978-0387949475}}
* 1999: ''Pseudoperiodic Topology'' (with [[Maxim Kontsevich]]), American Mathematical Society. {{isbn|978-0821820940}}

===Collected works===
* 2010: A. B. Givental; B. A. Khesin; J. E. Marsden; A. N. Varchenko; V. A. Vassilev; O. Ya. Viro; V. M. Zakalyukin (editors). ''Collected Works, Volume I: Representations of Functions, Celestial Mechanics, and KAM Theory (1957–1965)''. [[Springer Science+Business Media|Springer]]
* 2013: A. B. Givental; B. A. Khesin; A. N. Varchenko; V. A. Vassilev; O. Ya. Viro; (editors). ''Collected Works, Volume II: Hydrodynamics, Bifurcation Theory, and Algebraic Geometry (1965–1972)''. Springer.
* 2016: Givental, A.B., Khesin, B., Sevryuk, M.B., Vassiliev, V.A., Viro, O.Y. (Eds.). ''Collected Works, Volume III: Singularity Theory 1972–1979''. Springer.
* 2018: Givental, A.B., Khesin, B., Sevryuk, M.B., Vassiliev, V.A., Viro, O.Y. (Eds.). ''Collected Works, Volume IV: Singularities in Symplectic and Contact Geometry 1980–1985''. Springer.
* 2023: Alexander B. Givental, Boris A. Khesin, Mikhail B. Sevryuk, Victor A. Vassiliev, Oleg Ya. Viro (Eds.). ''Collected Works, Volume VI: Dynamics, Combinatorics, and Invariants of Knots, Curves, and Wave Fronts 1992–1995''. Springer.
* 2025: Boris A. Khesin, Mikhail B. Sevryuk, Victor A. Vassiliev (Eds.). ''Collected Works, Volume V: Symplectic Topology, Dynamics of Intersections, and Catastrophe Theory 1986–1991''. Springer.
* 2025: Boris A. Khesin, Mikhail B. Sevryuk, Victor A. Vassiliev (Eds.). ''Collected Works, Volume VII: Spaces and Singularities of Curves, Mathematical Trinities, and Mathematical Education 1996–1999''. Springer.

==See also==
{{Portal|Mathematics}}
*[[List of things named after Vladimir Arnold]]
*[[Geometric mechanics]]
*[[Tree-like curve]]

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

==Further reading==
* Khesin, Boris; Tabachnikov, Serge (Coordinating Editors). "[https://www.ams.org/notices/201203/rtx120300378p.pdf Tribute to Vladimir Arnold]", ''[[Notices of the American Mathematical Society]]'', March 2012, Volume 59, Number 3, pp.&nbsp;378–399.
* Khesin, Boris; Tabachnikov, Serge (Coordinating Editors). "[https://www.ams.org/notices/201204/rtx120400482p.pdf Memories of Vladimir Arnold]", ''Notices of the American Mathematical Society'', April 2012, Volume 59, Number 4, pp.&nbsp;482–502.
* {{cite book|editor1=Boris A. Khesin|editor2=Serge L. Tabachnikov|title=Arnold: Swimming Against the Tide|url=https://books.google.com/books?id=aBWHBAAAQBAJ|date=2014|publisher=American Mathematical Society|isbn=978-1-4704-1699-7}}
* {{cite journal |author1= Leonid Polterovich|author2= Inna Scherbak|date=7 September 2011|title=V.I. Arnold (1937–2010) |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=113 |issue=4 |pages=185–219 |doi=10.1365/s13291-011-0027-6 |author1-link= Leonid Polterovich}}
* {{cite journal |url=http://www.ems-ph.org/journals/newsletter/pdf/2015-06-96.pdf |journal=[[Newsletter of the European Mathematical Society|EMS Newsletter]] |issue=96 |date=June 2015 |pages=26–48 |title=Features: "Knotted Vortex Lines and Vortex Tubes in Stationary Fluid Flows"; "On Delusive Nodal Sets of Free Oscillations"}}

==External links==
{{Commons category}}
{{wikiquote}}
* [http://www.pdmi.ras.ru/~arnsem/Arnold/ V. I. Arnold's web page]
* [http://www.mi.ras.ru/~arnold/ Personal web page]
* [https://www.slmath.org/workshops/390/schedules/2714 V. I. Arnold lecturing on continued fractions] (2007, from [[Simons Laufer Mathematical Sciences Institute|SLMath]], formerly MSRI)
* [http://www.mccme.ru/ium/~arnold/ A short curriculum vitae]
* [https://www.karlin.mff.cuni.cz/~spurny/doc/articles/arnold.htm ''On Teaching Mathematics''] {{Webarchive|url=https://web.archive.org/web/20170428233041/http://pauli.uni-muenster.de/~munsteg/arnold.html |date=28 April 2017 }}, text of a talk from 1997 espousing Arnold's opinions on mathematical instruction
* [https://www.mathunion.org/fileadmin/IMU/Publications/Bulletins/1994_1999/39/Arnold.html ''Topology of Plane Curves, Wave Fronts, Legendrian Knots, Sturm Theory and Flattenings of Projective Curves'']
* [https://imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf Problems from 5 to 15], a text by Arnold for school students, available at the [http://www.imaginary.org IMAGINARY platform]
* {{MathGenealogy |id=17493}}
* [[Semyon Kutateladze|S. Kutateladze]], [https://web.archive.org/web/20190718234256/http://www.math.nsc.ru/LBRT/g2/english/ssk/arnold_e.html Arnold Is Gone] (June 4, 2010)
* [http://www.math.ru/lib/files/pdf/mehmat/mm3.pdf В.Б.Демидовичем (2009), МЕХМАТЯНЕ ВСПОМИНАЮТ 2: В.И.Арнольд, pp. 25–58] (in Russian)
* [https://zbmath.org/authors/?q=ai:arnold.vladimir-igorevich Author profile] in the database [[Zentralblatt MATH|zbMATH]]
* [[Leonid Polterovich]] and Inna Scherbakh (2011). "[https://www.mathematik.de/images/DMV/Jahresberichte/Jahresberichte_Archiv/2011.113Heft4.pdf V.I. Arnold (1937–2010)]" DOI 10.1365/s13291-011-0027-6
* S. M. Gusein-Zade, and A. N. Varchenko (2011) . "[https://www.ams.org/journals/mosc/2011-72-00/mosc-72-print-matter.pdf Vladimir Igorevich Arnold]", ''Transactions of the Moscow Mathematical Society''

{{Wolf Prize in Mathematics}}
{{Shaw Prize laureates}}

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