Editing Vladimir Arnold (section)

==Mathematical work==
{{see also|Stability of the Solar System}}
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}}
The problem is the following question: can every continuous function of three variables be expressed as a [[function composition|composition]] of finitely many continuous functions of two variables?  The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of [[Andrey Kolmogorov]]. Kolmogorov had shown in 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 in fact required, thus answering the Hilbert's question when posed 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 tongue|Liouville–Arnold theorem|Hilbert–Arnold problem}}
[[Jürgen Moser|Moser]] and Arnold expanded the ideas of [[Andrey Kolmogorov|Kolmogorov]] (who was inspired by questions of [[Henri Poincaré|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 of time, 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 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>Zaslavsky web map, by George Zaslavsky http://www.scholarpedia.org/article/Zaslavsky_web_map</ref>

===Singularity theory===
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 arXiv |eprint = math/0203260|last1 = Landsberg|first1 = J. M.|title = Representation theory and projective geometry|last2 = Manivel|first2 = L.|year = 2002}}</ref>

===Fluid dynamics===
{{see also|Arnold–Beltrami–Childress flow|
Beltrami vector field#Beltrami fields and complexity in fluid mechanics}}
In 1966, Arnold published "{{lang|fr|Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits}}", 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 effectively linked topics previously thought to be unrelated, and enabled mathematical solutions to many questions related to fluid flows and their 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 the year 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 http://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 the direction of 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 to be 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 |arxiv = math/0004134|doi = 10.1070/RM2000v055n04ABEH000315|bibcode = 2000RuMaS..55..735D|title = Topological properties of real algebraic varieties: Du coté de chez Rokhlin|journal = Russian Mathematical Surveys|volume = 55|issue = 4|pages = 735–814|year = 2000|last1 = Degtyarev|first1 = A. I.|last2 = Kharlamov|first2 = V. M.| s2cid=250775854 }}</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>

===Topology===
According to [[Victor Vassiliev]], Arnold "worked comparatively little on topology for topology's sake." And he was rather 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>http://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 revolutionized [[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>

=== Other ===
Arnold conjectured the existence of the [[gömböc]], a body with just one stable and one unstable [[Mechanical equilibrium|point of equilibrium]] when resting on a flat surface.<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 generalized 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>Ivan Izmestiev, [[Serge Tabachnikov]]. "Ivory’s theorem revisited", ''Journal of Integrable Systems'', Volume 2, Issue 1, (2017) https://doi.org/10.1093/integr/xyx006</ref>