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Vladimir Arnold

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Template:Short description Template:Use dmy dates Template:Family name hatnote Template:Infobox scientist Vladimir Igorevich Arnold (or Arnol'd; Template:Langx, Template:IPA; 12 June 1937 – 3 June 2010)<ref name=rsbm>Template:Cite journal</ref><ref>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 of integrable systems, and contributed to several areas, including geometrical theory of dynamical systems, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, differential equations, classical mechanics, differential-geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem.

His first main result was the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded three new branches of mathematics: topological Galois theory (with his student Askold Khovanskii), symplectic topology and KAM theory.

Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as Mathematical Methods of Classical Mechanics) and popular mathematics books, he influenced many mathematicians and physicists.<ref name="MacTutor">Template:MacTutor Biography</ref><ref>Template:Cite book</ref> Many of his books were translated into English. His views on education were particularly opposed to those of Bourbaki.

Biography

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File:Владимир Арнольд в 1963 г.jpg
Arnold in 1963.

Vladimir Igorevich Arnold was born on 12 June 1937 in Odesa, Soviet Union (now Odesa, Ukraine). His father was Igor Vladimirovich Arnold (1900–1948), a mathematician. His mother was Nina Alexandrovna Arnold (1909–1986, née Isakovich), a Jewish art historian.<ref name=obituary>Template:Citation</ref> While a school student, Arnold once asked his father on the reason why the multiplication of two negative numbers yielded a positive number, and his father provided an answer involving the field properties of real numbers and the preservation of the distributive property. Arnold was deeply disappointed with this answer, and developed an aversion to the axiomatic method that lasted through his life.<ref name="Arnold2007">Template:Cite book</ref> When Arnold was thirteen, his uncle Nikolai B. Zhitkov,<ref name="earlylife">Arnold: Swimming Against the Tide, p. 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 for mathematics, and he started to study by himself the mathematical books his father had left to him, which included some works of Leonhard Euler and Charles Hermite.<ref>Табачников, С. Л. . "Интервью с В.И.Арнольдом", Квант, 1990, Nº 7, pp. 2–7. (in Russian)</ref>

Arnold entered Moscow State University in 1954.<ref>Sevryuk, M.B. Translation of the V. I. Arnold paper “From Superpositions to KAM Theory” (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740). Regul. Chaot. Dyn. 19, 734–744 (2014). https://doi.org/10.1134/S1560354714060100</ref> Among his teachers there were A. N. Kolmogorov, I. M. Gelfand, L. S. Pontriagin and Pavel Alexandrov.<ref>Template:Citation</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>Template:Cite book</ref> This is the Kolmogorov–Arnold representation theorem.

Arnold obtained his PhD in 1961, with Kolmogorov as advisor.<ref>Template:Cite web</ref>

After graduating from Moscow State University in 1959, he worked there until 1986 (a professor since 1965), and then at Steklov Mathematical Institute.

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 said to have initiated the theory of symplectic topology as a distinct discipline. The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian intersections was also a motivation in the development of Floer homology.

In 1999 he suffered a serious bicycle accident in Paris, resulting in traumatic brain injury. He regained consciousness after a few weeks but had amnesia and for some time could not even recognize his own wife at the hospital.<ref>Template:Cite book</ref> He went on to make a good recovery.<ref>Polterovich and Scherbak (2011)</ref>

Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death. His PhD students include Rifkat Bogdanov, Alexander Givental, Victor Goryunov, Sabir Gusein-Zade, Emil Horozov, Yulij Ilyashenko, Boris Khesin, Askold Khovanskii, Nikolay Nekhoroshev, Boris Shapiro, Alexander Varchenko, Victor Vassiliev and Vladimir Zakalyukin.<ref name="mathgene">Template:MathGenealogy</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:

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Death

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Arnold died of acute pancreatitis<ref>Template:Cite news</ref> on 3 June 2010 in Paris, nine days before his 73rd birthday.<ref>Template:Cite news</ref> He was buried on 15 June in Moscow, at the Novodevichy Monastery.<ref> Template:Cite web</ref>

In a telegram to Arnold's family, Russian President Dmitry Medvedev stated:

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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 equations, and his many textbooks have proved 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 defense was that his books are meant to teach the subject to "those who truly wish to understand it" (Chicone, 2007).<ref>Carmen Chicone (2007), Book review of "Ordinary Differential Equations", by Vladimir I. Arnold. Springer-Verlag, Berlin, 2006. SIAM Review 49(2):335–336. (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 last century. He had very strong opinions on how this approach—which was most popularly implemented by the Bourbaki school in France—initially had a negative impact on French mathematical education, and then later on that of other countries as well.<ref>See [1] (archived from [2] Template:Webarchive) and other essays in [3].</ref><ref name="interview1">An Interview with Vladimir Arnol'd, by S. H. Lui, AMS Notices, 1991.</ref> He was very concerned about what he saw as the divorce of mathematics from the natural sciences in the 20th century.<ref>Template:Cite journal</ref> Arnold was very interested in the history of mathematics.<ref>Oleg Karpenkov. "Vladimir Igorevich Arnold"</ref> In an interview,<ref name="interview1" /> he said he had learned much of what he knew about mathematics through the study of Felix Klein's book Development of Mathematics in the 19th Century —a book he often recommended to his students.<ref>B. Khesin and S. Tabachnikov, Tribute to Vladimir Arnold, Notices of the AMS, 59:3 (2012) 378–399.</ref> He studied the classics, most notably the works of Huygens, Newton and Poincaré,<ref>Template:Citation.</ref> and many times he reported to have found in their works ideas that had not been explored yet.<ref>See for example: Arnold, V. I.; Vasilev, V. A. (1989), "Newton's Principia read 300 years later" and Arnold, V. I. (2006); "Forgotten and neglected theories of Poincaré".</ref>

Mathematical work

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Template:See also Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, 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

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Template:See also The problem is the following question: can every continuous function of three variables be expressed as a 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>Template:Cite web</ref>

Dynamical systems

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Template:See also Moser and Arnold expanded the ideas of Kolmogorov (who was inspired by questions of 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 systems) 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>Template:Cite book</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

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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 Etudes Scientifiques, which I frequented throughout the year 1965, profoundly changed my mathematical universe."<ref>Template:Cite web</ref> After this event, singularity theory became one of the major interests of Arnold and his students.<ref>Template:Cite web</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 Ak,Dk,Ek 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>Template:Cite book</ref><ref>Template:Cite arXiv</ref>

Fluid dynamics

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Template:See also In 1966, Arnold published "Template:Lang", in which he presented a common geometric interpretation for both the Euler's equations for rotating rigid bodies and the 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>Template:Cite book</ref><ref>Template:Cite news</ref><ref>IAMP News Bulletin, July 2010, pp. 25–26</ref>

Real algebraic geometry

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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 four-dimensional topology.<ref>Template:Cite book</ref> The conjecture was to be later fully solved by V. A. Rokhlin building on Arnold's work.<ref>Template:Cite book</ref><ref>Template:Cite journal</ref>

Symplectic geometry

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The Arnold conjecture, linking the number of fixed points of Hamiltonian symplectomorphisms and the topology of the subjacent manifolds, 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>"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

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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

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According to Marcel Berger, Arnold revolutionized plane curves theory.<ref>Template:Cite book</ref> He developed the theory of smooth closed plane curves in the 1990s.<ref>"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+, J− 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

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Arnold conjectured the existence of the gömböc, a body with just one stable and one unstable point of equilibrium when resting on a flat surface.<ref name=wolfram>Template:Cite web</ref><ref>Template:Cite book</ref>

Arnold generalized the results of Isaac Newton, Pierre-Simon Laplace, and 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>

Honours and awards

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File:Dmitry Medvedev 12 June 2008-10.jpg
Arnold (left) and Russia's president Dmitry Medvedev

The minor planet 10031 Vladarnolda was named after him in 1981 by Lyudmila Georgievna Karachkina.<ref>Template:Cite book</ref>

The Arnold Mathematical Journal, published for the first time in 2015, is named after him.<ref>Template:Citation.</ref>

The Arnold Fellowships, of the London Institute are named after him.<ref>Template:Cite web</ref><ref>Template:Cite news</ref>

He was a plenary speaker at both the 1974 and 1983 International Congress of Mathematicians in Vancouver and Warsaw, respectively.<ref>Template:Cite web</ref>

Fields Medal omission

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Even though Arnold was nominated for the 1974 Fields Medal, one of the highest honours a mathematician could receive, interference from the Soviet government led to it being withdrawn. Arnold's public opposition to the persecution of dissidents 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>Template:Cite encyclopedia</ref><ref>Template:Cite news</ref>

Selected bibliography

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Collected works

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  • 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
  • 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.

See also

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References

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Further reading

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