Vladimir Arnold

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Template:Short description Template:Family name hatnote Template:Use dmy dates Template:Use British English 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. 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 branches of mathematics: topological Galois theory (with his student Askold Khovanskii), 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) 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 opposed to those of Bourbaki.

A controversial and often quoted dictum of his is "Mathematics is the part of physics where experiments are cheap".<ref name=obituary /><ref>Template:Cite journal</ref><ref>Note: This phrase generated controversy and even parodies. He wrote his defense at Template:Cite journal</ref>

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>Gusein-Zade, S. M., Ilyashenko, Y. S., Khovanskii, A. G., Tsfasman, M. A. E., & Vassiliev, V. A. (2003). "Vladimir Igorevich Arnold". Moscow Mathematical Journal, Volume 3, Number 2, pp. 261-261.</ref>

Arnold received the inaugural Crafoord Prize in 1982 (with Louis Nirenberg), the Wolf Prize in 2001 and the Shaw Prize in 2008 (with Ludwig Faddeev).

Early life

Arnold in 1963

Vladimir Igorevich Arnold was born on 12 June 1937 in Odessa, Ukrainian SSR, Soviet Union (now Odesa, Ukraine). His father was Template:Ill (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, Template:Nee Isakovich), a Jewish art historian.<ref name=obituary>Template:Citation</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 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 his whole life.<ref name="Arnold2007">Template:Cite book</ref><ref>Template:Cite bookTemplate:Page needed</ref> When Arnold was thirteen, his uncle Nikolai B. Zhitkov,<ref name="earlylife">Template:Cite book</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>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>Template:Cite journal</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.

Mathematical work

Template:See also 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>Template:MathGenealogy</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 Hamiltonian symplectomorphisms and Lagrangian intersections 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 Template:Page needed</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, Alexander Varchenko, Victor Vassiliev and Vladimir Zakalyukin.<ref name="mathgene">Template:MathGenealogy</ref>

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

Template:See also Hilbert's thirteenth problem asks whether every continuous function of three variables can be expressed as a 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>Template:Cite web</ref>

Dynamical systems

Template:See also 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 systems) 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>Template:Cite book</ref>

In 1961, he introduced Arnold tongues; 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>Template:Cite journal</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

Template:See also 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 book</ref>

Fluid dynamics

Template:See also In 1966, Arnold published the paper "Template:Lang" ('On the differential geometry of infinite-dimensional Lie groups and its applications to the hydrodynamics of perfect fluids'), 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 linked topics previously thought to be unrelated, and enabled mathematical solutions to many questions related to flows and turbulence.<ref>Template:Cite book</ref><ref>Template:Cite news</ref><ref>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 four-dimensional topology.<ref>Template:Cite book</ref> The conjecture was later fully solved by V. A. Rokhlin building on Arnold's work.<ref>Template:Cite book</ref><ref>Template:Cite journal</ref>

Symplectic geometry

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> He also proposed the nearby Lagrangian conjecture, a still open problem in mathematics.<ref>Lisa Traynor (2024), "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 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>

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 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 point of equilibrium when resting on a flat surface. This conjecture was proved by Gábor Domokos in 2006.<ref name=wolfram>Template:Cite web</ref><ref>Template:Cite book</ref>

Arnold generalised the results of Isaac Newton, Pierre-Simon Laplace, and James Ivory on the shell theorem, showing it to be applicable to algebraic hypersurfaces.<ref>Template:Cite journal</ref>

Arnold's strange duality was one of the first examples of mirror symmetry (for K3 surfaces).<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 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>

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 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>Template:Cite journal (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 Bourbaki school in France—initially had a negative impact on French mathematical education and later in other countries.<ref>Template:Cite journal</ref><ref name="interview1">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 sciences in the 20th century.<ref>Template:Cite journal</ref> Arnold was very interested in the history of mathematics,<ref name="Karpenkov Vladimir Igorevich Arnold">Template:Cite report (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 Development of Mathematics in the 19th Century Template:Ndasha 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 works of Huygens, Newton and Poincaré,<ref>Template:Citation.</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), "Newton's Principia read 300 years later" and Template:Cite journal</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>Template:Cite book</ref> He went on to make a good recovery.<ref>Template:Cite journal</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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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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Honours and awards

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

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 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>

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</ref>

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
  • 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

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References

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Further reading

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