Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010)Mort d'un grand mathématicien russe, AFP (Le Figaro) 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. 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".Note: This phrase generated controversy and even parodies. He wrote his defense at The Arnold principle: "Discoveries are rarely attributed to the correct person" is also named after him.
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. He was one of the founders of the Independent University of Moscow.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.
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 in 2008 (with Ludwig Faddeev).
Early life
Vladimir Igorevich Arnold was born on 12 June 1937 in Odessa, Ukrainian SSR, Soviet Union (now Odesa, Ukraine). His father was (1900–1948), a mathematician known for his work in mathematical education and who learned algebra from Emmy Noether in the late 1920s. His mother was Nina Alexandrovna Arnold (1909–1986, Isakovich), a Jewish art historian.
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. When Arnold was thirteen, his uncle Nikolai B. Zhitkov, 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.Табачников, С. Л. . "Интервью с В.И.Арнольдом", Квант, 1990, Nº 7, pp. 2–7. (in Russian)
Arnold entered Moscow State University in 1954. Among his teachers there were A. N. Kolmogorov, I. M. Gelfand, L. S. Pontriagin and Pavel Alexandrov. 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. This is the Kolmogorov–Arnold representation theorem.
Mathematical work
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).
He became an academician of the Academy of Sciences of the Soviet Union (Russian Academy of Science since 1991) in 1990.Great Russian Encyclopedia (2005), Moscow: Bol'shaya Rossiyskaya Enciklopediya Publisher, vol. 2. 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.Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta (2009). Lagrangian Intersection Floer Theory Anomaly and Obstruction
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.
Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory. 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"."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)
Hilbert's thirteenth problem
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.
Dynamical systems
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.
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.
In 1964, he discovered Arnold diffusion, which he would later consider as his most important contribution to KAM theory."From Hilbert's Superposition Problem to Dynamical Systems" by Vladimir I. Arnold, published in the American Mathematical Monthly (Vol. 113, No. 7, August/September 2006, pp. 608–624)
In 1964, Arnold introduced the Arnold web, the first example of a stochastic web.Phase Space Crystals, by Lingzhen Guo https://iopscience.iop.org/book/978-0-7503-3563-8.pdfScholarpedia: "Zaslavsky web map", by George Zaslavsky http://www.scholarpedia.org/article/Zaslavsky_web_map
In 1974, Arnold proved the Liouville–Arnold theorem, now a classic result deeply geometric in character.
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.
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 Etudes Scientifiques, which I frequented throughout the year 1965, profoundly changed my mathematical universe." After this event, singularity theory became one of the major interests of Arnold and his students. 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".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
Fluid dynamics
In 1966, Arnold published the paper "" ('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.IAMP News Bulletin, July 2010, pp. 25–26
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",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 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. The conjecture was later fully solved by V. A. Rokhlin building on Arnold's work.
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."Arnold and Symplectic Geometry", by Helmut Hofer (in the book Arnold: Swimming Against the Tide)"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 He also proposed the nearby Lagrangian conjecture, a still open problem in mathematics.Lisa Traynor (2024), "Eliashberg’s contributions towards the theory of generating functions"
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.András Stipsicz, Frédéric Bourgeois, Vincent Colin (Editors) 2014. Contact and Symplectic Topology, p. 2, Springer
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."Topology in Arnold's work", by Victor Vassiliev
Theory of plane curves