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.^{[5]}^{[6]} Many of his books were translated into English. His views on education were particularly opposed to those of Bourbaki.
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.^{[4]} 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.^{[7]} When Arnold was thirteen, his uncle Nikolai B. Zhitkov,^{[8]} 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.^{[9]}
After graduating from Moscow State University in 1959, he worked there until 1986 (a professor since 1965), and then at Steklov Mathematical Institute.
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.^{[12]} He went on to make a good recovery.^{[13]}
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:
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.^{[14]}
Arnold died of acute pancreatitis^{[15]} on 3 June 2010 in Paris, nine days before his 73rd birthday.^{[16]} He was buried on 15 June in Moscow, at the Novodevichy Monastery.^{[17]}
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.^{[18]}
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).^{[19]}
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.^{[20]}^{[21]} Arnold was very interested in the history of mathematics.^{[22]} In an interview,^{[21]} 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.^{[23]} He studied the classics, most notably the works of Huygens, Newton and Poincaré,^{[24]} and many times he reported to have found in their works ideas that had not been explored yet.^{[25]}
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.^{[27]}
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.^{[28]}
In 1964, Arnold introduced the Arnold web, the first example of a stochastic web.^{[29]}^{[30]}
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."^{[31]} After this event, singularity theory became one of the major interests of Arnold and his students.^{[32]} 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_{k},D_{k},E_{k} and Lagrangian singularities".^{[33]}^{[34]}^{[35]}
In 1966, Arnold published "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 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.^{[36]}^{[37]}^{[38]}
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",^{[39]} 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.^{[40]} The conjecture was to be later fully solved by V. A. Rokhlin building on Arnold's work.^{[41]}^{[42]}
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.^{[43]}^{[44]}
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.^{[45]}^{[46]}
According to Marcel Berger, Arnold revolutionized plane curves theory.^{[47]} Among his contributions are the Arnold invariants of plane curves.^{[48]}
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."^{[58]}
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."^{[60]}
Shaw Prize in mathematical sciences (2008, with Ludwig Faddeev), "for their widespread and influential contributions to Mathematical Physics."^{[62]}^{[63]}
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.^{[69]}^{[70]}
1978: Ordinary Differential Equations, The MIT Press ISBN0-262-51018-9.^{[71]}^{[72]}^{[73]}
1985: Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. (1985). Singularities of Differentiable Maps, Volume I: The Classification of Critical Points Caustics and Wave Fronts. Monographs in Mathematics. Vol. 82. Birkhäuser. doi:10.1007/978-1-4612-5154-5. ISBN978-1-4612-9589-1.
1989 Арнольд, В. И. (1989). Гюйгенс и Барроу, Ньютон и Гук - Первые шаги математического анализа и теории катастроф. М.: Наука. p. 98. ISBN5-02-013935-1.
1989: (with A. Avez) Ergodic Problems of Classical Mechanics, Addison-Wesley ISBN0-201-09406-1.
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) ISBN3-7643-2383-3.^{[76]}^{[77]}^{[78]}
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.
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.
^Carmen Chicone (2007), Book review of "Ordinary Differential Equations", by Vladimir I. Arnold. Springer-Verlag, Berlin, 2006. SIAM Review49(2):335–336. (Chicone mentions the criticism but does not agree with it.)
^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".
^"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)
^Kazarinoff, N. (1 September 1991). "Huygens and Barrow, Newton and Hooke: Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals (V. I. Arnol'd)". SIAM Review. 33 (3): 493–495. doi:10.1137/1033119. ISSN0036-1445.
^Thiele, R. (1 January 1993). "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 of Applied Mathematics and Mechanics. 73 (1): 34. Bibcode:1993ZaMM...73S..34T. doi:10.1002/zamm.19930730109. ISSN1521-4001.
^Bernfeld, Stephen R. (1 January 1985). "Review of Catastrophe Theory". SIAM Review. 27 (1): 90–91. doi:10.1137/1027019. JSTOR2031497.
^Guenther, Ronald B.; Thomann, Enrique A. (2005). Renardy, Michael; Rogers, Robert C.; Arnold, Vladimir I. (eds.). "Featured Review: Two New Books on Partial Differential Equations". SIAM Review. 47 (1): 165–168. ISSN0036-1445. JSTOR20453608.
Khesin, Boris; Tabachnikov, Serge (Coordinating Editors). "Memories of Vladimir Arnold", Notices of the American Mathematical Society, April 2012, Volume 59, Number 4, pp. 482–502.