Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as the famous Mathematical Methods of Classical Mechanics) and popular mathematics books, he influenced many mathematicians and physicists. 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 Odessa, Soviet Union. His father was Igor Vladimirovich Arnold (1900–1948), a mathematician. His mother was Nina Alexandrovna Arnold (1909–1986, née Isakovich), a Jewish art historian. When Arnold was thirteen, an uncle who was an engineer told him about calculus and how it could be used to understand some physical phenomena, this contributed to spark 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.
In 1999 he suffered a serious bike accident in Paris, resulting in traumatic brain injury, and though he regained consciousness after a few weeks, he had amnesia and for some time could not even recognize his own wife at the hospital, but he went on to make a good recovery.
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.
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.
Popular mathematical writings
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 is that his books are meant to teach the subject to "those who truly wish to understand it" (Chicone, 2007).
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. Arnold was very interested in the history of mathematics. In an interview, 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. He liked to study the classics, most notably the works of Huygens, Newton and Poincaré, and many times he reported to have found in their works ideas that had not been explored yet.
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.
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.
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".
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.
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.
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.
Even though Arnold was nominated for the 1974 Fields Medal, which was then viewed as the highest honour 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.
1966: Arnold, Vladimir (1966). "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits" (PDF). Annales de l'Institut Fourier. 16 (1): 319–361. doi:10.5802/aif.233.
1978: Ordinary Differential Equations, The MIT Press ISBN 0-262-51018-9.
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. 82. Birkhäuser. doi:10.1007/978-1-4612-5154-5. ISBN 978-1-4612-9589-1.
1988: Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. (1988). Arnold, V. I; Gusein-Zade, S. M; Varchenko, A. N (eds.). Singularities of Differentiable Maps, Volume II: Monodromy and Asymptotics of Integrals. Monographs in Mathematics. 83. Birkhäuser. doi:10.1007/978-1-4612-3940-6. ISBN 978-1-4612-8408-6.
1988: Arnold, V.I. (1988). Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren der mathematischen Wissenschaften. 250 (2nd ed.). Springer. doi:10.1007/978-1-4612-1037-5. ISBN 978-1-4612-6994-6.
1989 Арнольд, В. И. (1989). Гюйгенс и Барроу, Ньютон и Гук - Первые шаги математического анализа и теории катастроф. М.: Наука. p. 98. ISBN 5-02-013935-1.
1989: (with A. Avez) Ergodic Problems of Classical Mechanics, Addison-Wesley ISBN 0-201-09406-1.
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) ISBN 3-7643-2383-3.
1991: Arnolʹd, Vladimir Igorevich (1991). The Theory of Singularities and Its Applications. Cambridge University Press. ISBN 9780521422802.
1995:Topological Invariants of Plane Curves and Caustics, American Mathematical Society (1994) ISBN 978-0-8218-0308-0
1998: "On the teaching of mathematics" (Russian) Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; translation in Russian Math. Surveys 53(1): 229–236.
2001: "Tsepniye Drobi" (Continued Fractions, in Russian), Moscow (2001).
2004: Teoriya Katastrof (Catastrophe Theory, in Russian), 4th ed. Moscow, Editorial-URSS (2004), ISBN 5-354-00674-0.
2004: Vladimir I. Arnold, ed. (15 November 2004). Arnold's Problems (2nd ed.). Springer-Verlag. ISBN 978-3-540-20748-1.
2004: Arnold, Vladimir I. (2004). Lectures on Partial Differential Equations. Universitext. Springer. doi:10.1007/978-3-662-05441-3. ISBN 978-3-540-40448-4.
2007: Yesterday and Long Ago, Springer (2007), ISBN 978-3-540-28734-6.
2013: Arnold, Vladimir I. (2013). Itenberg, Ilia; Kharlamov, Viatcheslav; Shustin, Eugenii I. (eds.). Real Algebraic Geometry. Unitext. 66. Springer. doi:10.1007/978-3-642-36243-9. ISBN 978-3-642-36242-2.
2014: V. I. Arnold (2014). Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians. American Mathematical Society. ISBN 978-1-4704-1701-7.
2015: Experimental Mathematics. American Mathematical Society (translated from Russian, 2015).
2015: Lectures and Problems: A Gift to Young Mathematicians, American Math Society, (translated from Russian, 2015)
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.
^Bartocci, Claudio; Betti, Renato; Guerraggio, Angelo; Lucchetti, Roberto; Williams, Kim (2010). Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles. Springer. p. 211. ISBN 9783642136061.
^Табачников, С. Л. . "Интервью с В.И.Арнольдом", Квант, 1990, Nº 7, pp. 2–7. (in Russian)
^Daniel Robertz (13 October 2014). Formal Algorithmic Elimination for PDEs. Springer. p. 192. ISBN 978-3-319-11445-3.
^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.)
^Note: The paper also appears with other names, as in http://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf
^A. G. Khovanskii; Aleksandr Nikolaevich Varchenko; V. A. Vasiliev (1997). Topics in Singularity Theory: V. I. Arnold's 60th Anniversary Collection (preface). American Mathematical Soc. p. 10. ISBN 978-0-8218-0807-8.
^Khesin, Boris A.; Tabachnikov, Serge L. (10 September 2014). Arnold: Swimming Against the Tide. p. 159. ISBN 9781470416997.
^Degtyarev, A. I.; Kharlamov, V. M. (2000). "Topological properties of real algebraic varieties: Du coté de chez Rokhlin". Russian Mathematical Surveys. 55 (4): 735–814. arXiv:math/0004134. Bibcode:2000RuMaS..55..735D. doi:10.1070/RM2000v055n04ABEH000315.
^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. ISSN 0036-1445.
^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. ISSN 1521-4001.
^Heggie, Douglas C. (1 June 1991). "V. I. Arnol'd, Huygens and Barrow, Newton and Hooke, translated by E. J. F. Primrose (Birkhäuser Verlag, Basel 1990), 118 pp., 3 7643 2383 3, sFr 24". Proceedings of the Edinburgh Mathematical Society. Series 2. 34 (2): 335–336. doi:10.1017/S0013091500007240. ISSN 1464-3839.
^Goryunov, V. V. (1 October 1996). "V. I. Arnold Topological invariants of plane curves and caustics (University Lecture Series, Vol. 5, American Mathematical Society, Providence, RI, 1995), 60pp., paperback, 0 8218 0308 5, £17.50". Proceedings of the Edinburgh Mathematical Society. Series 2. 39 (3): 590–591. doi:10.1017/S0013091500023348. ISSN 1464-3839.
^Bernfeld, Stephen R. (1 January 1985). "Review of Catastrophe Theory". SIAM Review. 27 (1): 90–91. doi:10.1137/1027019. JSTOR 2031497.
^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. ISSN 0036-1445. JSTOR 20453608.
^Groves, M. (2005). "Book Review: Vladimir I. Arnold, Lectures on Partial Differential Equations. Universitext". ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 85 (4): 304. Bibcode:2005ZaMM...85..304G. doi:10.1002/zamm.200590023. ISSN 1521-4001.