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Vladimir Arnold

Vladimir Igorevich Arnold (alternative spelling Arnol'd, Russian:Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematicsKAM theory, and topological Galois theory (this, with his student Askold Khovanskii).

Vladimir Arnold
Vladimir Arnold in 2008
Born(1937-06-12)12 June 1937
Died3 June 2010(2010-06-03) (aged 72)
Paris, France
NationalitySoviet Union, Russian
Alma materMoscow State University
Known forADE classification
Arnold's cat map
Arnold conjecture
Arnold diffusion
Arnold's rouble problem
Arnold's spectral sequence
Arnold tongue
ABC flow
Arnold–Givental conjecture
Gömböc
Gudkov's conjecture
Hilbert's thirteenth problem
KAM theorem
Kolmogorov–Arnold theorem
Liouville–Arnold theorem
Topological Galois theory
Mathematical Methods of Classical Mechanics
AwardsShaw Prize (2008)
State Prize of the Russian Federation (2007)
Wolf Prize (2001)
Dannie Heineman Prize for Mathematical Physics (2001)
Harvey Prize (1994)
RAS Lobachevsky Prize (1992)
Crafoord Prize (1982)
Lenin Prize (1965)
Scientific career
FieldsMathematics
InstitutionsParis Dauphine University
Steklov Institute of Mathematics
Independent University of Moscow
Moscow State University
Doctoral advisorAndrey Kolmogorov
Doctoral students

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.

Contents

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.

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.

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. 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 were also a major motivation in the development of Floer homology.

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.

Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death. As of 2006[update] he was reported to have the highest citation index among Russian scientists, and h-index of 40.

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.

Death

Arnold died of acute pancreatitis on 3 June 2010 in Paris, nine days before his 73rd birthday. His students include Alexander Givental, Victor Goryunov, Sabir Gusein-Zade, Emil Horozov, Boris Khesin, Askold Khovanskii, Nikolay Nekhoroshev, Boris Shapiro, Alexander Varchenko, Victor Vassiliev and Vladimir Zakalyukin.

He was buried on 15 June in Moscow, at the Novodevichy Monastery.

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

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.

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.

Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory.

Hilbert's thirteenth problem

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.

Dynamical systems

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.

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".

Fluid dynamics

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.

Real algebraic geometry

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", 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. The conjecture was to be 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.

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.

Theory of plane curves

Arnold revolutionized plane curves theory.

Other

Arnold conjectured the existence of the gömböc.

Arnold (left) and Russia's president Dmitry Medvedev

The minor planet 10031 Vladarnolda was named after him in 1981 by Lyudmila Georgievna Karachkina.

The Arnold Mathematical Journal, published for the first time in 2015, is named after him.

He was a plenary speaker at both the 1974 and 1983 International Congress of Mathematicians in Vancouver and Warsaw, respectively.

Fields Medal omission

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.

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.
  1. Khesin, Boris; Tabachnikov, Sergei (2018). "Vladimir Igorevich Arnold. 12 June 1937 – 3 June 2010". Biographical Memoirs of Fellows of the Royal Society. 64: 7–26. doi:10.1098/rsbm.2017.0016. ISSN 0080-4606.
  2. Vladimir Arnold at the Mathematics Genealogy Project
  3. Mort d'un grand mathématicien russe, AFP (Le Figaro)
  4. Gusein-Zade, Sabir M.; Varchenko, Alexander N (December 2010), "Obituary: Vladimir Arnold (12 June 1937 – 3 June 2010)"(PDF), Newsletter of the European Mathematical Society, 78: 28–29
  5. O'Connor, John J.; Robertson, Edmund F., "Vladimir Arnold", MacTutor History of Mathematics archive, University of St Andrews
  6. 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.
  7. Табачников, С. Л. . "Интервью с В.И.Арнольдом", Квант, 1990, Nº 7, pp. 2–7. (in Russian)
  8. Daniel Robertz (13 October 2014). Formal Algorithmic Elimination for PDEs. Springer. p. 192. ISBN 978-3-319-11445-3.
  9. Great Russian Encyclopedia (2005), Moscow: Bol'shaya Rossiyskaya Enciklopediya Publisher, vol. 2.
  10. Arnold: Yesterday and Long Ago (2010)
  11. Polterovich and Scherbak (2011)
  12. List of Russian Scientists with High Citation Index
  13. "Vladimir Arnold". The Daily Telegraph. London. 12 July 2010.
  14. Kenneth Chang (11 June 2010). "Vladimir Arnold Dies at 72; Pioneering Mathematician". The New York Times. Retrieved12 June 2013.
  15. "Number's up as top mathematician Vladimir Arnold dies". Herald Sun. 4 June 2010. Retrieved6 June 2010.
  16. "From V. I. Arnold's web page". Retrieved12 June 2013.
  17. "Condolences to the family of Vladimir Arnold". Presidential Press and Information Office. 15 June 2010. Retrieved1 September 2011.
  18. 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.)
  19. See [1] and other essays in [2].
  20. An Interview with Vladimir Arnol'd, by S. H. Lui, AMS Notices, 1991.
  21. Oleg Karpenkov. "Vladimir Igorevich Arnold"
  22. B. Khesin and S. Tabachnikov, Tribute to Vladimir Arnold, Notices of the AMS, 59:3 (2012) 378–399.
  23. Goryunov, V.; Zakalyukin, V. (2011), "Vladimir I. Arnold", Moscow Mathematical Journal, 11 (3).
  24. 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".
  25. Szpiro, George G. (29 July 2008). Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. Penguin. ISBN 9781440634284.
  26. "Archived copy"(PDF). Archived from the original(PDF) on 14 July 2015. Retrieved22 February 2015.CS1 maint: archived copy as title (link)
  27. "Resonance – Journal of Science Education | Indian Academy of Sciences"(PDF).
  28. 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
  29. Dirk Siersma; Charles Wall; V. Zakalyukin (30 June 2001). New Developments in Singularity Theory. Springer Science & Business Media. p. 29. ISBN 978-0-7923-6996-7.
  30. Landsberg, J. M.; Manivel, L. (2002). "Representation theory and projective geometry". arXiv:math/0203260.
  31. Terence Tao (22 March 2013). Compactness and Contradiction. American Mathematical Soc. pp. 205–206. ISBN 978-0-8218-9492-7.
  32. MacKay, Robert Sinclair; Stewart, Ian (19 August 2010). "VI Arnold obituary". The Guardian.
  33. IAMP News Bulletin, July 2010, pp. 25–26
  34. Note: The paper also appears with other names, as in http://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf
  35. 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.
  36. Khesin, Boris A.; Tabachnikov, Serge L. (10 September 2014). Arnold: Swimming Against the Tide. p. 159. ISBN 9781470416997.
  37. 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.
  38. "Arnold and Symplectic Geometry", by Helmut Hofer
  39. "Vladimir Igorevich Arnold and the invention of symplectic topology", by Michèle Audin
  40. "Topology in Arnold's work", by Victor Vassiliev
  41. 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
  42. A Panoramic View of Riemannian Geometry, by Marcel Berger
  43. Mackenzie, Dana (29 December 2010). What's Happening in the Mathematical Sciences. American Mathematical Soc. p. 104. ISBN 9780821849996.
  44. O. Karpenkov, "Vladimir Igorevich Arnold", Internat. Math. Nachrichten, no. 214, pp. 49–57, 2010. (link to arXiv preprint)
  45. Harold M. Schmeck Jr. (27 June 1982). "American and Russian Share Prize in Mathematics". The New York Times.
  46. "Book of Members, 1780–2010: Chapter A"(PDF). American Academy of Arts and Sciences. Retrieved25 April 2011.
  47. D. B. Anosov, A. A. Bolibrukh, Lyudvig D. Faddeev, A. A. Gonchar, M. L. Gromov, S. M. Gusein-Zade, Yu. S. Il'yashenko, B. A. Khesin, A. G. Khovanskii, M. L. Kontsevich, V. V. Kozlov, Yu. I. Manin, A. I. Neishtadt, S. P. Novikov, Yu. S. Osipov, M. B. Sevryuk, Yakov G. Sinai, A. N. Tyurin, A. N. Varchenko, V. A. Vasil'ev, V. M. Vershik and V. M. Zakalyukin (1997) . "Vladimir Igorevich Arnol'd (on his sixtieth birthday)". Russian Mathematical Surveys, Volume 52, Number 5. (translated from the Russian by R. F. Wheeler)
  48. American Physical Society – 2001 Dannie Heineman Prize for Mathematical Physics Recipient
  49. The Wolf Foundation – Vladimir I. Arnold Winner of Wolf Prize in Mathematics
  50. Названы лауреаты Государственной премии РФ Kommersant 20 May 2008.
  51. Lutz D. Schmadel (10 June 2012). Dictionary of Minor Planet Names. Springer Science & Business Media. p. 717. ISBN 978-3-642-29718-2.
  52. Editorial (2015), "Journal Description Arnold Mathematical Journal", Arnold Mathematical Journal, 1 (1): 1–3, doi:10.1007/s40598-015-0006-6.
  53. http://www.mathunion.org/db/ICM/Speakers/SortedByLastname.php
  54. Martin L. White (2015). "Vladimir Igorevich Arnold". Encyclopædia Britannica.
  55. Thomas H. Maugh II (23 June 2010). "Vladimir Arnold, noted Russian mathematician, dies at 72". The Washington Post. Retrieved18 March 2015.
  56. Review by Ian N. Sneddon (Bulletin of the American Mathematical Society, Vol. 2): http://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14755-2/S0273-0979-1980-14755-2.pdf
  57. Review by R. Broucke (Celestial Mechanics, Vol. 28): Bibcode:1982CeMec..28..345A.
  58. 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.
  59. 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.
  60. 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.
  61. 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.
  62. Bernfeld, Stephen R. (1 January 1985). "Review of Catastrophe Theory". SIAM Review. 27 (1): 90–91. doi:10.1137/1027019. JSTOR 2031497.
  63. 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.
  64. 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.
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Vladimir Arnold
Vladimir Arnold Language Watch Edit Vladimir Igorevich Arnold alternative spelling Arnol d Russian Vladi mir I gorevich Arno ld 12 June 1937 3 June 2010 3 4 1 was a Soviet and Russian mathematician While he is best known for the Kolmogorov Arnold Moser theorem regarding the stability of integrable systems he made important contributions in several areas including dynamical systems theory algebra catastrophe theory topology algebraic geometry symplectic geometry differential equations classical mechanics hydrodynamics and singularity theory including posing the ADE classification problem since his first main result the solution of Hilbert s thirteenth problem in 1957 at the age of 19 He co founded two new branches of mathematics KAM theory and topological Galois theory this with his student Askold Khovanskii Vladimir ArnoldVladimir Arnold in 2008Born 1937 06 12 12 June 1937 Odessa Ukrainian SSR Soviet UnionDied3 June 2010 2010 06 03 aged 72 Paris FranceNationalitySoviet Union RussianAlma materMoscow State UniversityKnown forADE classification Arnold s cat map Arnold conjecture Arnold diffusion Arnold s rouble problem Arnold s spectral sequence Arnold tongue ABC flow Arnold Givental conjecture Gomboc Gudkov s conjecture Hilbert s thirteenth problem KAM theorem Kolmogorov Arnold theorem Liouville Arnold theorem Topological Galois theory Mathematical Methods of Classical MechanicsAwardsShaw Prize 2008 State Prize of the Russian Federation 2007 Wolf Prize 2001 Dannie Heineman Prize for Mathematical Physics 2001 Harvey Prize 1994 RAS Lobachevsky Prize 1992 Crafoord Prize 1982 Lenin Prize 1965 Scientific careerFieldsMathematicsInstitutionsParis Dauphine University Steklov Institute of Mathematics Independent University of Moscow Moscow State UniversityDoctoral advisorAndrey KolmogorovDoctoral studentsAlexander Givental Victor Goryunov Sabir Gusein Zade Emil Horozov Boris Khesin 1 Askold Khovanskii Nikolay Nekhoroshev Boris Shapiro Alexander Varchenko Victor Vassiliev Vladimir Zakalyukin 2 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 5 6 Many of his books were translated into English His views on education were particularly opposed to those of Bourbaki Contents 1 Biography 1 1 Death 2 Popular mathematical writings 3 Work 3 1 Hilbert s thirteenth problem 3 2 Dynamical systems 3 3 Singularity theory 3 4 Fluid dynamics 3 5 Real algebraic geometry 3 6 Symplectic geometry 3 7 Topology 3 8 Theory of plane curves 3 9 Other 4 Honours and awards 4 1 Fields Medal omission 5 Selected bibliography 5 1 Collected works 6 See also 7 References 8 Further reading 9 External linksBiography EditVladimir 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 nee Isakovich a Jewish art historian 4 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 7 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 8 This is the Kolmogorov Arnold representation theorem 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 9 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 were also a major motivation in the development of Floer homology 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 10 but he went on to make a good recovery 11 Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death As of 2006 update he was reported to have the highest citation index among Russian scientists 12 and h index of 40 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 13 Death Edit Arnold died of acute pancreatitis 14 on 3 June 2010 in Paris nine days before his 73rd birthday 15 His students include Alexander Givental Victor Goryunov Sabir Gusein Zade Emil Horozov Boris Khesin Askold Khovanskii Nikolay Nekhoroshev Boris Shapiro Alexander Varchenko Victor Vassiliev and Vladimir Zakalyukin 2 He was buried on 15 June in Moscow at the Novodevichy Monastery 16 In a telegram to Arnold s family Russian President Dmitry Medvedev stated 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 17 Popular mathematical writings EditArnold 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 18 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 19 20 Arnold was very interested in the history of mathematics 21 In an interview 20 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 22 He liked to study the classics most notably the works of Huygens Newton and Poincare 23 and many times he reported to have found in their works ideas that had not been explored yet 24 Work EditSee also Arnold conjecture and Stability of the Solar System Arnold worked on dynamical systems theory catastrophe theory topology algebraic geometry symplectic geometry differential equations classical mechanics hydrodynamics and singularity theory 5 Hilbert s thirteenth problem Edit 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 Dynamical systems Edit See also Kolmogorov Arnold Moser theorem and Arnold diffusion Moser and Arnold expanded the ideas of Kolmogorov who was inspired by questions of Poincare 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 25 Singularity theory Edit In 1965 Arnold attended Rene 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 26 After this event singularity theory became one of the major interests of Arnold and his students 27 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 28 29 30 Fluid dynamics Edit In 1966 Arnold published Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a 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 31 32 33 Real algebraic geometry Edit 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 34 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 35 The conjecture was to be later fully solved by V A Rokhlin building on Arnold s work 36 37 Symplectic geometry Edit 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 38 39 Topology Edit 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 40 41 Theory of plane curves Edit Arnold revolutionized plane curves theory 42 Other Edit Arnold conjectured the existence of the gomboc 43 Honours and awards Edit Arnold left and Russia s president Dmitry Medvedev Lenin Prize 1965 with Andrey Kolmogorov 44 for work on celestial mechanics Crafoord Prize 1982 with Louis Nirenberg 45 for contributions to the theory of non linear differential equations Foreign Honorary Member of the American Academy of Arts and Sciences 1987 46 Elected a Foreign Member of the Royal Society ForMemRS of London in 1988 1 Lobachevsky Prize of the Russian Academy of Sciences 1992 47 Harvey Prize 1994 for basic contribution to the stability theory of dynamical systems his pioneering work on singularity theory and seminal contributions to analysis and geometry Dannie Heineman Prize for Mathematical Physics 2001 for his fundamental contributions to our understanding of dynamics and of singularities of maps with profound consequences for mechanics astrophysics statistical mechanics hydrodynamics and optics 48 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 49 State Prize of the Russian Federation 2007 50 for outstanding success in mathematics Shaw Prize in mathematical sciences 2008 with Ludwig Faddeev for their contributions to mathematical physics The minor planet 10031 Vladarnolda was named after him in 1981 by Lyudmila Georgievna Karachkina 51 The Arnold Mathematical Journal published for the first time in 2015 is named after him 52 He was a plenary speaker at both the 1974 and 1983 International Congress of Mathematicians in Vancouver and Warsaw respectively 53 Fields Medal omission Edit 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 54 55 Selected bibliography Edit1966 Arnold Vladimir 1966 Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a 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 Birkhauser 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 Birkhauser 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 Arnold V I 1989 Mathematical Methods of Classical Mechanics Graduate Texts in Mathematics 60 2nd ed Springer doi 10 1007 978 1 4757 2063 1 ISBN 978 1 4419 3087 3 56 57 1989 Arnold V I 1989 Gyujgens i Barrou Nyuton i Guk Pervye shagi matematicheskogo analiza i teorii katastrof M Nauka 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 Birkhauser Verlag 1990 ISBN 3 7643 2383 3 58 59 60 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 61 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 1999 with Valentin Afraimovich Bifurcation Theory And Catastrophe Theory Springer ISBN 3 540 65379 1 2001 Tsepniye Drobi Continued Fractions in Russian Moscow 2001 2004 Teoriya Katastrof Catastrophe Theory 62 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 63 64 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 Collected works Edit 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 See also Edit Mathematics portal List of things named after Vladimir Arnold Gomboc Independent University of Moscow Geometric mechanicsReferences Edit a b c Khesin Boris Tabachnikov Sergei 2018 Vladimir Igorevich Arnold 12 June 1937 3 June 2010 Biographical Memoirs of Fellows of the Royal Society 64 7 26 doi 10 1098 rsbm 2017 0016 ISSN 0080 4606 a b Vladimir Arnold at the Mathematics Genealogy Project Mort d un grand mathematicien russe AFP Le Figaro a b Gusein Zade Sabir M Varchenko Alexander N December 2010 Obituary Vladimir Arnold 12 June 1937 3 June 2010 PDF Newsletter of the European Mathematical Society 78 28 29 a b O Connor John J Robertson Edmund F Vladimir Arnold MacTutor History of Mathematics archive University of St Andrews 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 Tabachnikov S L Intervyu s V I Arnoldom Kvant 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 Great Russian Encyclopedia 2005 Moscow Bol shaya Rossiyskaya Enciklopediya Publisher vol 2 Arnold Yesterday and Long Ago 2010 Polterovich and Scherbak 2011 List of Russian Scientists with High Citation Index Vladimir Arnold The Daily Telegraph London 12 July 2010 Kenneth Chang 11 June 2010 Vladimir Arnold Dies at 72 Pioneering Mathematician The New York Times Retrieved 12 June 2013 Number s up as top mathematician Vladimir Arnold dies Herald Sun 4 June 2010 Retrieved 6 June 2010 From V I Arnold s web page Retrieved 12 June 2013 Condolences to the family of Vladimir Arnold Presidential Press and Information Office 15 June 2010 Retrieved 1 September 2011 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 See 1 and other essays in 2 a b An Interview with Vladimir Arnol d by S H Lui AMS Notices 1991 Oleg Karpenkov Vladimir Igorevich Arnold B Khesin and S Tabachnikov Tribute to Vladimir Arnold Notices of the AMS 59 3 2012 378 399 Goryunov V Zakalyukin V 2011 Vladimir I Arnold Moscow Mathematical Journal 11 3 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 Poincare Szpiro George G 29 July 2008 Poincare s Prize The Hundred Year Quest to Solve One of Math s Greatest Puzzles Penguin ISBN 9781440634284 Archived copy PDF Archived from the original PDF on 14 July 2015 Retrieved 22 February 2015 CS1 maint archived copy as title link Resonance Journal of Science Education Indian Academy of Sciences PDF 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 Dirk Siersma Charles Wall V Zakalyukin 30 June 2001 New Developments in Singularity Theory Springer Science amp Business Media p 29 ISBN 978 0 7923 6996 7 Landsberg J M Manivel L 2002 Representation theory and projective geometry arXiv math 0203260 Terence Tao 22 March 2013 Compactness and Contradiction American Mathematical Soc pp 205 206 ISBN 978 0 8218 9492 7 MacKay Robert Sinclair Stewart Ian 19 August 2010 VI Arnold obituary The Guardian IAMP News Bulletin July 2010 pp 25 26 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 cote de chez Rokhlin Russian Mathematical Surveys 55 4 735 814 arXiv math 0004134 Bibcode 2000RuMaS 55 735D doi 10 1070 RM2000v055n04ABEH000315 Arnold and Symplectic Geometry by Helmut Hofer Vladimir Igorevich Arnold and the invention of symplectic topology by Michele Audin Topology in Arnold s work by Victor Vassiliev 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 A Panoramic View of Riemannian Geometry by Marcel Berger Mackenzie Dana 29 December 2010 What s Happening in the Mathematical Sciences American Mathematical Soc p 104 ISBN 9780821849996 O Karpenkov Vladimir Igorevich Arnold Internat Math Nachrichten no 214 pp 49 57 2010 link to arXiv preprint Harold M Schmeck Jr 27 June 1982 American and Russian Share Prize in Mathematics The New York Times Book of Members 1780 2010 Chapter A PDF American Academy of Arts and Sciences Retrieved 25 April 2011 D B Anosov A A Bolibrukh Lyudvig D Faddeev A A Gonchar M L Gromov S M Gusein Zade Yu S Il yashenko B A Khesin A G Khovanskii M L Kontsevich V V Kozlov Yu I Manin A I Neishtadt S P Novikov Yu S Osipov M B Sevryuk Yakov G Sinai A N Tyurin A N Varchenko V A Vasil ev V M Vershik and V M Zakalyukin 1997 Vladimir Igorevich Arnol d on his sixtieth birthday Russian Mathematical Surveys Volume 52 Number 5 translated from the Russian by R F Wheeler American Physical Society 2001 Dannie Heineman Prize for Mathematical Physics Recipient The Wolf Foundation Vladimir I Arnold Winner of Wolf Prize in Mathematics Nazvany laureaty Gosudarstvennoj premii RF Kommersant 20 May 2008 Lutz D Schmadel 10 June 2012 Dictionary of Minor Planet Names Springer Science amp Business Media p 717 ISBN 978 3 642 29718 2 Editorial 2015 Journal Description Arnold Mathematical Journal Arnold Mathematical Journal 1 1 1 3 doi 10 1007 s40598 015 0006 6 http www mathunion org db ICM Speakers SortedByLastname php Martin L White 2015 Vladimir Igorevich Arnold Encyclopaedia Britannica Thomas H Maugh II 23 June 2010 Vladimir Arnold noted Russian mathematician dies at 72 The Washington Post Retrieved 18 March 2015 Review by Ian N Sneddon Bulletin of the American Mathematical Society Vol 2 http www ams org journals bull 1980 02 02 S0273 0979 1980 14755 2 S0273 0979 1980 14755 2 pdf Review by R Broucke Celestial Mechanics Vol 28 Bibcode 1982CeMec 28 345A 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 Birkhauser 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 Birkhauser 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 fur Angewandte Mathematik und Mechanik 85 4 304 Bibcode 2005ZaMM 85 304G doi 10 1002 zamm 200590023 ISSN 1521 4001 Further reading EditKhesin Boris Tabachnikov Serge Coordinating Editors Tribute to Vladimir Arnold Notices of the American Mathematical Society March 2012 Volume 59 Number 3 pp 378 399 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 Boris A Khesin Serge L Tabachnikov 2014 Arnold Swimming Against the Tide American Mathematical Society ISBN 978 1 4704 1699 7 Leonid Polterovich Inna Scherbak 7 September 2011 V I Arnold 1937 2010 Jahresbericht der Deutschen Mathematiker Vereinigung 113 4 185 219 doi 10 1365 s13291 011 0027 6 S2CID 122052411 Features Knotted Vortex Lines and Vortex Tubes in Stationary Fluid Flows On Delusive Nodal Sets of Free Oscillations PDF EMS Newsletter 96 26 48 June 2015 ISSN 1027 488X External links EditWikimedia Commons has media related to Vladimir Arnold Wikiquote has quotations related to Vladimir ArnoldV I Arnold s web page Personal web page V I Arnold lecturing on Continued Fractions A short curriculum vitae On Teaching Mathematics text of a talk espousing Arnold s opinions on mathematical instruction Problems from 5 to 15 a text by Arnold for school students available at the IMAGINARY platform Vladimir Arnold at the Mathematics Genealogy Project S Kutateladze Arnold Is Gone V B Demidovichem 2009 MEHMATYaNE VSPOMINAYuT 2 V I Arnold pp 25 58 Author profile in the database zbMATH Retrieved from https en wikipedia org w index php title Vladimir Arnold amp oldid 1050757909, wikipedia, wiki, book,

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