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Early life

Vladimir Igorevich Arnold was born on 12 June 1937 in Odesa, Ukrainian SSR, Soviet Union (now in Ukraine). His father was Igor Vladimirovich Arnold [ru] (1900–1948), a mathematician known for his work in mathematical education and who learned algebra from Emmy Noether in the late 1920s.⁹ His mother was Nina Alexandrovna Arnold (1909–1986, née Isakovich), a Jewish art historian.¹⁰ 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.¹¹¹² When Arnold was thirteen, his uncle Nikolai B. Zhitkov,¹³ 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.¹⁴

Arnold entered Moscow State University in 1954.¹⁵ Among his teachers there were A. N. Kolmogorov, I. M. Gelfand, L. S. Pontriagin and Pavel Alexandrov.¹⁶ 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.¹⁷ This is the Kolmogorov–Arnold representation theorem.

Mathematical work

See also: Stability of the Solar System

Arnold obtained his PhD in 1961, with Kolmogorov as his advisor (thesis: On The Representation of Continuous Functions of 3 Variables By The Superpositions of Continuous Functions of 2 Variables).¹⁸

He became an academician of the Academy of Sciences of the Soviet Union (Russian Academy of Science since 1991) in 1990.¹⁹ 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.²⁰

Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University until his death. His PhD students include 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.²¹

Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory.²² 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".²³

Hilbert's thirteenth problem

See also: Kolmogorov–Arnold representation theorem

The 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.²⁴

Dynamical systems

See also: Arnold diffusion and Arnold's cat map

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, and specifies what the conditions for this are.²⁵

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.²⁶

In 1964, Arnold introduced the Arnold web, the first example of a stochastic web.²⁷²⁸

In 1974, Arnold proved the Liouville–Arnold theorem, now a classic result deeply geometric in character.²⁹

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.³⁰

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."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 Ak,Dk,Ek and Lagrangian singularities".³³³⁴³⁵

Fluid dynamics

See also: Arnold–Beltrami–Childress flow, Beltrami vector field § Beltrami fields and complexity in fluid mechanics, and Euler–Arnold equation

In 1966, Arnold published the paper "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits" ('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.³⁶³⁷³⁸

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",³⁹ 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.⁴⁰ The conjecture was later fully solved by V. A. Rokhlin building on Arnold's work.⁴¹⁴²

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.⁴³⁴⁴ He also proposed the nearby Lagrangian conjecture, a still open problem in mathematics.⁴⁵

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.⁴⁶⁴⁷

Theory of plane curves

According to Marcel Berger, Arnold revolutionised plane curves theory.⁴⁸ He developed the theory of smooth closed plane curves in the 1990s.⁴⁹ Among his contributions are the introduction of the three Arnold invariants of plane curves: J+, J− and St.⁵⁰⁵¹

Other

Arnold conjectured the existence of the gömböc, a body with one stable and one unstable point of equilibrium when resting on a flat surface.⁵²⁵³

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.⁵⁴

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 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".⁵⁵

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.⁵⁶⁵⁷ He was very concerned about what he saw as the divorce of mathematics from the natural sciences in the 20th century.⁵⁸ Arnold was very interested in the history of mathematics,⁵⁹ and in an interview,⁶⁰ 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 –a book he often recommended to his students.⁶¹ He studied the works of Huygens, Newton and Poincaré,⁶² and reported finding ideas that had yet to be explored in the works of Newton and Poincaré.⁶³

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.⁶⁴ He went on to make a good recovery.⁶⁵

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.⁶⁶

Arnold died of acute pancreatitis⁶⁷ on 3 June 2010 in Paris, nine days before his 73rd birthday.⁶⁸ He was buried on 15 June in Moscow, at the Novodevichy Monastery.⁶⁹

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.⁷⁰

Honours and awards

• Moscow Mathematical Society Prize for Young Mathematicians (1958).⁷¹
• Lenin Prize (1965, with Andrey Kolmogorov),⁷² "for work on celestial mechanics."
• Crafoord Prize (1982, with Louis Nirenberg),⁷³ "for their outstanding achievements in the theory of non-linear differential equations."⁷⁴
• Elected member of the United States National Academy of Sciences in 1983.⁷⁵
• Foreign Honorary Member of the American Academy of Arts and Sciences (1987)⁷⁶
• Elected a Foreign Member of the Royal Society (ForMemRS) of London in 1988.⁷⁷
• Elected member of the American Philosophical Society in 1990.⁷⁸
• Lobachevsky Prize of the Russian Academy of Sciences (1992)⁷⁹
• 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."⁸⁰
• 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."⁸¹
• 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."⁸²
• State Prize of the Russian Federation (2007),⁸³ "for outstanding contribution to development of mathematics."
• Shaw Prize in mathematical sciences (2008, with Ludwig Faddeev), "for their widespread and influential contributions to Mathematical Physics."⁸⁴⁸⁵

The minor planet 10031 Vladarnolda was named after him in 1981 by Lyudmila Georgievna Karachkina.⁸⁶

The Arnold Mathematical Journal, published for the first time in 2015, is named after him.⁸⁷

The Arnold Fellowships, of the London Institute are named after him.⁸⁸⁸⁹

He was a plenary speaker at both the 1974 and 1983 International Congress of Mathematicians in Vancouver and Warsaw, respectively.⁹⁰

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.⁹¹⁹²

Selected bibliography

• 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 0-262-51018-9.⁹³⁹⁴⁹⁵
• 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. 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. Vol. 83. Birkhäuser. doi:10.1007/978-1-4612-3940-6. ISBN 978-1-4612-8408-6. S2CID 131768406.
• 1988: Arnold, V.I. (1988). Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 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. Vol. 60 (2nd ed.). Springer. doi:10.1007/978-1-4757-2063-1. ISBN 978-1-4419-3087-3.⁹⁶⁹⁷
• 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.^9899100
• 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,¹⁰¹ 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).
• 2002: "Что такое математика?" (What is mathematics?, in Russian) ISBN 978-5-94057-426-2.
• 2004: Teoriya Katastrof (Catastrophe Theory,¹⁰² 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. ¹⁰³
• 2004: Arnold, Vladimir I. (2004). Lectures on Partial Differential Equations. Universitext. Springer. doi:10.1007/978-3-662-05441-3. ISBN 978-3-540-40448-4.¹⁰⁴¹⁰⁵
• 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. Vol. 66. Springer. doi:10.1007/978-3-642-36243-9. ISBN 978-3-642-36242-2.¹⁰⁶
• 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)
• 1998: Topological Methods in Hydrodynamics (with Boris Khesin)¹⁰⁷

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.

See also

• Mathematics portal

• List of things named after Vladimir Arnold
• Independent University of Moscow
• Geometric mechanics

Further reading

• Khesin, 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

• V. I. Arnold's web page
• Personal web page
• V. I. Arnold lecturing on Continued Fractions
• A short curriculum vitae
• On Teaching Mathematics Archived 28 April 2017 at the Wayback Machine, text of a talk from 1997 espousing Arnold's opinions on mathematical instruction
• Topology of Plane Curves, Wave Fronts, Legendrian Knots, Sturm Theory and Flattenings of Projective Curves
• 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
• В.Б.Демидовичем (2009), МЕХМАТЯНЕ ВСПОМИНАЮТ 2: В.И.Арнольд, pp. 25–58
• Author profile in the database zbMATH
• Leonid Polterovich and Inna Scherbakh (2011). "V.I. Arnold (1937–2010)" DOI 10.1365/s13291-011-0027-6

References

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. /wiki/Boris_Khesin ↩

2. Mort d'un grand mathématicien russe, AFP (Le Figaro) http://www.lefigaro.fr/flash-actu/2010/06/03/97001-20100603FILWWW00719-mort-d-un-grand-mathematicien-russe.php ↩

3. 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 /wiki/Sabir_Gusein-Zade ↩

4. O'Connor, John J.; Robertson, Edmund F., "Vladimir Arnold", MacTutor History of Mathematics Archive, University of St Andrews /wiki/Edmund_F._Robertson ↩

5. 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. 9783642136061 ↩

6. 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 /wiki/Sabir_Gusein-Zade ↩

7. Euler M. (2004). "The role of experiments in the teaching and learning of physics" [JB]. Proceedings of the International School of Physics (Research on Physics Education), pp. 175–221. https://doi.org/10.3254/978-1-61499-012-3-175 https://doi.org/10.3254/978-1-61499-012-3-175 ↩

8. Note: This phrase generated controversy and even parodies. He wrote his defense at Arnold, V. I. (1999). "Mathematics and physics: mother and daughter or sisters?". Physics-Uspekhi. 42 (12): 1205–1217. Bibcode:1999PhyU...42.1205A. doi:10.1070/pu1999v042n12abeh000673. S2CID 250835608.. ↩

9. 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. /wiki/Boris_Khesin ↩

10. 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 /wiki/Sabir_Gusein-Zade ↩

11. Vladimir I. Arnold (2007). Yesterday and Long Ago. Springer. pp. 19–26. ISBN 978-3-540-28734-6. 978-3-540-28734-6 ↩

12. Rodin, A. (2014). Axiomatic Method and Category Theory. In Synthese Library. Springer International Publishing. https://doi.org/10.1007/978-3-319-00404-4 https://doi.org/10.1007/978-3-319-00404-4 ↩

13. Arnold: Swimming Against the Tide, p. 3 ↩

14. Табачников, С. Л. . "Интервью с В.И.Арнольдом", Квант, 1990, Nº 7, pp. 2–7. (in Russian) /wiki/Sergei_Tabachnikov ↩

15. Sevryuk, M.B. Translation of the V. I. Arnold paper “From Superpositions to KAM Theory” (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740). Regul. Chaot. Dyn. 19, 734–744 (2014). https://doi.org/10.1134/S1560354714060100 https://doi.org/10.1134/S1560354714060100 ↩

16. "An Interview with Vladimir Arnol'd" (PDF), Notices of the AMS, 44 (4): 432–438, April 1997 https://www.ams.org/journals/notices/199704/arnold.pdf ↩

17. Daniel Robertz (13 October 2014). Formal Algorithmic Elimination for PDEs. Springer. p. 192. ISBN 978-3-319-11445-3. 978-3-319-11445-3 ↩

18. Vladimir Igorevich Arnold at the Mathematics Genealogy Project https://mathgenealogy.org/id.php?id=17493 ↩

19. Great Russian Encyclopedia (2005), Moscow: Bol'shaya Rossiyskaya Enciklopediya Publisher, vol. 2. /wiki/Great_Russian_Encyclopedia ↩

20. Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta (2009). Lagrangian Intersection Floer Theory Anomaly and Obstruction [page needed] /wiki/Kenji_Fukaya ↩

21. Vladimir Arnold at the Mathematics Genealogy Project https://mathgenealogy.org/id.php?id=17493 ↩

22. O'Connor, John J.; Robertson, Edmund F., "Vladimir Arnold", MacTutor History of Mathematics Archive, University of St Andrews /wiki/Edmund_F._Robertson ↩

23. "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) ↩

24. Ornes, Stephen (14 January 2021). "Mathematicians Resurrect Hilbert's 13th Problem". Quanta Magazine. https://www.quantamagazine.org/mathematicians-resurrect-hilberts-13th-problem-20210114/ ↩

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. 9781440634284 ↩

26. Gérard, C.; Goldbeter, A. (2012). "The cell cycle is a limit cycle". Mathematical Modelling of Natural Phenomena. 7 (6): 126–166. doi:10.1051/mmnp/20127607. https://doi.org/10.1051%2Fmmnp%2F20127607 ↩

27. Phase Space Crystals, by Lingzhen Guo https://iopscience.iop.org/book/978-0-7503-3563-8.pdf https://iopscience.iop.org/book/978-0-7503-3563-8.pdf ↩

28. Zaslavsky web map, by George Zaslavsky http://www.scholarpedia.org/article/Zaslavsky_web_map /wiki/George_Zaslavsky ↩

29. 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. /wiki/Boris_Khesin ↩

30. 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. /wiki/Boris_Khesin ↩

31. "Archived copy" (PDF). Archived from the original (PDF) on 14 July 2015. Retrieved 22 February 2015.{{cite web}}: CS1 maint: archived copy as title (link) https://web.archive.org/web/20150714123033/https://www.math.upenn.edu/Arnold/Arnold-interview1997.pdf ↩

32. "Resonance – Journal of Science Education | Indian Academy of Sciences" (PDF). http://www.ias.ac.in/resonance/Volumes/19/09/0787-0796.pdf ↩

33. 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 http://www.maths.ed.ac.uk/~aar/papers/arnold15.pdf ↩

34. 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. 978-0-7923-6996-7 ↩

35. Landsberg, J. M.; Manivel, L. (2002). "Representation theory and projective geometry". arXiv:math/0203260. /wiki/ArXiv_(identifier) ↩

36. Terence Tao (22 March 2013). Compactness and Contradiction. American Mathematical Soc. pp. 205–206. ISBN 978-0-8218-9492-7. 978-0-8218-9492-7 ↩

37. MacKay, Robert Sinclair; Stewart, Ian (19 August 2010). "VI Arnold obituary". The Guardian. https://www.theguardian.com/science/2010/aug/19/v-i-arnold-obituary ↩

38. IAMP News Bulletin, July 2010, pp. 25–26 http://www.iamp.org/bulletins/old-bulletins/201007.pdf ↩

39. Note: The paper also appears with other names, as in http://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf http://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf ↩

40. 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. 978-0-8218-0807-8 ↩

41. Khesin, Boris A.; Tabachnikov, Serge L. (10 September 2014). Arnold: Swimming Against the Tide. American Mathematical Society. p. 159. ISBN 9781470416997. 9781470416997 ↩

42. 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. S2CID 250775854. /wiki/ArXiv_(identifier) ↩

43. "Arnold and Symplectic Geometry", by Helmut Hofer (in the book Arnold: Swimming Against the Tide) /wiki/Helmut_Hofer ↩

44. "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 http://www-irma.u-strasbg.fr/~maudin/Arnold.pdf ↩

45. Lisa Traynor (2024), "Eliashberg’s contributions towards the theory of generating functions" https://celebratio.org/Eliashberg_Y/article/1188/ ↩

46. "Topology in Arnold's work", by Victor Vassiliev /wiki/Victor_Vassiliev ↩

47. 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 https://www.ams.org/journals/bull/2008-45-02/S0273-0979-07-01165-2/S0273-0979-07-01165-2.pdf ↩

48. Berger, Marcel. A Panoramic View of Riemannian Geometry. pp. 24–25. /wiki/Marcel_Berger ↩

49. "On computational complexity of plane curve invariants", by Duzhin and Biaoshuai https://hosted.math.rochester.edu/ojac/vol9/90.pdf ↩

50. Extrema of Arnold's invariants of curves on surfaces, by Vladimir Chernov https://math.dartmouth.edu/~chernov-china/ https://math.dartmouth.edu/~chernov-china/ ↩

51. V. I. Arnold, "Plane curves, their invariants, perestroikas and classifications" (May 1993) ↩

52. Weisstein, Eric W. "Gömböc". MathWorld. Retrieved 29 April 2024. https://mathworld.wolfram.com/Gomboc.html ↩

53. Mackenzie, Dana (29 December 2010). What's Happening in the Mathematical Sciences. American Mathematical Soc. p. 104. ISBN 9780821849996. 9780821849996 ↩

54. Ivan Izmestiev, Serge Tabachnikov. "Ivory’s theorem revisited", Journal of Integrable Systems, Volume 2, Issue 1, (2017) https://doi.org/10.1093/integr/xyx006 /wiki/Serge_Tabachnikov ↩

55. 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.) ↩

56. See [1] (archived from [2] Archived 28 April 2017 at the Wayback Machine) and other essays in [3]. https://iopscience.iop.org/article/10.1070/RM1998v053n01ABEH000005/https://archive.today/20210331201831/https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html ↩

57. An Interview with Vladimir Arnol'd, by S. H. Lui, Notices of the AMS, 1997. https://www.ams.org/notices/199704/arnold.pdf ↩

58. Ezra, Gregory S.; Wiggins, Stephen (1 December 2010). "Vladimir Igorevich Arnold". Physics Today. 63 (12): 74–76. Bibcode:2010PhT....63l..74E. doi:10.1063/1.3529010. ISSN 0031-9228. /wiki/Physics_Today ↩

59. Oleg Karpenkov. "Vladimir Igorevich Arnold" https://arxiv.org/abs/1007.0688 ↩

60. An Interview with Vladimir Arnol'd, by S. H. Lui, Notices of the AMS, 1997. https://www.ams.org/notices/199704/arnold.pdf ↩

61. B. Khesin and S. Tabachnikov, Tribute to Vladimir Arnold, Notices of the AMS, 59:3 (2012) 378–399. /wiki/Boris_Khesin ↩

62. Goryunov, V.; Zakalyukin, V. (2011), "Vladimir I. Arnold", Moscow Mathematical Journal, 11 (3). https://www.ams.org/distribution/mmj/vol11-3-2011/vladimir-arnold.html ↩

63. 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". https://www.ams.org/journals/notices/198911/198911FullIssue.pdf ↩

64. Arnold, Vladimir I. (2007). Yesterday and Long Ago. Berlin ; New York : Moscow: Springer ; Phasis. p. V. ISBN 978-3-540-28734-6. OCLC 76794406. 978-3-540-28734-6 ↩

65. Polterovich and Scherbak (2011) ↩

66. "Vladimir Arnold". The Daily Telegraph. London. 12 July 2010. https://www.telegraph.co.uk/news/obituaries/science-obituaries/7886200/Vladimir-Arnold.html ↩

67. Kenneth Chang (11 June 2010). "Vladimir Arnold Dies at 72; Pioneering Mathematician". The New York Times. Retrieved 12 June 2013. https://www.nytimes.com/2010/06/11/science/11arnold.html ↩

68. "Number's up as top mathematician Vladimir Arnold dies". Herald Sun. 4 June 2010. Archived from the original on 14 June 2011. Retrieved 6 June 2010. https://web.archive.org/web/20110614172804/http://www.heraldsun.com.au/news/breaking-news/numbers-up-as-top-mathematician-vladimir-arnold-dies/story-e6frf7jx-1225875367896 ↩

69. "From V. I. Arnold's web page". Retrieved 12 June 2013. http://www.pdmi.ras.ru/~arnsem/Arnold/ ↩

70. "Condolences to the family of Vladimir Arnold". Presidential Press and Information Office. 15 June 2010. Retrieved 1 September 2011. http://eng.kremlin.ru/news/437#sel=3:1,5:29 ↩

71. O'Connor, John J.; Robertson, Edmund F., "Vladimir Arnold", MacTutor History of Mathematics Archive, University of St Andrews /wiki/Edmund_F._Robertson ↩

72. O. Karpenkov, "Vladimir Igorevich Arnold", Internat. Math. Nachrichten, no. 214, pp. 49–57, 2010. (link to arXiv preprint) https://arxiv.org/abs/1007.0688 ↩

73. Harold M. Schmeck Jr. (27 June 1982). "American and Russian Share Prize in Mathematics". The New York Times. https://www.nytimes.com/1982/06/27/us/american-and-russian-share-prize-in-mathematics.html ↩

74. "The Crafoord Prize 1982–2014" (PDF). The Anna-Greta and Holger Crafoord Fund. Archived from the original (PDF) on 26 January 2016. https://web.archive.org/web/20160126153013/http://www.kva.se/globalassets/priser/crafoord/2014/rattigheter/crafoordprize1982_2014.pdf ↩

75. "Vladimir I. Arnold". www.nasonline.org. Retrieved 14 April 2022. http://www.nasonline.org/member-directory/deceased-members/47101.html ↩

76. "Book of Members, 1780–2010: Chapter A" (PDF). American Academy of Arts and Sciences. Retrieved 25 April 2011. http://www.amacad.org/publications/BookofMembers/ChapterA.pdf ↩

77. 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. /wiki/Boris_Khesin ↩

78. "APS Member History". search.amphilsoc.org. Retrieved 14 April 2022. https://search.amphilsoc.org/memhist/search?creator=Vladimir+Arnold&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced ↩

79. 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) /wiki/Ludvig_Faddeev ↩

80. "Prize Winners – Harvey Prize". Technion. Retrieved 24 August 2024. https://harveypz.net.technion.ac.il/harvey-prize-laureates/ ↩

81. American Physical Society – 2001 Dannie Heineman Prize for Mathematical Physics Recipient http://www.aps.org/programs/honors/prizes/prizerecipient.cfm?last_nm=Arnol%27d&first_nm=Vladimir&year=2001 ↩

82. The Wolf Foundation – Vladimir I. Arnold Winner of Wolf Prize in Mathematics http://www.wolffund.org.il/index.php?dir=site&page=winners&cs=163&language=eng ↩

83. "Названы лауреаты Государственной премии РФ". www.kommersant.ru (in Russian). 20 May 2008. Retrieved 27 February 2025. https://www.kommersant.ru/doc/894018 ↩

84. "The 2008 Prize in Mathematical Sciences". Shaw Prize Foundation. Archived from the original on 7 October 2022. Retrieved 7 October 2022. https://web.archive.org/web/20221007154628/https://www.shawprize.org/laureates/mathematical-sciences/2008 ↩

85. "Arnold and Faddeev Receive 2008 Shaw Prize" (PDF). Notices of the American Mathematical Society. 55 (8): 966. 2008. Archived from the original (PDF) on 7 October 2022. Retrieved 8 October 2022. https://web.archive.org/web/20221007154713/https://www.ams.org/notices/200808/tx080800966p.pdf ↩

86. Lutz D. Schmadel (10 June 2012). Dictionary of Minor Planet Names. Springer Science & Business Media. p. 717. ISBN 978-3-642-29718-2. 978-3-642-29718-2 ↩

87. Editorial (2015), "Journal Description Arnold Mathematical Journal", Arnold Mathematical Journal, 1 (1): 1–3, Bibcode:2015ArnMJ...1....1., doi:10.1007/s40598-015-0006-6. /wiki/Bibcode_(identifier) ↩

88. "Arnold Fellowships". https://lims.ac.uk/arnold-fellowships/ ↩

89. Fink, Thomas (July 2022). "Britain is rescuing academics from Vladimir Putin's clutches". The Telegraph. https://www.telegraph.co.uk/opinion/2022/07/01/britain-rescuing-academics-vladimir-putins-clutches/ ↩

90. "International Mathematical Union (IMU)". Archived from the original on 24 November 2017. Retrieved 22 May 2015. https://web.archive.org/web/20171124141541/http://www.mathunion.org/db/ICM/Speakers/SortedByLastname.php ↩

91. Martin L. White (2015). "Vladimir Igorevich Arnold". Encyclopædia Britannica. Archived from the original on 2 April 2015. Retrieved 18 March 2015. https://web.archive.org/web/20150402102751/http://www.britannica.com/EBchecked/topic/1742993/Vladimir-Igorevich-Arnold ↩

92. Thomas H. Maugh II (23 June 2010). "Vladimir Arnold, noted Russian mathematician, dies at 72". The Washington Post. Retrieved 18 March 2015. https://www.washingtonpost.com/wp-dyn/content/article/2010/06/22/AR2010062205069.html ↩

93. Sacker, Robert J. (1 August 1975). "Ordinary Differential Equations". Technometrics. 17 (3): 388–389. doi:10.1080/00401706.1975.10489355. ISSN 0040-1706. https://www.tandfonline.com/doi/abs/10.1080/00401706.1975.10489355 ↩

94. Kapadia, Devendra A. (March 1995). "Ordinary differential equations, by V. I. Arnold. Pp 334. DM 78. 1992. ISBN 3-540-54813-0 (Springer)". The Mathematical Gazette. 79 (484): 228–229. doi:10.2307/3620107. ISSN 0025-5572. JSTOR 3620107. S2CID 125723419. https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/ordinary-differential-equations-by-v-i-arnold-pp-334-dm-78-1992-isbn-3540548130-springer/21D8730E2BDA13927F72B54866E2F4A7 ↩

95. Chicone, Carmen (2007). "Review of Ordinary Differential Equations". SIAM Review. 49 (2): 335–336. ISSN 0036-1445. JSTOR 20453964. https://www.jstor.org/stable/20453964 ↩

96. Review by Ian N. Sneddon (Bulletin of the American Mathematical Society, Vol. 2): https://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14755-2/S0273-0979-1980-14755-2.pdf https://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14755-2/S0273-0979-1980-14755-2.pdf ↩

97. Review by R. Broucke (Celestial Mechanics, Vol. 28): Bibcode:1982CeMec..28..345A. /wiki/Roger_A._Broucke ↩

98. 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. /wiki/Doi_(identifier) ↩

99. 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. /wiki/Bibcode_(identifier) ↩

100. 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. https://doi.org/10.1017%2FS0013091500007240 ↩

101. 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. https://doi.org/10.1017%2FS0013091500023348 ↩

102. Bernfeld, Stephen R. (1 January 1985). "Review of Catastrophe Theory". SIAM Review. 27 (1): 90–91. doi:10.1137/1027019. JSTOR 2031497. /wiki/Doi_(identifier) ↩

103. Sevryuk, Mikhail B. (1 June 2005). "Book Review: Arnold's problems". Bulletin of the American Mathematical Society. 43 (1). American Mathematical Society (AMS): 101–110. doi:10.1090/s0273-0979-05-01069-4. ISSN 0273-0979. https://doi.org/10.1090%2Fs0273-0979-05-01069-4 ↩

104. 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. /wiki/ISSN_(identifier) ↩

105. Groves, M. (2005). "Book Review: Vladimir I. Arnold, Lectures on Partial Differential Equations. Universitext". Journal of Applied Mathematics and Mechanics. 85 (4): 304. Bibcode:2005ZaMM...85..304G. doi:10.1002/zamm.200590023. ISSN 1521-4001. /wiki/Bibcode_(identifier) ↩

106. Gouvêa, Fernando Q. (15 August 2013). "Review of Real Algebraic Geometry by Arnold". MAA Reviews. Archived from the original on 23 March 2023. /wiki/Fernando_Q._Gouv%C3%AAa ↩

107. Review, by Daniel Peralta-Salas, of the book "Topological Methods in Hydrodynamics", by Vladimir I. Arnold and Boris A. Khesin https://ems.press/content/serial-article-files/28174 ↩