# Vladimir Arnold

Vladimir Igorevich Arnold (or Arnol'd; Russian: Влади́мир И́горевич Арно́льд, IPA:; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to several areas, including geometrical theory of dynamical systems, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, differential equations, classical mechanics, differential-geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem.
His first main result was the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded three new branches of mathematics: topological Galois theory (with his student Askold Khovanskii), symplectic topology and KAM theory.
Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as Mathematical Methods of Classical Mechanics) and popular mathematics books, he influenced many mathematicians and physicists. Many of his books were translated into English. His views on education were particularly opposed to those of Bourbaki.

## Biography

Vladimir Igorevich Arnold was born on 12 June 1937 in Odesa, Soviet Union (now Odesa, Ukraine). His father was Igor Vladimirovich Arnold (1900–1948), a mathematician. His mother was Nina Alexandrovna Arnold (1909–1986, née Isakovich), a Jewish art historian. While a school student, Arnold once asked his father on the reason why the multiplication of two negative numbers yielded a positive number, and his father provided an answer involving the field properties of real numbers and the preservation of the distributive property. Arnold was deeply disappointed with this answer, and developed an aversion to the axiomatic method that lasted through his life. When Arnold was thirteen, his uncle Nikolai B. Zhitkov, who was an engineer, told him about calculus and how it could be used to understand some physical phenomena. This contributed to sparking his interest 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.
Arnold entered Moscow State University in 1954. Among his teachers there were A. N. Kolmogorov, I. M. Gelfand, L. S. Pontriagin and Pavel Alexandrov. While a student of Andrey Kolmogorov at Moscow State University and still a teenager, Arnold showed in 1957 that any continuous function of several variables can be constructed with a finite number of two-variable functions, thereby solving Hilbert's thirteenth problem. This is the Kolmogorov–Arnold representation theorem.
Arnold obtained his PhD in 1961, with Kolmogorov as advisor.
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 was also a motivation in the development of Floer homology.
In 1999 he suffered a serious bicycle accident in Paris, resulting in traumatic brain injury. He regained consciousness after a few weeks but had amnesia and for some time could not even recognize his own wife at the hospital. 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. 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.
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. 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.

## 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 was 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. He was very concerned about what he saw as the divorce of mathematics from the natural sciences in the 20th century. 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 studied 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.

## Mathematical work

Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory. Michèle Audin described him as "a geometer in the widest possible sense of the word" and said that "he was very fast to make connections between different fields".

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

### 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 A$_k$,D$_k$,E$_k$ 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

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

### Other

Arnold conjectured the existence of the gömböc, a body with just one stable and one unstable point of equilibrium when resting on a flat surface.
Arnold generalized the results of Isaac Newton, Pierre-Simon Laplace, and James Ivory on the shell theorem, showing it to be applicable to algebraic hypersurfaces.

## Honours and awards

* Lenin Prize (1965, with Andrey Kolmogorov), "for work on celestial mechanics."
* Crafoord Prize (1982, with Louis Nirenberg), "for their outstanding achievements in the theory of non-linear differential equations."
* Elected member of the United States National Academy of Sciences in 1983.
* Foreign Honorary Member of the American Academy of Arts and Sciences (1987)
* Elected a Foreign Member of the Royal Society (ForMemRS) of London in 1988.
* Elected member of the American Philosophical Society in 1990.
* Lobachevsky Prize of the Russian Academy of Sciences (1992)
* 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."
* 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."
* 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."
* State Prize of the Russian Federation (2007), "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."
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.
The Arnold Fellowships, of the London Institute are 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, one of the highest honours a mathematician could receive, interference from the Soviet government led to it being withdrawn. Arnold's public opposition to the persecution of dissidents had led him into direct conflict with influential Soviet officials, and he suffered persecution himself, including not being allowed to leave the Soviet Union during most of the 1970s and 1980s.

## 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.
* 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.
* 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.
* 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, 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. Vol. 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)
* 1998: Topological Methods in Hydrodynamics

### 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.