## ⓘ Vladimir Arnold

Vladimir Igorevich Arnold 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 Hilberts thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics - KAM theory, and topological Galois theory.

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 anti-Bourbaki.

## 1. Biography

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, nee 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 Hilberts 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 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.

### * 1.1. Biography * 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 Arnolds 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 Arnolds 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.

## 2. 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 Arnolds 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 Kleins 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 Poincare, and many times he reported to have found in their works ideas that had not been explored yet.

## 3. Work

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

### * 3.1. Work * Hilberts 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 Hilberts question when posed for the class of continuous functions.

### * 3.2. Work * Dynamical systems

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.

### * 3.3. Work * Singularity theory

In 1965, Arnold attended Rene Thoms 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 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".

### * 3.4. Work * Fluid dynamics

In 1966, Arnold published Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications à lhydrodynamique des fluides parfaits ", in which he presented a common geometric interpretation for both the Eulers equations for rotating rigid bodies and the Eulers 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.

### * 3.5. Work * 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 Gudkovs 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 Arnolds work.

### * 3.6. 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.

### * 3.7. Work * Topology

According to Victor Vassiliev, Arnold "worked comparatively little on topology for topologys 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.

## 4. Honours and awards

- Lobachevsky Prize of the Russian Academy of Sciences 1992
- Lenin Prize 1965, with Andrey Kolmogorov, "for work on celestial mechanics."
- 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."
- Elected a Foreign Member of the Royal Society ForMemRS of London in 1988.
- 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."
- 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."
- Crafoord Prize 1982, with Louis Nirenberg, "for contributions to the theory of non-linear differential equations."
- Shaw Prize in mathematical sciences 2008, with Ludwig Faddeev, "for their contributions to mathematical physics."
- Foreign Honorary Member of the American Academy of Arts and Sciences 1987
- State Prize of the Russian Federation 2007, "for outstanding success in mathematics."

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.

### * 4.1. Honours and awards * 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. Arnolds 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.

## 5. Selected bibliography

- 1999: with Valentin Afraimovich Bifurcation Theory And Catastrophe Theory Springer ISBN 3-540-65379-1
- 2015: Lectures and Problems: A Gift to Young Mathematicians, American Math Society, translated from Russian, 2015
- 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.
- Lectures on Partial Differential Equations.
- Arnolʹd, Vladimir Igorevich 1991. The Theory of Singularities and Its Applications. Cambridge University Press. ISBN 9780521422802.
- 1985: with S. M. Gusein-Zade & A. N. Varchenko Singularities of Differentiable Maps, Volume I: The Classification of Critical Points, Caustics and Wave Fronts. Birkhauser.
- 1995: Topological Invariants of Plane Curves and Caustics, American Mathematical Society 1994 ISBN 978-0-8218-0308-0
- 2004: Teoriya Katastrof Catastrophe Theory, in Russian, 4th ed. Moscow, Editorial-URSS 2004, ISBN 5-354-00674-0.
- 1980: Mathematical Methods of Classical Mechanics, Springer-Verlag, ISBN 0-387-96890-3.
- 1988: Geometrical Methods In The Theory Of Ordinary Differential Equations, Springer-Verlag ISBN 0-387-96649-8.
- 1998: "On the teaching of mathematics" Russian Uspekhi Mat. Nauk 53 1998, no. 1319, 229–234; translation in Russian Math. Surveys 531: 229–236.
- 2015: Experimental Mathematics. American Mathematical Society translated from Russian, 2015.
- 1988: with S. M. Gusein-Zade & A. N. Varchenko Singularities of Differentiable Maps, Volume II: Monodromy and Asymptotics of Integrals. Monographs in Mathematics. Birkhauser.
- 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.
- 2007; Yesterday and Long Ago, Springer 2007, ISBN 978-3-540-28734-6.
- 2004: Vladimir I. Arnold, ed. 15 November 2004. Arnolds Problems 2nd ed. Springer-Verlag. ISBN 978-3-540-20748-1.
- 2001: "Tsepniye Drobi" Continued Fractions, in Russian, Moscow 2001.
- 1978; Ordinary Differential Equations, The MIT Press ISBN 0-262-51018-9.
- 1966: "Sur la geometrie differentielle des groupes de Lie de dimension infine et ses applications a lhydrodynamique des fluides parfaits" Annales de lInstitut Fourier 16: 319–361 doi:10.5802/aif.233
- Real Algebraic Geometry.

### * 5.1. Selected bibliography * Collected works

- 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.
- 2009: 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.