Vladimir Igorevich Arnold (or Arnol'd; Russian: Влади́мир И́горевич Арно́льд, IPA: [vlɐˈdʲimʲɪr ˈiɡərʲɪvʲɪtɕ ɐrˈnolʲt]; 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. In his later years he shifted his research interests, investigating discrete mathematics. His first main result was the solution of Hilbert's thirteenth problem in 1957 when he was 19. He co-founded three new branches of mathematics: topological Galois theory (with his student Askold Khovanskii), KAM theory (with Andrey Kolmogorov and Jürgen Moser) and symplectic topology. Arnold was also a populariser of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as Mathematical Methods of Classical Mechanics and Ordinary Differential Equations) and popular mathematics books, he influenced many mathematicians and physicists. Many of his books were translated into English. His views on education were opposed to those of Bourbaki. A controversial and often quoted dictum of his is "Mathematics is the part of physics where experiments are cheap". The Arnold principle: "Discoveries are rarely attributed to the correct person" is also named after him. Arnold worked at the Moscow State University from 1961 to 1986, at the Steklov Mathematical Institute since 1986, and at the Paris Dauphine University since 1993. He was one of the founders of the Independent University of Moscow. Arnold received many major prizes, including the inaugural Crafoord Prize in 1982 (with Louis Nirenberg), the Wolf Prize in Mathematics in 2001 and the Shaw Prize in 2008 (with Ludwig Faddeev).
Early life
Vladimir Igorevich Arnold was born on 12 June 1937 in Odessa, Ukrainian SSR, Soviet Union (now Odesa, Ukraine). His father was Igor Vladimirovich Arnold (1900–1948), a mathematician known for his work in mathematical education and who learned algebra from Emmy Noether in the late 1920s. His mother was Nina Alexandrovna Arnold (1909–1986, née Isakovich), a Jewish art historian. 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. 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 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. 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.
Mathematical work
Arnold obtained his PhD in 1961, with Andrey Kolmogorov as his advisor (thesis: On The Representation of Continuous Functions of 3 Variables By The Superpositions of Continuous Functions of 2 Variables). He became an academician of the Academy of Sciences of the Soviet Union (Russian Academy of Science since 1991) in 1990. 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. Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University until his death. He supervised 46 PhD students, including 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. 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
Hilbert's thirteenth 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.
Dynamical systems
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