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.
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 Jürgen Moser and Arnold expanded the ideas of Kolmogorov (who was inspired by questions of
Henri 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. 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. In 1964, Arnold introduced the
Arnold web, the first example of a stochastic web. In 1974, Arnold
proved the
Liouville–Arnold theorem, now a classic result deeply geometric in character. 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 the paper "" ('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.
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", 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. The
conjecture was 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. He also proposed the
nearby Lagrangian conjecture, a still open problem in mathematics. According to
Michèle Audin, the birth-date of symplectic topology was 27 October 1965, which is the day Arnold's paper "Sur une propriété topologique des applications globalement canoniques de la mécanique classique" was presented to the
Paris Academy of Sciences.
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.
Theory of plane curves According to
Marcel Berger, Arnold revolutionised
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.
Discrete mathematics In the last years of his life, Arnold's interests shifted to
discrete mathematics. He investigated
number theory and
combinatorics, producing around twenty papers on these topics, according to
Anatoly Vershik. Arnold's conjecture of a
matrix generalization of
Fermat's little theorem dates from this period.
Other In 1995, Arnold conjectured the existence of the
gömböc, a body with one stable and one unstable
point of equilibrium when resting on a flat surface. This conjecture was proved by
Gábor Domokos in 2006. In
classical mechanics, 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. In
algebraic geometry,
Arnold's strange duality was one of the first examples of
mirror symmetry (for
K3 surfaces). In
magnetohydrodynamics, Arnold and E. I. Korkina investigated in 1983 the dynamo property of the
ABC flow. The
Arnold complexity in
dynamical systems theory, and the
Arnold's stability theorems in
analysis of PDEs are named after him. ==Popular mathematical writings==