Collected Works, Volume 1: Representations of Functions, Celestial Mechanics, and KAM Theory 1957-1965

Vladimir Igorevich Arnold is one of the most influential mathematicians of our time. V. I. Arnold launched several mathematical domains (such as modern geometric mechanics, symplectic topology, and topological fluid dynamics) and contributed, in a fundamental way, to the foundations and methods in many subjects, from ordinary differential equations and celestial mechanics to singularity theory and real algebraic geometry. Even a quick look at a partial list of notions named after Arnold already gives an overview of the variety of such theories and domains: KAM (Kolmogorov–Arnold–Moser) theory, The Arnold conjectures in symplectic topology, The Hilbert–Arnold problem for the number of zeros of abelian integrals, Arnold’s inequality, comparison, and complexification method in real algebraic geometry, Arnold–Kolmogorov solution of Hilbert’s 13th problem, Arnold’s spectral sequence in singularity theory, Arnold diffusion, The Euler–Poincaré–Arnold equations for geodesics on Lie groups, Arnold’s stability criterion in hydrodynamics, ABC (Arnold–Beltrami–Childress) ?ows in ?uid dynamics, The Arnold–Korkina dynamo, Arnold’s cat map, The Arnold–Liouville theorem in integrable systems, Arnold’s continued fractions, Arnold’s interpretation of the Maslov index, Arnold’s relation in cohomology of braid groups, Arnold tongues in bifurcation theory, The Jordan–Arnold normal forms for families of matrices, The Arnold invariants of plane curves. Arnold wrote some 700 papers, and many books, including 10 university textbooks. He is known for his lucid writing style, which combines mathematical rigour with physical and geometric intuition. Arnold’s books on Ordinarydifferentialequations and Mathematical methodsofclassicalmechanics became mathematical bestsellers and integral parts of the mathematical education of students throughout the world.