VLADIMIR I. ARNOLD—Collected Works: Symplectic Topology, Dynamics of Intersections, and Catastrophe Theory 1986–1991(5 Vladimir I. Arnold - Collected Works)

This volume 5 of the Collected Works includes papers written by V.I. Arnold, one of the most outstanding mathematicians of all times, during the period from 1986 to 1991. Arnold’s work during this period covers symplectic topology, contact geometry and wave propagation, quasicrystals, dynamics of intersections, bifurcations, and catastrophe theory. He was seriously concerned with decaying mathematical education in Russia and worldwide — one can see this in several articles translated for this volume. Of particular interest are the sets of problems which Arnold collected under the name “Mathematical Trivium” — in his opinion, any math or physics university graduate should be able to solve any problem from that list. The reader will also enjoy perusing several interviews with Arnold, as well as his remarkable warm memories about Ya.B. Zeldovich and his teacher A.N. Kolmogorov. One of Arnold’s papers on catastrophe theory translated for this volume also contains a beautiful translation of E.A. Baratynsky’s poem made by A.B. Givental. The book will be of interest to the wide audience from college students to professionals in mathematics or physics and in the history of science.

Table of Contents:
1 The Sturm theorems and symplectic geometry.- 2 First steps of symplectic topology.- 3 First steps of symplectic topology.- 4 On some problems in symplectic topology.- 5 Contact structure, relaxation oscillations and singular points of implicit differential equations.- 6 Contact geometry and wave propagation.- 7 Contact geometry: the geometrical method of Gibbs’s thermodynamics.- 8 Hyperbolic polynomials and Vandermonde mappings.- 9 On surfaces defined by hyperbolic equations.- 10 On the interior scattering of waves, defined by hyperbolic variational principles.- 11 The ramified covering CP^2 → S^4, hyperbolicity, and projective topology.- 12 Quasicrystals, Penrose tilings, Markov partitions, stochastic web, and singularity theory.- 13 Remarks on quasicrystallic symmetries.- 14 Dynamics of complexity of intersections.- 15 Dynamics of intersections.- 16 Cardiac arrhythmias and circle mappings.- 17 Remarks on Poisson structures on the plane and on other powers of volume forms.- 18 Convex hulls and increasing the output of systems under a pulsating load.- 19 Spaces of functions with moderate singularities.- 20 𝐴-graded algebras and continued fractions.- 21 Topological and ergodic properties of closed 1-forms with incommensurable periods.- 22 Bifurcations and singularities in mathematics and mechanics.- 23 Singularities and bifurcations of potential flows.- 24 Singularities of the boundaries of spaces of differential equations.- 25 Some unsolved problems of the theory of differential equations and mathematical physics.- 26 Ten problems.- 27 Evolution processes and ordinary differential equations.- 28 The tercentennial of mathematical natural sciences and celestial mechanics.- 29 Kepler’s second law and the topology of Abelian integrals (according to Newton).- 30 The topological proof of transcendence of Abelian integrals in Newton’s “Mathematical Principles of Natural Philosophy”.- 31 Newton’s Principia read 300 years later.- 32 Meanders.- 33 A mathematical trivium.- 33a Comments on “A mathematical trivium” by V. Arnold.- 34 A mathematical trivium II.- 35 The catastrophe theory and new opportunities for application of mathematics.- 36 Catastrophe theory.- 37 Catastrophe theory.- 38 Conversation with Vladimir Igorevich Arnold (an interview with S. Zdravkovska).- 39 Arnold in his own words (an interview with S.L. Tabachnikov).- 40 Mathematics in the work of Ya.B. Zeldovich.- 41 YaB and mathematics.- 42 Remembering A.N. Kolmogorov.- 43 A few words on Andrei Nikolaevich Kolmogorov.- 44 A.N. Kolmogorov.- 45 On A.N. Kolmogorov.- 46 Look for talents!.- 47 Mathematics with a human face.- 48 Encyclopaedia of Mathematical Sciences, or mathematics with a human face.- 49 Preface to the Russian translation of the book by P.A. Griffiths “Exterior Differential Systems and the Calculus of Variations”.- 50 Preface to the Russian translation (in the form of a book) of the article by P. Scott “The geometries of 3-manifolds”.- 51 Preface to the Russian translation of the book by J.W. Bruce and P.J. Giblin “Curves and Singularities. A Geometrical Introduction to Singularity Theory”.- Acknowledgements.

About the Author :
Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors.