Vladimir I. Arnold - Collected Works - Spaces and Singularities of Curves, Mathematical Trinities, and Mathematical Education 1996-1999

This volume 7 of the "Collected Works" includes papers written by V.I. Arnold, one of the most outstanding mathematicians of all times,  during the period from 1996 to 1999. At that time Arnold was focusing on the description of various spaces of curves, higher-dimensional continued fractions, pseudoperiodic topology, and unifying ideas related to symplectization, complexification and mathematical trinities in topology and mathematics in general. The "Arnoldfest" conference celebrating 60th anniversary of V.Arnold took place at the Fields Institute and University of Toronto, Canada, in 1997,  and Arnold's lectures at that conference are included in this volume. In the 1990s Arnold got increasingly concerned with the decay of science and math education in many Western countries, and his publications fighting "the victorious march of the antiscientific revolution" (as he phrased it in one of his papers) are collected in this volume as well. Some of Arnold's writings stimulated others to write supplements with more detail, and the volume also includes the notes of A.M.Vershik, J.K.Moser, and B.A.Khesin. Finally, a glimpse of Arnold's personality can also be appreciated in a little gem, his short article devoted to resolving the mystery of the origin of the epigraph to "Eugene Onegin" by A.S. Pushkin, a famous XIXth century Russian poet.  

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. This volume completes the seven-volume project of "Collected Works of Vladimir Arnold", the first volume of which was published in 2009. Arnold's publications in the 2000s were also abundant, he turned to new topics, wrote several books, he gave numerous lectures, both research and educational. But what has already been published in these seven volumes is a treasure trove, an ocean of ideas, methods, and results, and this is now before the reader to explore. Happy voyage!

List of contents

1 Remarks on the enumeration of plane curves.- 2 Remarks on the extatic points of plane curves.- 3 On the number of flattening points on space curves.- 4 Remarks on the parabolic curves on surfaces and on the higher-dimensional Möbius-Sturm theory.- 5 Towards the Legendre Sturm theory of space curves.- 6 Topological problems of the theory of asymptotic curves.- 7 Simple singularities of curves.- 8 Topological classification of trigonometric polynomials and combinatorics of graphs with an equal number of vertices and edges.- 9 Topological classification of real trigonometric polynomials and cyclic serpents polyhedron.- 10 Topological problems of the theory of wave propagation.- 11 Topological content of the Maxwell theorem on multipole representation of spherical functions.- 12 Topologically necessary singularities on moving wavefronts and caustics.- 13 An appendix in the book Fewnomials by A.G. Khovanskii.- 14 Singularities of fractions and behavior of polynomials at infinity.- 15 Remarks concerning the Morse theory of a divergence-free vector field, the averaging method, and the motion of a charged particle in a magnetic field.- 16 On the problem of realization of a given Gaussian curvature function.- 17 Relatives of the quotient of the complex projective plane by the complex
conjugation.- 18 First steps of local symplectic algebra.- 19 First steps of local contact algebra.- 20 Higher-dimensional continued fractions.- 21 Weak asymptotics for the numbers of solutions of Diophantine problems.- 22 Mysterious mathematical trinities.- 23 The principle of topological economy in algebraic geometry.- 24 Translation of the V.I. Arnold paper "From superpositions to KAM theory".- 25 From Hilbert's superposition problem to dynamical systems.- 25a Recollections (by Jurgen Moser).- 26 Symplectization, complexification and mathematical trinities.- 27 Topological problems in wave propagation theory and topological economy principle in algebraic geometry.- 28 Catastrophe theory.- 29 Mathematics and physics: mother and daughter or sisters?.- 30 "Hard" ("rigid") and "soft" ("flexible") mathematical models.- 31 Repartitioning the world: Population and the powers of two.- 32 On some problems of pseudo-periodic topology.- 33 Preface to the book "Pseudoperiodic Topology".- 34 An interview with Vladimir Arnold (by S.H. Lui).- 35 Vershik work needs acknowledgement.- 35a On Arnold's Letter to the Notices of the AMS (by V.M. Vershik).- 36 The Russian edition of the works by David Hilbert.- 37 A disciple of the Moscow mathematical school has been awarded the Fields medal.- 38 About Vladimir Abramovich Rokhlin.- 39 Mathematics and mathematical education in the contemporary world.- 40 Mathematical illiteracy is more destructive than the fires of the inquisition.- 41 On teaching mathematics.- 42 Excerpts from the review of the textbook "Mathematical Analysis".- 43 The antiscientific revolution and mathematics.- 44 Answers to the questionnaire of the European Mathematical Society.- 45 International mathematical congress in Berlin.- 46 Preface to the book "International Congress of Mathematicians in Kyoto.- 47 Preface to the book "International Congress of Mathematicians in Zürich.- 48 Preface to the Russian translation of the book "Concrete Mathematics.- 49 On the epigraph to "Eugene Onegin".- 49a On Arnold's and Pushkin's puzzles (by Boris Khesin).- 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.

Summary

This volume 7 of the “Collected Works" includes papers written by V.I. Arnold, one of the most outstanding mathematicians of all times,  during the period from 1996 to 1999. At that time Arnold was focusing on the description of various spaces of curves, higher-dimensional continued fractions, pseudoperiodic topology, and unifying ideas related to symplectization, complexification and mathematical trinities in topology and mathematics in general. The “Arnoldfest" conference celebrating 60th anniversary of V.Arnold took place at the Fields Institute and University of Toronto, Canada, in 1997,  and Arnold's lectures at that conference are included in this volume. In the 1990s Arnold got increasingly concerned with the decay of science and math education in many Western countries, and his publications fighting “the victorious march of the antiscientific revolution" (as he phrased it in one of his papers) are collected in this volume as well. Some of Arnold's writings stimulated others to write supplements with more detail, and the volume also includes the notes of A.M.Vershik, J.K.Moser, and B.A.Khesin. Finally, a glimpse of Arnold's personality can also be appreciated in a little gem, his short article devoted to resolving the mystery of the origin of the epigraph to “Eugene Onegin" by A.S. Pushkin, a famous XIXth century Russian poet.  

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. This volume completes the seven-volume project of “Collected Works of Vladimir Arnold", the first volume of which was published in 2009. Arnold's publications in the 2000s were also abundant, he turned to new topics, wrote several books, he gave numerous lectures, both research and educational. But what has already been published in these seven volumes is a treasure trove, an ocean of ideas, methods, and results, and this is now before the reader to explore. Happy voyage!

Product details

• Authors: Vladimir I Arnold, Vladimir I. Arnold
• Assisted by: Victor A Vassiliev (Editor), Mikhail B Sevryuk (Editor), Boris A. Khesin (Editor), Mikhail B. Sevryuk (Editor), Victor A. Vassiliev (Editor)
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 01.01.2025
• EAN: 9783031774843
• ISBN: 978-3-0-3177484-3
• No. of pages: 502
• Illustrations: XVIII, 502 p. 143 illus., 6 illus. in color.
• Series: Vladimir I. Arnold - Collected Works
• Subjects: Natural sciences, medicine, IT, technology > Mathematics; Mathematik und Naturwissenschaften, Mathematics, Physics and Astronomy, wave propagation, caustics, Legendre Sturm Theory, mathematical education, singularities of curves