Vladimir Arnold - Collected Works: Singularity Theory 1972-1979

ISBN-13: 9783662496121 / Rosyjski / Twarda / 2016 / 509 str.

Vladimir I. Arnold; Alexander B. Givental; Boris Khesin

Opis

Volume III of the Collected Works of V.I. Arnold contains papers written in the years 1972 to 1979. The main theme emerging in Arnold's work of this period is the development of singularity theory of smooth functions and mappings. The volume also contains papers by V.I. Arnold on catastrophe theory and on A.N. Kolmogorov's school, his prefaces to Russian editions of several books related to singularity theory, V. Arnold's lectures on bifurcations of discrete dynamical systems, as well as a review by V.I. Arnold and Ya.B. Zeldovich of V.V. Beletsky's book on celestial mechanics.
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.

Spis treści

1  Modes and Quasimodes.- 2  Integrals of Rapidly Oscillating Functions and Singularities of Projections of Lagrangian Manifolds.- 3  Remarks on the Stationary Phase Method and Coxeter Numbers.- 4  Normal Forms of Functions near Degenerate Critical Points, the Weyl Groups A_k, D_k, E_k, and Lagrangian Singularities.- 5  Normal Forms of Functions in Neighbourhoods of Degenerate Critical Points.- 6  Critical Points of Functions and Classification of Caustics.- 7  Classification of Unimodal Critical Points of Functions.- 8  Classification of Bimodal Critical Points of Functions.- 9  Spectral Sequence for Reduction of Functions to Normal Form.- 10  Spectral Sequences for Reducing Functions to Normal Forms.- 11  Critical Points of Smooth Functions and Their Normal Forms.- 12  Local Normal Forms of Functions.- 13  Some Open Problems in Singularity Theory.- 14  On the Theory of Envelopes.- 15  Wave Front Evolution and Equivariant Morse Lemma.- 16  A Correction to: Wave Front Evolution and Equivariant Morse Lemma.- 17  A Conjecture on the Signature of the Quadratic Form of a Quasihomogeneous Singularity.- 18  On Contemporary Developments of I.G. Petrovskii's Works on Topology of Real Algebraic Varieties .- 19  Topology of Real Algebraic Varieties (with O.A. Oleinik).- 20  Bifurcations of Invariant Manifolds of Differential Equations and Normal Forms of Neighborhoods of Elliptic Curves.- 21  Loss of Stability of Self-Oscillations Close to Resonances and Versal Deformations of Equivariant Vector Fields.- 22  Some Problems in the Theory of Differential Equations.- 23  Bifurcations of Discrete Dynamical Systems (with A.P. Shapiro).- 24  Index of a Singular Point of a Vector Field, the Petrovskii-Oleinik Inequality, and Mixed Hodge Structures (in Russian).- 25  Index of a Singular Point of a Vector Field, the Petrovskii-Oleinik Inequalities, and Mixed Hodge Structures.- 26  Critical Points of Functions on a Manifold with Boundary, the Simple Lie Groups B_k, C_k, and F₄, and Singularities of Evolutes.- 27  Indices of Singular Points of 1-Forms on a Manifold with Boundary, Convolution of Invariants of Reflection Groups, and Singular Projections of Smooth Surfaces.- 28  Stable Oscillations with Potential Energy Harmonic in Space and Periodic in Time.- 29  The Loss of Stability of Self-Induced Oscillations near Resonances.- 30  Catastrophe Theory.- 31  Superposition of Algebraic Functions (with G. Shimura).- 32  The A-D-E Classifications.- 33  Real Algebraic Geometry (the 16th Hilbert Problem).- 34  Study of Singularities.- 35  Dynamical Systems and Differential Equations.- 36  Fixed Points of Symplectic Diffeomorphisms.- 37  Partial Differential Equations: What Is a Mathematical Equivalent to Physical ”Turbulence“?.- 38  The Beginning of a New Style in the Scientific Literature (a Review of V.V. Beletsky's Book "Essays on the Motion of Celestial Bodies", Moscow: Nauka Publishing House, 1972) (with Ya.B. Zeldovich).- 39 On the First All-Union Mathematical Student Olympiad (with A.A. Kirillov, V.M. Tikhomirov, and M.A. Shubin).- 40  A Regional Mathematical School in Syktyvkar (with A.M. Vershik, D.B. Fuks, and Ya.M. Eliashberg) (in Russian).- 41  Kolmogorov’s School.- 42 Preface to the Collection “Singularities of Differentiable Mappings” of Russian Translations of Papers in English and French.- 43  Preface to the Russian Translation of the Book “Introduction à l’étude topologique des singularités de Landau” by F. Pham.- 44 Preface to the Russian Translation of the Book “Singular Points of Complex Hypersurfaces” by J. Milnor.-  45 Preface to the Russian Translation of the Book “Differentiable Germs and Catastrophes” by Th. Bröcker and L. Lander.- 46  Preface to the Russian Translation of the Book “Stable Mappings and Their Singularities” by M. Golubitsky and V. Guillemin.

O autorze

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.