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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. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry.
List of contents
1 Variational principle for three-dimensional steady-state flows of an ideal fluid.- 2 On the Riemann curvature of diffeomorphism groups.- 3 Sur la topologie des écoulements stationnaires des fluides parfaits (in French).- 4 Conditions for non-linear stability of stationary plane curvilinear flows of an ideal fluid.- 5 On the topology of three-dimensional steady flows of an ideal fluid.- 6 On an a priori estimate in the theory of hydrodynamical stability.- 7 On the differential geometry of infinite-dimensional Lie groups and its application to the hydrodynamics of perfect fluids.- 8 On a variational principle for the steady flows of perfect fluids and its application to problems of non-linear stability.- 9 Characteristic class entering in quantization conditions.- 10 A note on Weierstrass auxiliary theorem.- 11 A letter to the editors (in Russian).- 12 The stability problem and ergodic properties for classical dynamical systems.- 13 Remark on the branching of hyperelliptic integrals asfunctions of the parameters.- 14 Singularities of smooth mappings.- 15 Braids of algebraic functions and the cohomology of swallowtails.- 16 Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of an ideal fluid.- 17 Remarks on singularities of finite codimension in complex dynamical systems.- 18 One-dimensional cohomologies of the Lie algebras of non-divergence vector fields, and rotation numbers of dynamic systems.- 19 The cohomology ring of the colored braid group.- 20 Trivial problems.- 21 Cohomology classes of algebraic functions invariant under Tschirnhausen transformations.- 22 Local Problems of Analysis.- 23 Algebraic unsolvability of the problem of stability and the problem of the topological classification of the singular points of analytic systems of differential equations (in Russian).- 24 On some topological invariants of algebraic functions.- 25 Topological invariants of algebraic functions II.- 26 Algebraic unsolvability of the problem of Ljapunov stability and the problem of the topological classification of singular points of an analytic system of differential equations.- 27 On matrices depending on parameters.- 28 Distribution of ovals of the real plane of algebraic curves of involutions of four-dimensional smooth manifolds and the arithmetic of integer valued quadratic forms.- 29 Versal families and bifurcations of differential equations (in Russian).- 30 Notes on the three-dimensional flow pattern of a perfect fluid in the presence of a small perturbation of the initial velocity field.- 31 Lectures on bifurcations in versal families.- 32 The topology of real algebraic curves (in Russian).- 33 The asymptotic Hopf invariant and its applications.- 34 Magnetic field in a moving conducting fluid.- 35 A magnetic field in a stationary flow with stretching in a Riemannian space.- 36 The large-scale structure of the universe I. General properties. One- and two-dimensional models.- 37 Elements of the large-scale structure of theuniverse.- 38 Some remarks on the antidynamo theorem.- 39 Steady-state magnetic field in a periodic flow.- 40 Evolution of a magnetic field under the action of transfer and diffusion.- 41 The growth of a magnetic field in a three-dimensional steady incompressible fluid.- 42 Evolution of a magnetic field under the action of drift and diffusion (in Russian).- 43 Exponential scattering of trajectories and its hydrodynamical applications.- 44 Kolmogorov's hydrodynamic attractors.- 45 Topological methods in hydrodynamics.- 46 Problemes mathématiques de l'hydrodynamique et de la magnétohydrodynamique (in English).- 47 Translator's Preface to J. Milnor's book "Morse Theory".- 48 Henri Poincaré: Selected Works in Three Volumes. Vol. I New Methods of Celestial Mechanics - Preface. From the editorial board. Comments.- 49 Comments on the paper "On a geometric theorem" by Henri Poincaré.
Summary
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. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry.
Additional text
“Volume II focuses on hydrodynamics, bifurcation theory and algebraic geometry. … Arnold was a very prolific writer. The publication of his collected works is a huge task and a great service to the mathematical community particularly since his papers appeared in various journals some of which are not easily accessible. … I have no doubt that these volumes will be of great benefit to the current and future generations of researchers and graduate students.” (Gerard Misiołek, zbMATH 1354.01038, 2017)
Report
"Volume II focuses on hydrodynamics, bifurcation theory and algebraic geometry. ... Arnold was a very prolific writer. The publication of his collected works is a huge task and a great service to the mathematical community particularly since his papers appeared in various journals some of which are not easily accessible. ... I have no doubt that these volumes will be of great benefit to the current and future generations of researchers and graduate students." (Gerard Misiolek, zbMATH 1354.01038, 2017)
