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Book DetailsFields Institute CommunicationsVolume: 24; 1999; 555 ppMSC: Primary 01; 14; 34; 57; 58; 76; 70
This volume presents articles originating from invited talks at an exciting international conference held at The Fields Institute in Toronto celebrating the sixtieth birthday of the renowned mathematician, Vladimir Arnold. Experts from the world over—including several from “Arnold's school”—gave illuminating talks and lively poster sessions. The presentations focussed on Arnold's main areas of interest: singularity theory, the theory of curves, symmetry groups, dynamical systems, mechanics, and related areas of mathematics.
The book begins with notes of three lectures by V. Arnold given in the framework of the Institute's Distinguished Lecturer program. The topics of the lectures are:
 From Hilbert's Superposition Problem to Dynamical Systems
 Symplectization, Complexification, and Mathematical Trinities
 Topological Problems in Wave Propagation Theory and Topological Economy Principle in Algebraic Geometry
Arnold's three articles include insightful comments on Russian and Western mathematics and science. Complementing the first is Jurgen Moser's “Recollections”, concerning some of the history of KAM theory.
Titles in this series are copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
ReadershipGraduate students and research mathematicians interested in singularity theory, the theory of curves, symmetry groups, dynamical systems, and mechanics; physicists and engineers interested in astronomy, mechanics, and bifurcation theory.

Table of Contents

Chapters

V. Arnold — From Hilbert’s superposition problem to dynamical systems

Jürgen Moser — Recollections

V. Arnold — Symplectization, complexification and mathematical trinities

V. Arnold — Topological problems in wave propagation theory and topological economy principle in algebraic geometry

Mark Alber, Gregory Luther, Jerrold Marsden and Jonathan Robbins — Geometry and control of threewave interactions

Edward Bierstone and Pierre Milman — Standard basis along a Samuel stratum, and implicit differentiation

James Damon — A global weighted version of Bezout’s theorem

Alexander Degtyarev and Viatcheslav Kharlamov — Real Enriques surfaces without real points and EnriquesEinsteinHitchin 4manifolds

W. Ebeling and S. GuseinZade — On the index of a vector field at an isolated singularity

David Ebin and Gerard Misiołek — The exponential map on $\mathcal {D}^s_\mu $

Michael Freedman — Zeldovich’s neutron star and the prediction of magnetic froth

Kenji Fukaya and Kaoru Ono — Arnold conjecture and GromovWitten invariant for general symplectic manifolds

Andrei Gabrielov — Multiplicity of a zero of an analytic function on a trajectory of a vector field

Alexander Givental — Singularity theory and symplectic topology

V. Goryunov and S. Lando — On enumeration of meromorphic functions on the line

H. Hofer and E. Zehnder — Pseudoholomorphic curves and dynamics

Yu. Ilyashenko and V. Kaloshin — Bifurcation of planar and spatial polycycles: Arnold’s program and its development

V. Kharlamov, S. Orevkov and E. Shustin — Singularity which has no $M$smoothing

Boris Khesin and Alexei Rosly — Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces

A. Khovanskii — Newton polyhedra, a new formula for mixed volume, product of roots of a system of equations

William Langford and Kaijun Zhan — Interactions of AndronovHopf and BogdanovTakens bifurcations

E. Mukhin and A. Varchenko — Solutions of the qKZB equation in tensor products of finite dimensional modules over the elliptic quantum group $E_{\tau ,\eta }sl_2$

S. Novikov — Schrodinger operators on graphs and symplectic geometry

Michael Rudnev and Stephen Wiggins — On the dominant Fourier modes in the series associated with separatrix splitting for an apriori stable, three degreeoffreedom Hamiltonian system

V. Vassiliev — Homology of $i$connected graphs and invariants of knots, plane arrangements, etc.

V. Vladimirov and K. Ilin — On Arnold’s variational principles in fluid mechanics

Sergei Yakovenko — On functions and curves defined by ordinary differential equations

Y. Yomdin — Global finiteness properties of analytic families and algebra of their Taylor coefficients


Reviews

The articles appearing in this volume illustrate diversity in mathematics stemming from a rather limited number of fundamental problems originally in Arnold's research ... communicates esteem and warm affection for Arnold by his colleagues and former students.
CMS Notes 
Provides an excellent record of the mathematical content of the conference, as well as an introduction to the mind of a great mathematician ... these articles provide a wonderful autobiographical retrospective together with a wealth of mathematical ideas. They are a joy to read.
SIAM Review


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This volume presents articles originating from invited talks at an exciting international conference held at The Fields Institute in Toronto celebrating the sixtieth birthday of the renowned mathematician, Vladimir Arnold. Experts from the world over—including several from “Arnold's school”—gave illuminating talks and lively poster sessions. The presentations focussed on Arnold's main areas of interest: singularity theory, the theory of curves, symmetry groups, dynamical systems, mechanics, and related areas of mathematics.
The book begins with notes of three lectures by V. Arnold given in the framework of the Institute's Distinguished Lecturer program. The topics of the lectures are:
 From Hilbert's Superposition Problem to Dynamical Systems
 Symplectization, Complexification, and Mathematical Trinities
 Topological Problems in Wave Propagation Theory and Topological Economy Principle in Algebraic Geometry
Arnold's three articles include insightful comments on Russian and Western mathematics and science. Complementing the first is Jurgen Moser's “Recollections”, concerning some of the history of KAM theory.
Titles in this series are copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Graduate students and research mathematicians interested in singularity theory, the theory of curves, symmetry groups, dynamical systems, and mechanics; physicists and engineers interested in astronomy, mechanics, and bifurcation theory.

Chapters

V. Arnold — From Hilbert’s superposition problem to dynamical systems

Jürgen Moser — Recollections

V. Arnold — Symplectization, complexification and mathematical trinities

V. Arnold — Topological problems in wave propagation theory and topological economy principle in algebraic geometry

Mark Alber, Gregory Luther, Jerrold Marsden and Jonathan Robbins — Geometry and control of threewave interactions

Edward Bierstone and Pierre Milman — Standard basis along a Samuel stratum, and implicit differentiation

James Damon — A global weighted version of Bezout’s theorem

Alexander Degtyarev and Viatcheslav Kharlamov — Real Enriques surfaces without real points and EnriquesEinsteinHitchin 4manifolds

W. Ebeling and S. GuseinZade — On the index of a vector field at an isolated singularity

David Ebin and Gerard Misiołek — The exponential map on $\mathcal {D}^s_\mu $

Michael Freedman — Zeldovich’s neutron star and the prediction of magnetic froth

Kenji Fukaya and Kaoru Ono — Arnold conjecture and GromovWitten invariant for general symplectic manifolds

Andrei Gabrielov — Multiplicity of a zero of an analytic function on a trajectory of a vector field

Alexander Givental — Singularity theory and symplectic topology

V. Goryunov and S. Lando — On enumeration of meromorphic functions on the line

H. Hofer and E. Zehnder — Pseudoholomorphic curves and dynamics

Yu. Ilyashenko and V. Kaloshin — Bifurcation of planar and spatial polycycles: Arnold’s program and its development

V. Kharlamov, S. Orevkov and E. Shustin — Singularity which has no $M$smoothing

Boris Khesin and Alexei Rosly — Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces

A. Khovanskii — Newton polyhedra, a new formula for mixed volume, product of roots of a system of equations

William Langford and Kaijun Zhan — Interactions of AndronovHopf and BogdanovTakens bifurcations

E. Mukhin and A. Varchenko — Solutions of the qKZB equation in tensor products of finite dimensional modules over the elliptic quantum group $E_{\tau ,\eta }sl_2$

S. Novikov — Schrodinger operators on graphs and symplectic geometry

Michael Rudnev and Stephen Wiggins — On the dominant Fourier modes in the series associated with separatrix splitting for an apriori stable, three degreeoffreedom Hamiltonian system

V. Vassiliev — Homology of $i$connected graphs and invariants of knots, plane arrangements, etc.

V. Vladimirov and K. Ilin — On Arnold’s variational principles in fluid mechanics

Sergei Yakovenko — On functions and curves defined by ordinary differential equations

Y. Yomdin — Global finiteness properties of analytic families and algebra of their Taylor coefficients

The articles appearing in this volume illustrate diversity in mathematics stemming from a rather limited number of fundamental problems originally in Arnold's research ... communicates esteem and warm affection for Arnold by his colleagues and former students.
CMS Notes 
Provides an excellent record of the mathematical content of the conference, as well as an introduction to the mind of a great mathematician ... these articles provide a wonderful autobiographical retrospective together with a wealth of mathematical ideas. They are a joy to read.
SIAM Review