VLADIMIR I. ARNOLD—Collected Works Dynamics, Combinatorics, and Invariants of Knots, Curves, and Wave Fronts 1992–1995

This volume 6 of the Collected Works comprises 27 papers by V.I.Arnold, one of the most outstanding mathematicians of all times, written in 1991 to 1995. During this period Arnold's interests covered Vassiliev’s theory of invariants and knots, invariants and bifurcations of plane curves,   combinatorics of Bernoulli, Euler and Springer numbers,   geometry of wave fronts, the Berry phase and quantum Hall effect.

The articles include a list of problems in dynamical systems, a discussion of the problem of (in)solvability of equations, papers on symplectic geometry of caustics and contact  geometry of wave fronts, comments on problems of A.D.Sakharov, as well as a rather unusual paper on projective topology. The interested reader will certainly enjoy Arnold’s 1994 paper on mathematical problems in physics with the opening by-now famous phrase “Mathematics is the name for those domains of theoretical physics that are temporarily unfashionable.”

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. The volume also includes translations of two interviews given by Arnold to the French and Spanish media. One can see how worried he was about the fate of Russian and world mathematics and science in general.

1 Bernoulli–Euler updown numbers associated with function singularities, their combinatorics and arithmetics.- 2 Congruences for Euler, Bernoulli and Springer numbers of Coxeter groups.- 3 The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups.- 4 Springer numbers and Morsification spaces.- 5 Polyintegrable flows.- 6 Bounds for Milnor numbers of intersections in holomorphic dynamical systems.- 7 Some remarks on symplectic monodromy of Milnor fibrations.- 8 Topological properties of Legendre projections in contact geometry of wave fronts [On topological properties of Legendre projections in contact geometry of wave fronts].- 9 Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques [On topological properties of Lagrangian projections in symplectic geometry of caustics].- 10 Plane curves, their invariants, perestroikas and classifications (with an appendix by F. Aicardi).- 11 Invariants and perestroikas of plane fronts.- 12 The Vassiliev theory of discriminants and knots.- 13 The geometry of spherical curves and the algebra of quaternions.- 14 Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect.- 15 Problems on singularities and dynamical systems.- 16 Sur quelques problèmes de la théorie des systèmes dynamiques [On some problems in the theory of dynamical systems].- 17 Mathematical problems in classical physics.- 18 Problèmes résolubles et problèmes irrésolubles analytiques et géométriques [Solvable and unsolvable analytic and geometric problems].- 19 Projective topology.- 20 Questions à V.I. Arnold (an interview with M. Audin and P. Iglésias) [Questions to V.I. Arnold].- 21 En todo matemático hay un ángel y un demonio (an interview with Marimar Jiménez) [In every mathematician, there is an angel and a devil].- 22 Will Russian mathematics survive?.- 23 Will mathematics survive? Report on the Zurich Congress.- 24 Why study mathematics? What mathematicians think about it.- 25 Preface to the Russian translation of the book by M.F. Atiyah “The Geometry and Physics of Knots”.- 26 A comment on one of A.D. Sakharov’s “Amateur Problems”.- 27 Comments on two of A.D. Sakharov’s “Amateur Problems”.- Acknowledgements.

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.