Vladimir I. Arnold

Ordinary Differential Equations.

Basic Concepts.- Basic Theorems.- Linear Systems.- Proofs of the Main Theorems.- Differential Equations on Manifolds.

Singularities of caustics and wave fronts

1 Symplectic geometry.- 1.1 Symplectic manifolds.- 1.2 Submanifolds of symplectic manifolds.- 1.3 Lagrangian manifolds, fibrations, mappings, and singularities.- 2 Applications of the theory of Lagrangian singularities.- 2.1 Oscillatory integrals.- 2.2 Lattice points.- 2.3 Perestroikas of caustics.- 2.4 Perestroikas of optical caustics.- 2.5 Shock wave singularities and perestroikas of Maxwell sets.- 3 Contact geometry.- 3.1 Wave fronts.- 3.2 Singularities of fronts.- 3.3 Perestroikas of fronts.- 4 Convolution of invariants, and period maps.- 4.1 Vector fields tangent to fronts.- 4.2 Linearised convolution of invariants.- 4.3 Period maps.- 4.4 Intersection forms of period maps.- 4.5 Poisson structures.- 4.6 Principal period maps.- 5 Lagrangian and Legendre topology.- 5.1 Lagrangian and Legendre cobordism.- 5.2 Lagrangian and Legendre characteristic classes.- 5.3 Topology of complex discriminants.- 5.4 Functions with mild singularities.- 5.5 Global properties of singularities.- 5.6 Topology of Lagrangian inclusions.- 6 Projections of surfaces, and singularities of apparent contours.- 6.1 Singularities of projections from a surface to the plane.- 6.2 Singularities of projections of complete intersections.- 6.3 Geometry of bifurcation diagrams.- 7 Obstacle problem.- 7.1 Asymptotic rays in symplectic geometry.- 7.2 Contact geometry of pairs of hypersurfaces.- 7.3 Unfurled swallowtails.- 7.4 Symplectic triads.- 7.5 Contact triads.- 7.6 Hypericosahedral singularity.- 7.7 Normal forms of singularities in the obstacle problem.- 8 Transformation of waves defined by hyperbolic variational principles.- 8.1 Hyperbolic systems and their light hypersurfaces.- 8.2 Singularities of light hypersurfaces of variational systems.- 8.3 Contact normal forms of singularities of quadratic cones.- 8.4 Singularities of ray systems and wave fronts at nonstrict hyperbolic points.

The cohomology ring of the colored braid group

The cohomology ring is obtained for the space of ordered sets of n different points of a plane.

Topological invariants of plane curves and caustics

Lecture 1: Invariants and discriminants of plane curves Plane curves Legendrian knots Lecture 2: Symplectic and contact topology of caustics and wave fronts, and Sturm theory Singularities of caustics and Sturm theory Singularities of wave fronts and the tennis ball theorem.

Sur la topologie des écoulements stationnaires des fluides parfaits

On considere les ecoulements stationnaires d’un fluide parfait, incompressible et non visqueux, dans un domaine borne D. On suppose que le vecteur vitesse n’est pas partout colineaire au vecteur rotation. On demontre alors que le domaine D est divise, par certaines surfaces et courbes, en un nombre fini de « cellules » ouvertes, fibrees en tores ou en cylindres engendres par des lignes de courant. Les lignes de courant sont fermees sur les cylindres, fermees ou denses sur les tores.

On some topological invariants of algebraic functions

There are some interesting connections between the theory of algebraic functions and Artin’s theory of braids. For instance, the space G n of polynomials of degree n not having multiple roots is the space K(π ,1) for the group B(n) of braids with n strings:

Dynamics of complexity of intersections

The topological complexity of the intersection of a submanifold, moved by a dynamical system, with a given submanifold of the phase space, can increase with time. It is proved that the Morse and Betti numbers of the transversal intersections “generically” grow at most exponentially, while for some special infinitely smooth systems the topological complexity of the intersections can become larger than any given function of time (for a growing sequence of integer time moments).

Huygens and Barrow, Newton and Hooke

model complex natural shapes, such as coastlines and mountains. Chaos and fractal geometry go handin-hand. Both fields deal with intricately shaped objects, and chaotic processes often produce fractal patterns. This fascinating book should be of interest to scientists, mathematicians, programmers and artists interested in the field of chaos and fractal geometry. This would also be a good textbook for students and teachers. For those readers new to these topics, the book starts gradually by introducing concepts of self-similarity, fractal snowflakes and fractal dimensions. Other topics discussed later in the book include pink noise, Brownian motion, Cantor sets, multifractals, iteration, bifurcation maps, the Mandelbrot set, Fibonacci numbers, percolation and cellular automata. Ample diagrams indicate the graphical results of the formulas and mathematical theories. My favorite sections discuss fractals in number theory-an area that may be new to readers familiar only with the standard Julia and Mandelbrot sets so richly illustrated on computer-art posters and T-shirts. The best section of all discusses the enigmatic Morse-Thue number sequence and its amazing fractal properties. This book has something for beginners and for advanced students of fractals and chaos.

Remarks on singularities of finite codimension in complex dynamical systems

There is an interesting connection between the theory of algebraic functions and Artin’s braid theory: the space G n of nth-degree polynomials not having multiple roots is the space K(π ,1) for the group B(n) of braids on n strands:

Topological invariants of algebraic functions II

In an earlier paper of the same name (2] the connection between algebraic functions and braids was used in order to calculate the cohomologies of braid groups. In this work the cohomologies of braid groups are used to confirm the non-representability of algebraic functions of a certain number of variables as superpositions of algebraic functions of a smaller number of variables.