This book describes recent progress in the topological study of plane curves. The theory of plane curves is much richer than knot theory, which may be considered the commutative version of the theory of plane curves. This study is based on singularity theory: the infinite-dimensional space of curves is subdivided by the discriminant hypersurfaces into parts consisting of generic curves of the same type. The invariants distinguishing the types are defined by their jumps at the crossings of these hypersurfaces. Arnold describes applications to the geometry of caustics and of wavefronts in symplectic and contact geometry. These applications extend the classical four-vertex theorem of elementary plane geometry to estimates on the minimal number of cusps necessary for the reversion of a wavefront and to generalizations of the last geometrical theorem of Jacobi on conjugated points on convex surfaces. These estimates open a new chapter in symplectic and contact topology: the theory of Lagrangian and Legendrian collapses, providing an unusual and far-reaching higher-dimensional extension of Sturm theory of the oscillations of linear combinations of eigenfunctions.

Readership

Graduate students and researchers in mathematics.

Reviews

"This book provides an attractive introduction to one of the most exciting and active fields of topology."

-- Mathematical Reviews

"An excellent introduction to the area of low-dimensional geometry in which a mathematician of any level ... would be able to find a source of interesting problems to solve ... the author opens up a new subject and encourages the reader to make his or her own contributions ... extremely readable."

-- Proceedings of the Edinburgh Mathematical Society

Table of Contents

• 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