Topological Invariants of Plane Curves and Caustics (University Lecture Series)

Topological Invariants of Plane Curves and Caustics (University Lecture Series)

Paperback

Special OrderSpecial Order item not currently available. We'll try and order for you.

Description

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.

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.

Product Details

  • ISBN13: 9780821803080
  • Format: Paperback
  • Number Of Pages: 60
  • ID: 9780821803080
  • ISBN10: 0821803085

Delivery Information

  • Saver Delivery: Yes
  • 1st Class Delivery: Yes
  • Courier Delivery: Yes
  • Store Delivery: Yes

Prices are for internet purchases only. Prices and availability in WHSmith Stores may vary significantly

Close