Topological Invariants of Plane Curves and Caustics (University Lecture Series) - V. I. Arnold, V. I. Arnol'd (1994)
ISBN 0821803085
Subject Curves on surfaces; Hamiltonian systems; Knot theory
Publisher American Mathematical Society
Publication Date September 1994
Format Paperback (244 x 170 mm)
Language e
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.
Personal Details
Collection Status In Collection
Index 204
Read It Yes
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Product Details
LoC Classification QA643 .A76 1994
Dewey 516.3/63
Cover Price $17.00
No. of Pages 60
"Dean Jacqueline B. Lewis memorial lectures."