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Topological Invariants of Plane Curves and Caustics : University Lecture Series - V. I. Arnold

Topological Invariants of Plane Curves and Caustics

University Lecture Series

Paperback Published: January 1995
ISBN: 9780821803080
Number Of Pages: 60

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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.

Invariants and discriminants of plane curves
Plane curves Legendrian knots
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
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780821803080
ISBN-10: 0821803085
Series: University Lecture Series : Book 5
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 60
Published: January 1995
Publisher: American Mathematical Society
Country of Publication: US
Dimensions (cm): 25.4 x 17.78  x 0.64
Weight (kg): 0.14

Earn 68 Qantas Points
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