Contents LECTURE 1 . INVARIANT S AN D DISCRIMINANT S O F PLAN E CURVE S 1 Preface t o Lectur e 1 3 CHAPTER 1 . Plan e Curve s 5 §1. Th e thre e basi c invariant s 5 §2. Propertie s o f th e basi c invariant s 8 §3. Computatio n o f basi c invariant s 1 0 §4. Extrema l curve s an d tree-lik e curve s 1 2 §5. Th e numerolog y 1 6 §6. Cobordism s 2 0 §7. Lon g curve s 2 2 CHAPTER 2 . Legendria n Knot s 2 5 §8. Fro m plan e curve s t o Legendria n knot s 2 5 §9. Th e spac e o f Legendria n curve s 2 7 §10. Th e basi c invarian t J + 2 9 §11. Th e Legendria n linkin g number s 3 1 §12. Calculatio n o f linkin g number s 3 2 LECTURE 2 . SYMPLECTI C AN D CONTAC T TOPOLOG Y O F CAUSTIC S AN D WAV E FRONTS, AN D STUR M THEOR Y 3 7 CHAPTER 3 . Singularitie s o f Caustic s an d Stur m Theor y 3 9 §13. Th e Lagrangia n collaps e an d th e las t geometrica l theore m o f Jacobi 3 9 §14. Th e fou r cus p theore m 4 1 §15. Stur m theor y an d Mors e theor y 4 3 §16. Proo f o f th e fou r cusp s theore m 4 4 CHAPTER 4 . Singularitie s o f Wave Front s an d th e Tenni s Bal l Theore m 4 7 §17. Th e wav e fronts reversa l 4 7 §18. Th e fron t reversa l fou r cusp s theore m 5 0 §19. Th e proof o f th e existenc e o f fou r cusp s o n a fron t 5 2 §20. Th e tenni s bal l theore m 5 3 References 5 9
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