本书描述了平面曲线拓扑研究中的最新进展。平面曲线理论比纽结理论更为丰富，后者可以视为平面曲线理论的交换形式。这个研究建立在奇点理论的基础上：无穷维的曲线空间通过判别超曲面而细分为由同型的泛曲线组成的各个部分。区分这些型的不变量则由在这些超曲面的交叉处的跃变定义。Arnold描绘了对于焦散曲线几何，以及辛几何和切触几何中的波前的应用。这些应用将初等平面几何的四顶点定理扩展为关于波前反演所必需的最小尖点数的估值，以及关于凸曲面共轭点的Jacobi几何大定理的推广。这些估值翻开了辛几何和切触几何的新篇章：拉格朗日和勒让德的衰退理论，它给出了对于特征函数线性组合振荡的Sturm理论的一个独特且意义深远的高维推广。

LECTURE 1. INVARIANTS AND DISCRIMINANTS OF PLANE CURVES
Preface to Lecture 1
CHAPTER 1. Plane Curves
1. The three basic invariants
2. Properties of the basic invariants
3. Computation of basic invariants
4. Extremal curves and tree-like curves
5. The numerology
6. Cobordisms
7. Long curves
CHAPTER 2. Legendrian Knots
8. From plane curves to Legendrian knots
9. The space of Legendrian curves
10. The basic invariant J+
11. The Legendrian linking numbers
12. Calculation of linking numbers
LECTURE 2. SYMPLECTIC AND CONTACT TOPOLOGY OF CAUSTICS AND WAVE
FRONTS, AND STURM THEORY
CHAPTER 3. Singularities of Caustics and Sturm Theory
13. The Lagrangian collapse and the last geometrical theorem of
Jacobi
14. The four cusp theorem
15. Sturm theory and Morse theory
16. Proof of the four cusps theorem
CHAPTER 4. Singularities of Wave Fronts and the Tennis Ball Theorem
17. The wave fronts reversal
18. The front reversal four cusps theorem
19. The proof of the existence of four cusps on a front
20. The tennis ball theorem
References