This booklet provides a motivating survey on the invariants and discriminants of plane curves and the symplectic and contact topology of caustics and wave fronts, and Sturm theory. It is illustrated by several instructive figures and tables displaying classifications in special cases. Proofs only appear in rare cases, but detailed references are given where they can be found.
The background for the exhibition is based on the singularity approach to the topological classification of objects, frequently associated with the author’s name, but going back to Poincaré according to the author’s own statement. In the first chapter the simplest results about the application of this approach to plane curves are surveyed. In the next chapter Legendrian knots are studied from this point of view. For details concerning both chapters see the author’s publications “Plane curves, their invariants, perestroikas, and classifications” in [Singularities and bifurcations, Adv. Sov. Math., 21, 33-91 (1994)] and “Invariants and perestroikas of plane fronts” Tr. Mat. Inst. Steklova (1994). Chapter 3 describes the symplectic topology version of the classical four vertices theorem. Several theorems and conjectures in symplectic topology are formulated which generalize this theorem. For details, the author’s report [Sur les propriétés topologiques des projections Lagrangiennes en géométrie symplectique des caustiques (Preprint No. 9320 CEREMADE, Cahiers Mathématiques de la Decision, Université Paris-Dauphine, 1-9, 14/6/93)] should be consulted in addition to the references mentioned above. In the last chapter, the same program is carried out for the case of contact geometry. For this, the author’s paper [St. Petersbg. Math. J. 6, No. 3, 439-452 (1995); translation from Algebra Anal. 6, No. 3, 1-16 (1994; Zbl 0827.58018)] may serve as a reference.