Topology of Plane Curves, Wave Fronts, Legendrian Knots, Sturm Theory and Flattenings of Projective Curves V.I. Arnold The theory of smooth (possibly selfintersecting) curves in the plane is parallel to knot theory (the last being a simplified, commutative version of the theory of plane curves). Strangely, the theory of plane curves invariants has not been developed until 1992, when the simplest three basic invariants (the "strangeness" St, counting the triple points crossings, and the "tangencies crossings counting" invariants J+ and J-) have been introduced, motivated by the symplectic and contact topology problems. In a sense these invariants are similar to the Vassiliev invariants of knot theory. Recently Viro has discovered their relation to the real algebraic geometry, then Schumakovich and Polyak have found the expressions of these new invariants in the spirit of the statistical physics and have related them to the Vassiliev knot invariants, while Lin and Wang have described their relations to the quantum field theory and to the Kontsevich and Bar Natan works on integral formulas for the knot invariants. In spite of the fast progress of the last two years in this domain (including the Aicardi's and Polyak's extension of the theory to the wave fronts, that is to the curves with cusps), the original problems, triggering all the theory, remain unsolved : these problems belong to the symplectic and contact topology of the Lagrangian and Legendrian mappings.
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