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.
Jacobi stated in Vorlesungen Ueber Dynamics that the curve formed by the conjugate points of a generic point on a generic ellipsoid has exactly four cusps.
As far as I know it had never been proved. However, the number four is really crucial here: is at least a minoration of the number of cusps. This fact is a brother of the four cusps theorem (saying that a plane convex closed curve has at least four extrema of the curvature).
Both the four conjugate points theorem and the four vertex theorem are in fact two particular cases of the same result in symplectic and contact topology. The infinitesimal version of this result is a theorem in Sturm theory, due to Hurwitz, Kellogg and Tabachnikov and saying that a Fourier series starting from higher order harmonics has at least as much zeroes, as the first harmonic presented in the series with a nonzero coefficient.
The simplest case of this Sturm type theorem is just the Morse inequality for the circle. The Sturm type theorems may thus be considered as the higher derivatives extensions of the Morse theory .
The symplectic topology extension of Morse theory replaces the functions by their multivalued versions- the Lagrange and Legendre manifolds. The corresponding extension of the Morse inequalities (called "the Arnold conjecture on the Lagrangian intersections and symplectic fixed points") goes back to 1965 and has been partially proved by Conley, Zehnder, Chaperon, Floer, Givental, Ono and many others (in different particular cases). Perhaps, the Floer homology is the best-known byproduct of this development.
The geometrical problems generalizing the four cusps property suggest that a similar "multivalued functions" version of the Sturm theory might be possible (the simplest example being the "tennis ball theorem": a simple closed curve, dividing the sphere into two parts of equal areas, has at least four inflection points).
In higher dimension this approach suggests (even at the infinitesimal level) a highly nontrivial version of what might be called "higherdimensional Sturm theory". It contains also the extensions of the Moebius theorem, saying that a projective plane curve, close to a projective line, has at least three inflection points.
The higherdimensional versions of this Moebius theorem are the minorations of the numbers of the flattening points and of the cusps of Gauss mappings of space curves. These results may also be formulated as (apparently new?) theorems on the Chebyshev systems of functions .
Steklov Mathematical Institute, Moscow and Universite Paris-Dauphine, Paris