Caltech/USC/UCLA Joint Topology Seminar

USC KAP414

The envelope of light rays reflected or refracted by a curved surface is called a caustic and generically has semi-cubical cusp singularities at isolated points. In generic families depending on one real parameter the cusps of the caustic will be born or die in pairs. At such an instance of birth/death the caustic traces a swallowtail singularity. This bifurcation is also known as the Legendrian Reidemeister I move. For families depending on more parameters or for front projections of higher dimensional Legendrians (or Lagrangians), the generic caustic singularities become more complicated. As the dimension increases the situation quickly becomes intractable and there is no explicit understanding or classification possible in the general case. In this lecture we will present a full h-principle (C^0-close, relative, parametric) for the simplification of higher singularities of caustics into superpostions of the familiar semi-cubical cusp. As a corollary we will obtain a Reidemeister type theorem for families of Legendrian knots in the standard contact Euclidean 3-space which depend on an arbitrary number of parameters. We will also explain the relation to Nadler's program for the arborealization of singularities of Lagrangian skeleta and give several other potential applications of the h-principle to symplectic and contact topology.