The simplification of caustics

We present a full $h$-principle (relative, parametric, $C^0$-close) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. The main ingredients in the proof are a refinement of the holonomic approximation lemma and the construction of a local wrinkling model for Lagrangian and Legendrian submanifolds. We give sample applications of our $h$-principle, including an Igusa-type theorem which states that higher singularities are unnecessary for the homotopy theoretic study of the space of Legendrian knots in the standard contact Euclidean 3-space. This last result can be understood as a generalization of the Reidemeister theorem for families of Legendrian knots parametrized by a space of arbitrarily high dimension.