Caustics of Lagrangian homotopy spheres with stably trivial Gauss map
The h-principle for the simplification of caustics (i.e. Lagrangian tangencies) reduces a geometric problem to a homotopical problem. In this talk I will explain the solution to this homotopical problem in the case of spheres. More precisely, I will show that the stably trivial elements of the nth homotopy group of the Lagrangian Grassmannian Un/OnUn/On, which lies in the metastable range, admit representatives with only fold type tangencies. By the h-principle, it follows that if DD is a Lagrangian distribution defined along a Lagrangian homotopy sphere LL, then there exists a Hamiltonian isotopy which simplifies the tangencies between LL and DD to consist only of folds if and only if DD is stably trivial. I will give two applications of this result, one to the arborealization program and another to the study of nearby Lagrangian homotopy spheres. Joint work with David Darrow (in the form of an undergraduate research project).