Lagrangian and Legendrian varieties and stability of their projections

The study of singular Lagrangian and Legendrian varieties was initiated about twenty-five years ago by Arnold when he was investigating singularities in the variational problem of obstacle bypassing [1]. The first examples of such varieties, open swallowtails, were related to the discriminants of the non-crystallographic Coxeter groups [8, 14]. Incorporating these examples into a general context, Givental [8] introduced the notion of stability of Lagrangian and Legendrian varieties with respect to perturbations of symplectic structure and Lagrangian or, respectively, Legendrian projection only, keeping the diffeomorphic type of the variety fixed. Later, in [13], it was shown that this stability notion has an explicit geometrical meaning in terms of generating families, versal deformations of function singularities and inducing mappings. The interest in theory of singular Lagrangian and Legendrian varieties has been growing recently due to its possible applications to Frobenius structures, D-modules and in other areas.