Sorbonne Université CNRS Paris Diderot

Séminaire Géométries et Topologie

The flexibility of caustics

Daniel Alvarez-Gavela - IAS

jeudi 17 janvier 2019 à 11:00

Starting with the work on immersion theory of Hirsch and Smale, many flavors of the problem of simplifying the singularities of smooth mappings have been studied in the mathematics literature. The over arching philosophy is that such a problem, a priori geometric in nature, nevertheless often reduces to the underlying homotopy theoretic problem (to which the tools of algebraic topology can then be applied). When this reduction is possible one says (following Gromov) that the problem abides by the h-principle. The flexibility of a problem refers to the extent to which an h-principle holds. In my PhD thesis it was established that an h-principle also holds for the problem of simplifying the singularities of Lagrangian and Legendrian fronts (also known as caustics). This result builds on previous work by Entov who adapted Eliashberg’s technique of surgery of singularities to the setting of caustics. Furthermore, in recent joint work with Eliashberg, Nadler and Starkston we prove a certain h-principle “without homotopical conditions” for the simplification of caustics. In this talk we will review the flexibility of singularities of smooth mappings and present our current understanding on the flexibility of caustics. Time permitting, we will discuss applications of the latter to symplectic and contact topology.

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