Loose Legendrian embeddings in high dimensional contact manifolds I

In this two-part talk, we will discuss the classification of loose Legendrians in high dimensional contact manifolds. The full classification of Legendrian embeddings up to isotopy is likely an intractable problem. Loose Legendrians are special, in that they are determined up to isotopy by their topological (rather than geometric) invariants. They are also very common, in that any Legendrian can be made loose by altering it in a small neighborhood of a point. To construct isotopies between loose Legendrians we use tools coming from the world of h-principles. In particular we use theorems about directed embeddings, holonomic approximation, and wrinkled embeddings to prove an isotopy theorem for Legendrians with unfurled swallowtail singularities, and finally show how the loose condition allows us to surger away those singularities. No prior knowledge of h-principles or contact geometry will be assumed, other than basics such as Darboux's theorem.