Invariant Distances on Legendrian Spaces

I will begin by motivating the study of invariant distances on spaces of Legendrians. I will then discuss two main results:
(a) the construction of a new unbounded invariant distance on the universal cover of many Legendrian isotopy classes ;
(b) the discreteness of any invariant distance on Legendrian isotopy classes.

In particular, we will see that (a) arises from contact rigidity, whereas (b) follows from contact flexibility.