Augmentations and Legendrians at the IAS

Lagrangian cobordisms between Legendrian submanifolds arise in Relative Symplectic Field Theory. In recent years, there has been much progress on answering qualitative questions such as: For a fixed pair of Legendrians, does there exist a Lagrangian cobordism? I will address two quantitative questions about Lagrangian cobordisms: For a fixed pair of Legendrians, what is the minimal "length" of a Lagrangian cobordism? What is the relative Gromov width of a Lagrangian cobordism? Regarding length, I will give examples of pairs of Legendrians where Lagrangian cobordisms are flexible in that the non-cylindrical region can be arbitrarily short; I will also give examples of other pairs of Legendrians where Lagrangian cobordisms are rigid in that there is a positive lower bound to their length. For the second quantitative measure, I will give some calculations and estimates of the relative Gromov width of particular Lagrangian cobordisms. This is joint work with Joshua M. Sabloff.