We show that there is a 1-parameter family of monotone Lagrangian tori in the cotangent bundle of the 3-sphere with the following property: every compact orientable monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the tori in the family. The proof involves studying a version of the wrapped Fukaya category of the cotangent bundle which includes monotone Lagrangians. Time permitting, we may also discuss an extension to other cotangent bundles. This is joint work with Mohammed Abouzaid.
In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold.