Abstract: 

Symplectic geometry is the geometry of phase spaces in classical mechanics, but it is notoriously unintuitive: more rigid than volume geometry, yet far more flexible than Riemannian geometry. Symplectic capacities are numerical invariants that help make this hidden rigidity visible by measuring symplectic size.

In this talk, I will first give a brief survey of symplectic capacities and the main ideas behind their computation. I will then focus on disk cotangent bundles, where a close connection with the Riemannian or Finsler geometry of the base emerges. I will illustrate this connection through several examples, including projective spaces, ellipsoids, and Katok metrics. If time permits, I will also indicate how techniques from billiard dynamics, pseudoholomorphic curves, and symplectic homology enter the picture.