Hofer–Zehnder capacity of disk cotangent bundles: from billiards to string topology

The Hofer–Zehnder capacity is a numerical symplectic invariant defined via Hamiltonian dynamics and closely related to the existence of periodic orbits. In this talk, I survey known and still missing results on the Hofer–Zehnder capacity of subsets of (co-)tangent bundles, with an emphasis on explicit computations for disk bundles. Lower bounds are typically obtained via Riemannian billiards, while upper bounds rely on Floer theoretic methods. More precisely, the Floer theory (symplectic homology) of cotangent bundles recovers the homology of the free loop space, and we will use operations from string topology to derive the desired upper bounds. So far, all explicit computations are restricted to highly degenerate settings, such as constant curvature metrics or, more generally, symmetric spaces. I will report on work in progress aimed at extending these techniques to less degenerate Riemannian metrics.