Symplectic rigidities from quantitative methods

Since Gromov’s celebrated non-squeezing theorem in the 1980s, symplectic geometry has revealed fundamental rigidities that distinguish it from Riemannian geometry. In this talk, I will present a series of results that establish rigidity phenomena for central objects in the field: Hamiltonian dynamics, Lagrangian submanifolds, and Liouville domains. Specifically, these rigidities are manifested in: the dynamical behavior of fixed points of Hamiltonian diffeomorphisms; obstructions to Lagrangian embeddings; and the large-scale geometry of the space of Liouville domains. Importantly, the techniques we employ represent some of the most advanced quantitative methods in modern symplectic geometry, including refined Floer homology, the shape invariant, and the Banach–Mazur distance. These results help stimulate and advance a rapidly evolving subject called quantitative symplectic geometry.