UC Berkeley

This thesis explores quantitative invariants in symplectic geometry. The first chapter is based on joint work with Julian Chaidez [Invent. Math. 229 (2022), 243–301], where we construct the first examples of dynamically convex domains that are not symplectomorphic to any convex domain. A key component of this work is a quantitative convexity criterion involving the Ruelle invariant.In the second chapter, based on the paper [arXiv:2206.07847], we establish a sharp dynamical characterization of convex domains in four-dimensional Euclidean space that admit a symplectic embedding into a given cylinder. More specifically, we demonstrate that the cylindrical capacity of a convex domain matches the minimum symplectic area of a disk-like global surface of section for the natural Reeb flow on the boundary of the domain. This contributes to progress toward the strong Viterbo conjecture regarding the equivalence of symplectic capacities on convex domains. The third chapter comprises a joint paper with Michael Hutchings [arXiv:2110.02463]. We show a quantitative smooth closing lemma for area preserving surface diffeomorphisms. Our proof relies on spectral invariants arising from periodic Floer homology. We also establish a new Weyl law for these spectral invariants.