UC Berkeley

Reeb flows are a rich, ubiquitous class of dynamical systems arising in symplectic geometry, which include billiard systems, many-body orbital systems, geodesic flows and many Hamiltonian flows. Convexity hypotheses play an important, albeit mysterious, role in the study of these flows. In this thesis, we discuss several new results in the study of convexity in symplectic geometry and Reeb dynamics.

In Chapter 1, we resolve a longstanding open problem on the intrinsic characterization of Reeb flows arising from Hamiltonian flows on the convex boundaries. Namely, we prove that dynamically convex Reeb flows, introduced by Hofer-Wysocki-Zehnder, are not all convex. Our proof uses a novel relation between Riemannian geometry and Reeb dynamics, and uses constructions of Abbondandolo-Bramham-Hryniewicz-Salomao.

In Chapter 2, we describe a powerful new framework for computationally modelling Reeb dynamics on the boundaries of convex polytopes. We apply this framework to provide new evidence and examples relating to the Viterbo conjecture, a major open problem in Reeb dynamics and quantitative symplectic geometry.

In Chapter 3, we study convex toric domains and toric surfaces. A longstanding conjecture in toric geometry states that the Gromov width is monotonic under inclusion of moment polytopes of closed toric varieties. We use methods from toric geometry and ECH to prove a generalization of this conjecture in dimension 4.