Collaborative Research: Symplectic Dynamics in Higher Dimensions and Floer Theory

Hamiltonian systems describe many classes of physical processes in which dissipative forces can be neglected. For example, planetary motion in celestial mechanics, the flow of an incompressible ideal fluid, and the motion of charged particles in electromagnetic fields are usually modeled as Hamiltonian dynamical systems. Periodic orbits, corresponding to recurring motions, are fundamentally important features of dynamical systems. Likewise, topological entropy is a key invariant of a dynamical system that measures its complexity and long-term unpredictability, with origins in physics and information theory. This project will develop new methods and tools for studying periodic orbits and topological entropy in Hamiltonian dynamical systems by incorporating ideas from topological data analysis, which extracts key structural information from complex data. This research, in turn, has the potential to advance topological data analysis and contribute to applications such as image and pattern recognition, machine learning and artificial intelligence. The project integrates research with the education and training of young scientists, contributing to higher education and the dissemination of knowledge both within the field and to the broader scientific community. At a more technical level, the main theme of the project is the interaction between quantitative features of filtered Floer theory, on the one hand, and Hamiltonian dynamics – in particular, topological entropy and periodic orbits – on the other. The project will investigate the topological entropy of compactly supported Hamiltonian diffeomorphisms and certain Reeb flows from the perspective of Floer theory through barcode entropy, which was introduced by the PI and her collaborators several years ago. More generally, this research will explore connections between barcode growth in filtered Floer and symplectic homology and the dynamical complexity of Hamiltonian diffeomorphisms and flows. In a similar vein, building on her recent work, including the proof of Ekeland’s multiplicity conjecture and a variant of the Hofer-Zehnder conjecture for Reeb flows, the PI will further develop the use of the dimension function of filtered Floer homology and barcode structures to study periodic orbits of Reeb and Finsler geodesic flows in higher dimensions. Achieving these goals will require the development of new techniques with broader applicability and will foster interactions with areas beyond symplectic geometry and dynamical systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.