Quantitative symplectic geometry and dynamics

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This project will facilitate development of new tools to study mathematical questions related to how physical systems evolve in time. A state of the system corresponds to a point in an even dimensional phase space, and the system evolves along an odd dimensional energy level in the phase space. A particular focus will be placed on understanding the existence and properties of periodic orbits, which describe behavior that repeats in time, especially in the case when the phase space is four dimensional and the energy level is three dimensional. The geometry of the phase space will also be studied, developing methods to determine when one dynamical system is equivalent to another one by a change of coordinates. Various research projects on these topics will provide research training for graduate students in the latest techniques in symplectic geometry and related areas of mathematics. The PI will also engage in multiple outreach activities. Specific projects include the following. Filtrations on embedded contact homology will be studied with the goal of proving in full generality that every contact form on a closed three-manifold has either two or infinitely many simple Reeb orbits. Knot filtered embedded contact homology will be studied with applications to symplectic cobordisms between transverse knots. Symplectic invariants of open domains arising from barcodes in equivariant symplectic homology will be used to classify some open domains up to symplectomorphism. Closing lemmas will be extended to more general vector fields than Reeb vector fields in three dimensions. Universal quantitative invariants will be developed with applications to symplectic embedding problems, constraints on Lagrangian submanifolds, refinements of the Arnold chord conjecture, elementary spectral invariants of contact manifolds, and comparisons between symplectic capacities. 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.