The primary goal of the proposed research is to investigate the phenomenon of existence of infinitely many periodic orbits for a variety of Hamiltonian dynamical systems and to understand the nature of the systems admitting finitely many periodic orbits. The proposal comprises several interconnected projects addressing these questions for certain classes of Hamiltonian diffeomorphisms and also for specific Hamiltonian systems such as magnetic flows. The PI will tackle these problems by employing methods from symplectic topology including Floer and quantum homological techniques, holomorphic curves, spectral invariants, Ljusternik-Schnirelman theory, as well as methods from differential geometry such as h-principles. The techniques utilized by the PI also have applications beyond the question of existence of infinitely many periodic orbits. In particular, the projects concerning the Poincaré recurrence, the Reeb flows and the coisotropic symplectic topology draw heavily on her recent works concerning periodic orbits. These projects have applications to measure-preserving and classical dynamical systems, and to some embedding problems in symplectic topology.
Hamiltonian systems constitute a broad class of physical systems where dissipative forces can be disregarded. For example, the planetary motion in celestial mechanics, the flow of an incompressible ideal fluid, and the motion of a charged particle in a magnetic field are usually treated as Hamiltonian systems. One general, but not universal, feature of such systems is that they tend to have numerous periodic orbits. Corresponding to the cyclic motion, this is the simplest dynamical phenomenon after equilibrium, and an investigation of periodic orbits of a system is crucial in understanding its global behavior. To give but a few applications, the knowledge of periodic orbits is crucial in astronomy, fluid dynamics (e.g., statistics of turbulent flow) or can be used to understand stability of solutions for large times. In all but the simplest cases, establishing existence of periodic orbits often requires advanced and powerful mathematical tools. For a broad class of Hamiltonian systems, the number of periodic orbits is known to be infinite and this is thought to be the case for many, but not all, Hamiltonian systems. The proposal focuses on the problem of existence of infinitely many periodic orbits for Hamiltonian dynamical systems in a variety of settings and on applications of the techniques used by the PI to attack this problem to some other related questions. The projects in the last part of the proposal concern a certain class of spaces which arise, for instance, in the study of Hamiltonian systems with symmetries. The proposed work is related to and has potential applications in mathematical physics, and geometric and quantum mechanics.
