Hamiltonian dynamics, Floer theory and symplectic topology

오용근 (University of Wisconsin)
2009. 11. 26.

In this lecture, I will convey subtle interplay between dynamics of Hamiltonian flows and La-grangian intersection theory via the analytic theory of Floer homology in symplectic geometry. I will explain how Floer homology theory (`closed string version') extracts some Morse theoretic invariants, so called `spectral invariants' of Hamiltonian flows which in turn leads to Entov-Polterovich's construction of partial symplectic quasi-states which detect intersection property of certain coisotropic objects. We then relate this study to the study of intersection property of Lagrangian fibers of toric manifolds and its Landau-Ginzburg-type potential carried out by Fukaya, Ohta, Ono and myself via Lagrangian Floer theory (`open string version'). If time permits, I will mention some implication of this study on the simpleness question of the area-preserving homeomorphism group of the two-sphere (and of the two-disc) in dynamical systems.