We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equi
pped with Hofers metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincar{e} recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.
We describe a method, based on hard contact topology, of showing the existence of semi-infinite trajectories of contact Hamiltonian flows which start on one Legendrian submanifold and asymptotically converge to another Legendrian submanifold. We disc
uss a mathematical model of non-equilibrium thermodynamics where such trajectories play a role of relaxation processes, and illustrate our results in the case of the Glauber dynamics for the mean field Ising model.
We relate the machinery of persistence modules to the Legendrian contact homology theory and to Poisson bracket invariants, and use it to show the existence of connecting trajectories of contact and symplectic Hamiltonian flows.
We show that for a special class of geometric quantizations with small quantum errors, the quantum classical correspondence gives rise to an asymptotic projective representation of the group of Hamiltonian diffeomorphisms. As an application, we get a
n obstruction to Hamiltonian actions of finitely presented groups.
The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions with geometry
and analysis. In particular, we present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, we discuss topological function theory which provides a new insight on oscillation of functions. The material should be accessible to readers with a basic background in algebraic and differential topology.
We introduce new invariants associated to collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants invo
lves various flavors of Floer theory. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
This paper is a fusion of a survey and a research article. We focus on certain rigidity phenomena in function spaces associated to a symplectic manifold. Our starting point is a lower bound obtained in an earlier paper with Zapolsky for the uniform n
orm of the Poisson bracket of a pair of functions in terms of symplectic quasi-states. After a short review of the theory of symplectic quasi-states, we extend this bound to the case of iterated Poisson brackets. A new technical ingredient is the use of symplectic integrators. In addition, we discuss some applications to symplectic approximation theory and present a number of open problems.
