Holomorphic curves into infinite dimensional almost K\"ahler manifolds and Hamiltonian dynamics

We study analysis over infinite dimensional manifolds consisted by sequences of almost K\"ahler manifolds. In particular we develop moduli theory of pseudo holomorphic curves into the spaces with high symmetry. As applications, we study Hamiltonian dynamics over the infinite dimensional manifolds, and induce some dynamical properties of Hamiltonian diffeomorphisms on such spaces.