Symplectic Geometry and Hamiltonian Dynamics

The method of pseudoholomorphic curves was introduced by Gromov [Gro85]. The equation of Gromov’s pseudoholomorphic curves is conformally invariant. Floer broke the conformal symmetry and combined Hamiltonian dynamics with the analysis of pseudoholomorphic curves by perturbing the equation by Hamiltonian vector fields [Flo89]. This amalgamation lies at the heart of modern development of symplectic topology via the Floer theory through the quantitative study of the natural filtered structure of Floer complex and Fukaya’s categorification of Floer homology [Fuk94]. A careful study of this filtration accompanies the development of a rather interesting calculus of Hamiltonian diffeomorphisms and Hamiltonians, which we call the Hamiltonian calculus.