Symplectic Invariants and Hamiltonian Dynamics

Part 1 Introduction: symplectic vectro spaces; symplectic diffeomorphisms and Hamiltonian vector fields in (R2n, omega-0); Hamiltonian vector fields and symplectic manifolds; periodic orbits on energy surfaces; existence of a periodic orbit on a convex energy surface; the problem of symplectic embeddings; symplectic classification of positive definite quadratic forms; the orbit structure near an equilibrium, Birkhoff normal form. Part 2 Symplectic capacities: definition and application to embeddings; rigidity of symplectic diffeomorphisms. Part 3 Existence of a capacity: definition of the capacity c-0; the minimax idea; the anlytical setting; the existence of a critical point; examples and illustrations. Part 4 Existence of closed characteristics: periodic solutions in energy surfaces; the characterisrtic line bundle of a hypersurface; hypersurfaces of contact type, the Weinstein conjecture; "classical" Hamiltonian systems; the torus and Herman's non-closing Lemma. Part 5 Compactly supported symplectic mappings: a special metric "d" for a group "D"; the action spectrum of a Hamiltonian map; a "universal" variational principle; a continuous section of the action spectrum bundle; an inequality between the displacement energy; comparison of the metric "d" on "D" with the "C0-metric"; fixed points and geodesics on "D". Part 6 The Arnold conjecture, Floer homology: the Arnold conjecture on symplectic fixed points; the model case of the torus; gradient-like flows on compact spaces; elliptic methods and symplectic fixed points; Floer's approach to Morse theory for the action functional; symplectic homology; generating functions of symplectic mappings in R2n; action-angle coordinates, the theorem of Arnold and Jost; embeddings of "H1/2(S1)" and smoothness of the action; the Cauchy-Riemann operator on the sphere; ellpitic estimates near the boundary and an application; the generalized Carleman similarity principle; the Brouwer degree; continuity property of the Alexander-Spanier cohomology.