Abstract Proposal: DMS 9802460 Principal Investigators: Gang Tian and Gang Liu The purpose of this project is to continue the investigation of Liu and Tian on the applications of their construction of relative virtual moduli cycles in Floer homology and the dynamics of Hamiltonian systems on symplectic manifolds. By using this construction, the principal investigators were able to extend Floer homology from the semi-positive case to all closed symplectic manifolds and consequently to solve the non-degenerate Arnold conjecture completely. By using certain refinements of this construction, they established a general relationship between non-vanishing of certain GW-invariants and the existence of closed orbits of Hamiltonian systems. As one of the applications of this, they solved a stabilized version of Weinstein conjecture. Base on these results, the principal investigators propose further investigations on the Arnold conjecture for degenerate case and some other unsettled cases of the Weinstein conjecture. Hamiltonian equations arise from classical mechanics, celestial mechanics and many other physical systems as fundamental equations governing the motions in such systems. The dynamics of Hamiltonian systems describes the evolution of the "classical" world. One of the important steps to understand the dynamics of Hamiltonian systems is to understand their simplest dynamic behavior, the periodical orbits. The basic questions here, known as the Arnold conjecture and the Weinstein conjecture, are about the existence and the number of closed orbits of Hamiltonian systems. Both of these conjectures have been considered as main guiding problems in the subject of symplectic topology. The methods developed by the investigators of this project to solve the non-degenerate Arnold conjecture and stabilized Weinstein conjecture have opened the door for investigating the Arnold conjecture for the degenerate case and the Weinstein conjectur e for general symplectic manifolds.
