Contact Floer Homology

点击翻译按钮获取中文摘要

AbstractAward: DMS-0072658Principal Investigator: Alexandre B. GiventalIn Symplectic Topology, the properties in question are usuallyformalized in the language of Floer homology theory and Gromov-Witten invariants. The proposed project is to extend andintertwine these two formalisms in the context of Contact Topology.The generalization exhibits natural relationships with the stringtheory and, moreover, shows how the structures of classical andquantum field theory emerge form the stringy properties ofcontact Floer homology. The project is conceived as a joint workof the P.I. with Ya. Eliashberg (Stanford University) andH. Hofer (Courant Institute) and is a part of the larger project"Symplectic Field Theory" started by the latter two authorsseveral years ago.The project originates from Symplectic and Contact Topologylaunched in the beginning of the twentieth century by A. Hurwitzand H. Poincare and studying profound geometrical properties ofclassical mechanical systems. The mechanical notions of positionand momentum coordinates lead to the geometric notion of asymplectic structure, in which the Hamiltonian formulation ofmechanics becomes extremely natural. Contact structures are theodd-dimensional counterparts to symplectic structures. Inaddition to their appearances in the study of subspaces ofsymplectic manifolds, contact structures have been of particularinterest in dimension three, where geometric ideas areilluminating some difficult questions in topology