Contact Floer Homology

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract Award: DMS-0072658 Principal Investigator: Alexandre B. Givental In Symplectic Topology, the properties in question are usually formalized in the language of Floer homology theory and Gromov- Witten invariants. The proposed project is to extend and intertwine these two formalisms in the context of Contact Topology. The generalization exhibits natural relationships with the string theory and, moreover, shows how the structures of classical and quantum field theory emerge form the stringy properties of contact Floer homology. The project is conceived as a joint work of the P.I. with Ya. Eliashberg (Stanford University) and H. Hofer (Courant Institute) and is a part of the larger project "Symplectic Field Theory" started by the latter two authors several years ago. The project originates from Symplectic and Contact Topology launched in the beginning of the twentieth century by A. Hurwitz and H. Poincare and studying profound geometrical properties of classical mechanical systems. The mechanical notions of position and momentum coordinates lead to the geometric notion of a symplectic structure, in which the Hamiltonian formulation of mechanics becomes extremely natural. Contact structures are the odd-dimensional counterparts to symplectic structures. In addition to their appearances in the study of subspaces of symplectic manifolds, contact structures have been of particular interest in dimension three, where geometric ideas are illuminating some difficult questions in topology