Much progress has been made in symplectic geometry, and more recently contact geometry, since the discovery of holomorphic curves in symplectic manifolds. Sullivan plans to continue calculating and applying two sets of invariants based on these curves: the contact homology of Legendrian submanifolds in contact manifolds and the periodic Floer homology of Riemann surface diffeomorphisms. The former is a special case of symplectic field theory. Although symplectic field theory is still not rigorously well-defined, Sullivan and others have completed the foundational analysis for their version of contact homology. Sullivan will extend this alternative version of contact homology to other manifolds. The ultimate goal is to develop a complete obstruction of Legendrian isotopy classes. The latter invariant is conjectured to agree with Seiberg-Witten-Floer homology. This project will develop the foundations of this theory, as well as broaden the existing set of computations. The investigator also hopes to work on applications of the periodic Floer homology computations, addressing the problems of existence and classification of symplectic structures on 4-manifolds.
Symplectic and contact geometry explain the physics of certain dynamical systems, such as the orbits of planets around the sun, the spin of a top, or the motion of a charged particle in a magnetic field. Such systems obey the least action principal, which among other things can mean that their total energy or momentum must be conserved. Many symplectic geometers study holomorphic curves, a reinterpretation of the least action principal, which has linked together several independently well-developed mathematical fields such as complex analysis and differential topology. The study of holomorphic curves has led to other ``physical" results, such as a generalization of Heisenberg's uncertainty principal. More recently, these curves are thought to appear in other areas of theoretical physics, like string theory.
