Gromov-Witten invariants of symplectic 4-manifolds

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This project aims to study the Gromov-Witten invariants of symplectic 4-manifolds, with special attention to Kahler surfaces. It consists of three projects. The PI (with Thomas H. Parker) showed that the GW theory of Kahler surfaces (with a smooth canonical divisor) can be reduced to the local GW theory of spin curves. The PI also showed that the local GW theory can be reduced to the dimension zero relative local GW theory. The goal of the first two projects (with Thomas H. Parker) is to completely describe the dimension zero relative local GW theory of spin curves. This will give a complete calculation of the GW invariants of Kahler surfaces (with a smooth canonical divisor). The goal of the third project is to extend aspects of a structure theorem for the GW invariants of Kahler surfaces to general symplectic 4-manifolds. This extension will provide a suitable notion of the number of the components of the canonical class which is a very important symplectic invariant for the topology of symplectic 4-manifolds. The extension will also remove the smoothness condition of canonical divisor in the previous projects.In recent years, some long-standing mathematics problems have been solved using the newly-developed theory of GW invariants. This theory provides an innovative method for counting certain fundamental geometric objects called holomorphic curves. The resulting counts, and the systematic methods for calculating them, have spurred significant advances in two classical fields of mathematics: symplectic geometry and enumerative geometry. The GW invariants also play an important role bridging mathematics and theoretical physics. This project seeks to develop new geometric techniques for calculating GW invariants using methods in partial differential equations, topology and algebraic geometry. The techniques and ideas developed in this project should be very useful in other contexts.