Proposal: DMS-9803554 Principal Investigator: Thomas Parker
This project involves analytic aspects of the theory of pseudo- holomorphic curves. The aim is to develop effective methods for computing Gromov-Witten invariants of symplectic manifolds and enumerative invariants of algebraic manifolds. The main thrust is a continuing project with E. Ionel on a gluing formula for Gromov invariants under the operation of `symplectic connect sum'. The idea is to collapse the connect sum to a singular manifold, keeping track of the holomorphic curves along the way. Parker and Ionel have already obtained a gluing formula in special cases; these imply the well-known recent formula of Caporaso-Harris. A general gluing formula should be an effective new tool for computing Gromov and enumerative invariants. A second part of the project seeks to for answer the following question. Suppose C is curve that is J-holomorphic for some non-generic J. If one perturbs J to a nearby generic J', how many J'-holomorphic curves are there close to C? Several well-known problems in enumerative geometry, some solved, some unsolved, reduce to this problem. Parker and Ionel have an approach based on the `Taubes obstruction bundle' . The last part of the project suggests using modified Gromov invariants to obtain invariants that count curves which exist only for special classes of almost complex structures.
One of the most basic problems in mathematics is to determine the solutions of a system of polynomial equations, and an important first step toward that goal is to determine the NUMBER of solutions. There is an explicit formula for the number of simultaneous solutions of a set of n polynomials in n variables. One can then ask for the number of solutions for n polynomials in n-1 variables. In this case there is a free parameter, so the locus of solutions will be a union of curves. How many? This question has been systematically studied for 100 years, but only a few special cases were solved. Then, around 1990, it was realized that these problems can be translated into symplectic geometry, and then tackeled using the powerful machinery of mathematical gauge theory. (Gauge theory, originally part of physics, has been the focus of many very fruitful interactions between mathematicians and physicists over the past twenty years; it includes Yang-Mills and Seiberg-Witten theory, and String theory). This `Gromov invariant' approach led quickly to formulas answering some of the original enumerative problems, and there are clear indications that there are more to be discovered. This project is aimed toward further developing the symplectic gauge theory in order to produce additional general formulas, and to meld these formulas into a coherent theory.
