Gromov Invariants and Enumerative Invariants

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Abstract Proposal: DMS 9802459 Principal Investigators: Tomasz Mrowka and Eleny-Nicoleta Ionel The first project is a continuation of joint work with T. Parker on gluing formulas for the Gromov invariants of symplectic connect sums. We have been working on a degeneration formula that describes what happens to holomorphic curves as one `pinches the neck' and explains how to compute the Gromov invariants of the symplectic connect sum from the limiting curves. This degeneration formula has several applications that we would like to explore. First, one can construct new `exotic' symplectic manifolds by cut-and-paste techniques and compute their Gromov invariants by the degeneration formula. This gives a practical technique for probing how much more complicated the symplectic classification of manifolds is compared to the smooth classification. Secondly, the degeneration formula leads to recursive formulas for the Gromov invariants; these could provide answers to several old unsolved problems in enumerative algebraic geometry. The final project suggests a method for extending Gromov-type invariants to `nongeneric' situations. This would provide more refined information about the symplectic manifold. Most of the problems in enumerative algebraic geometry are more than a hundred years old. The questions are easy to ask, but the progress in solving them using classical methods has been quite slow. Recently, the same kind of questions arised in two dimensional topological quantuum field theories from high energy physics. Inspired by these theories, new methods lead in the past couple of years to amazing progress in the field. The proposal explores two new ways of approaching these old problems that would further clarify the structure of the two dimensional topological quantuum field theories.