Gromov Invariants and Enumerative Invariants

摘要 提案：DMS 9802459主要研究者：Tomasz Mrowka和Eleny-Nicoleta Ionel 第一个项目是与T.帕克谈粘合 辛连通和的Gromov不变量公式。 我们有 一直在研究一个退化公式， 全纯曲线作为一个'捏脖子'，并解释如何计算 辛连通和的Gromov不变量 曲线. 这个退化公式有几个应用， 喜欢探索。 首先，可以构造新的“奇异”辛 流形的剪切和粘贴技术，并计算其Gromov不变量 退化公式。 这提供了一种实用的探测技术 流形的辛分类是多么复杂 与平滑分类相比。 其次，退化公式 导致Gromov不变量的递归公式;这些可以提供 解答了枚举代数几何中几个老的未解决的问题。 最后的项目提出了一种方法，用于扩展Gromov类型的不变量， “非一般性”情况。 这将提供更精确的信息， 辛流形 枚举代数几何中的大多数问题都不止一个 一百岁了。 这些问题很容易问，但在 用经典方法解决它们是相当缓慢的。 近日 在二维拓扑量子场中也出现了同样的问题 高能物理的理论。 受这些理论的启发，新方法 在过去的几年里，在这个领域取得了惊人的进展。 的 该提案探索了两种处理这些老问题的新方法， 进一步阐明了二维拓扑量子的结构 场理论

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.