Gromov Invariants and Enumerative Invariants

主要研究人员：Tomasz Mrowka和Eleny-Nicoleta Ionel。第一个项目是与T. Parker合作的关于格罗莫夫不变量的粘接公式的延续。我们一直在研究一个退化公式，它描述了当一个人“掐脖子”时全纯曲线的情况，并解释了如何从极限曲线中计算辛连接和的Gromov不变量。这个退化公式有几个我们想要探索的应用。首先，我们可以通过剪切粘贴技术构造新的“奇异的”辛流形，并通过退化公式计算它们的Gromov不变量。这为探索流形的辛分类比光滑分类复杂多少提供了一种实用的技术。其次，由退化公式导出Gromov不变量的递推公式；这些可以为枚举代数几何中几个未解决的老问题提供答案。最后的项目提出了一种将gromov型不变量扩展到“非泛型”情况的方法。这将提供关于辛流形的更精确的信息。列举代数几何中的大多数问题都有一百多年的历史了。这些问题很容易提出，但用经典方法解决这些问题的进展却相当缓慢。近年来，在高能物理的二维拓扑量子场论中也出现了类似的问题。受这些理论的启发，在过去的几年里，新方法在该领域取得了惊人的进展。该提案探索了两种解决这些老问题的新方法，这将进一步阐明二维拓扑量子场论的结构。

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.