Foundations of the theory of J-holomorphic curves

这个项目旨在重新研究辛几何中的一些基本的基础结构。辛不变量（通常是各种曲线的计数）是用扰动方程组的解来定义的，而寻找相干扰动所涉及的问题还没有完全解决。这项建议的一个主要部分是继续麦克达夫？的项目与Wehrheim，改造传统的方法（通过有限维减少），这个问题。到目前为止，他们已经解决了主要的拓扑问题，但关于各向同性和如何处理由胶合引起的光滑度不足的问题还没有详细解决，即使是在最简单的封闭曲线的情况下。一旦解决了这个最基本的问题，这个结构的许多重要变体也需要重新加工，例如当存在圆对称时会发生什么。麦克达夫还建议研究各种更多的几何问题。例如，在一个辛四流形中，一条非常非一般的全纯曲线的存在如何影响流形中的其他曲线？辛结构如何“弯曲”：是否存在一个具有孤立不动点的圆作用，其矩图定义了圆上的（奇异）纤维化？辛几何已经发展成为理解流形结构的一个非常重要的工具。这些空间就像物理学中的时空一样，局部看起来像我们熟悉的欧几里得空间，但可能具有全局扭曲。正如最近关于三维流形的庞加莱猜想的解决方案一样，事实证明，与其看普通的香草流形，我们应该给它们额外的结构：度量（这是测量距离的一种方式）或者可能是复杂或辛结构。后两者涉及进行二维测量（辛结构测量面积），并与物理学家首次提出的神秘镜像对称有关，最近给出了各种数学解释。在渴望与这些令人兴奋的物理学新思想合作的过程中，数学界一直在使用各种基础工具，但没有足够严格地设置它们。这已成为一个严重的问题。数学是通过直觉和想象力发展起来的，但是，由于结果不能通过实验来验证，没有仔细和有效的证明，人们只能进行猜测。这个项目在很大程度上是出于补救这一点的愿望。它提出了一个详细的改造基本结构（从拓扑和分析），允许一个计数对象在一个一致的方式。一旦完成，辛几何学家将能够以一种确定的方式来测试他们的论点的正确性。

This project seeks to rework some basic foundational constructions in symplectic geometry. Symplectic invariants (usually counts of curves of various kinds) are defined using solutions to perturbed systems of equations, and the issues involved in finding coherent perturbations are not yet fully worked out. A main part of this proposal is to continue McDuff?s project with Wehrheim that reworks the traditional approach (via finite dimensional reduction) to this problem. So far, they have resolved the main topological issues, but questions concerning isotropy and how to deal with the lack of smoothness caused by gluing have yet to be worked out in detail, even in the simplest case of closed curves. Once this most basic problem has been dealt with, many important variants of the construction also need reworking, for example what happens when there is a circle symmetry. McDuff also proposes to study a variety of more geometric questions. For example, how does the existence of a very non generic holomorphic curve in a symplectic four manifold affect the other curves in the manifold? How "bendable" are symplectic structures: is there a circle action with isolated fixed pointswhose moment map defines a (singular) fibration over a circle?Symplectic geometry has grown into a very important tool for understanding the structure of manifolds. These are spaces which, like the space-time of physics, locally look like familiar Euclidean space but might have global twisting. As in the recent solution to the Poincare conjecture concerning three dimensional manifolds, it has turned out that instead of looking at plain vanilla manifolds, one should give them extra structure: a metric (which is a way of measuring distance) or perhaps a complex or symplectic structure. The latter two involve making two-dimensional measurements (symplectic structures measure area), and are related by a mysterious mirror symmetry that was first suggested by physicists and recently given various mathematical interpretations. In its eagerness to work with these exciting new ideas from physics, the mathematical community has been using various foundational tools without setting them up with sufficient rigor. This has become a serious problem. Mathematics develops via intuition and imagination, but, because results cannot be verified by experimentation, without careful and valid proofs one is left with mere speculation. This project is largely motivated by the desire to remedy this. It proposes a detailed reworking of basic constructions (from topology and analysis) that allow one to count objects in a consistent way. Once completed, symplectic geometers will be able to move ahead with a sure way to test the correctness of their arguments.