Foundations of the theory of J-holomorphic curves

J-全纯曲线理论基础

• 批准号: 1308669
• 负责人: Dusa McDuff
• 金额: $ 25.05万
• 依托单位: Barnard College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308669&HistoricalAwards=false
• 关键词: Foundations; theory; holomorphic; curves

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.

这个项目试图重写辛几何中的一些基本基础结构。辛不变量(通常是各种曲线的计数)是使用摄动方程组的解来定义的，而寻找相干摄动所涉及的问题还没有完全解决。这项提议的一个主要部分是继续麦克达夫？S与魏赫海姆的项目，对解决这个问题的传统方法进行重新设计(通过有限维降维)。到目前为止，它们已经解决了主要的拓扑问题，但关于各向同性以及如何处理粘合造成的光滑性的问题尚未详细解决，即使是在闭合曲线的最简单情况下也是如此。一旦解决了这个最基本的问题，许多重要的结构变体也需要重新设计，例如，当存在圆对称时会发生什么。麦克达夫还提议研究各种更多的几何问题。例如，四辛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？辛结构有多“可弯曲”：有没有一个具有孤立不动点的圆作用，它的矩映射定义了圆上的(奇异)纤维？辛几何已经成为理解流形结构的一个非常重要的工具。这些空间，就像物理学的时空一样，局部看起来像熟悉的欧几里得空间，但可能有全球扭曲。就像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，人们应该给它们额外的结构，而不是普通的普通流形：度量(这是一种测量距离的方式)，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)，并与一种神秘的镜像对称有关，该对称最先由物理学家提出，最近给出了各种数学解释。在热衷于使用这些来自物理学的令人兴奋的新想法的过程中，数学界一直在使用各种基础工具，但没有对它们进行足够严格的设置。这已经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据，人们只能停留在猜测上。这个项目在很大程度上是由补救这一问题的愿望推动的。它建议对基本结构(从拓扑和分析)进行详细的重新设计，使人们能够以一致的方式计算对象。一旦完成，辛几何学家将能够以一种确定的方式前进，以测试他们论点的正确性。