Techniques in Symplectic Geometry and Applications

几何和拓扑学的中心目标是理解数学空间。数学空间可以由局部和全局信息来表征，例如它是如何弯曲的（局部）和它是如何连接的（全局）。作为一个子领域，辛几何研究一种特殊的空间，称为辛流形。这些空间在局部上都是相同的，但可以有各种各样的全局形状。主要的研究工具分为两类：代数和分析。代数工具是包裹全局信息和进行计算的框架，而分析工具是求解微分方程和构造代数框架的技术。该研究项目旨在改进现有技术并开发新工具来解决长期存在的难题，重点是分析方法。与此同时，该项目将丰富德克萨斯州农工大学的K-12外展（数学圈）计划，在德克萨斯州和中南美洲建立一个研究人员社区，并加强数学教师和学生之间的联系。在技术层面上，该项目涉及三个主题。首先，利用虚循环和绝热极限技术，建立了维滕规范线性σ模型与非线性σ模型之间的关系。与许多代数方法不同，该方法是解析的，允许扩展到代数方法尚未可用的开弦情况。其次，使用一种新的方法，PI计划证明在传统方法不适用的情况下的紧致性结果。其中包括Atiyah-Floer猜想中的一个紧性结果、关于完全相交拉格朗日子流形的逆手术的伪全纯曲线的紧性以及具有某些奇异拉格朗日边界条件的伪全纯曲线的紧性。第三，该项目旨在开发一种新的计数方法来定义整数上的Gromov-Witten不变量和Floer同调。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The central goal in geometry and topology is to understand mathematical spaces. A mathematical space can be characterized by both local and global information, such as how curved it is (local) and how connected it is (global). As a subfield, symplectic geometry studies a special kind of space called a symplectic manifold. These spaces are all the same locally but can have a variety of global shapes. The main tools of study fall into two categories: algebraic and analytic. The algebraic tools are the frameworks of wrapping up global information and making calculations, while the analytic tools are techniques for solving differential equations and for constructing the algebraic frameworks. This research project aims to improve existing techniques and develop new tools to solve longstanding difficult questions, with emphasis on analytic methods. At the same time, the project will enrich the K-12 outreach (Math Circle) program at Texas A&M University, build a community of researchers in Texas and Central South America, and enhance connections among mathematics faculty and students. On the technical level, the project involves three topics. First, using the technique of virtual cycle and adiabatic limit, the research aims to establish the relationship between Witten's gauged linear sigma model and the nonlinear sigma model. In contrast to many algebraic approaches, the method is analytic, allowing one to extend to the open-string case where algebraic methods are not yet available. Second, using a new method, the PI plans to prove compactness results in situations where traditional approaches do not apply. This includes a compactness result in Atiyah-Floer conjecture, compactness for pseudoholomorphic curves regarding reversed surgery for cleanly-intersecting Lagrangian submanifolds, and compactness for pseudoholomorphic curves with certain singular Lagrangian boundary conditions. Third, the project aims to develop a new counting method to define Gromov-Witten invariants and Floer homology over integers. The PI will also train PhD students and advise undergraduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.