Techniques in Symplectic Geometry and Applications

几何和拓扑学的中心目标是理解数学空间。数学空间可以通过局部和全局信息来表征，例如它的弯曲程度（局部）和连接程度（全局）。辛几何作为一个子领域，研究一种叫做辛流形的特殊空间。这些空间在局部都是相同的，但可以有各种各样的全局形状。主要的学习工具分为两类：代数和解析。代数工具是概括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的工具来解决长期存在的难题，重点是分析方法。与此同时，该项目将丰富德州农工大学的K-12外展（数学圈）项目，在德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，利用虚拟循环和绝热极限技术，建立了Witten量规线性模型与非线性模型之间的关系。与许多代数方法相比，该方法是解析的，允许扩展到代数方法尚不可用的开弦情况。其次，使用一种新方法，PI计划在传统方法不适用的情况下证明紧凑性结果。这包括Atiyah-Floer猜想的紧性结果，干净相交拉格朗日子流形关于反向手术的假全纯曲线的紧性，以及具有某些奇异拉格朗日边界条件的假全纯曲线的紧性。第三，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。PI还将培养博士生和指导本科生。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.