Techniques in Symplectic Geometry and Applications

几何学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信息和全局信息来表征，例如它是如何弯曲的(局部的)和如何连接的(全局的)。作为一个子域，辛几何研究一种特殊的空间，称为辛流形。这些空间在本地都是相同的，但可以有不同的全局形状。研究的主要工具分为两类：代数和分析。代数工具是包装全局信息和进行计算的框架，而分析工具是解微分方程和构建代数框架的技术。这一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，该项目将丰富德克萨斯农工大学的K-12扩展(数学圈)项目，在德克萨斯州和中南美洲建立一个研究人员社区，并加强数学教职员工和学生之间的联系。在技术层面，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Witten规范线性Sigma模型与非线性Sigma模型之间的关系。与许多代数方法不同的是，该方法是解析的，允许将其扩展到代数方法尚不可用的开弦情况。其次，PI计划使用一种新方法，在传统方法不适用的情况下证明紧凑性。这包括Atiyah-Floer猜想中的一个紧性结果，关于干净相交的拉格朗日子流形的逆运算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。PI还将培训博士生并为本科生提供建议。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.