Techniques in Symplectic Geometry and Applications

辛几何技术及其应用

• 批准号: 2345030
• 负责人: Guangbo Xu
• 金额: $ 24.97万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2345030&HistoricalAwards=false
• 关键词: Techniques; Symplectic; Geometry; Applications; Techniques; Symplectic; Geometry; Applications

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.

几何和拓扑的中心目标是了解数学空间。数学空间可以通过本地和全局信息来表征，例如它的弯曲程度（本地）以及它的连接方式（全局）。作为一个子场，符号几何学研究一种特殊的空间，称为象征歧管。这些空间在本地都是相同的，但可以具有多种全球形状。研究的主要工具分为两类：代数和分析。代数工具是包装全球信息和计算的框架，而分析工具是用于求解微分方程和构建代数框架的技术。该研究项目旨在改善现有技术并开发新的工具来解决长期存在的困难问题，重点是分析方法。同时，该项目将丰富德克萨斯A＆M大学的K-12外展（数学圈）计划，在德克萨斯州和中美洲建立研究人员社区，并增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，使用虚拟周期和绝热极限的技术，该研究旨在建立Witten测量的线性Sigma模型与非线性Sigma模型之间的关系。与许多代数方法相反，该方法是分析性的，可以扩展到尚不可用的代数方法的开弦情况。其次，使用新方法，PI计划在不适用传统方法的情况下证明紧凑性会导致紧凑性。这包括Atiyah-lloer猜想的紧凑性，伪酚晶状体曲线的紧凑性，用于逆转手术，以清洁拉格朗日的次级次级次数，以及具有某些单型lagrangian lagrangian lagrangian lagrangian lagrangian边界条件。第三，该项目旨在开发一种新的计数方法来定义格罗莫夫（Gromov-witten）的不变性和浮动同源性，而不是整数。 PI还将培训博士生并为本科生提供建议。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评论标准来评估值得支持的。