Lagrangian Floer Theory and Quantum Invariants of Symplectic Manifolds

辛几何是对经典力学基础的数学研究。尽管它起源于经典，但这一研究领域最近经历了以研究量子不变量为中心的爆炸性进展，这些不变量是使用称为全纯曲线的几何对象定义的。这些量子不变量不仅出现在几何分析中，而且出现在研究三维空间和四维时空的低维拓扑中，以及高能物理中的某些模型中。研究人员将研究有关这些量子不变量的基本问题，包括它们在被称为手术和对称性约化的数学运算下的行为。应用程序将在拓扑学和物理学方面引起人们的兴趣。研究人员还将继续他的中学几何教师的外展活动。具体地说，该项目研究量子不变量，如Fukaya范畴和量子K-理论在翻转、爆破和辛约化等操作下的行为。在项目的第一部分，研究人员将在辛流形上构造与手术相关的Fukaya类别的生成器，这些生成器随着辛结构的变化而自然产生。其中的每一个都产生了Fukaya范畴中的对象的集合，并推测是量子上同调的一个因素。这里的主要技术包括Abouzaid-Ganatra生成准则和使用稳定因子发展的Fukaya范畴的限制版辛场理论。在项目的第二部分，研究人员和合作者将构造从哈密顿群作用的拟映射Fukaya范畴到辛商的Fukaya范畴的Kirwan映射的同伦版本。这一研究将应用于辛商中拉格朗日函数的盘势。在第三个合作项目中，研究人员将研究K-理论Gromov-Witten不变量在跨墙情况下的行为，目的是证明在与辛商变化相关的初等双态变换的情况下，势一般是不变的。

Symplectic geometry is the mathematical study of the foundations of classical mechanics. Despite its classical origins, this research area has recently experienced an explosion of progress centered on the study of quantum invariants defined using geometric objects known as holomorphic curves. These quantum invariants have appeared not only in geometric analysis but also in low-dimensional topology, which studies three-dimensional space as well as four-dimensional space-time, and in certain models in high-energy physics. The investigator will study fundamental questions about these quantum invariants, including their behavior under mathematical operations known as surgery and symmetry reduction. Applications will be of interest in topology and physics. The investigator will also continue his outreach activities for middle-school geometry teachers.Specifically, the project studies the behavior of quantum invariants such as the Fukaya category and quantum K-theory under operations such as flips, blow-ups, and symplectic reduction. In the first part of the project, the investigator will construct generators of the Fukaya category associated to surgeries on the symplectic manifold that arise naturally as the symplectic structure is varied. Each of these gives rise to a collection of objects in the Fukaya category, and conjecturally a factor in the quantum cohomology. The main technique here involves the Abouzaid-Ganatra generation criterion and a restricted version of symplectic field theory for the Fukaya category developed using stabilizing divisors. In the second part of the project, the investigator and collaborators will construct a homotopy version of Kirwan's map from the quasimap Fukaya category of a Hamiltonian group action to the Fukaya category of the symplectic quotient. This study will have applications to the disk potentials of Lagrangians in symplectic quotients. In a third collaborative project, the investigator will study the behavior of K-theoretic Gromov-Witten invariants under wall-crossing and aims to prove that the potentials are unchanged, generically, in the case of crepant birational transformations associated to variation of symplectic quotient.