Lagrangian Floer Theory and Quantum Invariants of Symplectic Manifolds

辛几何是经典力学基础的数学研究。 尽管它的经典起源，这个研究领域最近经历了一个爆炸式的进展集中在研究量子不变量定义使用几何对象称为全纯曲线。 这些量子不变量不仅出现在几何分析中，也出现在研究三维空间和四维时空的低维拓扑中，以及高能物理学的某些模型中。 研究人员将研究有关这些量子不变量的基本问题，包括它们在称为手术和对称性约简的数学运算下的行为。 应用将在拓扑和物理感兴趣。研究者还将继续开展面向中学几何教师的推广活动。具体而言，该项目研究了量子不变量（如福谷范畴和量子K理论）在翻转、爆破和辛约化等操作下的行为。 在该项目的第一部分，研究者将构建与辛流形上的手术相关的福谷类别的生成器，这些手术随着辛结构的变化而自然产生。 每一个都产生了一个福谷范畴中的对象集合，并且在理论上是量子上同调中的一个因子。 这里的主要技术涉及的Abouzaid-Ganatra生成标准和限制版本的辛场论的福谷类别开发使用稳定因子。 在该项目的第二部分，研究者和合作者将构建一个同伦版本的Kirwan的映射从拟映射福谷类别的哈密顿群行动的福谷类别的辛商。 这项研究将有应用的辛共轭拉格朗日函数的磁盘潜力。 在第三个合作项目中，研究人员将研究K理论Gromov-Witten不变量在跨壁下的行为，并旨在证明在与辛商变化相关的crepant双有理变换的情况下，势能一般不变。

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.