Interactions Between Contact Geometry, Floer Theory and Low-Dimensional Topology

该项目由拓扑学计划和促进竞争研究的既定计划（EPSCoR）共同资助。它集中在两个数学领域的交叉点上，即接触几何和低维拓扑学。接触结构是特定空间上的几何物体，称为流形，在物理学中通过微分方程、光学和动力学自然产生。Eliashberg的开创性工作首次表明，接触结构在决定三维和四维流形的几何和拓扑性质方面起着至关重要的作用。接触几何在解决了几个长期存在的低维拓扑问题后，进入了复兴时期。正是在这种背景下，这个项目广泛地寻求更好地理解接触结构的特征如何决定它们所处空间的几何特性，或影响它们研究中使用的强大不变量。该项目将在几个领域产生直接影响，例如：低维拓扑，辛和接触拓扑，动力学和数学物理。PI还将花时间帮助指导研究生和博士后学者，帮助他们过渡到独立的研究人员。回想一下，接触结构可分为两类：紧绷或过度扭曲。了解哪些三流形支持紧密接触结构以及给定三流形支持的紧密接触结构的数量是现代接触几何的首要目标。因此，该项目的主要目标是开发能够确定紧密性和区分接触结构的有效且可计算的不变量。自从这些不变量出现以来，强有力的证据表明，三流形上的接触结构与其相关的花理论不变量之间存在着深刻的联系。例如，每个不同的flower同源性都支持一个能够检测紧密性和区分接触结构的不变量。该项目的一个重要目标是开发和探索这些不变量的改进，这些不变量同时更有效地检测紧密性，并且更容易计算。在一个平行的方向上，根据PI和其他人的研究，支撑弗洛尔理论的许多形式代数结构似乎反映了自然的几何特征和涉及接触结构的结构。反过来，这个项目的另一个关键目标是澄清花理论不变量的代数结构与自然接触几何现象和结构之间的联系和对应关系。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。

This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR). It centers on questions lying at the intersection of two fields of mathematics called contact geometry and low-dimensional topology. Contact structure are geometric objects on certain spaces known as manifolds that arise naturally in physics through differential equations, optics, and dynamics. Pioneering work of Eliashberg first showed that contact structures play an essential role in determining geometric and topological properties of three- and four-dimensional manifolds. Contact geometry has since entered a renaissance after featuring prominently in the resolution of several long-standing problems in low-dimensional topology. It is in this context that this project broadly seeks to better our understanding of how characteristics of contact structures determine either geometric properties of the spaces they live on, or influence powerful invariants used in their study. The project will have immediate impact in several fields, such as: low-dimensional topology, symplectic and contact topology, dynamics, and mathematical physics. The PI will also devote time to helping mentor graduate students and postdoctoral scholars as they transition to being independent researchers.Recall that contact structures fall into one of two categories: tight or overtwisted. Understanding which three-manifolds support tight contact structures and the number of tight contact structures supported by a given three-manifold are the paramount goals of modern contact geometry. Accordingly, a primary goal of this project is to develop effective and computable invariants capable of determining tightness and distinguishing contact structures. Since the inception of these invariants, strong evidence has steadily built suggesting deep connections between contact structures on three-manifolds and their associated Floer-theoretic invariants. For instance, each of the various Floer homologies support an invariants which are capable of detecting tightness and distinguishing contact structures. An important goal of this project is to develop and explore refinements of these invariants which are simultaneously more effective in detecting tightness, and are more easily computable. In a parallel direction, according to work on the PI and others, much of the formal algebraic structure underpinning Floer theory appears to mirror natural geometric characteristics and constructions involving contact structures. In turn, another key goal of this project is to clarify connections and correspondences between the algebraic structure of Floer-theoretic invariants and natural contact geometric phenomena and constructions.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.