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

该项目由拓扑学计划和既定的激励竞争研究计划(EPSCoR)共同资助。它以两个数学领域的交叉问题为中心，这两个领域称为接触几何和低维拓扑。接触结构是某些空间上的几何对象，称为流形，它通过微分方程式、光学和动力学在物理学中自然产生。Eliashberg的开创性工作首先表明接触结构在决定三维和四维流形的几何和拓扑性质中起着至关重要的作用。自那以后，接触几何在解决低维拓扑中几个长期存在的问题方面表现突出，进入了复兴阶段。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在空间的几何性质，或者如何影响在它们的研究中使用的强大的不变量。该项目将在几个领域产生立竿见影的影响，如：低维拓扑学、辛拓扑和接触拓扑学、动力学和数学物理。在研究生和博士后学者过渡到独立研究人员的过程中，PI还将花时间帮助他们指导他们。我们记得，联系结构分为两类：紧密或过度扭曲。了解哪些三流形支持紧接触结构，以及由给定的三流形支持的紧接触结构的数量是现代接触几何的最高目标。因此，这个项目的一个主要目标是开发有效的和可计算的不变量，能够确定紧密性和区分接触结构。自从这些不变量出现以来，已经有强有力的证据表明三维流形上的接触结构与它们相关的Floer理论不变量之间存在着深刻的联系。例如，各种Floer同源中的每一个都支持能够检测紧密性和区分接触结构的不变量。这个项目的一个重要目标是开发和探索这些不变量的精化，同时在检测紧密性方面更有效，并且更容易计算。在平行的方向上，根据PI和其他人的工作，支撑Floer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，该项目的另一个关键目标是澄清Floer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。该奖项反映了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。

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.