Link Floer Homology and Kleinian Groups

Link Floer 同调和 Kleinian 群

• 批准号: 2203237
• 负责人: Beibei Liu
• 金额: $ 15.6万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-05-15 至 2024-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203237&HistoricalAwards=false
• 关键词: Link; Floer; Homology; Kleinian; Groups

A link is a collection of disjoint circles that may be linked together embedded in a space of dimension three. One of the main topics in low-dimensional topology is to study the topological and geometric properties of the link, the three-dimensional space, and some four-dimensional spaces bounded by the three-space. This project aims to deepen understanding of these mathematical structures. The first part of the research concentrates on the study of three-manifolds obtained from links via the so-called "Dehn surgery" operation and the family of links appearing in algebraic geometry. Results are anticipated to advance the understanding of the complexity of three-manifolds and algebraic singularities in algebraic geometry. It will also provide topics that are suitable for undergraduate students' research. The second part of the research focuses on the topology, geometry, and dynamics of hyperbolic manifolds, which are important examples of Gromov hyperbolic spaces, negatively curved Hadamard manifolds, and symmetric spaces of non-compact type.The research consists of four specific projects about links and hyperbolic manifolds. The first aims to understand the possible obstructions for surgeries on 2-component links in the three-sphere. It focuses on the possibility of finding an infinite family of integer homology spheres that cannot be obtained by surgeries on 2-component links in the three-sphere. The second project is to understand the link Floer chain complex of algebraic links coming from the singularities of algebraic curves in the complex plane and provide potential applications in low dimensional topology. The third project is to study discrete isometry subgroups acting on hyperbolic spaces with small critical exponents and generalize the structure theorem for hyperbolic manifolds to negatively curved Hadamard manifolds. The fourth project concerns a counting question in hyperbolic manifolds, with the goal of determining whether the classical Bowen-Margulis measure and the spectral gap converge for a strongly convergent sequence of hyperbolic manifolds.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.

链接是不相交的圆的集合，它们可以链接在一起嵌入到三维空间中。低维拓扑的主要课题之一是研究连杆、三维空间以及由三维空间界定的某些四维空间的拓扑和几何性质。该项目旨在加深对这些数学结构的理解。研究的第一部分集中于研究通过所谓的“Dehn手术”操作从连杆获得的三流形以及代数几何中出现的连杆族。预计结果将促进对代数几何中三流形和代数奇点的复杂性的理解。它还将提供适合本科生研究的主题。研究的第二部分集中于双曲流形的拓扑、几何和动力学，它们是格罗莫夫双曲空间、负曲阿达玛流形和非紧型对称空间的重要例子。该研究包括四个具体项目：链接和双曲流形。第一个目的是了解三球体中二元链路手术可能存在的障碍。它关注的是寻找整数同调球的无限族的可能性，而这些整数同调球是无法通过对三球中的二元链接进行手术来获得的。第二个项目是理解来自复平面中代数曲线奇点的代数链接Floer链复形，并提供在低维拓扑中的潜在应用。第三个项目是研究作用于具有小临界指数的双曲空间的离散等距子群，并将双曲流形的结构定理推广到负弯曲哈达玛流形。第四个项目涉及双曲流形中的计数问题，目标是确定经典 Bowen-Margulis 测度和谱间隙是否收敛以获得双曲流形的强收敛序列。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。