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.

链接是一系列不交联圆圈的集合，可以将嵌入在第三维空间中的嵌入在一起。低维拓扑的主要主题之一是研究链接的拓扑和几何特性，三维空间以及一些由三个空间界定的四维空间。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在于通过所谓的“ Dehn手术”操作从链接获得的三个manifolds的研究，以及在代数几何形状中出现的链接家族。预计结果可以提高对代数几何形状中三序和代数奇异性的复杂性的理解。它还将提供适合本科生研究的主题。研究的第二部分侧重于双曲线歧管的拓扑，几何学和动力学，这是Gromov双曲空间的重要例子，负弯曲的Hadamard歧管以及非紧凑类型的对称空间。该研究由四个有关链接和链接和多重折线的特定项目组成。第一个旨在了解三球中2组分链路上可能的手术障碍。它着重于找到一个无限的整数同源性领域家族的可能性，该领域无法通过手术对三个球员的2组分链路获得。第二个项目是了解来自代数曲线在复杂平面中的奇异性的代数链接的链路链链复合物，并在低维拓扑的情况下提供了潜在的应用。第三个项目是研究作用于具有小关键指数的双曲线空间上的离散等轴测亚组，并将双曲线歧管的结构定理推广到负弯曲的Hadamard歧管。第四个项目涉及双曲线歧管中的一个计数问题，目的是确定经典的鲍恩·马古利斯（Bowen-Margulis）度量和光谱差距融合了强烈的双曲线歧管序列是否收敛序列。该奖项反映了NSF的法定任务，并认为通过基金会的知识优点和广泛的crietia crietia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia均值得通过评估。