Rigidity and Boundaries in Non-Positive Curvature

非正曲率的刚度和边界

• 批准号: 2204339
• 负责人: Emily Stark
• 金额: $ 21.1万
• 依托单位: Wesleyan University
• 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: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204339&HistoricalAwards=false
• 关键词: Rigidity; Boundaries; Non; Positive; Curvature

As long as mathematics has been studied, people have sought to understand the relationship between geometry and symmetry. The Euclidean plane is most familiar, closely followed by the sphere. It has long been known that one cannot periodically tile the plane using the same arrangement of shapes as one would the sphere. This can be understood mathematically through the geometric notion of curvature: the plane is flat while the sphere is positively curved. This project concerns the vast universe of non-Euclidean geometries with non-positive curvature. The PI will investigate asymptotic invariants and rigidity phenomena in this setting, while supporting student involvement and broadened participation in mathematics via mentoring and community outreach. This research concerns finitely generated groups and their large-scale geometry. The first project investigates graphical discreteness, a notion that unifies two distinct programs of study: rigidity phenomena and classifying lattice envelopes. The former has been a central problem in geometric group theory, while the latter was initiated with Mostow--Prasad Rigidity, which characterized Lie group envelopes of hyperbolic manifold groups. The PI will consider a diverse family of examples, including hyperbolic groups with Menger curve boundary and groups that split as graphs of groups. The second project focuses on hyperbolic groups with Menger compacta visual boundaries and will build techniques to study the quasi-conformal structures on these spaces. The third project aims to study relatively hyperbolic groups and their boundaries via analytic methods and quasi-conformal geometry.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.

只要对数学进行了研究，人们就试图了解几何与对称性之间的关系。欧几里得平面最熟悉，紧随其后是球体。长期以来，人们已经知道，不能使用与球体相同的形状排列来定期瓷砖平面。这可以通过曲率的几何概念来数学上理解：平面是平坦的，而球体则呈呈正弯。该项目涉及具有非阳性曲率的广阔的非欧几里得几何形状宇宙。 PI将在这种情况下调查渐近不变和僵化现象，同时通过指导和社区宣传来支持学生参与并扩大对数学的参与。 这项研究涉及有限生成的群体及其大规模的几何形状。第一个项目调查了图形离散性，这一概念统一了两个不同的研究程序：刚性现象和分类晶格信封。前者一直是几何群体理论中的一个核心问题，而后者则是用莫斯特（Mostow）刚度（prasad刚性）启动的，该刚度表征了双曲线歧管基团的谎言组信封。 PI将考虑各种各样的例子家族，包括具有Menger Curve边界的双曲线组和分裂成组图的群体。第二个项目的重点是具有Menger Compacta视觉边界的双曲线群，并将建立技术来研究这些空间上的准符合结构。第三个项目旨在通过分析方法和准符合形式的几何形状来研究相对双曲的群体及其边界。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响来通过评估来支持的。