Interactions between geometry, topology, number theory, and dynamics

几何、拓扑、数论和动力学之间的相互作用

• 批准号: 2303572
• 负责人: Nathan Dunfield
• 金额: $ 39.98万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2303572&HistoricalAwards=false
• 关键词: Interactions; between; geometry; topology; number

Topology is the study of objects up to stretching, and geometry the study of rigid bodies. Work on this project will lead to advances in our understanding of both subjects by combining surprising relationships between them and deep connections to other areas of mathematics and computer science. Both topology and geometry are playing an increasingly important role in applications such as data mining and engineering design, and this project includes collaboration with computer scientists as well as the development of open-source software for exploring aspects of these problems. Graduate students will be trained in this project. The first topic of this project is effective Mostow rigidity and torsion growth in homology, questions motivated in part by number theory and global analysis. The project will explore how topological and geometric invariants behave under towers of finite covers and other geometric limits, and also study the extent to which different topological, geometrical, and arithmetic notions of complexity coincide for hyperbolic manifolds. The second topic of the project is counting essential surfaces in hyperbolic 3-manifolds and, in particular, divining the basic structures of such counts. This will involve placing these questions into the setting of measured laminations as well as relating them to the work of Mirzakhani and to dynamical questions about orbits of integer points under a family of interval isometries.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.

拓扑学是对物体拉伸的研究，几何学是对刚体的研究。该项目的工作将通过结合这两门学科之间令人惊讶的关系以及与数学和计算机科学其他领域的深厚联系，促进我们对这两门学科的理解的进步。拓扑和几何在数据挖掘和工程设计等应用中发挥着越来越重要的作用，该项目包括与计算机科学家的合作以及开发开源软件来探索这些问题的各个方面。研究生将在该项目中接受培训。该项目的第一个主题是同调中的有效莫斯托刚度和扭转增长，这些问题部分是由数论和全局分析引发的。该项目将探索拓扑和几何不变量在有限覆盖塔和其他几何极限下的表现，并研究双曲流形的不同拓扑、​​几何和算术复杂性概念的重合程度。 该项目的第二个主题是计算双曲 3 流形中的基本表面，特别是预测此类计数的基本结构。这将涉及将这些问题置于测量层压的设置中，并将它们与 Mirzakhani 的工作以及有关间隔等距族下整数点轨道的动力学问题联系起来。该奖项反映了 NSF 的法定使命，并被认为值得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来提供支持。