Interactions between geometry, topology, number theory, and dynamics

拓扑学研究对象直至拉伸，几何学研究刚体。这个项目的工作将通过将它们之间令人惊讶的关系以及与数学和计算机科学的其他领域的深入联系结合起来，从而促进我们对这两个学科的理解。拓扑学和几何学在数据挖掘和工程设计等应用中都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。研究生将在这个项目中接受培训。这个项目的第一个主题是同调中的有效Mostow刚性和扭转增长，这些问题部分是由数论和整体分析引起的。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限制下的行为，并研究双曲流形不同的拓扑、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三维流形中的基本曲面，特别是预测这种计数的基本结构。这将涉及到将这些问题放入测量的分层的设置中，并将它们与Mirzakhani的工作以及关于区间等值线家族下的整点的轨道的动力学问题联系起来。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.