Facets of the Topology and Geometry of 3-Manifolds

拓扑学研究的是物体的弹性拉伸，而几何学研究的是刚体。该项目的目标是通过将这些领域之间的惊人关系与其他数学和计算机科学领域的深刻联系结合起来，来理解这些领域的某些基本问题。拓扑学和几何学在数据挖掘等应用中变得越来越重要，这个国家科学基金会资助的项目包括与计算机科学家合作，以及开发用于探索这些问题方面的软件，这些软件将通过网络免费提供给其他研究人员。本计画聚焦于有关三维流形的拓扑与几何的四个主题。第一个主题是有效的Mostow刚度和扭转增长，特别是理解拓扑和几何不变量在有限覆盖和其他几何极限下的表现。第二个主题是阐明三流形的heegard花同调、群有序性和紧叶之间的关系。第三个主题是构造一些非紧李群的Casson-Lin-Herald不变量。第四个主题是开发新的计算方法来探索关于三流形的各种问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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 elastic stretching, and geometry the study of rigid bodies. The goal of this project is to understand certain fundamental problems in these areas by combining surprising relationships between them with deep connections to other areas of mathematics and computer science. Both topology and geometry are becoming more important to applications such as data mining, and this National Science Foundation funded project includes collaboration with computer scientists as well as developing software for exploring aspects of these problems which will be freely available to other researchers via the web.This project focuses on four topics concerning the topology and geometry of three-dimensional manifolds. The first topic is effective Mostow rigidity and torsion growth, in particular understanding how topological and geometric invariants behave under towers of finite covers and other geometric limits. The second topic is to elucidate the relationships between Heegaard Floer homology, group orderability, and taut foliations for three-manifolds. The third topic is the construction of a Casson-Lin-Herald invariant for some noncompact Lie groups. The fourth topic is developing new computational methods for exploring a variety of questions about three-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.