Facets of low-dimensional topology

拓扑学是研究直到橡胶拉伸的对象，几何学是研究刚体。这个项目的目标是通过将这些领域之间令人惊讶的关系与数学和计算机科学的其他领域的深层联系结合起来，来理解这些领域的某些基本问题。拓扑学和几何学对数据挖掘等应用程序都变得越来越重要，这个项目包括与计算机科学家合作，以及开发软件来探索这些问题的各个方面，其他研究人员可以通过网络免费获得。这个项目集中在四个主题上。第一个是有效的Mostow刚性和扭转增长，即双曲三维流形的拓扑和几何性质可以独立改变的程度。二是在三维球面上寻找纽结亏格的计算复杂性，特别是能否在多项式时间内确定。第三个问题是，是否存在一个逐面单词双曲线组。最后一个主题是阐明3-流形的Heegaard-Floer同调、群可序性和张叶之间的关系。

Topology is the study of objects up to rubbery 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 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. The first is effective Mostow rigidity and torsion growth, specifically, the extent to which topological and geometric properties of hyperbolic 3-manifolds can be varied independently. The second is the computational complexity of finding the genus of a knot in the 3-sphere, in particular whether it can be determined in polynomial time. The third question is whether there exists a word-hyperbolic surface-by-surface group. The final topic is to elucidate the relationships between Heegaard-Floer homology, group orderability, and taut foliations for 3-manifolds.