Facets of low-dimensional topology

拓扑学是研究物体直至橡胶拉伸的学科，几何学是研究刚体的学科。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，该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。本项目主要关注四个主题。首先是有效的Mostow刚度和扭转增长，特别是双曲3-流形的拓扑和几何性质可以独立变化的程度。其次是在3球中寻找结的属的计算复杂度，特别是它是否可以在多项式时间内确定。第三个问题是，是否存在一个逐字双曲的曲面群。最后讨论了3流形的heegaard - flower同调、群序性和紧叶之间的关系。

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.