Connections in low-dimensional topology

三维流形是一个空间，在这个空间中，对象可以在三个不同的垂直方向上移动。我们居住的宇宙是一个三维流形，我们还不了解它的整体结构。多亏了瑟斯顿、佩雷尔曼和莫斯托的强大定理，我们确实知道流形的几何(角度、距离和曲率的测量)与它的拓扑(一个人可能在一个方向上前进并最终从另一个方向返回的不同方式的描述)密切相关。然而，我们还没有很好的定量了解几何测量如何决定拓扑结构，反之亦然。对这种关系的更深入的定量理解最终可以用来分析宇宙学数据和揭示宇宙的拓扑结构。这个项目试图在低维拓扑中的以下几个角度之间建立和加强联系：组合拓扑、双曲几何、量子不变量和群论。这一目标分为几个子项目。第一个子项目是基于圆周上纤维的组合数据建立双曲三维流形的显式组合模型。第二个子项目是使用量子不变量，如琼斯和有色琼斯多项式，以获得关于结点的边界斜率、纤维数据和双曲体积的信息。第三个子项目是使用来自三维三角剖分的想法来构建双曲群的度量模型。

A 3-manifold is a space where an object can move around in three distinct perpendicular directions. The universe that we inhabit is a 3-manifold whose global structure we do not yet understand. Thanks to powerful theorems by Thurston, Perelman, and Mostow, we do know that the geometry of a manifold (measurements of angles, distances, and curvature) is closely tied to its topology (an account of the different ways in which one may head off in one direction and eventually come back from another). However, we do not yet have a good quantitative understanding of exactly how geometric measurements determine topological structure and vice versa. A deeper quantitative understanding of this relationship can eventually be used to analyze cosmological data and shed light on the topology of the universe.This project seeks to build and strengthen connections among the following perspectives in low-dimensional topology: combinatorial topology, hyperbolic geometry, quantum invariants, and group theory. This goal splits into several sub-projects. The first sub-project is to build explicit combinatorial models for hyperbolic 3-manifolds based on the combinatorial data of a fibration over the circle. The second sub-project is to use quantum invariants such as the Jones and colored Jones polynomials to gain information about the boundary slopes of knots, fibration data, and hyperbolic volume. The third sub-project is to use ideas from 3-dimensional triangulations to build metric models for hyperbolic groups.