Connections in low-dimensional topology

三维流形是一个物体可以在三个不同的垂直方向上移动的空间。我们所居住的宇宙是一个三流形，其整体结构我们还不了解。由于瑟斯顿、佩雷尔曼和莫斯托的强大定理，我们确实知道了流形的几何（角度、距离和曲率的测量）与它的拓扑（一种对一个方向出发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，我们还没有一个很好的定量理解几何测量是如何决定拓扑结构的，反之亦然。对这种关系的更深入的定量理解最终可以用于分析宇宙学数据，并阐明宇宙的拓扑结构。本项目旨在建立和加强低维拓扑学中以下观点之间的联系：组合拓扑学、双曲几何、量子不变量和群论。这个目标分成几个子项目。第一个子课题是建立双曲型3流形的显式组合模型，该模型是基于圆上振动的组合数据。第二个子项目是使用量子不变量，如琼斯多项式和有色琼斯多项式来获得关于结的边界斜率、纤维化数据和双曲体积的信息。第三个子项目是使用三维三角形的概念来建立双曲群的度量模型。

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.