Connections in low-dimensional topology

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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.