Statement : Jessica

A 3–manifold is often described combinatorially, for example by a knot diagram, or by gluing a collection of polyhedra, or by attaching handles to a given manifold. Figure 1 shows a few examples. Just over a decade ago, Perelman posted the outlines of a proof of the Geometrization Theorem on the ArXiv [56, 57], and with his work and that of others it is now known that all 3–manifolds decompose into pieces that admit a geometric structure (see, for example [7, 14, 40, 51]). In an appropriate sense, the most common geometric structure is hyperbolic, i.e. admitting a metric with constant sectional curvature −1 [67]. For closed or finite volume 3–manifolds, the hyperbolic structure is unique [53, 58]. Therefore, it is known that a combinatorial description of a hyperbolic 3–manifold uniquely determines the geometry of that manifold. However, what is still unknown in general is, given a combinatorial description of a hyperbolic 3–manifold, how does one determine geometric information about that 3–manifold? Or given other invariants of a 3–manifold, including quantum invariants, how are they related to the hyperbolic structure? Much of my research has focused on these types of questions. In recent years, the investigation of these problems has been called “Effective Geometrization,” or “WYSIWYG Topology: What you see is what you get,” indicating that salient properties of the combinatorics of a manifold seem to have relevance to the geometry as well. These types of problems are of major importance in the field. While our knowledge of 3–manifold geometry, quantum topology, and combinatorial 3–manifolds has progressed rapidly in recent years, we are still often unable to apply resulting theorems to related problems encountered in other settings. What is needed is a dictionary between the geometry and other invariants.