Since Thurston's work on the geometrization conjecture, the study of hyperbolic structures on surfaces and 3-manifolds has been a central topic in low-dimensional topology and geometry. Thurston made a number of conjectures that have been a driving force in the field, not only due to their intrinsic interest but because the study of these conjectures has produced new mathematics that has had wide ranging impact in many fields of mathematics. In the previous decade many of Thurston's original conjectures have been solved, introducing both new techniques and new problems into the field. The principal investigator will use these new tools to answer further questions about hyperbolic 3-manifolds and in particular will study the topology of spaces of hyperbolic 3-manifolds attempting to answer questions such as if these spaces are locally connected and if not where locally connectivity fails. One area where the study of hyperbolic 3-manifolds has had great influence is in the study of mapping class groups. This is perhaps the simplest example of a naturally occurring group that is not a lattice in a Lie group and has been much studied by geometric group theorists. The PI will apply many of the tools developed in the study of hyperbolic 3-manifolds, such as the curve complex, to study problems about the mapping class group.
Three-manifolds are a central object of mathematical study for several reasons. The most obvious is that the space we live in is a 3-manifold and one basic goal of mathematics is to gain a better understanding of the real world. On a more abstract level, 3-manifolds have a deep and intricate structure that connects to many other areas of mathematics. This project will focus on hyperbolic 3-manifolds. When looking at the simplest examples it first appears that hyperbolic 3-manifolds are quite rare. However, a deeper analysis shows that, in a natural sense, hyperbolic 3-manifolds are the most prevalent type of 3-manifolds. While 3-manifolds themselves are topological objects to understand hyperbolic 3-manifolds them one is quickly led to algebra, geometry and analysis. By studying hyperbolic 3-manifolds we learn more about mathematics as a whole.
