Culler and Shalen are continuing to investigate consequences of their work on the character variety of a knot group. This involves studying the relationship between the boundary slopes of a knot and the topological properties of essential surfaces that realize the slopes. It also includes their ongoing project with Dunfield and Jaco about smallish knots in non-Haken manifolds, which is part of a program to prove the Poincare Conjecture. A joint project by Agol, Culler and Shalen, concerning the construction of covering spaces of a triangulated 3-manifold by an inductive process, is also relevant to this program. Agol is also continuing his work on geometric finiteness of geometrically defined subgroups of knot groups, volume estimates for non-fibered Haken manifolds and hyperbolic orbifolds, and the complexity of algorithms in 3-manifold theory.
A fundamental problem in many areas of mathematics is to classify all examples of a certain type of mathematical object. The objects of study in this proposal are 3-manifolds, which are mathematical models of 3-dimensional spaces. Since our universe is a 3-dimensional space, the classification of 3-manifolds is directly related to our understanding of nature itself. The classification problem for 3-manifolds is far from solved, but the work of many mathematicians over the last 25 years has at least produced a conjectural answer. Remarkably, the conjectures, if true, will provide a unification of the most classical, rigid kind of geometry---both the Euclidean version first studied by the ancient Greeks and the non-Euclidean kind that constituted an exciting discovery in the 19th century---and topology, a subject devoted to studying much more flexible geometric structures, which until recently had developed quite independently of the more classical theories. The work supported by this grant forms part of the effort to verify the conjectured geometric classification of 3-manifolds.
