Two of the most successful tools for investigating the topology of a three-dimensional manifold are the use of non-Euclidean geometric structures and the study of embedded surfaces. Marc Culler and I discovered an interaction between these two techniques involving the variety of representations of the fundamental group and actions of the fundamental group on trees. I will describe the basic idea and mention some recent applications.

Lecture II: December 13, 2000 (15:30)

If K is a knot in a 3-manifold M, one can associate with K a set of rational numbers, called boundary slopes of K, which encode information about geometrically meaningful surfaces in the complement of K. Marc Culler and I showed that if M is assumed to have a cyclic fundamental group, and if K is nontrivial in a suitable sense, then restrictions are imposed on the set of boundary slopes of K. I will discuss the proof of this result, which is an application of the ideas presented in my first talk, "Three-manifolds and trees." I will also explain the role of the result in our program for proving the Poincare Conjecture.

Lecture III: December 17, 2000 (09:30)

The result that I discussed in my second talk, "Boundary slopes of knots..." is the beginning of a program for proving the Poincare Conjecture. To carry out the program would require finding a knot in a rather general kind of 3-manifold (a non-Haken manifold) for which one has good control over the set of geometrically meaningful surfaces (essential surfaces) in the knot exterior. I will describe recent joint work with Marc Culler, Nathan Dunfield and William Jaco which represents progress on this kind of question. Unlike the algebraic and hyperbolic-geometric techniques used in the proof of the result of "Boundary slopes...," the recent results are largely combinatorial in nature, involving triangulations of 3-manifolds and the theory of normal and almost normal surfaces.