9626676 Shalen One objective of this research project is to understand as much as possible about the incompressible surfaces in the exterior of a knot in a homotopy 3-sphere. This is an important aspect of classical knot theory and has potential applications to the Poincare conjecture. A second goal is to improve the known bounds for the distance between various types of exceptional Dehn fillings of a 3-manifold with torus boundary. This amounts to achieving a quantitative understanding of the phenomenon that a generic Dehn surgery on a hyperbolic knot produces a hyperbolic manifold. A third objective is to improve some of the known lower bounds for volumes of hyperbolic manifolds under various topological restrictions. Such bounds give a quantitative understanding of the Mostow Rigidity Theorem, which implies that the volume of a hyperbolic manifold is a topological invariant. Topology is the branch of mathematics that deals with properties of geometric objects that are so universal that they are unaffected by any distortion of the shape of the object. A simple example of such a property is the winding number of a closed circuit in the plane. If the circuit does not pass through the origin, then it winds around the origin a certain number of times, and that number is not affected by any distortion of the circuit (just so long as it never passes through the origin during its deformation). This was the main idea in Gauss's first proof that every polynomial equation has a solution in the complex numbers and was the origin of the subject of topology. The geometric objects being studied by Culler and Shalen are 3-dimensional spaces, one example of which is our physical universe. (The geometric and topological properties of the physical universe are not yet understood, but modern physics has shown that it is definitely not the 3-dimensional Euclidean space that we study in high school.) This research focuses on various numerical quantities that are determined by the geometric properties of the space. The goal is to understand how the fundamental topological properties of the space are reflected in these quantities. In very general terms, this is analogous to the process by which the chemical composition of a star can be deduced by studying its spectrogram. ***
