This project is in the area of knot theory and the topology of3-dimensional manifolds. The investigators will study 3-manifolds, using two points of view to analyze surfaces in a 3-manifold. The first method uses combinatorial techniques combined with the concept of 'thin position' of a knot. This will be used to consider some old problems in knot theory, such as the cabling conjecture, as well as the classification problem for Heegaard splittings of Seifert fibered spaces. Thin position surfaces serve as a combinatorial version of the minimax surfaces used by Pitts-Rubinstein in their investigations of Heegaard splittings, and they will be used to attack problems similar to the recognition problem for the 3-sphere, recently solved by Rubinstein. The second method uses techniques of smooth minimal surface theory and hyperbolic geometry to investigate the topology of 3-manifolds. One goal is to show that any 3-manifold admits only a finite number of Heegaard splittings of each genus, a result recently established for Haken manifolds by Johannson. Another goal is to show thatnon-Haken 3-manifolds are determined by their fundamental groups. Finally, the investigators will explore the topological consequences that follow from imposing curvature conditions on a manifold and its boundary. The geometry of three dimensions is the geometry of the world we live in, so one might expect that the mathematical study of this area would have particularly broad applications. This is indeed the case, and the area leads to applications in physics, in the study of the differential equations governing physical phenomena, to group theory, and to many other branches of mathematics. Mathematicians study, in all dimensions, the geometric objects called "manifolds," which have properties similar to those of the space we live in. Manifolds in dimension three exhibit unique features which make their study a flourishing area of current research. The objective of this project is to study certain categories of these 3-manifolds, in pursuit of the overall goal of understanding these geometric objects. Using techniques developed from studying knotted curves and the complexity of their raveling, and from the theory of soap films and minimal surfaces, the investigators aim to contribute to the classification problem for3-manifolds and to understand specific classes of 3-manifolds in great detail.
