9504438 Scharlemann Cooper, Long and Scharlemann focus on 3-manifolds. One of Cooper's projects is to utilize the relationship between finite foliations and geometrically finite surfaces in hyperbolic 3-manifolds. Another is to continue developing the properties of the A-polynomial for knots that arise from representation theory. A third involves the theory of buildings applied to representations of the braid group. Long's specific projects concern the study of finite foliations and the resulting dynamical systems, as well as the application of these ideas to hyperbolic 3-manifolds. He also is working on problems in algebraic geometry and the use of the degree conjecture to prove Property P, and on the finite dimensional linear representations of the braid groups. Scharlemann's main interest is the stabilization problem for Heegaard splittings. Success would have important implications for the general classification problem for 3-manifolds. There are connections to knot theory as well, via the notion of "tunnel number" for a knot. One of the most basic observations about the world around us, apparent almost from our birth, is that it is 3-dimensional. So it is of interest to understand spaces with precisely this property: anyone living in the space would see their world as 3-dimensional. Such spaces are called "3-manifolds," and this project aims to increase our understanding of them. 3-manifolds support interesting phenomena. One of these phenomena is "knotting," in which an intrinsically simple object like a garden-hose (or a DNA string) can be maneuvered so that its positioning in space is quite complex. More generally, objects like chemical molecules, usually thought of abstractly as "graphs" (much like tinkertoy models), can be put into a 3-manifold in extraordinarily complex ways if one thinks of the "sticks" as made of rubber which can be knotted and interweaved. Tools which are being developed to understand 3-manifolds help us und erstand knotting and, conversely, understanding knotting (a second principal aim of the project) helps us to understand 3-manifolds. For example, the "Heegaard splittings" mentioned above refer to a technique in which all the complexity of a general 3-manifold is absorbed into a thick graph. Then information about the graph gives information about the 3-manifolds. A more disciplined type of knotting, called braiding, occurs, for example, in the trajectories of fluid or plasma flow in a Tokomak-type torus. Since this knotting is more disciplined, more tools are available for understanding and classifying such knotting. Hence the interest in braid theory and its connections to dynamical systems. ***
