The proposer's recent research has centered on knot theory and the theory of 3-manifolds, particularly Heegaard splittings of 3-manifolds and related notion of unknotting tunnels. The main approach might humorously be called neo-classical, for it focuses on the behavior of surfaces contained in the 3-manifolds (a classical approach) but in a more sophisticated way. For example, in combinatorial arguments on surface intersections, graph-theoretic lines of argument (on the arcs of intersection) have proven to be almost magically effective in distilling geometric information; in Gabai's theory of sutured 3-manifolds, intersection with a `parameterizing surface is the chief mechanism for transmitting information from the end of the hierarchy back to the beginning; the minimax principle usually called ``thin position" exploits the intersection between a stationary surface and a moving surface to understand the topology of the entire manfold. Comparing intersections between two sets of moving surfaces (using the tools of Cerf theory) can be even more effective as in, for example, the Gordon-Luecke solution to the knot complement conjecture or the proposer's work with Rubinstein on explicit bounds for Heegaard stabilization. The problems addressed in this proposal have been chosen not solely for their intrinsic interest but also because the hope is that techniques developed in their solution would point the way to the solution of grander problems and, more generally, give insight to the general structure of knotting phenomena (broadly construed) and the structure of 3-manifolds, particularly structure that is on the edge of and interacts with natural questions about 4-manifolds. An underlying theme of this proposal is that there is much to be learned about knot theory and the theory of 3-manifolds via a geometric approach that is more flexible than differential geometry but more structured than simple combinatorics. This medium ground is characterized by a use of thin position arguments, in which sweep-outs of 3-space by level planes or sweep-outs of 3-manifolds by Heegaard surfaces can reveal structure that might be difficult to find by other techniques.
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 objects with this property: anyone living in the object would see their world as 3-dimensional. Such objects are called 3-manifolds, and the broad goal of this research proposal is to increase our understanding of them. Three-manifolds support interesting phenomena. One of these phenomena is knotting, in which a simple object like a garden-hose (or a string of DNA) can be maneuvered so that its positioning in space is quite complex. More generally, objects like chemical molecules can be put in a 3-manifold in extraordinarily complex ways if one thinks of their parts as made of rubber which can be knotted and interweaved. Tools which are being developed to understand 3-manifolds help us understand knotting and, conversely, understanding knotting helps us understand 3-manifolds. One success of this ongoing research program is the proposer's solution (with Abigail Thompson) of the graph-planarity problem: There is a simple criterion, algorithmic in execution, which determines whether a knotted graph in 3-space (e. g. a chemical molecule) can be isotoped to lie in a plane, and so in fact is unknotted.
