The principal investigator, a low dimensional topologist, is utilizing a tool he developed with Peter Teichner (called ``grope cobordism") to find topological applications and interpretations for an important class of invariants called Vassiliev invariants. In particular he and Teichner are pursuing the question of whether Vassiliev invariants detect knottedness. The principal investigator is also looking for more powerful versions of the theory of Vassiliev invariants, hopefully giving rise to four dimensional (concordance) information. Some of this is joint with Jacob Mostovoy and Ted Stanford. Working jointly with Matt Horak and Karen Vogtmann, the principal investigator is studying the mapping class group of surfaces via a tool developed by Harer, Penner, and elucidated by Kontsevich (graph homology). Graph homology is very similar to the algebra that appears in the context of Vassiliev invariants. The principal investigator is also studying the group of outer automorphisms of a free group via graph homology. Finally, the principal investigator, working jointly with Ryan Budney, Kevin Scannell, and Dev Sinha, is pursuing a program for grounding Vassiliev invariants in classical homotopy theory and differential topology. This program has already borne fruit in terms of geometric interpretations of the simplest Vassiliev invariant.
The yoga behind the theory of topological invariants is that they provide useful, computable topological information. When the number of dimensions is large, complete answers to topological questions can often be found using algebraic invariants. In low dimensions, such as 3 and 4, the situation is more complicated. The invariants still exist, but they are much weaker, and usually do not give complete answers. On the other hand, there are lots of invariants in low dimensions which don't have high dimensional analogues, and which are poorly understood topologically. For example, there is a polynomial you can associate to any knotted loop (such as a piece of DNA or a singularity in spacetime) which will not change even if the loop is pushed and pulled into a new shape. This polynomial, called the Jones polynomial, has a definition which eludes any topological interpretation and whose topological applications have been modest. The principal investigator is working to understand precisely the connection of this polynomial (and other similar objects) with topology, and this will lead to more effective methods for answering questions in low dimensional topology and beyond.
