Invariants in Low Dimensional Topology and Geometric Group Theory

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The aim of this proposal is to advance fundamental understanding in low dimensional topology, geometric group theory, as well as logic and computer science. Three projects in this proposal target a very promising and active area of low dimensional topology, namely the study of Vassiliev invariants of classical knots. The aim is to, on the one hand, expose more internal structure, but on the other hand to make them more relevant to topological questions, such as whether a classical knot can be the boundary of a disk in the four dimensional ball. Somewhat related to this area is the Schneiderman-Teichner theory of intersections of two dimensional spheres in 4-manifolds. Together with Schneiderman and Teichner, the P.I. hopes to fill in a missing step and complete the theory to a complete classification scheme. Another cluster of projects revolves around the algebraic object called "graph homology," which has relevance to the study of invariants of manifolds, the study of the mapping class group of surfaces, the study of the important group Out(F_n), and to the deformation quantization problem. The P.I. proposes to make a deeper study of the general theory of graph homology, on the one hand, while on the other analyzing the specific examples of relevance to topology. More specifically the P.I. will continue a program, joint with Vogtmann, for enlarging the known set of rational homology classes for Out(F_n) from 2 to infinity. Lastly, probing the interface between topology and computer science, the P.I. proposes to take some steps toward distinguishing P from NP, utilizing ideas of Mike Freedman.This proposal concerns fundamental research in several core areas of mathematics, including topology, group theory and computer science. The projects in the proposal aim at making significant breakthroughs in our understanding of some of the most rapidly developing subdisciplines. A common theme to the projects is the elucidation of connections between seemingly disparate fields, such as topology and computer science. For example, the P.I. will explore ideas of Mike Freedman to apply topology to the study of how efficiently alogrithms can solve problems. The proposed research offers ample oppurtunity to meaningfully involve undegraduates and graduate students.