Invariants of 3-Manifolds and Their Applications

Barnard College, New York NY

Investigators

Abstract

DMS-0206464 Walter Neuman Part of this project concerns a continuation of the investigator's successful application of topological methods to the study of singularities of complex surfaces and the topology of plane affine curves. One issue to be addressed is explicit analytic description of singularities with given topology. The general question is notoriously difficult, but the investigator and J. Wahl have identified a remarkably broad class of topologies where significant advance can be made. Another issue is the study of the global topology of two-variable polynomials, as well as classification issues, which are both of intrinsic interest and also have potential to contribute to the resolution of the famous Jacobian Conjecture. A significant tool in this work are the splice diagrams of Eisenbud and the investigator and a recent generalization of them. In a rather different direction, the investigator will study invariants of 3-dimensional manifolds that relate to number theory and algebra, namely the so-called character and eigenvalue varieties and the Bloch invariant. Connections are emerging between these invariants that will inform both topology and algebra/number theory. Algebraic Geometry, which is essentially the study of the zero sets of families of polynomials, has always been an important area of fundamental research, and is also important in such diverse applications as control theory and communications. Such zero-sets arise, for instance, as the parameter spaces of processes and the control spaces of automata. The singularities (places where the nature of the topology changes), are of fundamental significance. This project has two thrusts: to understand better the link between the analytic description and the topology of singularities, and to investigate invariants of low dimensional manifolds that link topology with other fields.