Invariants of 3-Manifolds and Their Applications

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DMS-0206464Walter NeumanPart of this project concerns a continuation of the investigator'ssuccessful application of topological methods to the study ofsingularities of complex surfaces and the topology of plane affinecurves. One issue to be addressed is explicit analytic description ofsingularities with given topology. The general question is notoriouslydifficult, but the investigator and J. Wahl have identified aremarkably broad class of topologies where significant advance can bemade. Another issue is the study of the global topology oftwo-variable polynomials, as well as classification issues, which areboth of intrinsic interest and also have potential to contributeto the resolution of the famous Jacobian Conjecture. A significanttool in this work are the splice diagrams of Eisenbud and theinvestigator and a recent generalization of them. In a ratherdifferent direction, the investigator will study invariants of3-dimensional manifolds that relate to number theory and algebra,namely the so-called character and eigenvalue varieties and the Blochinvariant. Connections are emerging between these invariants that willinform both topology and algebra/number theory.Algebraic Geometry, which is essentially the study of the zero sets offamilies of polynomials, has always been an important area offundamental research, and is also important in such diverseapplications as control theory and communications. Such zero-setsarise, for instance, as the parameter spaces of processes and thecontrol spaces of automata. The singularities (places where the natureof the topology changes), are of fundamental significance. Thisproject has two thrusts: to understand better the link between theanalytic description and the topology of singularities, and toinvestigate invariants of low dimensional manifolds that link topologywith other fields.