The Geometry and Topology of Harmonic Maps to R-Trees

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AbstractAward: DMS-9971860Principal Investigator: Richard A. WentworthThe Principal Investigator's main interest is the study ofgeometric structures arising from equivariant harmonic maps toR-trees, especially where the domain is a compact 3-manifold or asymplectic 4-manifold. Harmonic maps to non-positively curvedmetric spaces have had many applications to rigidity-typequestions. At the same time, group actions on R-trees (aparticular example of such a metric space target) have played animportant role in combinatorial group theory and 3-manifoldtopology. This proposal seeks to understand the relationshipbetween the analytic and topological aspects of these problems.The research will explore the interaction between harmonic mapsand incompressible surfaces, taut foliations, and essentiallaminations in 3-manifolds. A second part of this project willstudy the notion of an SL(2,C) Casson invariant.Mathematicians have long been interested in classifying thepossible shapes of geometric spaces. Of particular relevance tothe physical world are spaces of dimensions three and four.These are, perhaps surprisingly, also intimately related to thestudy of knots. Classically, the methods used for research inthis area have come from algebra, topology, and analytic functiontheory. More recently, in part motivated from the physics ofvortices, new techniques have been developed involving maps whichminimize energy. This project seeks to explore the topologicalramifications of this approach.