The Geometry and Topology of Harmonic Maps to R-Trees

摘要：dms -9971860首席研究员：Richard a . wentworth首席研究员的主要兴趣是研究由等变调和映射tor树引起的几何结构，特别是当域是紧3流形或渐辛4流形时。非正曲线度量空间的调和映射在刚性型问题中有许多应用。同时，r树上的群作用（特别是这种度量空间目标）在组合群论和3流形拓扑中起着重要的作用。本提案旨在了解这些问题的分析和拓扑方面之间的关系。本研究将探讨3流形中调和映射与不可压缩曲面、紧叶理和基本层化之间的相互作用。这个项目的第二部分将研究SL(2，C) Casson不变量的概念。数学家长期以来一直对几何空间的可能形状进行分类很感兴趣。与物理世界特别相关的是三维和四维空间。也许令人惊讶的是，这些也与结的研究密切相关。经典地，用于这一领域研究的方法来自代数、拓扑和解析泛函理论。最近，部分出于涡旋物理学的动机，人们开发了一些新技术，其中包括将能量最小化的地图。本项目旨在探索这种方法的拓扑分支。

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.