Moduli of Vector Bundles on Riemann Surfaces and Applications to Topology

摘要建议：DMS-9803606首席研究员：Georgios Daskalopoulos首席研究员的主要兴趣是分析方法在拓扑和几何问题中的应用。更确切地说，在第一部分的建议，principalinvestigator，动机的存在表示的基本群体的结补，建议之间的联系yang-mills理论黎曼曲面和Teichmueller理论。第二部分主要研究曲面和三维流形的基本群的特征标簇的某些紧化。调查relatesthis行动的团体对树木和不可压缩表面上的三维流形。在提案的其余部分，主要研究者提出了一些关于物理学中出现的模空间的度量的问题，这项工作的主要动机是通过研究三维流形的基本群的表示来理解三维流形的拓扑问题。本项目的结果将促进我们对黎曼曲面上的规范理论、调和映射、Teichmueller理论和三维流形拓扑学等问题的认识。此外，这门学科的工作将促进数学和物理学某些方面之间的进一步联系。

AbstractProposal: DMS-9803606Principal Investigator: Georgios DaskalopoulosThe principal investigator's main topic of interest is the applicationof analytic methods to topological and geometric problems. Moreprecisely, in the first part of the proposal the principalinvestigator, motivated by the existence of representations offundamental groups of knot complements, suggests a connection betweenYang-Mills theory on Riemann surfaces and Teichmueller theory. In thesecond part, the principal investigator proposes to study certaincompactifications of character varieties of fundamental groups ofsurfaces and three dimensional manifolds. The investigator relatesthis to actions of groups on trees and incompressible surfaces onthree dimensional manifolds. In the rest of the proposal, theprincipal investigator suggests certain questions about the metric ofthe monopole moduli space which appear in physics.The main motivation of this work is to understand questions about thetopology of three dimensional manifolds by studying representations oftheir fundamental groups. The results in this project will advance ourknowledge in questions relating gauge theory on Riemann surfaces,harmonic maps, Teichmueller theory and three dimensional manifoldtopology. In addition, work in this subject will facilitate furtherconnections between certain aspects of mathematics and physics.