Moduli of Vector Bundles on Riemann Surfaces and Applications to Topology

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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.