Proposal: DMS-9803606 Principal Investigator: Georgios Daskalopoulos
The principal investigator's main topic of interest is the application of analytic methods to topological and geometric problems. More precisely, in the first part of the proposal the principal investigator, motivated by the existence of representations of fundamental groups of knot complements, suggests a connection between Yang-Mills theory on Riemann surfaces and Teichmueller theory. In the second part, the principal investigator proposes to study certain compactifications of character varieties of fundamental groups of surfaces and three dimensional manifolds. The investigator relates this to actions of groups on trees and incompressible surfaces on three dimensional manifolds. In the rest of the proposal, the principal investigator suggests certain questions about the metric of the monopole moduli space which appear in physics.
The main motivation of this work is to understand questions about the topology of three dimensional manifolds by studying representations of their fundamental groups. The results in this project will advance our knowledge in questions relating gauge theory on Riemann surfaces, harmonic maps, Teichmueller theory and three dimensional manifold topology. In addition, work in this subject will facilitate further connections between certain aspects of mathematics and physics.
