Gauge Theory, Harmonic Maps to Singular Spaces and Applications to Topology

Brown University, Providence RI

Investigators

Abstract

DMS-0204191 PI: Georgios Daskalopoulos The PI proposes three projects. In the first, the PI proposes to study regularity of harmonic maps into the completion of Teichmueller space. The two main applications are on symplectic Lefschetz pencils and on superrigidity of lattices in the mapping class group. In the second project the PI proposes a proof of higher dimensional partial analogues of the conjectures of Atiyah and Bott on Yang-Mills equations over Riemann surfaces. These are nontrivial generalizations to higher dimensions of the PI's previous work on the connection between Yang-Mills and the theory of holomorphic vector bundles. Finally in the third project the PI proposes some questions about harmonic maps from three-dimensional manifolds to trees which generalize the well-known theory of harmonic maps from surfaces. The motivation for the first project is to use harmonic maps in order to understand analytically the results of Kaimanovich-Masur and Farb-Masur on the superrigidity of lattices of rank at least 2 in the mapping class group. In the process the PI realized that the methods should also shed light in the rank one case. The second project shows that there are still very close connections between the analysis of the Yang-Mills equations and algebraic geometry. These connections have been exploited in great detail by several people in the case of Riemann surfaces but very little is known in higher dimensions. The PI's work sheds some light in this direction.