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

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This proposal contains three projects. In the first, we propose to study harmonic maps from polyhedral domains to arbitrary nonpositively curved metric spaces. The main application is in geometric superrigidity, geometric group theory, character varieties and Hodge theory. In the second project we propose a continuation of our previous work on the Yang-Mills flow on Kaehler manifolds. Several conjectures are stated about the convergence and the blow-up set of the Yang-Mills flow in terms of the Harder-Narasimhan stratification of the initial condition. These conjectures have already been proved by the PI in the case of Kaehler surfaces. Finally in the third project we propose a variety of examples (mostly infinite dimensional) that could shed some light in the direction of Kirwan surjectivity for hyperkaehler quotients. These examples include Higgs bundles, quiver varieties and vector bundles on K3 surfaces among others. In this direction we propose some serious research in pure mathematics that would encourage graduate students to write Ph. D Theses in the fields of geometric analysis, gauge theory and topology. These are very important fields because of their connection with other fields in geometry and mathematical physics. It is also a very good direction for interdisciplinary studies between mathematics and physics, something that the PI has pursued both with undergraduate and graduate students