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

该提案包含三个项目。首先，我们提出研究从多面体域到任意非正弯曲度量空间的调和映射。主要应用于几何超刚性、几何群论、特征变异和霍奇理论。在第二个项目中，我们建议继续我们之前关于Kaehler流形上的Yang-Mills流的工作。从初始条件的hard - narasimhan分层出发，对杨-米尔斯流的收敛性和爆破集提出了若干猜想。这些猜想已经在Kaehler曲面的情况下被PI证明了。最后，在第三个项目中，我们提出了各种各样的例子（大多数是无限维的），这些例子可以为超凯勒商的Kirwan满射方向提供一些启示。这些例子包括希格斯束、颤抖束和K3表面上的矢量束等。在这个方向上，我们建议在纯数学方面进行一些严肃的研究，鼓励研究生在几何分析、规范理论和拓扑领域撰写博士论文。这些是非常重要的领域，因为它们与几何和数学物理中的其他领域有联系。这也是数学和物理之间的跨学科研究的一个很好的方向，这是PI在本科生和研究生中所追求的

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