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

规范理论、奇异空间调和图以及拓扑应用

• 批准号: 0604930
• 负责人: Georgios Daskalopoulos
• 金额: $ 12.4万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604930&HistoricalAwards=false
• 关键词: Gauge; Theory; Harmonic; Maps; Singular

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

这份提案包含三个项目。首先，我们研究了从多面体区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、特征标簇和Hodge理论。在第二个项目中，我们建议继续我们以前关于Kaehler流形上的Yang-Mills流的工作。根据初始条件的Hard-Narasimhan层结，提出了关于Yang-Mills流的收敛和爆破集的几个猜想。这些猜想已经在Kaehler曲面的情况下被PI证明。最后，在第三个项目中，我们提出了各种例子(大部分是无限维的)，这些例子可以为超Kaehler商的Kirwan满射性的方向提供一些启发。这些例子包括Higgs丛、箭图簇和K3曲面上的向量丛等。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，鼓励研究生撰写几何分析、规范理论和拓扑学领域的博士论文。这些都是非常重要的领域，因为它们与几何和数学物理中的其他领域有联系。对于数学和物理之间的跨学科研究来说，这也是一个非常好的方向，PI一直在本科生和研究生中追求这一点