Harmonic Maps Between Singular Spaces, Gauge Theory, and Applications

奇异空间、规范理论和应用之间的调和图

• 批准号: 1608764
• 负责人: Georgios Daskalopoulos
• 金额: $ 35.52万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608764&HistoricalAwards=false
• 关键词: Harmonic; Maps; Between; Singular; Spaces

Most objects in nature are non-smooth; in other words, they have singularities. Mathematics has had tremendous success in dealing with smooth objects--for example, surfaces or more generally manifolds. In this project, the PI proposes the study of calculus on singular spaces and in particular the study of maps between such spaces that are optimal in the sense of minimizing energy. From this, the PI will deduce properties of symmetries of such spaces that are of interest in mathematics and applications. The project also has an education component in which the PI plans to advise graduate students.The PI will study several questions dealing with harmonic maps between singular spaces. The main goal is to prove that, despite the singularities of the ambient spaces, harmonic maps have sufficient regularity so that geometric rigidity holds. Several applications to geometric group theory, character varieties, measured foliations, and Teichmuller theory will be pursued. These questions are a continuation of the PI's work on geometric super rigidity via harmonic maps.

自然界中的大多数物体都是不光滑的；换句话说，它们有奇点。数学在处理光滑物体方面取得了巨大的成功，例如，表面或更普遍的流形。在这个项目中，PI建议研究奇异空间上的微积分，特别是研究这些空间之间的映射，这些映射在最小化能量的意义上是最优的。由此，PI将推导出在数学和应用中感兴趣的这些空间的对称性的性质。该项目还有一个教育部分，PI计划在其中为研究生提供建议。PI将研究几个与奇异空间之间的调和映射有关的问题。主要的目标是证明,尽管环境空间的奇异点,谐波地图有足够的规律性,几何刚度。几个应用几何群论，特征变异，测量叶，和Teichmuller理论将进行。这些问题是PI通过谐波映射研究几何超刚性工作的延续。