Harmonic maps between singular spaces, Gauge theory and applications

奇异空间之间的调和映射、规范理论及其应用

• 批准号: 1308708
• 负责人: Georgios Daskalopoulos
• 金额: $ 19.85万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308708&HistoricalAwards=false
• 关键词: Harmonic; maps; between; singular; spaces

The PI proposes to study several questions on 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 Teichmueller theory are being proposed. These questions are a continuation of the PI's work on geometric superrigidity via harmonic maps.Most objects in nature are non-smooth, in other words they have singularities. Mathematics is primarily designed to deal with smooth objects, for example something like the surface of the plane or the sphere or more generally objects without corners or complicated edges. 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, he can deduce properties of symmetries of such spaces (groups) that are of interest in mathematics and applications.

PI提出研究奇异空间间调和映射的几个问题。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。提出了在几何群论、特征变异、测量叶理和Teichmueller理论等方面的应用。这些问题是PI通过谐波映射研究几何超刚性工作的延续。自然界中的大多数物体都是非光滑的，换句话说，它们都有奇点。数学主要是用来处理光滑的物体，比如平面或球体的表面，或者更一般的没有角或复杂边缘的物体。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，他可以推导出这些空间（群）的对称性性质，这些性质在数学和应用中都很有意义。