Harmonic Maps Between Singular Spaces, Gauge Theory, and Applications

自然界中的大多数物体都是非光滑的；换句话说，它们具有奇点。数学在处理光滑对象方面取得了巨大的成功--例如，曲面或更一般的流形。在这个项目中，PI建议研究奇异空间上的微积分，特别是研究这种空间之间的映射，这些映射在能量最小化意义上是最优的。由此，PI将推导出在数学和应用中感兴趣的此类空间的对称性的性质。该项目也有一个教育部分，其中PI计划为研究生提供建议。PI将研究几个问题，涉及奇异空间之间的调和映射。主要目的是证明，尽管环境空间具有奇性，调和映射仍具有足够的正则性，使得几何刚性成立。本课程将探讨几何群论、特征变量、测量的叶层和泰希穆勒理论的几个应用。这些问题是PI通过调和映射研究几何超刚性的继续。

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.