Harmonic maps between singular spaces, Gauge theory and applications

PI提出研究奇异空间之间调和映射的几个问题。主要目标是证明，尽管周围空间的奇异性，调和映射有足够的规律性，使几何刚性举行。几何群论的几个应用，字符品种，衡量叶理和Teichmueller理论正在提出。这些问题是PI通过调和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，它们具有奇点。数学主要用于处理光滑的物体，例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提出了奇异空间上的微积分的研究，特别是在最小化能量的意义上最优的空间之间的映射的研究。从这一点，他可以推导出这些空间（群）的对称性的性质，这些性质在数学和应用中很有意义。

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.