Harmonic maps between singular spaces, Gauge theory and applications

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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.