Harmonic Maps Between Singular Spaces, Gauge Theory, and Applications

Brown University, Providence RI

Investigators

Abstract

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.