Appears in collection : Not Only Scalar Curvature Seminar

Harmonic maps are critical points for the energy and existence and compactness results for harmonic maps have played a major role in the advancement of geometric analysis. Gromov-Schoen and Korevaar-Schoen developed a theory of harmonic maps into metric spaces with non-positive curvature in order to address rigidity problems in geometric group theory. In this talk we consider harmonic maps into metric spaces with upper curvature bounds. We will define these objects, state some key results, and highlight their application to rigidity and uniformization problems. We finish the talk by discussing some recently determined Bochner inequalities for maps from (possibly) singular domains.