Harmonic maps into singular spaces

In the 90's, Gromov and Schoen introduced the theory of harmonic maps into singular spaces, in particular Euclidean buildings, in order to understand p-adic superrigidity. The study was quickly generalized in a number of directions by a number of authors. This talk will focus on the work initiated by Korevaar and Schoen on harmonic maps into metric spaces with curvature bounded above in the sense of Alexandrov. I will describe the variational characterization of harmonic maps into such spaces, some analytic consequences, and in particular a Bochner formula capturing the role of both the domain and target curvatures