Harmonic maps between singular spaces

After reviewing briefly the classical theory of harmonic maps between smooth manifolds, I will describe some recent results related to harmonic maps with free boundary, emphasizing on two different approaches based on recent developments by Da Lio and Riviere. This latter approach allows in particular to give another formulation which is well-suited for such maps between singular spaces. After the works of Gromov, Korevaar and Schoen, harmonic maps between singular spaces have been instrumental to investigate super-rigidity in geometry. I will report on recent results where we introduce a new energy between singular spaces and prove a version of Takahashi’s theorem (related to minimal immersions by eigenfunctions) on RCD spaces.