On the regularity of Alexandrov spaces

Alexandrov spaces with curvature bounded from below, finite Hausdorff dimension, and equipped with the corresponding Hausdorff measure are particular instances of metric measure spaces with Ricci curvature bounded from below (namely RCD spaces). In this talk, I will give results on the regularity of an Alexandrov space considered as a singular Riemannian manifold. These results are based on joint work with Luigi Ambrosio.