After recalling the two main approaches to the theory of synthetic Ricci bounds, I will present the theory of "tamed spaces”, which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature seen from an Eulerian point of view. In particular, the approach is based on the analysis of singular perturbations of Dirichlet forms by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality generalizing the well-known Bakry-Emery curvature-dimension condition. Among other things, we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup as well as consequences in terms of functional inequalities. Examples of tamed spaces include in particular Riemannian manifolds with either interior singularities or singular boundary behavior.This is a joint work with Matthias Erbar, Theo Sturm, and Luca Tamanini.
