Appears in collection : Not Only Scalar Curvature Seminar

We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to $C^0$ metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$ initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from $C^0$ initial data.