Remarks about Synthetic Upper Ricci Bounds for Metric Measure Spaces
  We discuss various characterizations of synthetic upper Ricci bounds for
metric measure spaces in terms of heat flow, entropy and optimal transport. In
particular, we present a characterization in terms of semiconcavity of the
entropy along certain Wasserstein geodesics which is stable under convergence
of mm-spaces. And we prove that a related characterization is equivalent to an
asymptotic lower bound on the growth of the Wasseretein distance between heat
flows. For weighted Riemannian manifolds, the crucial result will be a precise
uniform two-sided bound for \begin{eqnarray*}\frac{d}{dt}\Big|_{t=0}W\big(\hat
P_t\delta_x,\hat P_t\delta_y\big)\end{eqnarray*} in terms of the mean value of
the Bakry-Emery Ricci tensor $\mathrm{Ric}+\mathrm{Hess}\, f$ along the
minimizing geodesic from $x$ to $y$ and an explicit correction term depending
on the bound for the curvature along this curve.
