We study the heat equation on time-dependent metric measure spaces and its dual as gradient flows for the energy and for the Boltzmann entropy, resp. Monotonicity estimates for transportation distances and for squared gradients will be shown to be equivalent to the so-called dynamical convexity of the Boltzmann entropy on the Wasserstein space which is the defining property of super-Ricci flows.

Moreover, we discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. These concepts allow us e.g. to prove that every Ricci bounded metric measure space which is a N-cone must be $R^{N+1}$.