(i) The novel concept of dynamical convexity allowed us to merge two cutting-edge developments (the study of `static' mm-spaces with synthetic Ricci bounds and the study of Ricci flows for `smooth' Riemannian manifolds) and to develop a theory of super Ricci flows for metric measure spaces. Moreover, we analyzed in great detail the heat flow on time-dependent metric measure spaces and characterized super-Ricci flows in terms of contraction properties of heat flows and numerous functional inequalities on such time-dependent, singular spaces. Even in the smooth time-dependent case and also in the static case, several of these characterizations were unknown before.
The gradient flow perspective on the space of probability measures was successfully studied for the heat flow in various other cases, including quantum heat semigroup, heat flow with Neumann boundary conditions and – mostly surprisingly – also heat flow with Dirichlet boundary conditions.
(ii) With “tamed spaces”, a novel direction of research for metric measure spaces with distribution-valued Ricci bounds was established. This in particular covers the important case of measure-valued curvature bounds arising from the curvature of the boundary in the analysis of the Neumann heat flow on non-convex domains. To deal with the non-convexity, the innovative “convexification method” has been developed. Moreover, novel transformation formulas for the parameters K and N in the curvature-dimension condition CD(K,N) under conformal transformations and under time changes of the underlying metric measure spaces turned out to be of great importance.
Furthermore, first and second order calculus on metric measure spaces could be developed further on spaces with synthetic lower Ricci bounds, and various novel concepts of synthetic upper Ricci bounds were presented and analyzed in detail. As important applications, new rigidity results could be obtained.
(iii) Probabilistic representations turned out to provide deep insights into the effects of variable curvature for gradient estimates and for transport estimates. This in particular also applies to the Neumann heat flow, taking into account also the effects of the (“singular”) curvature of the boundary. Of particular interest is the construction of optimally coupled pairs of Brownian motions. Furthermore, fundamental results concerning the equivalence of Eulerian and Lagrangian formulations of the curvature-dimension condition on metric measure spaces with variable Ricci bounds have been obtained. A more detailed analysis based on a pathwise calculus was presented on manifolds with Kato-class Ricci bounds.
Related results concern the short time asymptotic of the heat content, again depending in a sophisticated way on the curvature of the boundary.
Numerous further important results at the crossroad of probabilistic objects and curvature concepts were obtained, among them limit theorems on Poisson spaces, spectral estimates on random Riemannian manifolds, and conformally invariant Liouville quantum geometries.
