(joint work with K. Kuwada and K.Th. Sturm) Bochner's inequality is one of the fundamental estimates in geometric analysis on Riemannian manifolds. It states that
2-\langle\nabla u, \nabla\Delta u\rangle\ge K \cdot
for each smooth function $u$ on a Riemannian manifold provided K is a lower bound for the Ricci curvature on and N is an upper bound for the dimension.
The main result I present in this talk is a Bochner inequality on infinitesimally Hilbertian metric measure spaces satisfying the (reduced) curvature-dimension condition. Moreover, also the converse is true: an appropriate version of the Bochner inequality (for the canonical gradient and Laplacian) on an infinitesimally Hilbertian mms will imply the reduced curvature-dimension condition. Our approach relies on the new so-called entropic curvature-dimension condition which encodes bounds on the curvature and dimension through refined convexity properties of the Boltzmann entropy (and not the Renyi entropy). Besides that, I will present also new, sharp Wasserstein-contraction results for the heat flow as well as Bakry-Ledoux type gradient estimates for the heat semigroup, each of which are equivalent to the (reduced) curvature-dimension condition.