We introduce the notions of a path complex and its homologies. Particular cases of path homologies are simplicial homologies and digraph homologies. We state and prove some properties of path homologies, in particular, the K¨unneth formulas for Cartesian product and join, which happen to be true at the level of chain complexes.

We prove that there is a unique way to construct a geometric scale of Hilbert spaces interpolating between two given spaces. We investigate what properties of operators, other than boundedness, are preserved by interpolation. We show that self-adjointness is, but subnormality and Krein subnormality are not. On the way to this last result, we establish a representation theorem for cyclic Krein subnormal operators.

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We introduce and analyze lower ($Ricci$) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$for metric measure spaces $ {\left( {M,d,m} \right)} $ . Our definition is based on convexity properties of the relative entropy $ Ent{\left( { \cdot \left| m \right.} \right)} $ regarded as a function on the$L$_{2}-Wasserstein space of probability measures on the metric space $ {\left( {M,d} \right)} $ . Among others, we show that $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$if and only if $ Ric_{M} {\left( {\xi ,\xi } \right)} $ ⩾$K$ $ {\left| \xi \right|}^{2} $ for all $ \xi \in TM $ .