Functional analysis and metric geometry

Aim of the talk is to present some aspects of the important role that functional analysis has in the context of metric geometry. I shall discuss both the case of synthetic description of lower Ricci curvature bounds, where this role is by now well understood, and some potential applications to the world of lower sectional curvature bounds, where it might potentially lead to the solution of long-standing open problems including giving the definition of sectional curvature tensor.