What is the scalar curvature on non-smooth spaces?

Abstract:

Since the scalar curvature appears in the term of Einstein field equations, the study of it becomes also important in general relativity. One of the open question about it is how to define the non-negative scalar curvature on non-smooth spaces. We will propose two definitions: one is the MV-scalar curvature on a closed topological manifold and the other one is n-volumic scalar curvature on a compact metric measure space.