Quantitative K-theory, positive scalar curvature, and band width

We develop two connections between the quantitative framework of operator $K$-theory for geometric $C^*$-algebras and the problem of positive scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a refinement of the well-known obstruction of Rosenberg to positive scalar curvature on closed spin manifolds coming from the higher index of the Dirac operator. We show that on a manifold with uniformly positive scalar curvature, the propagation at which the ind...