The analytic-surgery sequence is a long exact sequence of K-theory groups which combines topological information (the K-homology of manifolds), index-theoretic information (the K-theory of group C*-algebras), and secondary index information (the analytic structure group of Higson-Roe). We give a new definition of the terms based entirely on algebras of pseudodifferential operators and their K-theory. We use this to develop systematically maps to an exact sequence of non-commutative de Rham homology / cyclic homology. Via pairings with cyclic cohomology classes, this gives rise to new numeric secondary index invariants (higher rho numbers) with explicit formulas and calculation tools. We use this for geometric applications. In particular, we derive new information about the moduli space of Riemannian metrics of positive scalar curvature, where we give new lower bounds on the number of its components. In this talk, a focus will be on the issues of constructing suitable cyclic cocycles to detect structural information. This is joint work of Paolo Piazza, Thomas Schick, Vito Zenobi.