Higher-rank graphs (k-graphs) are a combinatorial model for C*-algebras; indeed, much of the structure of k-graph C*-algebras is encoded in the underlying combinatorial data of the k-graph.  However, different k-graphs can give rise to isomorphic or Morita equivalent C*-algebras.  In this talk, we present several ways to modify the structure of a k-graph which preserve the Morita equivalence class of the associated C*-algebra.  Our constructions are inspired by the analogous work for graph C*-algebras of Bates and Pask, as well as by the textile system approach to describing k-graphs.  This is joint work with C. Eckhardt, K. Fieldhouse, D. Gent, I. Gonzales, and D. Pask.

Given a Hilbert C*-correspondence over C(X), one can construct the associated Cuntz-Pimsner algebra. In the case that the correspondence is full, nonperiodic, and minimal, the resulting C*-algebra is simple and unital. An example of such a correspondence is obtained by taking the right Hilbert C(X)-module of continuous sections of a vector bundle over X and twisting the left multiplication by a minimal homeomorphism. As is the case of crossed products by minimal homeomorphisms, one can identify "orbit-breaking" C*-subalgebras. In this talk I will discuss these Cuntz--Pimsner C*-algebras and their orbit-breaking subalgebras, their relationship to one another, and when they can be classified by Elliott invariants. This is joint work with Adamo, Archey, Forough, Georgescu, Jeong and Viola.

In this talk I will explain how to define so-called “higher genera” for a proper cocompact action of a Lie group satisfying some additional assumptions. These higher genera are numbers, whose definition is inspired by index theory for Dirac type operators in Noncommutative Geometry, and generalize the signature and A-hat genus for compact manifolds. The main theorem states that the higher signatures are equivariant homotopy invariants and that vanishing of the higher A-hat numbers is a necessary condition for the existence of an invariant spin structure together with a metric of positive scalar curvature. This talk is based on joint work with P. Piazza.