Abstract: Operator algebraists are so gregarious that we'll interact with any field. One of the most fruitful cross-fertilizations for us is with combinatorics, specifically (directed) graphs. Graph algebras are C*-algebras associated to graphs (what else?), and are intensively studied because they comprise a broad class of C*-algebras, whose properties can be seen from the structure of the graphs. There is a satisfying theory of coverings of graphs, completely parallel with the topological theory of covering spaces. In particular, if a group G acts freely on a graph E, then the quotient is also a graph, and the Gross-Tucker theorem shows how to reconstruct E from E/G. This has implications for the C*-algebras: the C*-algebra of E/G is Morita equivalent (a very good thing) to the crossed product of C*(E) by the action of G. After an elementary description of all this, I'll introduce a topological version, where the coverings are replaced by principal G-bundles. This is joint work with Valentin Deaconu and Alex Kumjian.
This talk will be accessible to graduate students.