Research

In low dimensions, this has been applied to verify Matui's HK conjecture that for certain ample groupoids the K-theory groups are a direct sum of the homology groups of the same parity. In higher dimensions, it is more difficult to transfer homological data to K-theoretic data directly from the spectral sequence, as we don't know how the pieces are put together to build the K-theory groups. However, a morphism of spectral sequences is able to convert an isomorphism of homological data into an isomorphism of K-theory groups even in higher dimensions. My work allows us to build a morphism of ABC spectral sequences from a proper groupoid correspondence, and we obtain a functorial version of the above diagram.

Applications 

Wielding a tool designed to turn proper groupoid correspondences into morphisms of spectral sequences, I am on the market for interesting examples of groupoid correspondences! Please get in touch with me if you come across any. 

One interesting example of a groupoid correspondence comes from an inverse semigroup S. This acts both on its set of non-zero idempotents and on the space of filters of these idempotents, from which we obtain two groupoids. Each idempotent corresponds to a compact open subset of the space of filters, which allows us to construct a proper groupoid correspondence. The induced morphism of spectral sequences enables us to prove a K-theory formula for the inverse semigroup C*-algebra C*(S), of the kind found in this paper by Xin Li.