Calculating K-theory of substitution tiling C*-algebras using dual tilings

Greg Maloney (University of Newcastle)

Friday 14th March, 2014 15:55-16:40 504 Joseph Black Building

Abstract

(joint work with Franz Gähler and John Hunton)

When studying aperiodic tilings, rather than looking at a single tiling one typically considers a topological space, the points of which are tilings. Under standard assumptions, this topological space is pathological, in the sense that the orbit of any point under the translation action is dense. One method of dealing with this pathology is to build a C*-algebra from the space; this C*-algebra is the groupoid C*-algebra of the translation groupoid of the topological space.
In the special case that the tilings arise from a substitution, Anderson and Putnam have described a method that, in low dimensions, allows us to calculate the K-theory of this algebra. But this calculation can be difficult: some interesting examples are impossible to calculate without the aid of a computer, and other interesting examples are impossible to calculate even with the aid of a computer. I will describe a modification to the Anderson-Putnam method that enables us to calculate the K-theory in cases that were previously intractable, and I will mention some results that have been found using this method.

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