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MP: Theoretische und Mathematische Grundlagen der Physik

MP XIV: HV XIV

MP XIV.1: Invited Talk

Friday, March 21, 1997, 11:45–12:30, HS 118

Topological Theory of Tilings and Gap-Labelling — •Johannes Kellendonk — Technische Universität Berlin, Fachbereich Mathematik, Sekr. MA 7-2, 10623 Berlin

We introduce the foundations of a theory which is based on the building blocks of tilings, i.e. their finite patches, and aims at the computation of topological invariants for tilings.

The patches of a tiling determine an inverse semigroup. The groupoid associated to the tiling, and hence, also the associated C*-algebra, can be constructed from that semigroup. But the advantage lies in the possibility of expressing the topological invariants of the groupoid (groupoid cohomology) and those of the C*-algebra (K-groups) in terms of the semigroup. We focus our attention to the invariant given by the group of coinvariants and show how to obtain it for certain kind of selfsimilar tilings. This yields a solution of the Gap-Labelling (in Bellissard’s K-theoretical formulation) for particles moving in such a tiling.