Topological methods for aperiodic tilings Ian F. Putnam (University of Victoria), Lorenzo Sadun (University of Texas, Austin) October 13 to October 17, 2008 1 Overview of the Field The study of aperiodic tilings began with the work of Hao Wang in the 1950’s. It was energized by examples given by Raphael Robinson in the 1960’s and, more famously, Roger Penrose in the 1970’s. See [Se, GS] for overviews of the subject. Penrose’s example was striking because it admitted rotational symmetries which are impossible in periodic tilings. Perhaps the most significant development was in the early 1980’s, when physical materials, now called quasi-crystals, were discovered which possessed the same rotational symmetries forbidden in periodic structures and yet displayed a high degree or regularity [SBGC]. The field since then has been characterized by an interesting mix of a wide variety of mathematical and physical subjects. Among the former are discrete geometry, harmonic analysis, ergodic theory, operator algebras and topology. This workshop was designed to highlight recent progress in the areas of topology and ergodic theory [Sa1]. Ergodic theory and dynamical systems are natural tools for the study of aperiodic patterns. Rather than study a single aperiodic pattern, dynamical systems prefers to study a collection of such objects which are globally invariant under translations (and perhaps other rigid motions of the underlying space as well). In fact, many of the examples are exactly that: a class of objects with very similar properties. At the same time, it is very natural to construct a class from a single pattern by looking at all other patterns which have exactly the same collection of local data. More specifically, given an aperiodic pattern P , look at all P ′ such that the intersection of P ′ with any bounded region appears somewhere in P . Finally, this approach of looking at a collection of patterns grew naturally in physical models for quasi-crystals. The key step in this approach is establishing some kind of finiteness condition of the collection. Looking at such systems from a probabilistic point of view, this means finding a finite measure which is invariant under the translation action. In most examples, this measure not only exists, but is unique. From a topological point of view, the aim is to find some natural topology in which the action is continuous and the collection is compact. Compactness is the analyst’s analogue of finite. Such topologies were easily developed - they are natural extensions of topologies on shift spaces which are standard in symbolic dynamics. To elaborate a little, suppose one considers point sets in a Euclidean space which are uniformly dense (for some fixed positive radius R, the R-balls around all points cover the space) and uniformly discrete (for some fixed radius r, the r-balls around the points are pairwise disjoint). Such a set is called a Delone set, but here we require the same two constants to be good for all elements of the collection. A neighbourhood base for the topology is as follows. For a fixed point set P , open set U in the underlying space and positive constant ǫ, look at all other points sets P ′ such that P ′ ∩ U is within ǫ of the restriction of P ∩ U . (This is not quite correct because of what is happening near the boundary of U , but it will suffice for the moment.) This topology first appeared as a tool, since it makes the collections of interest compact and hence many standard results of dynamics may apply. Later, it was observed by a number of people that the local structure 1