PREFACE ix Although the physicist is interested in a single quasicrystal (or a single tiling), mathematicians like to define spaces. From the tiling T we construct a space ΩT of tilings that have the same properties of T . If T has desirable properties (like finite local complexity, repetitivity, and well-defined patch frequencies), then ΩT has corresponding properties (compactness, minimal- ity as a dynamical system, and unique ergodicity), and we can ask the following mathematical questions: • M1. What is the topology of ΩT ? What does the neighborhood of a point of ΩT look like? What are the (Cech) ˇ cohomology groups of ΩT ? • M2. There is a natural action of the group Rd of translations on ΩT . This makes ΩT into a dynamical system, with d commuting flows. What are the ergodic measures on ΩT ? For each such measure, what is the spectrum of the generator of translations (think: momentum operator) on L2(ΩT )? This is called the dynamical spectrum of ΩT . • M3. From the action of the translation group on ΩT , one can construct a C∗ algebra. What is the K-theory of this C∗-algebra? Remarkably, each math question about ΩT answers a physics question about a material modeled on T . M1 answers P3, M2 answers P1, and M3 (in large part) answers P2. Far from being a pointless mathematical abstraction, tiling spaces are important! This book is the story of the first mathematical question, and the answers we have gleaned so far. In chapter 1 we consider a variety of interesting tilings and the construction of the corresponding tiling spaces. In chapter 2 we explore the local structure of ΩT and its realization as an inverse limit space. In chapter 3 we introduce the ˇ Cech cohomology of ΩT and show how the answer to the third physics question is tied to the first ˇ Cech cohomology. In chapter 4 we study the rotational properties of tilings — what made quasicrystals and the Penrose tiling so amazing in the first place! In chapter 5 we introduce “pattern-equivariant cohomology”, a beautiful realization of tiling cohomology in terms of properties of an individual tiling. In this way we come full circle, from tilings to tiling spaces and back to individual tilings. The material in the first four chapters is basically set in place, as is some of the material of chapter 5. Chapters 6 and 7, however, are cutting-edge research. The reader may have some diﬃculty with these chapters, both because the concepts aren’t as neatly prescribed as the earlier topics, and because the calculations require more advanced algebraic topology. Chapter 6 is devoted to “tricks of the trade”, recently developed calcula- tional techniques that are powerful but are not generally known. Chapter 7 is about tilings without the simplifying assumption of finite local complexity. Until recently, such tilings were thought to be beyond our understanding, but that is rapidly changing.

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