Electronic ISBN:  9781470403515 
Product Code:  MEMO/159/758.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 159; 2002; 120 ppMSC: Primary 52; Secondary 19; 37; 46; 55; 82;
This memoir develops, discusses and compares a range of commutative and noncommutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smaller dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general — any compact set which is the closure of its interior — while in the second half we concentrate on the socalled canonical patterns. The topological invariants used are various forms of \(K\)theory and cohomology applied to a variety of both \(C^*\)algebras and dynamical systems derived from such a pattern.
The invariants considered all aim to capture geometric properties of the original patterns, such as quasiperiodicity or selfsimilarity, but one of the main motivations is also to provide an accessible approach to the the \(K_0\) group of the algebra of observables associated to a quasicrystal with atoms arranged on such a pattern.
The main results provide complete descriptions of the (unordered) \(K\)theory and cohomology of codimension 1 projection patterns, formulæ for these invariants for codimension 2 and 3 canonical projection patterns, general methods for higher codimension patterns and a closed formula for the Euler characteristic of arbitrary canonical projection patterns. Computations are made for the AmmannKramer tiling. Also included are qualitative descriptions of these invariants for generic canonical projection patterns. Further results include an obstruction to a tiling arising as a substitution and an obstruction to a substitution pattern arising as a projection. One corollary is that, generically, projection patterns cannot be derived via substitution systems.ReadershipGraduate students and research mathematicians interested in convex and discrete geometry.

Table of Contents

Chapters

General introduction

I. Topological spaces and dynamical systems

II. Groupoids, $C$*algebras, and their invariants

III. Approaches to calculation I: Cohomology for codimension one

IV. Approaches to calculation II: Infinitely generated cohomology

V. Approaches to calculation III: Cohomology for small codimension


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This memoir develops, discusses and compares a range of commutative and noncommutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smaller dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general — any compact set which is the closure of its interior — while in the second half we concentrate on the socalled canonical patterns. The topological invariants used are various forms of \(K\)theory and cohomology applied to a variety of both \(C^*\)algebras and dynamical systems derived from such a pattern.
The invariants considered all aim to capture geometric properties of the original patterns, such as quasiperiodicity or selfsimilarity, but one of the main motivations is also to provide an accessible approach to the the \(K_0\) group of the algebra of observables associated to a quasicrystal with atoms arranged on such a pattern.
The main results provide complete descriptions of the (unordered) \(K\)theory and cohomology of codimension 1 projection patterns, formulæ for these invariants for codimension 2 and 3 canonical projection patterns, general methods for higher codimension patterns and a closed formula for the Euler characteristic of arbitrary canonical projection patterns. Computations are made for the AmmannKramer tiling. Also included are qualitative descriptions of these invariants for generic canonical projection patterns. Further results include an obstruction to a tiling arising as a substitution and an obstruction to a substitution pattern arising as a projection. One corollary is that, generically, projection patterns cannot be derived via substitution systems.
Graduate students and research mathematicians interested in convex and discrete geometry.

Chapters

General introduction

I. Topological spaces and dynamical systems

II. Groupoids, $C$*algebras, and their invariants

III. Approaches to calculation I: Cohomology for codimension one

IV. Approaches to calculation II: Infinitely generated cohomology

V. Approaches to calculation III: Cohomology for small codimension