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Topological Invariants for Projection Method Patterns
 
Alan Forrest , Glasgow, Scotland
John Hunton University of Leicester, Leicester, England
Johannes Kellendonk Cardiff University, Cardiff, Wales
Front Cover for Topological Invariants for Projection Method Patterns
Available Formats:
Electronic ISBN: 978-1-4704-0351-5
Product Code: MEMO/159/758.E
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $37.20
Front Cover for Topological Invariants for Projection Method Patterns
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  • Front Cover for Topological Invariants for Projection Method Patterns
  • Back Cover for Topological Invariants for Projection Method Patterns
Topological Invariants for Projection Method Patterns
Alan Forrest , Glasgow, Scotland
John Hunton University of Leicester, Leicester, England
Johannes Kellendonk Cardiff University, Cardiff, Wales
Available Formats:
Electronic ISBN:  978-1-4704-0351-5
Product Code:  MEMO/159/758.E
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $37.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1592002; 120 pp
    MSC: Primary 52; Secondary 19; 37; 46; 55; 82;

    This memoir develops, discusses and compares a range of commutative and non-commutative 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 so-called 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 self-similarity, 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 Ammann-Kramer 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.

    Readership

    Graduate 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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Volume: 1592002; 120 pp
MSC: Primary 52; Secondary 19; 37; 46; 55; 82;

This memoir develops, discusses and compares a range of commutative and non-commutative 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 so-called 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 self-similarity, 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 Ammann-Kramer 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.

Readership

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
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