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Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The underlying mathematics, known as the theory of aperiodic order, is the subject of this comprehensive multi-volume series. This first volume provides a graduate-level introduction to the many facets of this relatively new area of mathematics. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. In particular, the authors provide a systematic exposition of the mathematical theory of kinematic diffraction. Numerous illustrations and worked-out examples help the reader to bridge the gap between theory and application. The authors also point to more advanced topics to show how the theory interacts with other areas of pure and applied mathematics.

Aperiodicity --- Crystallography, Mathematical --- Aperiodic tilings --- Quasicrystals --- Pavage (mathématiques) --- Quasicristaux --- Mathematics --- Mathématiques --- Mathématiques. --- Aperiodic tilings. --- Quasi-crystals --- Condensed matter --- Crystals --- Aperiodic point sets --- Sets, Aperiodic point --- Discrete geometry --- Point set theory --- Tiling (Mathematics) --- Mathematics.

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What is order that is not based on simple repetition, that is, periodicity? How must atoms be arranged in a material so that it diffracts like a quasicrystal? How can we describe aperiodically ordered systems mathematically? Originally triggered by the – later Nobel prize-winning – discovery of quasicrystals, the investigation of aperiodic order has since become a well-established and rapidly evolving field of mathematical research with close ties to a surprising variety of branches of mathematics and physics. This book offers an overview of the state of the art in the field of aperiodic order, presented in carefully selected authoritative surveys. It is intended for non-experts with a general background in mathematics, theoretical physics or computer science, and offers a highly accessible source of first-hand information for all those interested in this rich and exciting field. Topics covered include the mathematical theory of diffraction, the dynamical systems of tilings or Delone sets, their cohomology and non-commutative geometry, the Pisot substitution conjecture, aperiodic Schrödinger operators, and connections to arithmetic number theory.

Mathematics. --- Convex and Discrete Geometry. --- Dynamical Systems and Ergodic Theory. --- Operator Theory. --- Number Theory. --- Global Analysis and Analysis on Manifolds. --- Differentiable dynamical systems. --- Global analysis. --- Operator theory. --- Discrete groups. --- Number theory. --- Mathématiques --- Dynamique différentiable --- Théorie des opérateurs --- Groupes discrets --- Théorie des nombres --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Aperiodic tilings. --- Aperiodicity. --- Aperiodic point sets --- Sets, Aperiodic point --- Dynamics. --- Ergodic theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Convex geometry. --- Discrete geometry. --- Chaotic behavior in systems --- Discrete geometry --- Point set theory --- Tiling (Mathematics) --- Number study --- Numbers, Theory of --- Algebra --- Functional analysis --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry . --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Combinatorial geometry