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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

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Teoria de grafs --- Anàlisi combinatòria --- Topologia --- Anàlisi de xarxes (Planificació) --- Arbres (Teoria de grafs) --- Matroides --- Teoria de Ramsey --- Xarxes de Petri --- Teoria de l'aparellament --- Functional analysis. --- Graph theory. --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Graph theory --- Graphs, Theory of --- Theory of graphs --- Combinatorial analysis --- Topology --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Extremal problems

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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.

Number theory --- Algebraic geometry --- Differential geometry. Global analysis --- Geometry --- Operator theory --- Ergodic theory. Information theory --- Discrete mathematics --- Mathematics --- Classical mechanics. Field theory --- Geology. Earth sciences --- analyse (wiskunde) --- topologie (wiskunde) --- discrete wiskunde --- statistiek --- wiskunde --- getallenleer --- geofysica --- geometrie --- dynamica --- informatietheorie

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These proceedings represent the current state of research on the topics 'boundary theory' and 'spectral and probability theory' of random walks on infinite graphs. They are the result of the two workshops held in Styria (Graz and St. Kathrein am Offenegg, Austria) between June 29th and July 5th, 2009. Many of the participants joined both meetings. Even though the perspectives range from very different fields of mathematics, they all contribute with important results to the same wonderful topic from structure theory, which, by extending a quotation of Laurent Saloff-Coste, could be described by 'exploration of groups by random processes'. Contributors: M. Arnaudon A. Bendikov M. Björklund B. Bobikau D. D'Angeli A. Donno M.J. Dunwoody A. Erschler R. Froese A. Gnedin Y. Guivarc'h S. Haeseler D. Hasler P.E.T. Jorgensen M. Keller I. Krasovsky P. Müller T. Nagnibeda J. Parkinson E.P.J. Pearse C. Pittet C.R.E. Raja B. Schapira W. Spitzer P. Stollmann A. Thalmaier T.S. Turova R.K. Wojciechowski

Operational research. Game theory --- stochastische analyse --- kansrekening

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The interplay of geometry, spectral theory and stochastics has a long and fruitful history, and is the driving force behind many developments in modern mathematics. Bringing together contributions from a 2017 conference at the University of Potsdam, this volume focuses on global effects of local properties. Exploring the similarities and differences between the discrete and the continuous settings is of great interest to both researchers and graduate students in geometric analysis. The range of survey articles presented in this volume give an expository overview of various topics, including curvature, the effects of geometry on the spectrum, geometric group theory, and spectral theory of Laplacian and Schrödinger operators. Also included are shorter articles focusing on specific techniques and problems, allowing the reader to get to the heart of several key topics.

Geometric analysis --- Graph theory --- Manifolds (Mathematics)