Mathematics of Aperiodic Order

What is order which 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 by now become a well-established and strongly evolving field of mathematical research. It is closely tied to a surprising variety of branches of mathematics and physics. The book offers an overview over the state of the art in the field of aperiodic order. It comprises carefully selected surveys which are written by leading researchers. Since the book is written for a readership of non-experts which have a general background in mathematics, theoretical physics or computer science, it will serve as a highly accessible first hand source of information to anybody interested in this rich and exciting field. Topics covered in the book 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.