Diffraction theory for aperiodic point sets in Lie groups
The study of aperiodic point sets in Euclidean space is a classical topic in harmonic analysis,
combinatorics and geometry. Aperiodic point sets in R^3 are models for quasi-crystals, and in
this context it is of interest to study their diffraction measure, i.e. the way they scatter an
incoming laser or x-ray beam. By a classical theorem of Meyer, every sufficiently regular
aperiodic point set in a Euclidean space is a shadow of a periodic one in a larger locally
compact abelian group. The diffraction of these "model sets" can be computed in terms of a
certain group of irrational rotations of an associated torus.
In this talk, I will review the classical theory of diffraction of Euclidean model sets and then
explain how the theory generalizes to model sets in arbitrary (non-abelian) locally compact groups.
We will explain the construction of new examples of different flavours, and how the classical
theory has to be modified in order to accomodate these new examples. We will focus on the case
of model sets in groups admitting a Gelfand pair, since for these the (spherical) diffraction
theory is particularly accessible.
No previous knowledge of model sets or diffraction theory is assumed.
This is based on joint work with Michael Bjorklund and Felix Pogorzelski.