In order to analyze mathematical and physical systems it is often necessary to assume some form of order, e.g. perfect symmetry or complete randomness. Fortunately, nature seems to be biased towards such forms of order as well. During the second half of the 20th century, a new paradigm of "aperiodic order" was suggested. Instances of aperiodic order were discovered in different areas of mathematics, such as harmonic analysis and diophantine approximation (Meyer), tiling theory (Wang, Penrose) and additive combinatorics (Freiman, Erdös-Szemeredi); after some initial resistance is has now been accepted that aperiodic order also exists in nature in the form of quasicrystals.
Together with Michael Björklund we have proposed a general mathematical framework for the study of aperiodic structures in metric spaces, based on the notion of an "approximate lattices". Roughly speaking, approximate lattices generalize lattices in the same way that approximate subgroups (in the sense of Tao) generalize subgroups. Approximate lattices in Euclidean space are essentially the "harmonious sets" of Meyer (a.k.a. mathematical quasi-crystals), but there are interesting examples in other geometries, such as symmetric spaces, Bruhat-Tits buildings or nilpotent Lie groups. It turns out that with every approximate lattice one can associate a dynamical system, which replaces the homogeneous space associated with a lattice - thus the study of approximate lattices can be considered as "geometric group theory enriched over dynamical systems".
In this talk I will (1) give an overview over the basic framework of approximate lattices and geometric approximate group theory; (2) illustrate the framework by formulating Meyer's theory of harmonious sets in this language; (3) time permitting, discuss some recent structure theory of approximate lattices in nilpotent Lie groups and applications to Bragg peaks in the Schrödinger spectrum of magnetic quasicrystals.
Based on joint works with Michael Björklund (Chalmers), Matthew Cordes (ETH), Felix Pogorzelski (Leipzig) and Vera Tonić (Rijeka).