Abstract: 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. In joint work with Michael Björklund (Chalmers University) we developed a framework for studying aperiodic structures in geometry.
Perfect symmetry in geometry can be modelled algebraically using "geometric actions" of groups. Similarly, we propose to model aperiodic structures in geometry using geometric actions of approximate groups in the sense of Tao. Geometric actions of groups lead to uniform lattices, and similarly geometric actions of approximate groups lead to “uniform approximate lattices”.
After a general introduction to approximate group and approximate lattices I will focus on approximate lattices in the Heisenberg group, which are relevant for modelling quasi-crystals under the influence of a magnetic field. I will explain how the notion of “sharp Bragg peaks” in the diffraction picture of quasi-crystals, which lead to their experimental discovery (D. Shechtman, Nobel Prize in Chemistry, 2011), can be generalized to the case of the Heisenberg group. I will hint at the underlying methods, which come from different areas of mathematics, such as group theory, dynamical systems, ergodic theory, representation theory and harmonic analysis. Joint work with Michael Björklund.