06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Peter Müller (Ludwig-Maximilians-Universität München), Felix Pogorzelski (Universität Leipzig)
14:00 Peter Müller (Ludwig-Maximilians-Universität München): Localisation for Delone operators: results from wonderland and beyond
15:00 Tea and Coffee Break
15:30 Felix Pogorzelski (Universität Leipzig): What is a quasicrystal?
Peter Müller (Ludwig-Maximilians-Universität München): Localisation for Delone operators: results from wonderland and beyond
Delone sets form an important class of point sets in Euclidean space as they cover the full range from periodic lattices to random point sets. We study spectral properties of Schrödinger operators associated with Delone sets. These operators arise in physical applications and describe electric properties of solids whose atoms are located at the positions of the Delone set.
We give an introductory overview of the by now well established theory of Anderson localisation for random operators and apply it to obtain a probabilistic description of certain families of Delone sets for which the associated operators exhibit Anderson localisation at the bottom of the spectrum.
We also present a topological description of these families based on both Simon’s wonderland theorem and “hard” analysis.
Felix Pogorzelski (Universität Leipzig) What is a quasicrystal?
There is no rigorous mathematical definition of a quasicrystal. In spaces with some group translation the latter term usually refers to well-scattered point sets (Delone sets) that are not periodic but display long range symmetries.
One important class are model sets that arise from a simple cut-and-project method. Model sets have already been studied by Yves Meyer in the 70's, i.e. some time before Dan Shechtman's discovery of physical alloys with non-periodic molecular structure in 1982 (Nobel prize for chemistry in 2011).
In this talk we focus on aperiodic point sets which are not too far from crystals in a dynamical sense. Those sets show equidistribution phenomena which can be studied via the concept of unique ergodicity in dynamical systems theory. We explain these connections and mention both classical and recent results. Further, we describe new examples in non-commutative spaces and sketch some future research challenges.