Hyperuniformity: a hidden long-range order in random geometric systems

Kurzfassung

A random geometric system can be both locally similar to complete spatial randomness and globally homogeneous like a lattice. Such an anomalous suppression of large-scale density fluctuations is known as hyperuniformity, and it has profound implications for the mathematical and physical properties of the system. In these two lectures, we will take a closer look at the definitions and properties of hyperuniform random geometric systems, highlighting some of the subtleties of this fascinating field of research. We will discuss prominent examples as well as physical implications and relations to different fields of mathematics. Moreover, we will touch upon the connections to other types of novel short- and long-range order, like rigidity or quasicrystal symmetries.