On the metric spaces of lattices and periodic point sets.

12^th December 2023, 14:00
Alexey Garber
University of Texas Rio Grande Valley, USA

Abstract

In the talk I will describe some properties of the family of discrete point sets in R^d as the metric space equipped with the bottleneck distance. Mainly, I will provide different conditions, under which the spaces of lattices or periodic point sets are Lipschitz or coarsely (non)embeddable into Hilbert space. The various conditions are phrased in terms of density, packing radius, covering radius, the cardinality of the motif, and the diameter of the unit cell. While the main motivation will be to provide some structure for the smaller family of lattices and periodic point sets, I will also review some known results for quasiperiodic point sets and connections to geometry of numbers, and aperiodic order. The talk is based on a joint work with Žiga Virk and Nicolò Zava.