A discussion of equivalences and invariants of periodic crystals.

25^th March 2022, 14:00
[open table]

Abstract

Periodic lattices and more general periodic point sets model any solid crystalline materials and can be studied modulo various equivalence relations. Historically, all early crystals looked very symmetric and were distinguished by their symmetries. This traditional classification splits all periodic crystals into 219 space-group types (230 if mirror images are distinguished). The recent review by Massimo Nespolo, Mois Aroyo, and Bernd Souvignier gave excellent examples of potential confusion even between "groups" and "group types". Since many similar words have different meanings in closely related areas such as crystallography and mathematics (algebra and geometry), it makes sense to discuss the basic concepts, for example, lattices, equivalences, invariants, abstract groups and their linear (matrix) representations.