Abstract: The following application has motivated us to develop new Computational Geometry and Topology methods, involving Brillouin zones and periodic k-fold persistent homology: We model crystals by (infinite) periodic point sets, i.e. by the union of several translates of a lattice, where every point represents an atom. Two periodic point sets are equivalent if there is a rigid transformation from one to the other. A periodic point set can be represented by a finite cutout s.t. copying this cutout infinitely often in all directions yields the periodic point set. The fact that these cutouts are not unique creates problems when working with them. Therefore, material scientists would like to work with a complete, continuous invariant instead. In this talk, I will present two continuous invariants that are at least generically complete: Firstly, the density fingerprint, computing the probability that a random ball of radius r contains exactly k points of the periodic point set, for all positive integers k and positive reals r. And secondly, the persistence fingerprint, which is the sequence of order k persistence diagrams, newly defined for infinite periodic point sets, for all positive integers k. Joint work with Herbert Edelsbrunner, Alexey Garber, Vitaliy Kurlin, Georg Osang, Janos Pach, Morteza Saghafian, Phil Smith, Mathijs Wintraecken.