Before revision:  The key new problem in Periodic Geometry is an isometry classification of periodic point sets. A required complete invariant should continuously change under point perturbations, because atoms always vibrate in real crystals. The main objects of Periodic Topology are embeddings of curves in a thickened plane that are invariant under lattice translations. Such periodic knots were classified in the past up to continuous deformations (isotopies) that keep a fixed lattice structure, hence are realized in a fixed thickened torus. The more practical equivalence is a periodic isotopyin a thickened plane without fixing a lattice basis.
After revision:  The key new problem in Periodic Geometry is to unambiguously parameterize this space of isometry classes by continuous coordinates that allow a complete reconstruction of any crystal. The major part of this manuscript reviews the recently developed isometry invariants to resolve the above problem: (1) density functions computed from higher order Voronoi zones, (2) distance-based invariants that allow ultra-fast visualizations of huge crystal datasets, and (3) the complete invariant isoset (a DNA-type code) with a first continuous metric on all periodic crystals. The main goal of Periodic Topology is to classify textiles up to periodic isotopy, which is a continuous deformation of a thickened plane without fixing a lattice basis.