Explore the geometry and topology of periodic point sets, including crystals, in this comprehensive lecture. Delve into Computational Geometry and Topology tools like Brillouin zones and order k persistent homology, and their applications in material science for creating unique and continuous crystal fingerprints. Examine the challenges of defining persistent homology for periodic point sets, and discover a new invariant definition along with its advantages, disadvantages, and potential improvements. Learn about periodic crystals, finite representations, equivalence relations, distance computation, and crystal comparison methods. Investigate packing and covering radii, density fingerprints, persistence fingerprints, and the concept of features per volume. Gain insights into capturing non-cubic cycles and alternative distance measures between crystals, while considering future research directions in this field.