Geometry and Topology of Periodic Point Sets, for Example Crystals

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.

Intro
Overview
What is a (periodic) crystal?
What is a periodic crystal not?
Periodic point set
Finite representation not unique
Define equivalence relation
Computing distance
Comparing crystals
Why do we want a fingerprint function?
Packing and covering radii
Proof Sketch: Computability using Brillouin zones
The density fingerprint and the persistence fingerprint!
Persistent homology
Order k persistence
Definition (Persistence fingerprint function )
Definition: Infinitely many holes
Definition: Torus
Definition: Features per volume
Computing: Features per volume
Definition (Persistence fingerprint function 4)
Capturing non-cubic cycles
Future Work
Bibliography
Alternative distance between crystals