Geometric Data Science develops continuous parameterizations on moduli spaces of data objects up to important equivalences. The key example is a finite or periodic set of unlabeled points considered up to rigid motion or isometry preserving inter-point distances. Periodic point sets model all solid crystalline materials (periodic crystals) with zero-size points at all atomic centers. A periodic point set is usually given by a finite motif of points (atoms or ions) in a unit cell (parallelepiped) spanned by a linear basis. The underlying lattice can be generated by infinitely many bases. Even worse, the set of possible motifs for any periodic set is continuously infinite.

This typical ambiguity of data representation was recently resolved by generically complete and continuous isometry invariants: Pointwise Distance Distributions (PDD) of periodic point sets. The near-linear time algorithm for PDD invariants was tested on more than 200 billion pairwise comparisons of all 660K+ periodic crystals in the world's largest collection of real materials: the Cambridge Structural Database.

The huge experiment above took only two days on a modest desktop and detected five pairs of isometric duplicates. In each pair, the crystals are truly isometric to each other but one atom is replaced with a different atom type, which seems physically impossible without perturbing distances to atomic neighbors. Five journals are now investigating the integrity of the underlying publications that claimed these crystals.

The more important conclusion is the Crystal Isometry Principle meaning that all real periodic crystals have unique geographic-style locations in a common continuous Crystal Isometry Space (CRISP). This space is parameterized by complete isometry invariants and continuously extends Mendeleev's table of elements to all crystals.

The relevant publications are in NeirIPS 2022, MATCH 2022, SoCG 2021. The latest paper in arxiv:2207.08502 defined complete isometry invariants with continuous computable metrics on any finite sets of unlabeled points in a Euclidean space. Many papers are co-authored with colleagues at Liverpool Materials Innovation Factory and inked at http://kurlin.org/research-papers.php#Geometric-Data-Science.
Speaker(s): Vitaliy Kurkin (University of Liverpool)