# SQUIDS Seminar: Vitaliy Kurlin - Geometric Data Science: old challenges and new solutions

Dates: | 26 April 2023 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Current University students |

Speaker: | Dr Vitaliy Kurlin |

Join us for this research seminar, part of the SQUIDS (Statistics, quantification of uncertainty, inverse problems and data science) seminar series.

Abstract: Geometric Data Science (GDS) studies moduli spaces of data objects up to important equivalences. The key example is a finite or periodic set of unlabelled points up to rigid motion or isometry preserving inter-point distances. The best known result in the finite case was the SSS theorem saying that triangles are completely classified by a triple of their sides. There was no similar continuous invariant even for 4 points in the plane partially due to a 4-parameter family of 4-point clouds that have the same 6 pairwise distances. One major result is a complete and continuous invariant for clouds of m unlabeled points in any Euclidean space, to appear in CVPR 2023.

The classification problem for periodic point sets, which model all crystalline materials, is substantially harder and cannot be reduced to a finite cloud because the smallest pattern (a unit cell) of a periodic crystal is discontinuous under tiny (or thermal) vibrations of atoms. This ambiguity was resolved by generically complete and continuous Pointwise Distance Distributions (PDD). The near-linear time algorithm for PDD invariants was tested via more than 200 billion pairwise comparisons of all 660K+ periodic crystals in the world's largest collection of real materials: the Cambridge Structural Database. This experiment took only two days on a modest desktop and detected five pairs of geometric 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 geometry. Five journals are investigating the integrity of the underlying articles.

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), published in NeurIPS 2022. Hence complete invariants form a DNA-style code or materials genome that parametrises a continuous map of CRISP including all known and not yet discovered crystals. All relevant papers are co-authored with many Liverpool colleagues, see http://kurlin.org/research-papers.php#Geometric-Data-Science.