Partner Event with Columbia Engineering: Applied Physics and Applied Mathematics (APAM); and Materials Science and Engineering

The Crystal Isometry Principle, developed by Vitaliy Kurlin and his team, revolutionizes how we classify the millions of known crystal structures beyond their traditional groups. By using a novel method called the Pointwise Distance Distribution, this principle uniquely identifies crystals based on their ability to be rigidly matched in 3-dimensional space, offering a much finer classification. This breakthrough in crystallography allows for a more precise understanding of periodic structures, opening new avenues for study. This talk will detail the current work in development by Kurlin and his team at the Materials Innovation Factory, University of Liverpool, UK. 

Speaker

Vitaliy Kurlin, Professor, Computer Science, University of Liverpool

Hosted By:

  • Yuan Yang, Associate Professor of Materials Science and Engineering, Applied Physics and Applied Mathematics, Columbia University
  • Simon Billinge, Professor of Materials Science and Applied Physics and Applied Mathematics, Columbia University
  • Qiang Du, Fu Foundation Professor of Applied Mathematics, Columbia University

Event Details

Friday, October 11, 2024 (11:00 AM – 12:00 PM ET)

Location: Mudd Building – Room 214

Register

Talk Information

The Crystal Isometry Principle

Full Abstract: For hundreds of years, crystals were studied almost exclusively by discrete tools such as symmetry groups. The classification of 230 space groups into 230 types was a great achievement at the end of the 19th century. In 2024, the Cambridge Structural Database (CSD) and other major datasets contain altogether more than 2 million experimental structures, while simulated crystals emerge in even  greater numbers. This scale requires a much finer (stronger) classification of all known periodic crystals into more than 230 classes. There is no practical sense to distinguish crystals that can be ideally matched by rigid motion. But we need to distinguish crystals that cannot be ideally matched (are not rigidly equivalent). Indeed, if we call ‘the same’ any crystals whose all atoms can be matched up to a small perturbation, sufficiently many perturbations can geometrically deform any crystal to any other (of the same composition if we keep atomic types). 

This continuum fallacy (a version of the sorites paradox) is resolved by the following new definition [1]: A crystal structure is an equivalence class of all periodic crystals that can be rigidly matched by rigid motion in 3-dimensional space. Such a rigid class contains infinitely many crystals represented by infinitely many different CIFs, all encoding the same periodic arrangement of atoms. Any slight perturbation of a single atom produces a crystal in a different rigid class, so there are infinitely many rigid classes, some of which can be very close due to noise, while others are very distant from each other. A classification under isometry is only slightly weaker than under rigid motion because mirror images can be distinguished by a sign of orientation. 

We developed a generically complete invariant descriptor that is preserved under isometry and continuously changes under any perturbation [2]. This Pointwise Distance Distribution PDD(S;k) is a matrix in which each row for any atom in an asymmetric unit of a periodic crystal S contains distances from the fixed atom to its k-nearest neighbors in increasing order. Within 1 hour on a modest desktop computer, PDD(100;k) distinguished all real periodic crystals in the CSD and justified the Crystal Isometry Principle saying that all these periodic crystals live at unique locations in a common continuous space.

[1] Anosova, O., Kurlin, V., Senechal, M. The importance of definitions in crystallography. IUCrJ, v.11(4), 453-463, 2024.

[2] Widdowson, D., Kurlin, V. Resolving the data ambiguity for periodic crystals. NeurIPS, v.35, 24625-24638, 2022.

Bio: Prof Vitaliy Kurlin is a universal scientist, mathematician by training, now leading the Data Science Theory and Applications group in the Materials Innovation Factory, Liverpool, UK. He completed a PhD at Moscow State University (2003) and won a Marie Curie International Incoming Fellowship (2005-2007), Royal Academy of Engineering Industry Fellowship (2021-2023), and Royal Society APEX fellowship (2023-2025). His group is developing a new area of Geometric Data Science whose key results include the Crystal Isometry Principle (NeurIPS 2022) and a polynomial-time extension of the SSS theorem classifying triangles under congruence to any clouds of unordered points (CVPR 2023), see details at http://kurlin.org.