Dan Widdowson's talk at MACSMIN 2021 : Mathematics and Computer Science for Materials Innovation
Channel: MACSMIN conference 2021
Date: 2021-09-16

Title. Pointwise Distance Distributions of a periodic point set.
Abstract. The fundamental model of a periodic structure is a periodic set of points considered up to rigid motion or isometry in Euclidean space. The recent work (http://kurlin.org/research-papers.php#SoCG2021) defined the new isometry invariants (density functions), which are continuous under perturbations of points and complete for generic sets in dimension 3. The new work (http://arxiv.org/abs/2108.04798) introduces much faster invariants called higher order Pointwise Distance Distributions (PDD). The new PDD invariants are simpler represented by numerical matrices and are also continuous under perturbations important for applications. Completeness of PDD invariants is proved for distance-generic sets in any dimension, which was also confirmed by distinguishing all 229K known molecular organic structures from the world's largest Cambridge Structural Database. This huge experiment took only seven hours on a modest desktop due to the proposed algorithm with a near linear or small polynomial complexity in terms of key input sizes. Most importantly, the above completeness allows one to build a common map of all periodic structures, which are continuously parameterized by PDD and explicitly reconstructible from PDD.