Geometric and Topological Methods for Applications to Materials and Data Skeletonisation

Crystal Structure Prediction (CSP) aims to speed up functional materials discovery by using supercomputers to predict whether an input molecule can form stable crystal struc- tures with desirable properties. The process produces large datasets where each entry is a simulated arrangement of copies of the input molecule to form a crystal. However, these datasets have little structure themselves, and it is the aim of this thesis to contribute towards simplifying and analysing such datasets. Crystals are unbounded collections of atoms or molecules, extending infinitely in the space they lie within. As such, rigorously quantifying the geometric similarity of crystal structures, and even just identifying identical structures, is a challenging problem. To solve it, we seek a continuous, complete, isometry classification of crystals. Consequently, by modelling crystals as periodic point sets, we introduce the density fingerprint, which is invariant under isometries, Lipschitz continuous, and complete for an open and dense space of crystal structures. Such a classification will be able to identify and remove near- duplicates from these large CSP datasets, and potentially even guide future searches. We describe how this fingerprint can be computed using periodic higher Voronoi zones. This geometric concept of concentric regions around a fixed centre characterises relative positions of points from the centre in a periodic point set. We present an algorithm to compute these zones in addition to proving key structural properties. We later discuss research into skeletonisation algorithms, proving theoretical guarantees of the homological persistent skeleton (HoPeS), subsequently formulating and performing an experimental comparison of HoPeS with other relevant algorithms. Such algorithms, if effectively used, can be applied to large datasets including those produced by CSP to reveal the shape of the data, helping to highlight regions of interest and branches that merit further study