Prof Vitaliy Kurlin: mathematics & computer science

MACSMIN 2021 : Mathematics and Computer Science for Materials Innovation

The MACSMIN logo includes the basic examples of the rock-salt cubic crystal, the benzene ring, and a blue wave containing a local maximum and a local minimum.

• The 2nd MACSMIN on 15-17 Septemeber was online because of Covid rules, see videos below.
• The conference was supported by the Sir Henry Royce Institute for advanced materials research and innovation, and by the IUCr commission on Mathematical and Theoretical Crystallography.
• The conference followed the annual meeting of the Centre for TDA on 13-15 September 2021.
• Other meetings of the MACSMIN conference series: 2023, 2022, 2020.
• Wednesday afternoon 15th September 2021 : all UK times
  • 13.50-14.00 Brief opening by Vitaliy Kurlin
  • 14.00-14.40 Phil Smith. Higher index Voronoi zones of periodic point sets. Video
    Abstract. Higher index Voronoi zones of a periodic point set are concentric regions around a fixed centre, where the k-th Voronoi zone consists of all points in Euclidean space that have the centre as their k-th nearest neighbour. As periodic point sets model all solid crystalline materials (crystals), higher Voronoi zones substantially simplified the computation of density functions and helped detect a missing crystal in the Cambridge Structural Database. Despite important applications in crystallography, there was no publicly available algorithm to compute higher Voronoi zones beyond simple lattices or the case k=1 of Voronoi domains. We describe a practical algorithm to compute all Voronoi zones up to a given index k in dimensions two or three for any periodic point set, generalising the case of lattices.
  • 14.50-15.30 Sergey Aksenov. Topological analysis of zeolite-like compounds with mixed frameworks and crystal structure prediction using the approach of OD (“order-disorder”) theory. Abstract (pdf). Video
  • 15.40-16.20 Larry Andrews and Herbert Bernstein.
    A new, efficient algorithm for measuring the “distance” between unit cells. Abstract (pdf). Video
  • 16.30-17.10 Graeme Day. Molecular crystal energy landscape analysis using the threshold algorithm. Video
    Abstract. The prediction of crystal structures is usually performed by searching the energy landscape for local minima, each of which is assumed to correspond to a potential crystal structure. To understand the stability of these predicted structures and possible transitions between structures, we require further information: the energy barriers between structures. We have implemented the Monte Carlo threshold algorithm to sample the energy landscape of molecular crystals and find the energies at which local minima become connected. The presentation will present the method and results that we obtain for a series of polymorphic molecular crystals, using disconnectivity graphs to summarise the structure of the lattice energy landscapes.
• Thursday morning 16th September 2021 : tutorials about space groups and the new area of Periodic Geometry
  • 8.30-9.10 Bernd Souvignier. Order in the crystal bin: space-group hierarchy explained.
    Abstract. Space groups are classified, according to different criteria, into types, classes, systems and families. Depending on the specific research topic, some of these concepts will be more relevant to the everyday crystallographer than others. Unfortunately, incorrect use of the classification terms often leads to misunderstandings. This talk presents the rationale behind the different classification levels, based on the paper in Journal of Applied Crystallography (2018).
  • 9.20-10.10 Vitaliy Kurlin. The Crystal Isometry Principle justifies a continuous map of periodic crystals. Video
    Abstract. More than 150 years ago Mendeleev put all known chemical elements into the same periodic table despite their differences to better understand their similarities. The recent advances in the new area of Periodic Geometry show that replacing any atom by its centre loses no information about a crystal structure, because all resulting periodic point sets remain non-isometric for all known crystals in the Cambridge Structural Database. This Crystal Isometry Principle (CRISP) justifies a map of all known and not yet discovered crystals continuously parameterised by complete isometry invariants. The IUCr congress included the related introductory talks: A unique and continuous code of all periodic crystals (16 min) and Introduction to invariant-based machine learning for periodic crystals (20 min).
  • 10.20-11.00 Matt Bright. Continuous metrics on the space of isometry classes of lattices. Video
    Abstract. A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. Motivated by rigid crystal structures, we consider lattices up to rigid motion or isometry that preserves inter-point distances. Then all isometry classes of lattices form a continuous space. In dimensions two and three this space had several parametrisations, but all past parameters were discontinuous under perturbations in singular cases. We introduce new continuous coordinates (root forms) on the space of lattices and also new metrics satisfying all metric axioms and continuity under all perturbations. The root forms allow visualisations of hundreds of thousands of real crystal lattices from the Cambridge Structural Database for the first time.
  • 11.10-11.50 Dan Widdowson. Pointwise Distance Distributions of a periodic point set. Video
    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 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 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.
• Thursday afternoon 16th September 2021
  • 14.00-14.40 Egon Schulte. Skeletal Polyhedra, Complexes, and Symmetry. Video
    Abstract. The study of highly symmetric structures in Euclidean 3-space has a long and fascinating history tracing back to the early days of geometry. With the passage of time, various notions of polyhedral structures have attracted attention and have brought to light new exciting figures intimately related to finite or infinite groups of isometries. A radically new, skeletal approach to polyhedra was pioneered by Grunbaum in the 1970's building on Coxeter's work. A polyhedron is viewed not as a solid but rather as a finite or infinite periodic geometric edge graph in space, equipped with additional polyhedral super-structure imposed by the faces. Since the mid 1970's there has been a lot of activity in this area. The lecture surveys the present state of the ongoing program to classify discrete polyhedral structures in ordinary space by symmetry, where the degree of symmetry is defined via distinguished transitivity properties of the geometric symmetry groups. These skeletal figures exhibit fascinating geometric, combinatorial, and algebraic properties and include many new finite polyhedra as well as many new periodic structures with crystallographic symmetry groups.
  • 14.50-15.30 Nikolai Dolbilin. Local Groups in Delone Sets: Results and Conjectures. Abstract (pdf)
  • 15.40-16.20 Lev Sarkisov. Computational tools for the era of materials informatics. Abstract (docx). Video
  • 16.30-17.10 Ti-Yen Lan. Multi-target detection and its application to the cryo-EM reconstruction problem. Video
    Abstract. Multi-target detection consists of estimating a signal from many of its copies embedded at unknown locations in a large, noisy measurement. The number of observations needed for accurate estimation, an efficient recovery algorithm, and application to cryo-electron microscopy will be discussed.
• Friday morning 17th September 2021
  • 8.30-9.10 Mois Aroyo. Structural relationships based on space-group symmetry relations. Abstract (rtf).
  • 9.20-10.50 a round table discussion of research challenges related to the Royce Institute
  • 11.00-11.40 Pawel Dlotko. New invariants of embedded trees and graphs. Video
    Abstract. In this talk we will present a collection of invariants of embedded trees and graphs. Most of the talk will focus on so-called Sholl descriptors of embedded trees. Those neuroscience-motivated descriptors allow attachment, to every embedded rooted tree, a function from nonnegative reals to a metric space. A number of such functions is considered, each focusing on different aspects of morphology of a tree so that all together, they cover most of the important aspects of tree structure. The value of this function, at a given x >= 0 is the value of a given descriptor computed on a connected component of a tree, containing the root, restricted to a distance no greater than x from the root. Subsequently, if the time permits, we will propose a number of topological invariants of graphs. In this talk we will focus on neuroscience--motivated examples, but will discuss the possibilities of use of this machinery to analyze material-science data. This is joint work with Ahmad Farhat, Sadok Kallel and Reem Khalil.

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