**MACSMIN 2020** : 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 1st MACSMIN on 7-8 September 2020 was online because of Covid rules, see videos below.
- Later meetings of the MACSMIN conference series: 2023, 2022, 2021.
- Following the successful series of the past meetings, the first MACSMIN conference was online on 7-8 September 2020 jointly with the 15th meeting of the Applied Geometry and Topology network funded by the London Mathematical Society.
- Participants: 13 speakers (2 female), 8 local students (2 female), 2 external students (1 female), 7 others (2 female).
- To cover many time zones in the world, we hosted short 25-min talks with 5 min breaks between 14.00-17.30 UK time.
**Monday 7th September 2020**between 14.00-17.30 UK time- 13.55-14.00 A brief opening by Vitaliy Kurlin (Materials Innovation Factory, Liverpool)
- 14.00-14.25 Ron Lifshitz (Condensed Matter Physics, Tel Aviv University)

Title: Aperiodic Crystals: How is that even possible? Video

Abstract: For over a century it was understood that crystals were a form of matter in which the atomic constituents were ordered periodically. In the late 1960s it started to become evident that long-range order survives, even when the periodicity is removed via incommensurate modulations, or the intergrowth of two or more crystals with incommensurate periodicities. Yet, it was only in 1982, with Shechtman's discovery of quasicrystals, that the periodicity paradigm had to be fully abandoned and the notion of crystallinity had to be redefined. As time permits, I will briefly explain some of the basic notions of aperiodic crystals: What does it mean to have long-range order without periodicity? What does an aperiodic crystal look like? How can we model such crystals? What do we really mean when we say that a crystal has a certain rotational symmetry? Can we still talk about point groups, space groups, and systematic extinctions? What happens to Goldstone modes and topological defects? Further reading:

R. Lifshitz, "What is a crystal?" Z. Kristallogr. 222 (2007) 313.

R. Lifshitz, "Quasicrystals: A matter of definition." Foundations of Physics 33 (2003) 1703.

R. Lifshitz, "Symmetry breaking and order in the age of quasicrystals." Isr. J. Chem. 51 (2011) 1156.

R. Lifshitz, "Theory of color symmetry for periodic and quasiperiodic crystals." Rev. Mod. Phys. 69 (1997) 1181. - 14.30-14.55 Uwe Grimm (Aperiodic order and quasicrystals, Open University, UK)

Title: A new approach to diffraction for inflation-invariant model sets. Video

Abstract: I will introduce the standard example in aperiodic order, the Fibonacci chain, which possesses both an inflation and a cut and project (model set) description. For model sets, the diffraction can be computed as an integral over the characteristic function of the window, which works well as long as the window is simple, as in the Fibonacci case where the window is just an interval. The inflation structure can be used to derive an alternative approach to compute the diffraction, based on a renormalisation argument that exploits the self-similarity of the structure. It provides a method that can be applied to model sets with a more complex window, in particular in the common situation where the window has a fractal boundary. Some examples will be shown, which are obtained for two-dimensional generalisations of the Fibonacci system. - 15.00-15.25 Jean-Guillaume Eon (Graph theory in crystallography, Instituto de Quimica, Brazil).

Title : Combinatorial aspects of Lowenstein's avoidance rule in aluminosilicates. Abstract (pdf). - 15.30-16.00 break
- 16.00-16.25
Graeme Day (Computational Chemistry, Southampton, UK)

Title: An introduction to molecular crystal structure prediction. Video

Abstract:The talk will outline the global energy minimisation approach to crystal structure prediction and its applications to molecular crystals. I will review the methods used in exploring the enegy landscape for stable crystal structures, and for assessing their relative thermodynamic stabilities, highlighting the complexity of the search space and challenges in obtaining reliable energy ranking of structures. The talk will highlight the current challenges in the field of crystal structure prediction in both the generation of structures and analysis of datasets. - 16.30-16.55 Angeles Pulido (Materials Science team, Cambridge Crystallographic Data Centre)

Title: Role of computed solid form landscapes in functional materials design and development.

Abstract: I will illustrate the consequences of flawed materials development in areas such as drug development; and how differences in molecule arrangement in the solid-state might lead to drastic changes in materials properties. We will discuss which is the role predicted solid form landscapes can play into computer-aid functional materials design, and how computational approaches might assist in minimizing risks during the materials development processes. We will look at some of the approaches used to understand similarities and differences between the different solid forms of a given molecule and why it's still so challenging to compare organic molecular crystals. Also, I would like to visit the long-standing challenge of differentiating between putative and synthetically accessible crystal forms on a solid form landscape, and how computational approaches could inform experiments. - 17.00-17.25 Herbert Bernstein and Larry Andrews (Mathematical Crystallography, Ronin Institute, US).
Video

Title : Metric spaces for comparing lattices; walking on a ripply surface in six or more dimensions. Slides. **Tuesday 8th September 2020**between 14.00-17.30 UK time- 14.00-14.25 Herbert Edelsbrunner (Computational Geometry, IST Austria)

Title. The Density Fingerprint of a Periodic Set. Video

Abstract. Adding a lattice to a finite set, we get a periodic set, which in 3 dimensions models crystalline materials. The k-th density function maps a radius r to the portion of space covered by exactly k balls of radius r centered at the points. We propose the vector of density functions as a fingerprint of a periodic set. It is clearly invariant under rigid motions, it is stable under perturbations (talk by Teresa Heiss), and we still have some faint hope that it is complete. Furthermore, there are fast algorithms computing it for a given periodic set (talk by Phil Smith). - 14.30-14.55 Phil Smith (Materials Innovation Factory, Liverpool).

Title. Computations of Density Fingerprint Via Brillouin Zones. Video

Abstract. Continuing on the topic of density functions, this talk will focus on their computations. In particular, we will introduce the concept of Brillouin zones, explain their link to density functions and thus how they can be used to provide a more elegant way of computing these functions than any brute-force approach. We will finish by looking at how computations of density functions have been applied to a real-world crystal dataset and the encouraging results of this. - 15.00-15.25 Teresa Heiss (Computational Geometry, IST Austria)

Title. Stability of the Density Fingerprint. Video

Abstract. In my talk I will prove Lipschitz stability of the density fingerprint (defined in the previous talk by Herbert Edelsbrunner). In other words, perturbing every atom of a crystal by at most delta will change the density fingerprint by at most a Lipschitz constant times delta when choosing an appropriate metric between density fingerprints. Stability is an important property to make a crystal-invariant useful in real life (where the position of atoms can be measured only up to a certain precision): Two different (noisy) measurements of the same crystal should get similar invariants assigned. Joint work with Herbert Edelsbrunner, Vitaliy Kurlin, Phil Smith and Mathijs Wintraecken. - 15.30-16.00 break
- 16.00-16.25 Matt Bright (Materials Innovation Factory, Liverpool).

Title. Introduction to Periodic Topology.

Abstract. We consider an infinite collection of lines and circles whose embedding in three dimensions maps to itself under periodic translations in one, two or three linearly independent directions. Finite representation of such structures requires selection of a unit cell, which is not unique. Topological classification should therefore be up to both deformation of the ambient space and changes in unit cell selection - this notion of periodic isotopy will be the key tool used to study real world examples of periodic structures such as textiles and crystals. - 16.30-16.55 Vitaliy Kurlin (Materials Innovation Factory, Liverpool).

Title. Introduction to Periodic Geometry. Video

Abstract. A periodic crystal is modeled as a periodic set of zero-sized points in 3-space. Such a periodic set is usually given by a unit cell (a parallelepiped defined by 3 edge-lengths and 3 angles) and a motif of points with fractional coordinates in this cell. Representing a crystal as a unit cell plus a motif is highly ambiguous. Hence a reliable comparison of rigid crystals should be based on isometry invariants that are preserved under any rigid motions, hence are independent of a unit cell and a motif. The talk will state conditions for a stable and complete isometry classification of periodic crystals that can accelerate materials discovery. - 17.00-17.25 Marco Mosca (Materials Innovation Factory, Liverpool).

Title. Average Minimum Distances are stable isometry invariants of a periodic crystal. Video

Abstract. The talk introduces the infinite sequence of isometry invariants called the Average Minimum Distances (AMDs). The k-th AMD is the k-th distance from a point to its k-th nearest neighbour within the infinite crystal, averaged over all points in a given motif. The AMDs are provably stable under perturbations and have continuously quantified similarities between 5679 crystals based on a single T2 molecule in the 2017 Nature paper.

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