**MIF++** is the interdisciplinary seminar organised by the DSTA group

- Vision of the MIF++ seminar : enrich materials science with mathematics.
- Style of MIF++ seminars : who is welcome to participate and contribute.
- MIF++ timetable : every two weeks on average, on zoom from March 2020.
- Annual conference MACSMIN is a focused 1-week version of MIF++ seminars.

### Vision of the MIF++ seminar : enrich materials science with mathematics

- The Materials Innovation Factory (MIF) is the £82M research institute, which was officially opened in 2018.
- The vision is to provide collaborative opportunities for colleagues on the interfaces of mathematics and materials.
- Before Covid, the MIF++ seminar was run for 2 hours including a talk combined with lunch and informal discussions.
- The lunch was free and should be well-deserved, so it comes one hour after the start to make all discussions easier.
- The aim is to connect colleagues, who are experts from different areas and would like to start a joint research project.
- Since March 2020, the MIF++ seminar is on zoom for 45-50 min plus 10-15 min for questions, no free lunch online, sorry.

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### Style of network seminars and who is welcome to participate and contribute

- An initial presentation should be self-contained and understandable to non-experts, aimed at first year undergraduates.
- A
*chemist*(in a broad sense including chemical engineering and physical chemistry) could first explain basic concepts and then introduce a real problem, where computational methods can potentially help, and describe the state-of-the-art. - A
*computational expert*(computer science, mathematics, electrical engineering) could similarly start from basic concepts and explain their methods on toy examples, also describe types of real data that can be processed by their methods. - Since September 2021 our group presents advances in a new area of Geometric Data Science every month on average.

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### Network seminars (in the reverse chronological order) : 2023 | 2022 | 2021 | 2020 | 2019

If you would like to participate in the MIF++ seminar, please e-mail.

**24 February 2023**14.00-15.00 UK time.- Luigi Vitagliano (Institute of Biostructures and Bioimaging, Naples, Italy)
**Title**. TBA.**Abstract**. TBA

**20 February 2023**11.00-12.00 UK time.- Nadav Dym (Technion's Faculty of Mathematics, Israel)
**Title**. TBA.**Abstract**. TBA

**17 February 2023**14.00-15.00 UK time.- Jonathan Balasingham (Liverpool Computer Science department, UK)
**Title**. Graph representation of crystals using Pointwise Distance Distributions.**Abstract**. TBA

**10 February 2023**14.00-15.00 UK time.- Danny Ritchie (Liverpool Materials Innovation Factory, UK)
**Title**. Exploiting Powder X-ray Diffraction techniques for automated phase isolation in the laboratory.**Abstract**. A chemical composition consisting of n elements can be represented as Euclidean, where the components give the relative amount of each element. Consider an experimentally prepared composition s that has been heated such that it reaches thermodynamic equilibrium. At equilibrium it will decompose into a set of phases {p}. A ubiquitous method for the identification of new phases is to trial an arbitrary s in the hope that a p is formed which is new. The problem that this work aims to address is how to find the composition of the new p. The {p} are separated into the new phase, denoted u, and the known phases, denoted {k}. Powder X-Ray Diffraction (PXRD) is an experimental technique that can be used to estimate the relative ratios of {k}, and so the average composition of the {k} can be calculated; K. The importance of this is that s will be a weighted average of u and K, and so u will lie on the line segment extending from s away from K. Unfortunately the estimation of the relative ratios from PXRD is not exact, and so rather than being a point, K must be represented as a random variable. Assuming the distribution of K to be Gaussian, the induced probability density for u is estimated by a projection of K through s.

**3 February 2023**14.00-15.00 UK time. Video (58 min) Slides (6M)- Gabriel Stoltz (CERMICS, Ecole des Ponts ParisTech, France).
**Title**. Coarse-graining and efficiently sampling with autoencoders.**Abstract**. A coarse-grained description of atomistic systems in molecular dynamics is provided by reaction coordinates. These nonlinear functions of the atomic positions are a basic ingredient to compute more efficiently average properties of the system of interest, such as free energy profiles. However, reaction coordinates are often based on an intuitive understanding of the system, and one would like to complement this intuition or even replace it with automated tools. One appealing tool is autoencoders, for which the bottleneck layer provides a low dimensional representation of high dimensional atomistic systems. In order to have an efficient numerical method, autoencoders should be combined with importance sampling techniques based on adaptive biasing methods. The algorithm then iterates between an update of the reaction coordinate, and free energy biasing. I will discuss some mathematical foundations of this method, and present illustrative applications for biophysical systems, including alanine dipeptide and chignolin. Some on-going extensions to more demanding systems, namely HSP90, will also be hinted at.

**27 January 2023**14.00-15.00 UK time- Jack Wang (Rice University, US).
**Title**. RetMol: Retrieval-based Controllable Molecule Generation.**Abstract**. Generating new molecules with specified chemical and biological properties via generative models has emerged as a promising direction for drug discovery. However, existing methods require extensive training/fine-tuning with a large dataset, often unavailable in real-world generation tasks. In this work, we propose a new retrieval-based framework for controllable molecule generation. We use a small set of exemplar molecules, i.e., those that (partially) satisfy the design criteria, to steer the pre-trained generative model towards synthesizing molecules that satisfy the given design criteria. We design a retrieval mechanism that retrieves and fuses the exemplar molecules with the input molecule, which is trained by a new self-supervised objective that predicts the nearest neighbor of the input molecule. We also propose an iterative refinement process to dynamically update the generated molecules and retrieval database for better generalization. Our approach is agnostic to the choice of generative models and requires no task-specific fine-tuning. On various tasks ranging from simple design criteria to a challenging real-world scenario for designing lead compounds that bind to the SARS-CoV-2 main protease, we demonstrate our approach extrapolates well beyond the retrieval database, and achieves better performance and wider applicability than previous methods.

**13 January 2023**14.00-15.00 UK time

(joint with the journal club at AstraZeneca).- Vitaliy Kurlin (Liverpool Materials Innovation Factory, UK).
**Title**. Mapping the spaces of crystalline materials and proteins.**Abstract**. Same or different? This question remained open for many real objects including periodic crystals and proteins. Since crystal structures are determined in a rigid form, there is little sense to distinguish them modulo rigid motion (a composition of translations and rotations). Considering the chirality or sign of orientation, it suffices to distinguish crystals modulo isometry, which is any transformation (for example, mirror reflection) maintaining inter-point distances. Geometric Data Science develops complete invariants that are DNA-style descriptors uniquely identifying classes of real objects modulo rigid motion, isometry, or other important equivalences. Such invariants practically distinguished all (660+ thousand) periodic crystals in the Cambridge Structural Database via 200+ billion pairwise comparisons. This experiment was completed over two days on a modest desktop, while traditional RMSD comparisons are estimated to require 34+ thousand years. The new invariants unexpectedly detected five pairs of duplicate structures that are geometrically identical, but one atomic identity was replaced with a different one (Cd with Mn in the pair HIFCAB vs JEPLIA). Since such a replacement seems physically impossible without perturbing geometry, five journals are investigating the integrity of the underlying publications. The more important conclusion is that all known and undiscovered crystals live in a common space of isometry classes of periodic point sets parameterized by complete invariants playing the role of geographic-style coordinates in this crystal universe whose first maps appeared in Acta Cryst A, v.79, p.1-13. A similar rigorous approach to proteins detected unexpected coincidences in the Protein Data Bank.

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**16 December 2022**14.00-15.00 UK time. Video (57 min)- Semen Gorfman (Tel Aviv University, Israel).
**Title**. The algorithms of target transformation of basis vectors of crystal lattice.**Abstract**(pdf with bio). The crystal lattice is the mathematical object that describes the long-range order (three-dimensional periodicity) of crystal structures. The lattice is set by non-coplanar vectors (also known as basis vectors), whose linear combinations with the integer coordinates describe the position of individual lattice points. While the lattice parameters (the lengths and the angles between the basis vectors) are often used as one of the fingerprints of a crystal structure, the choice of basis vectors is not ambiguous. Specifically, one and the same lattice can be created by an infinitely large number of sets of basis vectors. Transformation of basis vectors is the key mathematical operation that appears to be useful for the comparison of polymorphs crystal structures, phase transitions analysis, the understanding of twinning laws, indexing of X-ray diffraction peaks, and lattice reduction. While such operations are often built into larger computer programs and serve as only small blocks, the need for stand-alone algorithms of specific transformation of basis vectors of a crystal lattice still exist. The goal of this presentation is to introduce a simple numerical approach for the transformation of lattice basis vectors to a specific target: in the first case, one of the new basis vectors is aligned to a predefined lattice direction. In the second case (for a three-dimensional lattice) two of the basis vectors are brought to the lattice “plane” with specific Miller indices. Two, three and multidimensional versions of the algorithms will be presented. The applications for the simulation of zone planes and the transformation of a unit cell setting will be shown.

**9 December 2022**14.00-15.00 UK time. Video (52 min)- Guo-Wei Wei (Michigan State University, US).
**Title**. Mathematics and AI are revolutionizing Biosciences.**Abstract**(doc with bio). Mathematics underpins fundamental theories in physics such as quantum mechanics, general relativity, and quantum field theory. Nonetheless, its success in modern biology, namely cellular biology, molecular biology, biochemistry, genomics, and genetics, has been quite limited. Artificial intelligence (AI) has fundamentally changed the landscape of science, technology, industry, and social media in the past few years and holds a great promise for discovering the rules of life. However, AI-based biological discovery encounters challenges arising from the structural complexity of macromolecules, the high dimensionality of biological variability, the multiscale entanglement of molecular, cell, tissue, organ, and organism networks, the nonlinearity of genotype, phenotype, and environment coupling, and the excessiveness of genomic, transcriptomic, proteomic, and metabolomic data. We tackle these challenges mathematically. Our work focuses on reducing the complexity, dimensionality, entanglement, and nonlinearity of biological data. We have introduced persistent cohomology and various topological Laplacians, including evolutionary Hodge Laplacian, persistent Laplacian, persistent sheaf Laplacian, and persistent path Laplacian to model complex, heterogeneous, multiscale biological systems and thus significantly enhance AI's ability to handle biological data. Using our mathematical AI approaches, my team has been the top winner in D3R Grand Challenges, a worldwide annual competition series in computer-aided drug design and discovery for years. By further integrating with millions of genomes isolated from patients, we reveal the natural selection mechanisms of SARS-CoV-2 evolution and accurately forecast emerging SARS-CoV-2 variants.

**18 November 2022**14.00-15.00 UK time. Video (42 min).- April Lynne D. Say-awen (Ateneo de Manila University, Philippines).
**Title**. On the Frequency Module of the Hull of a Primitive Substitution Tiling.**Abstract**. This talk is based on a joint paper with Dirk Frettlöh and Ma. Louise Antonette N. De Las Peñas where a method will be presented on how to arrive at the frequency module of the hull of a primitive substitution tiling. The frequency module of a tiling T is the minimal Z-module that contains the absolute frequency of each patch of T. The method involves introducing a new substitution tiling T' that is derivable from T.

**Keywords**: frequency module, primitive substitution tilings, aperiodic tilings, quasicrystals

**21 October 2022**14.00-15.00 UK time. Video (55 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Density functions of periodic sequences.**Abstract**. The past work at SoCG 2021 introduced an infinite sequence of density functions, which are continuous isometry invariants of a periodic point set. For any integer k>=0, the k-th density function of a variable radius measures the fractional volume of the ambient space covered by exactly k balls centred at all given points. These density functions are highly non-trivial even in dimension 1 for periodic sequences of points in the line. The new work at DGMM 2022 fully describes the density functions of any periodic sequence and their symmetry properties. The explicit description confirms coincidences of density functions for a pair of non-isometric sequences that were previously computed via finite samples.

**14 October 2022**12.00-13.00 UK time. Video (40 min).- José Miguel Hernández Lobato (University of Cambridge, UK).
**Title**. Meta-learning Adaptive Deep Kernel Gaussian Processes for Molecular Property Prediction.**Abstract**. We propose Adaptive Deep Kernel Fitting with Implicit Function Theorem (ADKF-IFT), a novel framework for learning deep kernel Gaussian processes (GPs) by interpolating between meta-learning and conventional deep kernel learning. Our approach employs a bilevel optimization objective where we meta-learn generally useful feature representations across tasks, in the sense that task-specific GP models estimated on top of such features achieve the lowest possible predictive loss on average. We solve the resulting nested optimization problem using the implicit function theorem (IFT). We show that our ADKF-IFT framework contains previously proposed Deep Kernel Learning (DKL) and Deep Kernel Transfer (DKT) as special cases. Although ADKF-IFT is a completely general method, we argue that it is especially well-suited for drug discovery problems and demonstrate that it significantly outperforms previous SOTA methods on a variety of real-world few-shot molecular property prediction tasks and out-of-domain molecular property prediction and optimization tasks.

**7 October 2022**14.00-15.00 UK time. Video (55 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Degree-k Voronoi domains of periodic point sets.**Abstract**. The talk is based on a joint paper with Phil Smith. Degree-k Voronoi domains of a periodic point set are concentric regions around a fixed centre consisting of all points in Euclidean space that have the centre as their k-th nearest neighbour. Periodic point sets generalise the concept of a lattice by allowing multiple points to appear within a unit cell of the lattice. Thus, periodic point sets model all solid crystalline materials (periodic crystals), and degree-k Voronoi domains of periodic point sets can be used to characterise the relative positions of atoms in a crystal from a fixed centre. We describe the first algorithm to compute all degree-k Voronoi domains up to any degree k>0 for any two or three-dimensional periodic point set.

**17 June 2022**14.00-15.00 UK time. Video (59 min). Slides (pdf)- Michele Ceriotti (EPFL, Switzerland).
**Title**. A unified theory of atom cloud representations for machine learning.**Abstract**. Simulations of matter at the atomic scale are precious to provide a mechanistic understanding of chemical processes, and to design molecules and materials with predictive accuracy. As with many fields of science, machine learning have become an essential part of the modeling toolbox, with many frameworks having become well-established, and many more being developed in new research directions.

The most effective frameworks incorporate fundamental physical principles, such as symmetry, locality, and hierarchical decompositions of the interactions between atoms, in the construction of the ML model.

I will discuss a general framework that unifies several of the most recent developments in the field, including the representation of structures in terms of systematically-convergent atom-centered correlations of the neighbor density, as well as equivariant message-passing schemes that build automatically descriptors with equivalent information content.

I will discuss a few examples of the implications of these fundamental findings, for both chemical machine learning and in general for problems that require a description of three-dimensional objects in terms of point clouds, and present some examples that highlight the limitations of some common approaches and point to strategies to overcome them.

**10 June 2022**15.00-16.00 UK time. Video (45 min).- Jan Rohlíček (Institute of Physics of the Czech Academy of Sciences).
**Title**. CrystalCMP : automatic comparison of molecular structures.**Abstract**. This article describes new developments in the CrystalCMP software. In particular, an automatic procedure for comparison of molecular packing is presented. The key components are an automated procedure for fragment selection and the replacement of the angle calculation by root-mean-square deviation of atomic positions. The procedure was tested on a large data set taken from the Cambridge Structural Database (CSD) and the results of all the comparisons were saved as an HTML page, which is freely available on the web. The analysis of the results allowed estimation of the threshold for identification of identical packing and allowed duplicates and entries with potentially incorrect space groups to be found in the CSD.

**26 May 2022**14.00-15.00 UK time. Video (67 min). Slides (pdf)- Ryoko Tomiyasu (Kyushu University, Japan).
**Title**. Reduction theory for determining the Bravais type of unit-cell parameters containing observation errors.**Abstract**. In crystallography, the parameters of the unit cells or crystal structures recovered from experimental data are required to be output using their conventional cells. Namely, the lattice-basis reduction must be applied, and the software also transforms the parameters of the primitive cells into those of the conventional cells, based on their symmetries (centering types).

The influence of observational errors cannot be ignored during this algebraic calculation. In 2012, the speaker provided an algorithm for 3D lattices containing observational errors; similar results have been obtained for 2D lattices. The algorithm is new in the sense that all the candidates for nearly reduced bases were explicitly given to speed up the algorithm, and we proved based on the Venkov reduction that it works correctly for parameters with errors of various sizes, ranging from rounding errors to observation errors. The Eisenstein and Selling reductions (known as the Niggli and Delaunay reductions in crystallography) can be regarded as a special case of the Venkov reduction. The code has been used in the author's research in geometry of numbers, and in the ab-initio indexing software CONOGRAPH for PXRD and EBSD distributed by and for experimental scientists.

In the course of the research, the speaker had the opportunity to study the history of the reduction theory in crystallography, which will be mentioned as well as the background in mathematics.

**20 May 2022**14.00-15.00 UK time. Video (52 min)- Cyril Cayron (EPFL, Switzerland).
**Title**. A method to reduce N-dimensional lattices and solve N-dimensional Bézout's identities.**Abstract**. In order to predict deformation twins in metals and minerals, simple shears on different integral (reticular) planes should be determined. Calculating the shear vectors associated with a given shear plane requires to find a reduced unit cell attached to this plane. Different connected mathematical problems thus emerge: Bézout's identities, integer relations, and cell/lattice reduction. Quite ironically, instead of applying a pre-established lattice reduction method such as LLL, it appeared that simple shearing itself (hyperplanar shearing in dimension N > 3) can be used to reduce the unit cell, and solve the associated N-dimensional Bézout's identity. When applied recursively on different integral planes, hyperplane shearing also permits to reduce N-dimensional lattices by minimizing their metric "rhombicity". The talk will explain the different steps of the method and their geometrical meanings. Some examples with real crystals (N = 3) will be given to explain how the reduction method permits determine the Bravais lattice from diffraction measurements.

**13 May 2022**14.00-15.00 UK time. Video (44 min)- James Cumby (University of Edinburgh, UK).
**Title**. Encoding chemical similarity using pairwise distances.**Abstract**. In order to design new materials that fulfil a desired function, we require tools to rapidly predict physical properties from the underlying atomic structure. Statistical machine learning is accelerating advances in this area, but a key limitation remains; accurately predicting properties requires an effective measure of similarity between crystal structures. Here, we show that a representation based on grouped pairwise inter-atomic distances (GRID) combined with earth mover's distance (EMD) gives an accurate reflection of chemical space, and can be used to predict multiple physical properties using a simple regression model.

**6 May 2022**14.00-15.00 UK time. Video (55 min)- Emre S. Tasci (Hacettepe University, Turkey).
**Title**. Using Group Theory to Relate and Compare Structures.**Abstract**. Application of group theory in crystallography and solid state physics brings powerful impositions that drastically limit the possible outcomes or intermediate pathways to manageable potential cases. The talk will begin by a brief introduction of group theory and symmetry, along with symmetry related concepts such as symmetry operators, site symmetries, Wyckoff positions, group settings, group-subgroup relations and will continue on how to relate given structures along with quantitative means of comparison. The presentation will employ tools served by the Bilbao Crystallographic Server (BCS) [https://www.cryst.ehu.es/] such as WYCKPOS, SUBGROUPGRAPH, MINSUP, WYCKSPLIT, PSEUDO, COMPSTRU and STRUCTURE RELATIONS, all of which are freely available at BCS.

**29 April 2022**14.00-15.00 UK time. Video (55 min)- Gregory McColm (University of South Florida, US).
**Title**. Using Symmetry-Based Equivalence Relations in Classifying Crystallographic Structures**Abstract**. One way to classify geometric figures - including figures of interest to crystallographers - is to fix a hierarchy of groups acting on the underlying geometric space, and from these define a hierarchy of equivalence relations arising from these groups. We look at some of the more popular groups acting on Euclidean spaces, with some kinds of figures popular among crystallographers. Time permitting, we look at some variants within this classification system.

**26 April 2022**14.00-15.00 UK time (joint with LIV.DAT seminar). Video (44 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. The Crystal Isometry Principle.**Abstract**. Any solid crystalline material (periodic crystal) consists of periodically translated unit cells of atoms or molecules. Since crystal structures are determined in a rigid form, the strongest and most practical equivalence of crystals is rigid motion (a composition of translations and rotations) or isometry (also including reflections). Periodic crystals were classified by coarser equivalences such as space groups into 219 types (230 if mirror images are distinguished). However, the world's largest Cambridge Structural Database (CSD) of 1.1M+ existing crystals requires finer classifications.

The Data Science Theory and Applications group at the Liverpool Materials Innovation Factory developed generically complete and continuous isometry invariants for periodic sets of atomic centres. Computing these invariants for all 660K+ periodic crystals (full 3D structure; no disorder) in the CSD through 200 billion+ pairwise comparisons over two days on a modest desktop detected five pairs of "identical needles in a haystack". For example, the CSD crystals HIFCAB and JEPLIA are truly isometric, but one atom of Cadmium is replaced by Manganese, which should inevitably perturb a local geometry of atoms. As a result, five journals are now investigating the data integrity of the underlying publications.

These experiments justified the Crystal Isometry Principle, which states that any periodic crystal is uniquely determined by its geometry of atomic centres without chemical information. Then all known and undiscovered periodic crystals live in a common Crystal Isometry Space (CRISP), so that Mendeleev’s periodic table representing individual elements, categorised by two discrete parameters (atomic number and group), can be extended into a continuous space for all solid crystalline materials. For instance, diamond and graphite, both consisting purely of carbon, have different locations in CRISP.

**8 April 2022**14.00-15.00 UK time. Video (59 min).- Larry Andrews (Ronin Institute, US).
**Title**. Comparing crystal lattices.**Abstract**. Comparing crystal lattices is necessary for several uses: databases of unit cell parameters, Bravais lattice type determination, clustering of cells from serial crystallography at XFELs and synchrotrons elimination of duplicate and near duplicates in crystal structure prediction. The talk will include a brief, simple review of linear algebra. The mechanics of measuring the difference of pairs of unit cells will be discussed ("distance" between lattices). That leads naturally to a discussion of how to determine Bravais lattice types.

**1 April 2022**14.00-15.00 UK time. Slides (pdf, 1.4M).- Open table with MaThCryst colleagues and other crystallographers.
**Title**. A discussion of continuous invariants and metrics on crystals.**Abstract**. During the great discussion on 25th March with many crystallographers across the world, we agreed that the strongest possible equivalence of periodic crystals is rigid motion (or isometry), which cannot really change any crystal stucture. We also decided to continue next week in a similar format. The next step is to understand which invariants can reliably distinguish crystals up to isometry and how to avoid arbitrary thresholds to continuously quantify the similarity of crystals. One motivation is to automatically identify near duplicates that appear in experimental databases and among simulated structures obtained as different approximations of the same energy minimum in Crystal Structure Prediction.

**25 March 2022**14.00-15.00 UK time. Slides (pdf, 1M).- Open table with MaThCryst colleagues and other crystallographers.
**Title**. A discussion of equivalences and invariants of periodic crystals.**Abstract**. Periodic lattices and more general periodic point sets model any solid crystalline materials and can be studied modulo various equivalence relations. Historically, all early crystals looked very symmetric and were distinguished by their symmetries. This traditional classification splits all periodic crystals into 219 space-group types (230 if mirror images are distinguished). The recent review by Massimo Nespolo, Mois Aroyo, and Bernd Souvignier gave excellent examples of potential confusion even between "groups" and "group types". Since many similar words have different meanings in closely related areas such as crystallography and mathematics (algebra and geometry), it makes sense to discuss the basic concepts, for example, lattices, equivalences, invariants, abstract groups and their linear (matrix) representations.

**11 March 2022**14.00-15.00 UK time. Video (60 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Continuous metrics and chiral distances for 2-dimensional lattices.**Abstract**. The paper "Mathematics of 2-dimensional lattices" (the latest pdf, an early version in arxiv:2201.05150) introduced continuous metrics on the spaces of 2-dimensional lattices considered up to isometry, rigid motion, similarity (with uniform scaling) in the plane. These spaces contain the non-oblique Bravais classes (hexagonal, tetragonal (square), rectangular, centered-rectangular) as low-dimensional strata. A continuous distance from a lattice to the subspace of non-oblique lattices can be considered as a real-valued chiral measure for a deviation of a given lattice from its closest higher-symmetry neighbor. We will discuss explicit formulae of metrics and chiral distances, and their computations for millions of crystal lattices extracted from the Cambridge Structural Database in the joint work with Matt Bright and Andy Cooper.

**4 March 2022**14.00-15.00 UK time.- James Darby (University of Cambridge, UK).
**Title**. Compressing local atomic neighbourhood descriptors.**Abstract**. Many atomic descriptors are currently limited by their unfavourable scaling with the number of chemical elements S. For instance, the length of body-ordered descriptors typically scales as (NS)^v where v+1 is the body-order and N is the number of radial basis functions used in the density expansion. We introduce two distinct approaches which can be used to overcome this scaling for the Smooth Overlap of Atomic Positions (SOAP) power spectrum. Firstly, we show that the power spectrum is amenable to lossless compression with respect to both S and N, so that the descriptor length can be reduced from O(N^2 S^2) to O(NS). Secondly, we introduce a generalized SOAP kernel, where compression is achieved through the use of the total, element agnostic density, in combination with radial projection. The ideas used in the generalized kernel are equally applicably to any other body-ordered descriptors and we demonstrate this for the Atom Centered Symmetry Functions (ACSF). Finally, both compression approaches are shown to offer comparable performance to the original descriptor across a variety of numerical tests.

**28 January 2022**14.00-15.00 UK time. Video (54 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Geographic-style maps of crystal lattices from the Cambridge Structural Database.**Abstract**. The Crystal Isometry Principle (first presented at MACSMIN 2021) says that all real non-equivalent crystals should be geometrically recognizable by their periodic structures of atomic centres without any chemical data. As a consequence, all known and not yet discovered periodic crystals live in a common Crystal Isometry Space (CRISP) similarly to all chemical elements in Mendeleev's table. The space CRISP can be continuously and unambiguously parameterised by complete isometry invariants, which found unexpected duplicates in the Cambridge Structural Database (CSD), now under investigation by five journals. The talk will present visual maps of CRISP for hundreds of thousands of crystal lattices from the CSD. These maps support the natural preference for higher symmetry crystals and experimentally justify the continuity of CRISP. The latest versions of the key papers are linked at the Geometric Data Science page.

**14 January 2022**14.00-15.00 UK time.- Michael White (University of Manchester, UK).
**Title**. Digital Fingerprinting of Microstructures.**Abstract**. A statistical framework is systematically developed for compressed characterisation of a population of images, which includes some classical computer vision methods as special cases. The focus is on materials microstructure. The ultimate purpose is to rapidly fingerprint sample images in the context of various high-throughput design/make/test scenarios. This includes, but is not limited to, quantification of the disparity between microstructures for quality control, classifying microstructures, predicting materials properties from image data and identifying potential processing routes to engineer new materials with specific properties. Here, we consider microstructure classification and utilise the resulting features over a range of related machine learning tasks, such as supervised, semi-supervised, and unsupervised learning.

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**10 December 2021**14.00-15.00 UK time. Video (55 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Visual maps of the isometry space of 3-dimensional lattices.**Abstract**. Since crystal structures are determined in a rigid form, 3-dimensional periodic crystals and their lattices should be studied up to rigid motion or isometry, which are compositions of translations, rotations, and reflections. The resulting space of isometry classes of lattices is infinite and was previously studied mostly by discrete invariants, which split this continuous space into a finite number of strata or low-dimensional subspaces representing high-symmetry lattices. We extend the continuously parameterised map of all lattices in dimension 2 (joint with Matt Bright and Andy Cooper) to dimension 3 and also introduce easy metrics on lattices, which are continuous under perturbations of lattice bases.

**26 November 2021**14.00-15.00 UK time.- Andreas Alpers (Liverpool Mathematical Sciences).
**Title**. Tomographic Imaging of Crystalline Structures: Mathematics & Materials Science.**Abstract**. Since the invention of the first CT scanner in the 1970s, tomography has evolved into a powerful imaging tool in modern medicine. By computer processing of scans acquired from multiple angles, tomography can provide 3D images of inner parts without having to cut the patient. Of course, tomography has also been found to be attractive in materials science. Here, however, new fundamental challenges arise as, for instance, crystalline objects are non-continuous, data is (or can be) acquired only from few angles, and data can be very noisy. It therefore becomes increasingly important to develop problem-specific reconstruction algorithms. As I will discuss in my talk, this can lead into areas of mathematics previously thought to be unrelated to materials science. And, vice versa, the materials science imaging tasks can motivate new developments in mathematics. I will focus in my talk on this interplay between mathematics and materials science, drawing on two examples from my research: the tomographic reconstruction of polycrystals from diffraction data and the reconstruction of nanowires from electron tomography data.

**12 November 2021**14.00-15.00 UK time. Video (67 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Continuous metrics on isometry classes of 2-dimensional lattices.**Abstract**. A periodic lattice is an infinite set of all integer linear combinations of basis vectors in a Euclidean space. A natural equivalence on lattices is rigid motion (a composition of translations and rotations) or isometry (also including reflections). All 2-dimensional lattices can be parameterised by three real parameters, but there was no easy metric on isometry classes of lattices, which is also continuous under basis perturbations. Using a recent parameterisation of 2-dimensional lattices by root forms, we introduce metrics that are invariant up to isometry or up to rigid motion, which preserves orientation. The latter case also quantifies chirality by measuring a distance from a chiral lattice to its closest mirror-symmetric neighbour. The talk is based on the updated dimension 2 paper joint with Matt Bright and Andy Cooper.

**29 October 2021**14.00-15.00 UK time.- Steff Farley (Loughborough University Mathematical Sciences).
**Title**. Quantitative Data Integration with Computational Models of Dewetting Processes.**Abstract**. The self-assembly of nanostructures has been of growing interest in materials science, with particular advancements in the development of computational models that describe this self-assembly. So far, however, the utility of these models have been limited by the absence of methods to integrate real experimental data with numerical simulations or the experimental and simulation conditions that generate them. We simulate images of the resulting nanostructures using two models, a kinetic Monte Carlo (KMC) model and a dynamical density functional theory (DDFT) model, and attempt to relate a dataset of 2625 real atomic force microscope (AFM) images with these models. We propose a map to a feature space of modified Minkowski functionals that meaningfully characterise the geometry of this image space as the first step of this task. These coordinate statistics show some promise of allowing us to successfully carry out the inverse problem but are insufficient for this task when used alone. We propose that drawing from the methods of Riemannian geometry in combination with Approximate Bayesian Computation should be considered as a more suitable approach.

**15 October 2021**14.00-15.00 UK time. Video (58 min).- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Complete isometry invariants of periodic lattices in dimensions 2 and 3.**Abstract**. Motivated by crystal structures determined in a rigid form, we study lattices up to rigid motion or isometry, which is a composition of translations, rotations and reflections. The resulting Lattice Isometry Space (LISP) consists of infinitely many isometry classes of lattices. In dimensions 2 and 3, we parameterise this continuous space by root forms (consisting of three and six numbers) and introduce new metrics on root forms satisfying all metric axioms and continuity under perturbations. These parameterisations help visualise hundreds of thousands of crystal lattices in the Cambridge Structural Database. The talk is based on the dimension 2 and dimension 3 papers with Matt Bright and Andy Cooper.

**9 July 2021**14.00-15.00 UK time.- Daniel Widdowson (Liverpool Materials Innovation Factory).
**Title**. Invariant-based visualisation of large crystal datasets.**Abstract**. Any solid crystalline material can be represented by a periodic set of points at atomic centres. Crystals or periodic point sets can be distinguished in the strongest possible way up to rigid motion or isometry (a composition of rotations and translations). The talk will discuss continuous isometry invariants that reliably quantify similarities between periodic point sets. These AMD invariants produce distance matrices for any crystal dataset, which may come from Crystal Structure Prediction or be more diverse and include different chemical compositions. Invariant-based distance matrices can be visualised in the form of a tree map showing all crystals connected to their closest neighbours. The AMD invariants are so fast that a map of all 229K molecular organic crystals from the Cambridge Structural Database was produced in less than nine hours on a modest desktop, see the last image in the AMD paper.

**26 March 2021**14.00-15.00 UK time.- Phil Smith (Liverpool Materials Innovation Factory).
**Title**. The Density Fingerprint of a Periodic Point Set.**Abstract**. Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction. The talk is based on the paper with H.Edelsbrunner, M.Wintraecken, T.Heiss, V.Kurlin, accepted at SoCG 2021.

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**17 December 2020**15.15-16.45 UK time, joint with the Computational Mathematics Seminar.- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Introduction to Periodic Geometry for applications in Crystallography and Materials Discovery.**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 discuss a stable isometry classification of periodic crystals. Most past invariants of crystals such as symmetry groups are discontinuous under atomic vibrations. The new classification establishes foundations of periodic geometry. Based on joint papers with Andy Cooper, Angeles Pulido, Herbert Edelsbrunner, Mathijs Wintraecken, Teresa Heiss, Phil Smith, Marco Mosca, Dan Widdowson, e.g. arxiv:2009.02488.

**22 October 2020**15.00-16.00 (UK), 7.00-8.00 (California).- Larry Andrews (Ronin Institute, US).
**Title**. The Rise and Fall of the V7 Metric.**Abstract**. Having a measure of the difference between two lattices has several applications: databases of unit cells, clustering of images from serial crystallography, planning of epitaxial processes. Originally, unit cells were compared and cataloged using their sorted unit cell edge lengths. This technique has several problems. The first complete measure for lattices was the V7 metric (Andrews, Bernstein, Pelletier, 1980). It was complete, simple to understand, and in database retrievals it would never have false negative results. However, it was discovered to have a deficiency. The creation of V7, problems with it, and the evolution of methods for more correct distance measures will be discussed.

**10 July 2020**13.30-15.00 UK time.- Marco Mosca (Liverpool Materials Innovation Factory).
**Title**. Distance-based isometry invariants of crystals.**Abstract**. The talk will consider the isometry classification of periodic sets motivated by the Crystal Structure Prediction. A periodic sets of points models any solid crystalline material (a crystal) and represents any atom or a molecular centre by a zero-sized point. Since most crystals are rigid bodies, the natural equivalence on periodic sets is a rigid motion (a composition of translations and rotations) or an isometry, which preserves distances between points. Traditionally crystals were distinguished by discrete invariants such as symmetry groups that are unstable under atomic vibrations. However, large datasets of simulated crystals contain too many nearly identical crystals obtained as slightly different approximations to local minima of an energy function. The new distance-based isometry invariants are provably stable under noise. The experiments on the T2 dataset from the 2017 Nature paper show that the new invariants distinguish simulated crystals better than past tools and can automatically match experimental crystals to their closest simulated versions.

**19 May 2020**10.30-12.00 UK time.- Vitaliy Kurlin (Liverpool Materials Innovation Factory).
**Title**. Isometry classification of periodic crystals.**Abstract**. The Crystal Structure Prediction (CSP) tools output too many simulated crystals, including many nearly identical local minima of an energy, because there was no reliable way to quantify the similarity between crystals. Since most solid crystalline materials are rigid, the similarity problem is now rigorously stated as a classification modulo rigid motions or isometries that preserve inter-point distances. Most descriptors, especially if they are based on ambiguous unit cells, are either not isometry invariants or are unstable under atomic vibrations, hence can not reliably quantify crystal similarities. The talk introduces three approaches to build stable-under-noise and complete isometry invariants of periodic crystals.

**3 February 2020**11-13 (lunch and discussion from about 12).- Uwe Grimm (Open University School of Mathematics and Statistics).
- Location : the boardroom on the ground floor in the MIF.
**Title**. An introduction to aperiodically ordered systems and their diffraction.**Abstract**. The talk will present an introductory overview on aperiodically ordered systems obtained from cut and project or inflation methods. The key methods and results will be illustrated by means of examples and pictures, leaving aside much of the mathematical details. For cut and project sets, the diffraction is well understood, and the general result and its application to examples will be discussed. In contrast, inflation systems are still only partially understood, and paradigms with different spectral type are presented. Finally, I will briefly discuss recent results on the Fourier transform of Rauzy fractals and the point spectrum of unimodular Pisot inflation tilings, which uses both the inflation and the projection structure.

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**13 December 2019**: Tom Hasell (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the 3rd floor meeting room in the MIF.
**Title**. Making polymers from elemental sulfur**Abstract**. Sulfur is an industrial by-product, removed as an impurity in oil-refining. This has led to vast unwanted stockpiles of sulfur, and resulted in low bulk prices. Sulfur is therefore a promising alternative feedstock to carbon for polymeric materials. Sulfur normally exists as S8 rings – a small molecule with poor physical properties. On heating, these sulfur rings can open and polymerise to form long chains. However, because of the reversibility of sulfur bonds, these polymers are not stable, and decompose back to S8 over time, even at room temperature. Inverse vulcanisation has made possible the production of high sulfur polymers, stabilised against depolymerisation by crosslinking [1]. These high sulfur-materials show excellent potential as low cost water filters to remove mercury [2-4]. Heavy metal contamination exists in the waste streams of many industries, and mercury is of particular concern for human health. Alternative crosslinkers for inverse vulcanisation, from industrial by-products or bio-renewable sources, can be used to reduce the cost and improve the properties of the resultant polymers [4,5]. Polymers made from sulfur also have many other intriguing properties and applications in optics, electronics, insulation, and antimicrobial materials. We recently reported a catalytic route that reduces the required reaction time, temperature, and by-products – and allows otherwise unreactive crosslinkers to be used [6].

[1] Nat. Chem. 2013, 5, 518–524. [2] Chem. Commun., 2016, 52, 5383 - 5386. [3] J. Mater. Chem. A. 2017,5, 11682-11692 [4] J. Mater. Chem. A., 2017, 5, 18603 [5] Chem.-Eur. J. 2019, 25, 10433-10440 [6] Nat. Commun. 2019, 10, 647

**4 November 2019**: Linjiang Chen (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the 3rd floor meeting room in the MIF.
**Title**. Functional materials discovery: ‘Seeing’ structures and functions?**Abstract**. We have recently demonstrated a strategy for how new, functional molecular materials can be discovered. By carrying out a priori prediction of both the crystal structure and its functional properties, energy–structure–function (ESF) maps are created to aid researchers, without computational expertise, in realizing several remarkable porous materials promising for different possible applications.

As large-scale computational screening studies become routinely carried out, new opportunities have arisen for accelerating materials discovery by taking advantage of the availability of big data. It, however, remains a challenge to properly understand and use the vast amount of data generated by simulations. In my talk, I will show how we use computation to accelerate serendipitous discovery of new functional materials, as well as the prowess of data-driven machineries becoming available. I will outline our current focuses on and approaches to the following topics:- 1. ESF mapping to identify synthetic targets with desired functional properties;
- 2. Interactive visualization of high-dimensional structure–property relationships;
- 3. Smart navigation of the ESF space with machine learning.

**21 October 2019**: Georg Osang (IST Austria Computational Geometry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the boardroom on the ground floor in the MIF.
**Title**. The Multi-cover Persistence of Euclidean Balls**Abstract**. Persistent homology has become a popular tool to analyse various kinds of data, in particular in material sciences. Specifically, persistence of discrete point sets has recently been used to analyse sphere packing data, to shed light on structures arising in sphere packings at different packing densities. We generalize this notion and introduce higher-order persistence of discrete point sets. We address computational challenges and show how this notion can deal with noisy point samples. In the setting of sphere packings we show that this notion can also capture a wider variety of local structures, and in particular can distinguish between the hexagonal close packing and the face centered cubic lattice packing, two structures know to have optimal packing density in 3 dimensions.

**1 July 2019**: John Claridge (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the 3rd floor meeting room in the MIF.
**Title**: Challenges and opportunities in condensed matter science.**Abstract**. Solid state chemistry, is the study of the synthesis, structure and properties of solid state materials and particularly how these are related to each other. It is obviously closely linked to condensed matter physics, mineralogy, crystallography, ceramics, metallurgy, materials science and electronics. We'll begin with some simple illustrations of these relationships, before looking at how structural/crystallographic methods have advanced in order to both characterise and describe the materials we can now make, and where I believe there are limitations in our current tools and understanding, with examples taken from my own work and more generally, and why they are important.

**13 May 2019**: Vanessa Robins (ANU Applied Mathematics).- Time : 11-12.15 (with lunch at VGM from 12.30).
- Location : the boardroom on the ground floor in the MIF.
**Title**: An introduction to the mathematical description and computational enumeration of periodic tilings and nets.**Abstract**. Periodic tilings and nets are simplified models of atomic arrangements in crystals. Systematic methods for describing and enumerating possible crystal structures are based on factoring out either the translational periodicity or the full crystallographic symmetry. The mathematical version of “factoring out” gives us an object called an orbifold. I will explain how orbifolds encapsulate the essential elements of a crystallographic group and provide an effective basis for the enumeration of hypothetical crystalline structures. The presentation will be informal and largely based on pictures and examples, hopefully accessible to non-specialists.

**29 April 2019**: Vladimir Gusev (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the boardroom on the ground floor in the MIF.
**Title**: New approaches to the crystal structure prediction

**1 April 2019**: Chris Collins (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the boardroom on the ground floor in the MIF.
**Title**: Computationally driven materials discovery**Abstract**. In the previous seminar by Dr D. Antypov, we discussed the general concepts of energy calculations and crystal structure prediction (CSP). In this session are to discuss how CSP is used in practice in the search for new inorganic materials. We present how the use of ‘probe’ structures accelerates the experimental exploration of composition space. A probe structure is a hypothetical crystal structure for a given composition, it need not be the ground state. However, it needs to be complex enough that it can contain enough co-ordination chemistry to produce an energy which is representative of potential single phases. When probe structures are generated across a range of compositions we can then construct the convex hull, and look for minima – regions of compositional space in which new compounds are likely to be found. We will discuss the methods for creating probe structures. The development of a method is split into two streams: how can we best represent crystal structures in a computer? And how can we search through the possible configuration space? We will then present our latest codes developed for probe structure prediction (Monte Carlo – Extended Module Materials Assembly (MC-EMMA[1]) and the Flexible Unit Structure Engine (FUSE[2]), along with example phase fields computed using them and show how probe structure generation for composition prediction is a powerful, predictive tool to accelerate the discovery of materials.

1. C. Collins, M. S. Dyer, M. J. Pitcher, G. F. S. Whitehead, M. Zanella, P. Mandal, J. B. Claridge, G. R. Darling and M. J. Rosseinsky, Nature, 2017, 546, 280-284.

2. C. Collins, G. R. Darling and M. J. Rosseinsky, Faraday Discussions, 2018, 211, 117-131.

**11 March 2019**: Dmytro Antypov (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the 3rd floor meeting room in the MIF.
**Title**: Introduction to crystal structure prediction**Abstract**. Crystalline solids such as metal oxides and perovskites are functional materials used in solar cells, batteries and many other devices. These materials are often composed from 3, 4 or 5 different chemical elements arranged in a periodic three dimensional structure. Such chemical diversity on the one hand allows for fine-tuning of material properties but on the other hand makes the identification and synthesis of stable compounds difficult. To accelerate the design of such materials we use computation to predict the combinations of the constituent chemical elements that will lead to stable crystalline structures. In this talk I will explain how this chemistry problem is formalised to have a tractable computational solution and show some examples.

**25 February 2019**: Phillip Maffettone (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the 3rd floor meeting room in the MIF.
**Title**: Describing solids, and challenges for high-throughput analysis.**Abstract**. The talk will introduce reciprocal space and the reciprocal lattice as the inverse of the direct lattice, building on the previous introduction describing periodic structures. The reciprocal lattice plays a fundamental role in understanding wave mechanics in crystalline materials, and is implemented in most analytical studies of materials through the theory of diffraction. These concepts will be presented with a focus on their experimental utility, necessary limitations, and potential implications for high-throughput automation. In particular, we will explore the question: When can we meaningfully invert the diffraction pattern and the pair-wise information it contains, and how can we automate this inversion?

**11 February 2019**: Mike Gaultois (Liverpool Chemistry).- Time : 11-13 (lunch and discussion from about 12).
- Location : the 3rd floor meeting room in the MIF.
**Title**: Searching for new magnetic materials.**Abstract**. Magnetic materials have fascinated humans since the antiquity, but only recently since the 1800's with the work of Oersted and Maxwell did we begin to develop a deeper understanding of magnetism. Magnetic materials are now involved in many technological applications, many of which are limited by the material performance. One area of enquiry is the search for materials that can make hard permanent magnets, and another area of enquiry is the search for materials that conduct electricity without resistance (i.e., superconductors). This discussion is interested in exploring how we can best computationally predict (through first principles, or otherwise) the magnetic properties of a given material. We will describe what others have done to search for new materials using machine learning techniques, and discuss potential areas of interaction between MIF++ where advances can be made.

**28 January 2019**: Vitaliy Kurlin (Liverpool Materials Innovation Factory).- Time : 11-13 (lunch and discussion from about 12).
- Location : the boardroom on the ground floor in the MIF.
**Title**: Mathematical challenges for solid crystalline materials.**Abstract**. The talk will introduce periodic structures that model all solid crystalline materials, for example molecular crystals, covalent organic frameworks and inorganic crystals based on isolated atoms or ions. The key problems of materials discovery will be stated in mathematical terms: what structures should be equivalent and how to quantify a similarity between crystals. We will discuss several requirements for potentials solutions that should work for real crystals.

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