Prof Vitaliy Kurlin: mathematics & computer science

MACSMIN 2025: 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.

• Dates: 8-12 September 2025 in a hybrid form in the Materials Innovation Factory, Liverpool, UK.
• MACSMIN 2025 is a satellite of the 35th European Crystallographic Meeting, Poznan, Poland.
• Conference organizers: Vitaliy's Data Science group including Olga Anosova, Dan Widdowson, Will Jeffcott, Yury Elkin, and Jonathan (Teddy) McManus. To participate for free, please e-mail.
• Tentative timetable. Tutorials on 9.9.2025 will present the latest advances in Geometric Data Science for crystals (Dan Widdowson), proteins (Olga Anosova), and molecules (Yury Elkin).
• We plan 8-10 September in person with a conference dinner and 11-12 September on zoom.
• All talks will be aimed for a broad audience of scientists including PhD students and postdocs.
• The conference is funded by the LMS through the network Applied Geometry and Topology.
• Travel information for Liverpool (UK): venue, accommodation, trains, and flights.
• Related meetings: mini-symposium Computational Data Science of Nanostructures (6 hours) at the SIAM annual meeting on July 28 - August 1, 2025 in Montreal: part 1, part 2, part 3.
• The AMS session on Open Problems in Geometric Data Science at JMM 2026, Washington DC.
• The ICERM workshop Rigidity Theory meets Geometric Data Science for applications in chemistry on 6-10 July 2026, Brown University, Providence (US).
• The regular MIF++ seminar is a continuous version of the annual MACSMIN.
• Past meetings of the MACSMIN series: 2024, 2023, 2022, 2021, 2020.

Timetable (very tentative, all UK times)   Monday   Tuesday   Wednesday   Thursday   Friday

• Monday 8th September (in person and on zoom)
  • 08.45-09.00 Opening, history, and vision of MACSMIN by Vitaliy Kurlin (Liverpool MIF, UK)
  • • 09.00-09.50 Janos Pach (Rényi Institute, Budapest and EPFL, Lausanne)
    • Title. Mysteries about crossing numbers.
    • Abstract. For any graph G, the crossing number cr(G) of G is defined as the smallest number k such that G can be drawn in the plane with k crossing points between its edges. The pairwise crossing number pair-cr(G) of G is defined in a similar way, except that in this case we want to minimize the number of crossing pairs of edges. Obviously, pair-cr(G) is at most cr(G). It is one of the most annoying unsolved problems for graph embeddings whether there exists a graph for which these two parameters do not coincide. Since computing these parameters are NP-complete problems, it is difficult to test the inequality even for small graphs. After giving a whirlwind survey of the topic, we discuss some recent develop.
  • • 10.00-10.50 Isaac Sugden (Cambridge Crystallographic Data Centre, UK)
    • Title. Metal steric descriptors (TBC).
  • • 11.00-11.50 Thérèse Malliavin (Université de Lorraine, CNRS, France)
    • Title. Investigating a Local Approach for Calculating Protein Conformations.
    • Abstract. Protein structure prediction is generally based on the use of local conformational information coupled with long- range distance restraints. Such restraints can be derived from the knowledge of a template structure or the analysis of protein sequence alignment in the framework of models arising from the physics of disordered systems. The accuracy of approaches based on sequence alignment, however, is limited in the case where the number of aligned sequences is small. Here, we derive protein conformations using only local conformations knowledge by means of the interval Branch-and-Prune algorithm. The computation efficiency is directly related to the knowledge of stereochemistry (bond angle and ω values) along the protein sequence and, in particular, to the variations of the torsion angle ω. The efficiency of predicting torsion angles ω is then evaluated for various pre-processing of input data-sets.
      Reference: da Rocha W, Liberti L, Mucherino A, Malliavin TE. Influence of Stereochemistry in a Local Approach for Calculating Protein Conformations. J Chem Inf Model. 2024 Dec 9;64(23):8999-9008. doi: 10.1021/acs.jcim.4c01232
  • 12.00-14.00 lunch break
  • • 14.00-14.50 Graeme Day (University of Southampton, UK)
    • Title. Predictive crystallography at scale: learning from an unprecedentedly large collection of crystal structure prediction landscapes.
    • Abstract. The talk will discuss the results of scaling crystal structure prediction calculations to over 1000 molecules [1], in an effort at validating global lattice energy minimisation for crystal structure prediction and creating high quality date for learning relationships between molecular structure, crystal structure and properties. I will discuss assessment of the dataset, insights that the data provides on crystallisation behaviour and our future outlook for how such a dataset will be valuable for machine learning methods applied to molecular organic crystals.
      [1] Taylor, Butler and Day, Faraday Discussions, 2025,256, 434-458.

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• Tuesday 9th September (in person and on zoom)
  • • 09.00-09.50 Yury Elkin (University of Liverpool, UK)
    • Title. Complete invariants of molecular graphs with continuous metrics in polynomial time.
    • Abstract. Geometric graphs are graphs with unordered vertices and straight-line edges embedded in a Euclidean space. These graphs represent many real structures including molecular graphs whose rigid shapes (equivalence classes under Euclidean motion) determine their chemical properties. Atomic vibrations and experimental noise motivate developing invariant descriptors that are preserved under any rigid motion (a composition of translations and rotations) and continuously change under perturbations of vertices. We developed a complete invariant that can be inverted back to a geometric graph, uniquely under rigid motion, and has a Lipschitz continuous metric satisfying all axioms. For a fixed dimension, the invariant and metric are computable in a polynomial time of the number of unordered vertices and hence avoiding exponentially many permutations. The new invariant distinguishes all chemically different molecules in big databases QM9 and GEOM of 130K+ and 31M+ conformations, respectively, within a few hours on a modest computer.
  • • 10.00-10.50 Olga Anosova (University of Liverpool, UK)
    • Title. A classification of protein backbones by complete and bi-continuous invariants in linear time. /li>
    • Abstract. Proteins are large biomolecules that regulate all living organisms and consist of one or several chains. The primary structure of a protein chain is a sequence of amino acid residues whose three main atoms (alpha-carbon, nitrogen, and carbonyl carbon) form a protein backbone. The tertiary structure of a protein chain is a geometric graph represented by atomic positions in 3-dimensional space. Because different geometric graphs often have distinct functional properties, it is important to continuously quantify differences in rigid graphs of protein backbones. Unfortunately, many widely used similarities of proteins fail axioms of a distance metric and discontinuously change under tiny perturbations of atoms. This work develops a complete invariant that identifies any protein backbone in 3-dimensional space, uniquely under rigid motion. This invariant is Lipschitz bi-continuous in the sense that it changes up to a constant multiple of a maximum perturbation of atoms, and vice versa. The new invariant detected thousands of (near-)duplicates in the Protein Data Bank, whose presence skews machine learning predictions. The talk is based on the paper in MATCH 2025.
  • • 11.00-11.25 Jack Gallimore (University of Liverpool, UK)
    • Title. Geometry based method for identifying hydrogen bonds and classifying helices.
    • Abstract. Understanding secondary structure in proteins is fundamental to structural biology, yet current algorithms such as DSSP rely on fixed geometric cutoffs and manually tuned heuristics, making them both opaque and discrete. We propose a geometry-based framework that identifies hydrogen bonding using statistically justified continuous regions in geometric space, without rigid thresholds. By analyzing millions of residues across 100,000 PDB structures, we define regions in the geometric distribution of interatomic distances and angles that correspond to hydrogen bonding. These regions, rather than fixed cutoffs, naturally reflect the spatial variation observed in experimental data and enable us to robustly detect secondary structure, particularly helices. This method offers a more interpretable and adaptable alternative for structural annotation, with the potential to unify the classification of helices, including 3₁₀ and π-helices.
  • • 11.30-11.55 Surya Majumder (University of Liverpool, UK)
    • Title. Quantifying continuous asymmetry with isometry invariants.
    • Abstract. Symmetry plays a central role in understanding the structure and properties of crystals. However, many crystals are not perfectly symmetric, and quantifying such deviations in a continuous way remains an open problem. In this work, we present a geometric approach to characterise this asymmetry using the Pointwise Deviation from Asymptotic (PDA), an isometry invariant that measures how much distances to atomic neighbours deviate from their asymptotic behaviour in periodic point sets. The resulting framework yields a real-valued, geometry-based measure of asymmetry that extends beyond classical crystallographic symmetry groups. We demonstrate the method on both experimental and predicted molecular crystals, particularly by highlighting regularities in experimental crystals that indicate potential synthesisability.
  • 12.00-14.00 lunch break
  • • 14.00-14.50 Vitaliy Kurlin (University of Liverpool, UK)
    • Title. Geometric Data Science established three principles of Atomic Geometry.
    • Abstract. The Crystal Isometry Principle (CRISP, first presented at MACSMIN 2021) says that any real periodic crystal (under fixed ambient conditions) is uniquely determined by the geometry of only atomic centres without atomic types such as chemical elements, charges, etc. The CRISP confirms our physical intuition saying that changing an atomic type should inevitably perturb inter-atomic interactions and hence relative atomic positions, which was experimentally verified on the Cambridge Structural Database (NeurIPS 2022, CGD 2024) and later on other databases (IUCrJ 2024, SR 2025).
      
      MACSMIN 2024 presented the Principle of Molecular Rigidity making a similar conclusion for molecules instead of materials: any (3D conformation of a) real molecule can be uniquely reconstructed with all atomic types from its cloud of atomic centres under rigid motion. This conclusion required complete and continuous invariants of all finite clouds of unordered points in a Euclidean space (CVPR 2023) and was verified on the databases QM9 and PDB (MATCH 2025).
      
      This talk will finalize the hierarchy of continuous invariants of crystals from the simplest to complete with polynomial-time metrics (PR 2025) and will present the 3rd principle: the precise enough geometry of a real atomic neighbourhood based on a very few numerical invariants characterises any chemical element. The emerging area of Atomic Geometry provides continuous versions (geographic-style maps) of Mendeleev's table and can reduce all chemistry problems to geometry.
  • • 15.00-15.50 Daniel Widdowson (University of Liverpool, UK)
    • Title. Crystal Geomap: a visualisation & comparison tool for crystals.
    • Abstract. The number of experimentally measured crystal structures has risen exponentially in recent decades. Crystals are often classified by space group, which can be the same for distinct structures and different for similar ones; yet most continuous descriptors such as density can coincide for dissimilar crystals (a false positive). The Pointwise Distance Distribution (PDD) is a continuous descriptor (isometry invariant) akin to a crystal’s genetic code with no false positives amongst real organic crystals. PDDs contain enough information to reconstruct most crystals while comparisons take only milliseconds and are fast enough to find all matches in a database of a million structures in a few minutes, a task requiring years for comparison tools such as COMPACK. In 2021 we used invariants to find duplicates in the Cambridge Structural Database (CSD), 5 of which were investigated by curators. Since then the experiment’s speed and scope have grown considerably, culminating in a list of over 380,000 cross-references between the CSD and COD (Open Crystallography Database), comprising 80% of the COD. The PDD of a crystal is considered its coordinate in a universal ‘crystal space’, motivating us to create a visualisation and comparison tool based on our invariants. Our ‘Crystal Geomap’ app allows a user to compare or cluster crystals, to export or plot on chosen x and y axes where all crystals are seen in one space with physically meaningful coordinates.
  • • 16.00-16.25 Jonathan McManus (University of Liverpool, UK)
    • Title. Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets.
    • Abstract. The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values. The talk is based on the paper whose early draft is arxiv:2410.23288, see the latest pdf.
  • 16.30-16.55 (TBC) William Jeffcott (University of Liverpool, UK)

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• Wednesday 10th September (in person and on zoom)
  • • 10.00-10.50 (TBC) Mike Glazer (Oxford University, UK)
    • Title. Perovskite structures – a personal journey.
    • Abstract. Perovskites are some of the most important crystalline materials because of their vast range of variations and properties. In this talk I shall review the history of structure determination and phase transitions of perovskite materials. This begins in the early 1940's, a time when there were hardly any publications on these materials and when perovskites such as barium titanate were considered to be strategic materials for the war effort. Since then, the amount of interest in the structures of perovskites and their related materials has become stratospheric, so much so that it is now impossible to keep up with the literature. It is clear that there is still a considerable future for research into these important structures, as new variations based on the crystal structures, and therefore their physical properties, are still being made.
  • • 12.00-14.00 lunch break
  • • 14.00-14.50 Simon J. L. Billinge (Columbia University, US)
    • Title. When is a structure a new structure? Towards new measures of structural similarity.
    • Abstract. For 100 years, since the development of x-ray crystallography, we have been able to study atomic structure. This knowledge is the foundation for much of our understanding of material properties. Despite these successes, challenges remain. I will describe some advances in AI/ML that are helping us take a completely new look at solving structure. As well as helping us solve conventional unsolved problems, these new methods are helping us to think about structure in new ways, causing us to ponder questions whose answers seemed hitherto obvious, such as "what is the structure of this material?", "when do two materials have the same structure and when do they have a different structure?", and "when is a defect a defect?". Many of these questions don't have good answers, but defining the problem is often a first step. In many cases, the problem is the problem, not the solution.
  • • 15.00-15.50 Xiaocheng Shang (University of Birmingham, UK)
    • Title. Accurate and efficient numerical methods for molecular and particle simulation using adaptive thermostats.
    • Abstract. I will discuss how we can use the so-called adaptive thermostats, which rely on a negative feedback loop, to develop accurate and efficient numerical methods for molecular and particle simulations, focusing on applications at mesoscales.

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• Thursday 11th September (only on zoom)
  • • 14.00-14.50 Marjorie Senechal (Smith College, US)
    • Title. Alan Mackay and the Local Theorem.
    • Abstract. Alan Lindsay Mackay (1926 - 2025) is known to crystallographers for thinking - and working - outside the box they call the unit cell. To turn the subject inside out, he drew on his wide range of interests and expertise. In this talk we consider the (possible) influence of the mathematician B. N. Delone.
  • • 15.00-15.50 Greg McColm (University of South Florida, US)
    • Title. Using Category Theory to Describe Complex Structures.
    • Abstract. Category theory is a branch of abstract algebra used to describe complex structures and systems in mathematics, science, and engineering. We present a brief and accessible introduction to some of the basics of the subject and show how they can be applied to describe complex crystal structures. Given a periodic graph or complex, a category can indicate how substructures can be embedded within it, and (“dually”) indicate the overall structure of the graph or complex. We go through some examples of real and hypothetical crystal structures.
  • • 16.00-16.50 Peter Olver (University of Minnesota, US)
    • Title. Reconstruction and signatures of 3D bodies.
    • Abstract. I will present a solution to the reconstruction problem for three-dimensional bodies from their two-dimensional projections based on the method of envelopes. Then the induced action of the Euclidean group on the body's projected outlines is analyzed using moving frames, leading to a complete classification of the outline differential invariants and the associated outline signature of the body.

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• Friday 12th September (only on zoom)
  • • 09.00-9.50 Fabio Pietrucci (Sorbonne Université, France)
    • Title. Exploring the structural landscape of materials with topology-based collective variables.
    • Abstract. will present a few similar approaches [1-4] aimed at exploring (meta)stable structures of molecules, clusters or extended materials: First, a suitable space of collective variables - invariant upon permutation of identical atoms - is defined, starting from the matrix of interatomic contacts (adjacency matrix). Second, biasing forces are applied along such variables, within finite-temperature molecular dynamics, to accelerate barrier crossing and explore in a reasonable computer time many different structures. Of course, the collective variables can also be employed to identify structures in a trajectory obtained, e.g., by brute force. A key question in this field is: how much information do we want to retain, when adopting some collective variables, and how much do we want to neglect? Contrary to machine learning force fields, for instance, here it is not optimal to resolve each distinct topology (particularly in liquids and disordered solids). I will present applications to different materials, from crystal nucleation to amino acid synthesis and decomposition.
      [1] F. Pietrucci and W. Andreoni, Phys. Rev. Lett. 107, 085504 (2011).
      [2] F. Pietrucci, A.M. Saitta, Proc. Natl. Acad. Sci. U. S. A. 112, 15030 (2015).
      [3] S. Pipolo, M. Salanne, G. Ferlat, S. Klotz, A.M. Saitta, F. Pietrucci, Phys. Rev. Lett. 119, 245701 (2017).
      [4] A. France-Lanord, H. Vroylandt, M. Salanne, B. Rotenberg, A.M. Saitta, F. Pietrucci, J. Chem. Theory Comput. 20, 3069 (2024).
  • 10.00-10.50 (TBC) Sergei Grudinin (Grenoble Alpes University, France)
  • • 13.00-13.50 Jason R. Hattrick-Simpers (University of Toronto, Acceleration Consortium and Vector Institute for AI)
    • Title. Methods for Generating Unbiased and Robust A.I.’s for Autonomous Materials Discovery.
    • Abstract. Since the publication of the Mission Innovation Materials Acceleration Platform, AI is increasingly responsible for driving automated experimental and computational campaigns. There have been multiple case studies for which autonomy was demonstrated to successfully drive materials optimization or discovery and the world of scientific robots has moved from science fiction to reality. However, within the AI community, it is well understood that AI models carry the biases of their creators, which can have serious implications for model deployment. These models may behave unpredictably, even within the bounds of their training data. Using specific case studies, I will illustrate how such biases arise in materials science and outline steps to mitigate them, leading to more robust models. In particular, I will discuss our recent work on: (1) identifying and mitigating search space bias through model disagreement, (2) quantitatively demonstrating how little basis there is for our biases about dataset completeness, and (3) developing new metrics for evaluating model-data synergies. Finally, I will briefly touch on broader efforts in self-driving lab development within the University of Toronto’s Acceleration Consortium.
  • • 14.00-14.50 Alex Chernatynsky (Missouri University of Science and Technology, US)
    • Title. A symmetry-oriented crystal structure prediction method for crystals with rigid bodies.
    • Abstract. We have developed an efficient crystal structure prediction (CSP) method for desired chemical compositions, specifically suited for compounds featuring recurring molecules or rigid bodies. We applied this method to two metal chalcogenides: Li3PS4 and Na6Ge2Se6, treating PS4 as a tetrahedral rigid body and Ge2Se6 as an ethane-like dimer rigid body. Initial trials not only identified the experimentally observed structures of these compounds but also uncovered several novel phases, including a new stannite-type Li3PS4 structure and a potential stable structure for Na6Ge2Se6 that exhibits significantly lower energy than the observed phase, as evaluated by density functional theory calculations. We compared our results with those obtained using USPEX, a popular CSP package leveraging genetic algorithms. Both methods predicted the same lowest energy structures in both compounds. However, our method demonstrated better performance in predicting metastable structures. The method is implemented with Python code which is available at https://github.com/ColdSnaap/sgrcsp.git.
  • • 15.00-15.50 Shubham Sharma (Pfizer, US)
    • Title. Addressing the challenges of polymorphism at Pfizer (TBC).
  • • 16.00-16.50 Andrea Darù (University of Chicago, US)
    • Title. Computational tools for the prediction and generation of novel reticular framework materials.
    • Abstract. This talk will present recent work on the computational prediction of metal-organic and covalent organic frameworks (MOFs and COFs), with addition of experimental insights. I will focus on the study "Symmetry is the Key to the Design of Reticular Frameworks" (Adv. Mater., 10.1002/adma.202414617), which introduces a computational tool that predicts feasible reticular framework topologies based solely on the connectivity and symmetry of isolated nodes and linkers. Unlike high-throughput methods, this approach prioritizes accuracy by generating a small, targeted set of candidate structures, optimizing computational resources. The tool is at https://rationaldesign.pythonanywhere.com. The talk will conclude with experimental perspectives on the applicability of this method to known and unknown synthesizable structures.

Travel information: venue, accommodation, trains, and flights

• All talks in person will be in the ground floor boardroom in the Materials Innovation Factory (MIF), Liverpool, UK. Address: 51 Oxford street, building 807 in the grid cell F5 on the campus map. The building has a secure entrance, so we will let the reception know about MACSMIN participants. The MIF is 15 min on foot from the Liverpool Lime Street station.
• If you contact us in advance, we can help with booking hotels. One option is the Liner hotel in a quiet street close to the Liverpool Lime Street main rail station. Explore other good hotels and attractions on the website visit Liverpool.
• The city has the Liverpool John Lennon airport with convenient buses to the centre. The larger Manchester airport has the train station with direct 90-min trains to the Liverpool Lime Street station. Check flights to nearby airports at Skyscanner.

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