**MACSMIN 2024** : 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 : 9-13 September 2024 in a hybrid form in the Materials Innovation Factory, Liverpool, UK.
- 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.
- We invite to participate in-person and have a modest budget to cover travel and accommodation.
- MACSMIN has the MIF++ scientific style and encourages rigorous results justified by proofs.
- The conference is funded by the LMS through the network Applied Geometry and Topology.
- All talks will be aimed for a broad audience of scientists, see the final timetable.
- Travel information for Liverpool (UK): venue, accommodation, trains, flights.
- Past meetings of the MACSMIN series: 2023, 2022, 2021, 2020.

### Timetable (final, all UK times) Monday Tuesday Wednesday Thursday Friday

**Monday 9th September**(only on zoom)**13.45-14.00**Opening, history, and vision of MACSMIN by Vitaliy Kurlin (Liverpool MIF, UK)**14.00-14.50**Marjorie Senechal (Smith College, US) Video (42 min) Slides (on Google drive)

Title. Geometry and crystallography: a conversation.

Abstract. Throughout the century from Fedorov's enumeration of the five convex parallelohedra in 1885 to Schectman's 1982 discovery of aperiodic crystals, "geometrical crystallography" comprised lattices, point sets, symmetry groups, tilings, and other topics in discrete geometry. Today the field includes continuous models and measures. Models include "soft packings" in which cells overlap, or have soft corners, or are skeletal rather than solid. Indeed, the skeletal Fedorov five, together with the tilings they form, transform continuously into one! Though the older discrete classifications still distinguish periodic crystals, comparing the millions of known structures in materials databases requires continuous distance metrics. In this talk we imagine a conversation between crystallographers of 1924 and 2024.**15.00-15.50**Gregory McColm (University of South Florida, US)

Title. Geometric Families of (Edge Transitive) Crystal Nets.

Abstract. A covalent crystal may be represented as a geometric graph whose vertices are atoms or molecular building blocks and whose edges are bonds or ligands. If edge transitive, such a crystal net may be derived from a voltage graph that encodes the vertex symmetries and edge displacements occurring within the crystal net. Say that two voltage graphs are homologous if they are isomorphic and have corresponding vertex symmetries, and this homological equivalence relation partitions the space of voltage graphs into topological spaces corresponding to topological spaces of the corresponding crystal nets. For any one equivalence class of voltage graphs, almost all of its crystal nets are of a common dimension and symmetry group, although there may be subspaces of crystal nets of greater symmetry, lower dimension, or some other reduction. We consider several examples of spaces of edge transitive crystal nets.**16.00-16.50**Simon J. L. Billinge (Columbia University, US)

Title. Materials genomics, materials heredity and the structure definition problem.

Abstract. We have recently been thinking, somewhat whimsically, about the materials genome. The materials genome initiative (MGI) is a US government initiative from 2011 to speed up, and lower the cost of, the discovery of new advanced materials, and their deployment in technologies, by applying data analytic methods inspired by biological genomics. Of course, materials do not have a genome. However, if we generalize the concept of a genome as a 1D discrete quantity that codes for the 3-dimensional arrangement of atoms, then we do have quantities that could serve as this generalized genome. At MACSMIN 2023 I introduced our thinking on this idea in a general sense. In this talk I will briefly reiterate the main idea, and then discuss, developments in our thinking over the past year. Two new concepts emerge from this framing. One is the idea of the materials ribosome. The ribosome is the cellular machinery that converts a DNA code into protein molecules which then fold to unique 3D structures. For our materials genome, this would be machinery, either physical or computational, that can convert our materials genome code into a 3D structure. The other concept is that of heredity, whereby we can find new materials similar to existing ones by making small mutations to our materials genome. I will discuss how this way of thinking can guide us in our search for genomic codings for materials structure. Something that has emerged from this thinking is a realization that the field of crystallography doesn't have clear definitions of what constitutes a structure. I call this the "structure definition problem" and I suggest that, as a community it would be most beneficial if we could address this problem and formalize definitions for what is meant by materials structure.

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**Tuesday 10th September**(only on zoom)**14.00-14.50**Peter Michor (University of Vienna, Austria) Slides (pdf, 1M)

Title. Closed surfaces with different shapes that are indistinguishable by the square root normal form.

Abstract. The Square Root Normal Field (SRNF), introduced by Jermyn et al. in 2012, provides a way of representing immersed surfaces in R^{3}, and equipping the set of these immersions with a ``distance function" (to be precise, a pseudometric) that is easy to compute. Importantly, this distance function is invariant under reparametrizations (i.e., under self-diffeomorphisms of the domain surface) and under rigid motions of R^{3}. Thus, it induces a distance function on the shape space of immersions, i.e., the space of immersions modulo reparametrizations and rigid motions of R^{3}.

In this talk, we give examples of the degeneracy of this distance function, i.e., examples of immersed surfaces (some closed and some open) that have the same SRNF, but are not the same up to reparametrization and rigid motions. We also prove that the SRNF does distinguish the shape of a standard sphere from the shape of any other immersed surface, and does distinguish between the shapes of any two embedded strictly convex surfaces.**15.00-15.50**Andy Zhang (Princeton University, US)

Title. Method of moments for cryo-EM images: from molecular similarity to reconstruction (based on this paper).

Abstract. Kam’s method is a statistical method-of-moments approach applied to the cryo-EM reconstruction problem that uses low-order moments of the 2-D projection images to reconstruct a 3-D structure. A notable advantage to this method is its ability to bypass angular assignment, which is typically a large computational burden.

In this talk, we will first discuss the uses of Kam’s method for 3-D reconstruction, and how its limitations can be addressed by using effective priors. We will then introduce two metrics that are inspired Kam’s method. We define a rotationally invariant metric between two molecular structures which does not require 3-D alignment. Further, we create a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. If time permits, we will also discuss future directions for Kam’s method.**16.00-16.50**Richard D. James (University of Minnesota, US)

Title. Objective molecular dynamics in classical and quantum mechanics.

Abstract. In the simplest case Objective Structures (OS) are monatomic structures in which each atom “sees the same environment” up to orthogonal transformation and translation. Familiar examples are buckyballs, single-walled carbon nanotubes (all chiralities), phosphorene, and parts of many viruses. We give a general introduction to OS: its expression in the Periodic Table, and generalizations to molecular structures and quasicrystals.

As it turns out, the idea behind OS is much more about invariance than structure. Objective Molecular Dynamics (OMD) exploits that invariance. OMD is an exact method of molecular dynamics (MD) in which only a finite number of atoms are simulated but all atoms filling all of space satisfy exactly the equations of molecular dynamics. From the perspective of dynamical systems the method exploits a little studied invariant manifold of the MD equations. We briefly describe several recent studies using this method: simulation of phase change (gas to liquid), use of the method to find a generalization of the Navier-Stokes equations for high rates, the simulation of dislocation interaction, the simulation of the high-rate extension of carbon nanotubes. We finish by showing that a version of OMD holds exactly in non adiabatic quantum mechanics (classical MD for the nuclei coupled to full QM for the electrons). This is a summary of joint work with Gunjan Pahlani, Ananya Renuka Balakrishna and Shivam Sharma.

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**Wednesday 11th September**(on zoom, in person at the Materials Innovation Factory in the ground floor boardroom)**9.00-9.50**John Helliwell (University of Manchester, UK)

Title. The Scientific Truth, the Whole Truth and Nothing But the Truth (based on this book).

Abstract. There is a massive thrust on Open Science from Governments, Organisations and Funding Agencies. Puzzlingly Trust is not emphasised. Peer review of articles with underpinning data is a tradition of crystallography including community agreed validation checks as standard to try and ensure Trust. Exemplars are IUCr’s Checkcif and PDB’s Validation Report. The question arises once editor (and their referees) and database validator are content with an article and underpinning data what are the standard uncertainties on atomic coordinates and atomic displacement parameters? Expansion of digital data archiving allows raw data preservation reaching close to objectivity in science. In law the truth, the whole truth and nothing but the truth is sought, why not in science as well?!**10.00-10.50**Olga Anosova (University of Liverpool, UK)

Title. Geographic-style invariant coordinates of protein backbones.

Abstract. A protein is a large biomolecule consisting of one or several long chains of amino acid residues. A protein backbone is an ordered chain of three main atoms per residue embedded into 3D space. Any rigid motion (a composition of translations and rotations) changes atomic coordinates but preserves the underlying structure of a protein backbone and hence all its functional properties. Protein backbones were traditionally represented by only two torsion angles (per residue) but their rigid shapes require nine invariants for three atoms per residue. We develop a complete invariant that always suffices for a unique reconstruction of any protein backbone in 3D space. This invariant detected thousands of rigidly equivalent backbones in the Protein Data Bank (PDB), some of which require corrections in primary sequences.**11.00-11.50**William Jeffcott (University of Liverpool, UK)

Title. Continuous invariant-based maps of the Protein Data Bank (PDB).

Abstract. In the past, the PDB was visualised by using two or three moments of inertia of proteins. At the level of amino acid residies, the PDB was also visualised by the Ramachandran plot of two torsion angles. We present continuous maps of the latest PDB version at the level of chains and residues using complete invariants consisting of 9 invariants for three main atoms per residue. The resulting maps show large variations of geometric invariants, e.g. bond lengths and angles.**12.00-13.20**lunch (covered by the organisers)**13.30-14.20**Daniel Rigden (University of Liverpool, UK).

Title. Using deep learning predictions reveals a large number of register errors in PDB deposits.

Abstract. The accuracy of the information in the Protein Data Bank (PDB) is of great importance for the myriad downstream applications that make use of protein structural information. Despite best efforts, the occasional introduction of errors is inevitable, especially where the experimental data are of limited resolution. We have previously established a novel protein structure validation approach based on spotting inconsistencies between the residue contacts and distances observed in a structural model and those computationally predicted by methods such as AlphaFold 2. It is particularly well-suited to the detection of register errors. Importantly, the new approach is orthogonal to traditional methods based on stereochemistry or map-model agreement, and is resolution-independent. Here we identify thousands of likely register errors by scanning 3-5Å resolution structures in the PDB. Unlike most methods, application of our approach yields suggested corrections to the register of affected regions which we show, even by limited implementation, lead to improved refinement statistics in the vast majority of cases. A few limitations and confounding factors such as fold-switching proteins are characterised, but we expect our approach to have broad application in spotting potential issues in current accessions and, through its implementation and distribution in CCP4, helping ensure the accuracy of future deposits.**14.20-14.50**discussion of open problems in structural biology**15.00-15.50**Alexander Movchan (University of Liverpool, UK)

Title. Modelling of waves in semi-infinite metamaterials clusters.

Abstract. The talk presents a class of models for waves in two-dimensional metamaterials systems containing clusters of resonators. Such waves are dispersive, and they are characterised by the dynamic anisotropy. Special attention is given to the wave localisation and lattice Green's functions. For semi-infinite clusters of resonators, the approach based on the Floquet theory would not be applicable. Instead, we derive and analyse the functional equations of the Wiener-Hopf type. In particular, the connection is drawn between the kernels of the Wiener-Hopf equations and the quasiperiodic lattice Green's functions for infinite doubly periodic clusters. The analytical results are accompanied by the numerical simulations and examples.

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**Thursday 12th September**(on zoom and in person at the Materials Innovation Factory in the ground floor boardroom)**9.00-9.50**Yury Elkin (University of Liverpool, UK)

Title. Complete and continuous invariants of molecular graphs.

Abstract. Though molecules are often represented by compositions, 1-dimensional strings, or molecular diagrams in the plane, many molecular properties are determined by 3-dimensional molecular structures. In this talk, we model any molecule as an embedded graph whose vertices and edges are atoms and covalent bonds, respectively. Such molecular graphs are considered the same (equivalent) if they can be completely matched by rigid motion, which is a composition of translations and rotations in 3D. If molecules cannot be exactly matched, it is important to quantify their difference by a distance metric satisfying the basic axioms. We developed new invariant descriptors that (1) completely distinguish any non-equivalent graphs in a Euclidean space, (2) sufficient to reconstruct any Euclidean graph uniquely modulo rigid motion, (3) Lipschitz continuous so that perturbations of all vertices up to epsilon change the invariant up to a constant multiple of epsilon in a suitable distance metric, and (4) for a fixed dimension n, computable in polynomial time of the number of unordered vertices. The graph invariants distinguished all 130K+ molecular graphs in the QM9 database.**10.00-10.50**Vitaliy Kurlin (University of Liverpool, UK)

Title. Complete, bi-continuous, and realisable invariants of atomic clouds.

Abstract. The most fundamental model of a molecule is a cloud of atoms even without chemical bonds, which are loosely defined depending various thresholds. It makes sense to consider atoms as unordered zero-sized points because many chemically different molecules have close geometric shapes and hence similar properties. Such atomic clouds also appear as local environments in large biomolecules or solid materials. The strongest equivalence on atomic clouds in practice is rigid motion in 3D. Indeed, any composition of translations and rotations changes a coordinate representation of an atomic cloud but preserves its rigid shape and hence properties. We develop new invariants that are (1) complete due to theoretically distinguishing all rigidly non-equivalent clouds, (2) realisable in the sense that any sampled invariant gives rise to a well-defined cloud, (3) bi-continuous under perturbations and hence can be considered geographic-style coordinates on the space of all rigid clouds, (4) computable in polynomial time in the number of points for a fixed dimension of ambient Euclidean space. These invariants distinguished all 130K+ atomic clouds in the QM9 database.**11.00-11.50**David Leigh FRS (University of Manchester, UK).

Title. Molecular Ratchets and Kinetic Asymmetry: Giving Chemistry Direction (based on this review).

Abstract.Over the last two decades ratchet mechanisms have transformed the understanding and design of stochastic molecular systems—biological, chemical and physical—in a move away from the mechanical macroscopic analogies that dominated thinking regarding molecular dynamics in the 1990s and early 2000s (e.g. pistons, springs, etc), to the more scale-relevant concepts that underpin out-of-equilibrium research in the molecular sciences today. Ratcheting has established molecular nanotechnology as a research frontier for energy transduction and metabolism, and has enabled the reverse engineering of biomolecular machinery, delivering insights into how molecules ‘walk’ and track-based synthesisers operate, how the acceleration of chemical reactions enables energy to be transduced by catalysts (both motor proteins and synthetic catalysts), and how dynamic systems can be driven away from equilibrium through catalysis. The recognition of molecular ratchet mechanisms in biology, and their invention in synthetic systems, is proving significant in areas as diverse as supramolecular chemistry, systems chemistry, dynamic covalent chemistry, DNA nanotechnology, polymer and materials science, molecular biology, heterogeneous catalysis, endergonic synthesis, the origin of life, and many other branches of chemical science. Put simply, ratchet mechanisms give chemistry direction. Kinetic asymmetry, the key feature of ratcheting, is the dynamic counterpart of structural asymmetry (i.e. chirality). Given the ubiquity of ratchet mechanisms in endergonic chemical processes in biology, and their significance for behaviour and function from systems to synthesis, it is surely just as fundamentally important. This Review charts the recognition, invention and development of molecular ratchets, focussing particularly on the role for which they were originally envisaged in chemistry, as design elements for molecular machinery. Different kinetically asymmetric systems are compared, and the consequences of their dynamic behaviour discussed. These archetypal examples demonstrate how chemical systems can be driven inexorably away from equilibrium, rather than relax towards it.**12.00-13.20**lunch (covered by the organisers)**13.30-14.00**tour of the Materials Innovation Factory**14.10-15.00**Anthony Nixon (Lancaster University, UK)

Title. A tour through rigidity theory.

Abstract. A bar-joint framework is a discrete structure made of fixed length bars connected at universal joints. Mathematically the framework is a combination of a graph and a realisation mapping the vertices into Euclidean space. Rigidity theory concerns the analysis of the rigidity and flexibility properties of such frameworks. In this talk I will survey the basic mathematical theory with a focus on past, present and future applications of rigidity theory to materials science, biology and engineering.**15.10-16.00**Daniel Widdowson (University of Liverpool, UK)

Title. Ultra-fast detection of (near-)duplicate structures across major crystal databases.

Abstract. The exponentially growing number of known crystals calls for a comparison method that is both fast and discerning. The common comparisons or approximate matches of unit cells are practically ambiguous. Indeed, not only can different structures share a common cell, but the same (rigidly equivalent) structure can be described with different unit cells and almost any displacement of atoms discontinuously changes the size of a primitive cell [1].

We developed new structural invariants that resolve the ambiguity problem by being independent of a unit cell, provably continuous under atomic displacements, and having a theoretical strength that allows us to uniquely reconstruct any generic crystal from its periodic lattice and a sufficiently large structural invariant [2].

The key advantage of the new invariants is their hierarchical nature and ultra-fast speed which allowed us to detect all near-duplicates among all (more than 2 million) periodic structures from the major structural databases CSD (Cambridge Structural Database), ICSD (Inorganic Crystal Structure Database), COD (Crystallography Open Database), and MP (Materials Project) within 9 minutes on a typical desktop computer, equivalent to more than 4 trillion comparisons.

[1] D.Widdowson et al. Average Minimum Distances of periodic point sets. MATCH, v.87 (3), 529-559, 2022.

[2] D.Widdowson, V.Kurlin. Resolving the data ambiguity for periodic crystals. NeurIPS, v.35, 24625-24638, 2022.**16.05-16.30**Jonathan (Teddy) McManus (University of Liverpool, UK)

Title. Computing the bridge length of periodic point sets.

Abstract. The recent isometry classification of periodic point sets used a complete invariant isoset whose size essentially depends on the bridge length, defined as the maximum `jump' that is needed 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.**16.35-17.00**Jonathan Balasingham (University of Liverpool, UK)

Title. Accelerating Material Property Prediction using Generically Complete Isometry Invariants (based on this paper).

Abstract. Periodic material or crystal property prediction using machine learning has grown popular in recent years as it provides a computationally efficient replacement for classical simulation methods. A crucial first step for any of these algorithms is the representation used for a periodic crystal. While similar objects like molecules and proteins have a finite number of atoms and their representation can be built based upon a finite point cloud interpretation, periodic crystals are unbounded in size, making their representation more challenging. In the present work, we adapt the Pointwise Distance Distribution (PDD), a continuous and generically complete isometry invariant for periodic point sets, as a representation for our learning algorithm. The PDD distinguished all (more than 660 thousand) periodic crystals in the Cambridge Structural Database as purely periodic sets of points without atomic types. We develop a transformer model with a modified self-attention mechanism that combines PDD with compositional information via a spatial encoding method. This model is tested on the crystals of the Materials Project and Jarvis-DFT databases and shown to produce accuracy on par with state-of-the-art methods while being several times faster in both training and prediction time.**18.00-20.00**conference dinner (covered by the organisers, TBC)

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**Friday 13th September**(on zoom and in person at the Materials Innovation Factory in the ground floor boardroom)**9.00-9.50**Cristian Micheletti (SISSA, Trieste, Italy)

Title. Designed self-assembly of molecular knots, links and topological gels.

Abstract. Supramolecular constructs with complex topologies are of great interest across soft-matter physics, biology and chemistry, and hold much promise as metamaterials with unusual mechanical properties. A particularly challenging problem is how to rationally design, and subsequently realize, these structures and the precise interlockings of their multiple molecular strands. Here we report on the combined use of theory and simulations to obtain complex supramolecular constructs via programmed self-assembly. Specifically, by controlling the geometry of the self-assembled monomers we show that the assembly process can be directed towards "privileged", addressable topologies of molecular knots, and extended linked structures, such as Olympic gels and catenanes. We conclude presenting an overview of the unique static and dynamical properties of linear catenanes. The talk will cover results based on publications [1-5].

[1] E. Orlandini and C.Micheletti, J. Phys. Condensed Matter, 34, 013002 (2022).

[2] M. Marenda, E. Orlandini, and C. Micheletti, Nat. Commun. 9, 3051 (2018).

[3] G. Polles, E. Orlandini, and C. Micheletti, ACS Macro Lett.5 , 931 (2016).

[4] G. Polles, D. Marenduzzo, E. Orlandini, and C. Micheletti, Nat. Commun. 6, 6423 (2015).

[5] M. Becchi, R. Capelli, C. Perego, G.M. Pavan, and C. Micheletti, Soft Matter 18, 8106 (2022).

For an actual hands-on demonstration of the designed self-assembly of "macroscopic" trefoil knots, see this video.**10.00-10.50**Artem Mishchenko (University of Manchester, UK)

Title. Machine learning-driven discovery and classification of flat band materials.

Abstract. Flat bands are electronic energy states in crystalline materials where the electron energy remains nearly constant across a range of momentum values, appearing as horizontal lines in band structure diagrams. They are of great interest because they can lead to enhanced electron-electron interactions and exotic quantum phenomena, making them promising for novel technological applications such as quantum computing, high-temperature superconductivity, and next-generation electronics. High-throughput computational databases provide access to hundreds of thousands of computed materials, where many materials with flat bands are awaiting discovery. However, identifying and categorising flat band materials across vast computational databases remains challenging. We present a comprehensive machine learning framework that combines supervised and unsupervised techniques to efficiently discover and classify flat band materials. We developed a convolutional neural network (CNN) based approach to identify flat bands from electronic structure images in the 2DMatPedia (2D materials) and the Materials Project (3D materials) databases. We coupled CNN with density-based clustering of structural fingerprints to reveal families of flat band materials, uncovering new lattice structures beyond known paradigms. We then extended our approach beyond structural fingerprints, by training a convolutional autoencoder (CAE) to encode band structures into compact electronic fingerprints. Unsupervised clustering of these fingerprints efficiently mapped the electronic property spaces of flat band materials. To further advance the field, we propose a novel metric inspired by quantum geometry to quantify the triviality of flat band materials. This metric provides crucial insights into the nature of the flat bands and their potential for hosting exotic quantum phenomena.

In this talk, I will overview our recent progress in the field of autonomous materials discovery, including our current results and future directions. Our hybrid framework provides a powerful, generalisable tool to rapidly screen computational databases and identify promising flat band candidates. In the future, we aim to extend our framework to predict synthesis conditions for promising flat band materials and to incorporate experimental feedback for continuous learning and improvement of our models.**11.00-11.50**Lucas Foppa (NOMAD lab at the Fritz-Haber-Institute, Berlin, Germany)

Title. From Prediction to Action: Critical Role of Performance Estimation for Machine-Learning-Driven Materials Discovery.

Abstract. Materials discovery driven by statistical property models is an iterative decision process, during which an initial data collection is extended with new data proposed by a model-informed acquisition function--with the goal to maximize a certain "reward" over time, such as the maximum property value discovered so far. While the materials science community achieved much progress in developing property models that predict well on average with respect to the training distribution, this form of in-distribution performance measurement is not directly coupled with the discovery reward. This is because an iterative discovery process has a shifting reward distribution that is over-proportionally determined by the model performance for exceptional materials. We demonstrate this problem using the example of bulk modulus maximization among double perovskite oxides. We find that the in-distribution predictive performance suggests random forests as superior to Gaussian process regression, while the results are inverse in terms of the discovery rewards. We argue that the lack of proper performance estimation methods from pre-computed data collections is a fundamental problem for improving data-driven materials discovery, and we propose a novel such estimator that, in contrast to naïve reward estimation, successfully predicts Gaussian processes with the "expected improvement" acquisition function as the best out of four options in our demonstrational study for double perovskites. Importantly, it does so without requiring the over thousand ab initio computations that were needed to confirm this prediction.**12.00-13.20**lunch (covered by the organisers)

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### Travel information : venue, accommodation, trains, 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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