Applied Geometry and Topology (AGT) network funded by the LMS in the UK
Next meeting of the Applied Geometry and Topology network
 The next meeting is on 21st September 2023 in the Materials Innovation Factory at the University of Liverpool.
 If you would like to offer a talk at the next meeting, please email the title and abstract to one of local organisers.
 In October 2022, the London Mathematical Society has extended the funding for another year until September 2023.
 The London Mathematical Society partially covers local travel and accommodation of participants of the AGT meetings.
 Before 2022, the network was called Applied Algebraic Topology, originally organised in 2015 by Michael Farber.
Organisers of the Applied Geometry and Topology network
 Michael Farber (m.farber at qmul.ac.uk), Queen Mary University of London
 Ginestra Bianconi (ginestra.bianconi at gmail.com), Queen Mary University of London
 Jacek Brodzki (j.brodzki at soton.ac.uk), University of Southampton
 Jelena Grbic (j.grbic at soton.ac.uk), University of Southampton
 Vitaliy Kurlin (vitaliy.kurlin at gmail.com), University of Liverpool.
Past meetings of the Applied Geometry and Topology network
2023 : Liverpool (21 September)  Queen Mary (2021 January)
2022 : Liverpool (hybrid, joint with a satellite of the ECM33)  Queen Mary (in person)
2021 : Liverpool (online, joint with the annual TDA meeting)
2020 : Liverpool (online, joint with the MACSMIN conference)
2019 : ATI London  Liverpool  Queen Mary London
2018 : Liverpool  Southampton  Queen Mary London
 The 20th meeting on 21st September 2023
in Liverpool Materials Innovation Factory (ground floor boardroom).
 12.3013.45 Lunch with participants at the Waterhouse cafe in the Victoria Gallery and Museum.
 14.0014.50 Primoz Skraba (Queen Mary University of London)
Title. A universal nulldistribution for topological data analysis.
Abstract. One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams arising from data. Despite much effort and its many successful applications, this is largely an open problem. We present a surprising discovery: normalized properly, persistence diagrams arising from random pointclouds obey a universal probability law. Our statements are based on extensive experimentation on both simulated and real data, covering pointclouds with vastly different geometry, topology, and probability distributions. Our results also include an explicit wellknown distribution as a candidate for the universal law. We demonstrate the power of these new discoveries by proposing a new hypothesis testing framework for computing significance values for individual topological features within persistence diagrams, providing a new quantitative way to assess the significance of structure in data.  15.0015.50 Isaac Sugden (Cambridge Crystallographic Data Centre)
Title. Computational and Data Science at the CCDC: ranking coformers and studying Aromatic interactions in the CSD.
Abstract. Computational and data science techniques are critical to understanding and shaping the next generation of materials and solidstate pharmaceuticals. This talk will focus on the projects that we, the Cambridge Crystallographic Data Centre (CCDC) are engaged with to modernise our core responsibility – the Cambridge Structural Database (CSD), the world’s collection of organic, coordination compound and MOF crystal structure data. There will be short discussion of some pertinent projects: tools for functional materials research, an autoencoder for chemical feature extraction, the use of the PDD methods within the CCDC, and experimental PXRD pattern matching with theoretical structures. The main focus of the talk will be on 2 projects that make use of the CSD: predicting what coformer will form a multicomponent form with a given API using a transformer neural network model, trained on the CSD, and an informatics tool for studying aromatic interactions within the CSD, seeing applications that bridge the pharmaceutical and functional materials worlds.  16.0016.50 Miloslav Torda (Liverpool Materials Innovation Factory)
Title. Maximally Dense Crystallographic Symmetry Group Packings: An Information Geometric Perspective.
Abstract. Stochastic relaxation is a wellestablished technique in machine learning and artificial intelligence used to tackle complex optimization landscapes. Here, we employ stochastic relaxation to address the challenge of discovering the densest packing of closed subsets of ndimensional Euclidean space, subject to constraints imposed by the Crystallographic Symmetry Group (CSG). To this end, we introduce the Entropic Trust Region Packing Algorithm (ETRPA), which is an instance of the natural gradient learning method with adaptive selection quantile fitness rewriting. Since CSGs induce a toroidal topology on the configuration space, we perform the ETRPA search on a parametric family of probability measures defined on an ndimensional torus. We examine the information geometry of the ETRPA through its equivalence with the generalized proximal method and characterize the algorithm via local dual geodesic flows that maximize multiinformation, a measure of stochastic dependence in complex systems. Consequently, ETRPA’s theoretical foundation in evolutionary dynamics, statistical physics, and recurrent neural computing can be interpreted in terms of group equivariant geometric learning, providing a deeper understanding of the algorithm. This work is motivated by the problem of molecular Crystal Structure Prediction, which involves predicting a synthesizable periodic structure based on a molecule's chemical composition and specific pressuretemperature conditions.
 The 19th meeting Applied topology and Robot Motion Planning was on 2021 January 2023 in Queen Mary London.
 The 18th meeting was joint with MACSMIN on 59 September 2022, organised by Vitaliy Kurlin in Liverpool.
 The 17th meeting was online on 31 January  1 February 2022, organised by Michael Farber (Queen Mary).
 The 16th meeting was online and joint with the annual meeting of the centre for TDA on 1315 September 2021.
 The 15th meeting was joint with the Maths and Computer Science for Materials Innovation on 89 September 2020.
 The 14th meeting was joint with the workshop Mathematics and Data organised by Michael Farber (Queen Mary) at the Alan Turing Institute on 12th September 2019 in London, UK.
 10:30 – 10:35 Introduction and welcome – Michael Farber (The Alan Turing Institute)
 10:35 – 11:20 Expander graphs  Nati Linial (Hebrew University of Jerusalem
 11:20 – 12:05 Local view of combinatorial structures and applications to data analysis – Chaim EvenZohar (The Alan Turing Institute)
 12:05 – 12:50 On the expressiveness of comparison queries – Shay Moran (Princeton University)
 13:45 – 14:30 Topological complexity and its applications – John Oprea (Cleveland State University)
 14:30 – 15:15 Classical information theory of networks – Ginestra Bianconi (Queen Mary University of London and The Alan Turing Institute)
 15:30 – 16:15 An introduction to tropical geometry and connections to deep learning – Felipe Rincon (Queen Mary University of London)
 16:15 – 17:00 The simplicity revolutions in the era of complexity and the unreasonable effectiveness of small neural ensembles in highdimensional brain – Alexander Gorban (University of Leicester)
 The 13th meeting on Friday 30th August 2019 was organised by Vitaliy Kurlin in the Materials Innovation Factory.
 9.3010.30 Herbert Edelsbrunner (IST Austria)
Title. Tripartition of and hole systems in a polyhedral complex
Abstract. We prove that for every polyhedral complex, K, and every dimension, p, there is a partition of the pcells into a maximal ptree, a maximal pcotree, and the remaining pcells defining the pth homology of K. As an application, we consider the manipulation of the hole structure in geometric shapes, using the tripartition to facilitate the opening and closing of holes. In a concrete application, we let K be the Delaunay mosaic of a finite set, and we extract a partial order on the filtration induced by the radius function, whose cuts define the subcomplexes that can be constructed with this method. Joint work with Katharina Oelsboeck.  11.0012.00
Argyrios Deligkas (MIF, Liverpool)
Title. Crystal Structure Prediction via Oblivious Local Search
Abstract. We study Crystal Structure Prediction, one of the major problems in computational chemistry. This is essentially a continuous optimization problem, where many different, simple and sophisticated, methods have been proposed and applied. The simple searching techniques are easy to understand, usually easy to implement, but they can be slow in practice. On the other hand, the more sophisticated approaches perform well in general, however almost all of them have a large number of parameters that require fine tuning and, in the majority of the cases, chemical expertise is needed in order to properly set them up. In addition, due to the chemical expertise involved in the parametertuning, these approaches can be biased towards previouslyknown crystal structures. Our contribution is twofold. Firstly, we formalize the Crystal Structure Prediction problem, alongside several other intermediate problems, from a theoretical computer science perspective. Secondly, we propose an oblivious algorithm for Crystal Structure Prediction that is based on local search. Oblivious means that our algorithm requires minimal knowledge about the composition we are trying to compute a crystal structure for. In addition, our algorithm can be used as an intermediate step by {\em any} method. Our experiments show that our algorithms outperform the naive basin hopping, a standard, well studied, algorithm for the problem. Joint work with Dmytro Antypov, Vladimir Gusev, Matthew J. Rosseinsky, Paul G. Spirakis, Michael Theofilatos.  14.0015.00 Mr Georg Osang (IST Austria)
Title. The Multicover 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 higherorder 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.  15.1016.10 Ms Teresa Heiss (IST Austria)
Title. A topological approach to comparing crystals (periodic point sets)
Abstract. As the atoms in crystals are arranged periodically, crystalline materials can be modeled by periodic point sets. Two periodic point sets are considered equivalent if there is a rigid motion from one to the other. A periodic point set can be represented by a finite cutout s.t. copying this cutout in all directions yields the periodic point set. As these cutouts are not unique, comparing periodic point sets for similarity is difficult. We would therefore like to work with a complete, continuous invariant instead of working with the periodic point set itself. We conjecture that the sequence of order k persistence diagrams (will be defined in Georg Osang's talk) for all positive integers k is such a complete, continuous invariant of equivalence classes of periodic point sets.  16.2017.00 Mr Phil Smith (MIF, Liverpool)
Title. Working towards a geometric classification of crystals.
Abstract. An important problem in materials science is to be able to cluster similar crystal structures together, which will help speed up the process of materials discovery. This requires a classification of crystals quantifying how similar two structures are. We are approaching this problem from a geometric angle, looking to classify crystals based on their geometry. To this end, we introduce packing functions which computes the proportion of a unit cell of a crystal that is covered by n balls centred at the atoms of the crystal. These functions reveal critical distances dependent on the crystal's geometry, and we hope we can use this information to form a metric on crystal space that can be used to cluster datasets of crystal structures.
 9.3010.30 Herbert Edelsbrunner (IST Austria)
 The 12th meeting was on Monday 4th February 2019 in Queen Mary University of London.
 10.0010.45 Primoz Skraba (QMUL) Computing Persistence in Parallel
 10.4511.30 Jon Woolf (Liverpool) Stratified Homotopy Theory
 11.4512.30 Tomaso Aste (UCL) Learning Clique Forests for Probabilistic Modeling
 14.00  14.45 Tahl Nowik (Bar Ilan University) Random knots
 14.55  15.40 Ian Leary (Southampton) Examples for Brown’s question(s) on dimensions of groups
 The 11th meeting was organised by Vitaliy Kurlin as a satellite workshop at MFCS on Friday 31st August 2018 in the Materials Innovation Factory (3rd floor meeting room, building 807, square 5F on the campus map), Liverpool, UK.
 14.0014.45 Hubert Wagner (IST Austria)
Title. Computing persistent homology of images with Cubicle.
Abstract. Persistent homology is gaining popularity for analyzing data coming from medical imaging, astrophysics and material science. I will focus on novel techniques for computing persistent homology of multidimensional images. In particular, I will address the eternal question: "To use, or not to use (discrete Morse theory)". A new software package, Cubicle, will be showcased and compared with existing packages.  15.1516.00 Ms Katharina Oelsboeck (IST Austria)
Title. Shape Reconstruction with Holes.
Abstract. We want to reconstruct the shape of a point clould, with focus on the holes of the resulting model. In many cases, the alpha complex of appropriate scale gives a good reconstruction. However, in some applications the holes of the model are important and there is no scale of the alpha complex that gives a satisfactory result.We define operations to change the birth and death of holes in a filtered simplicial complex, i.e., they open or close holes in a subcomplex of a fixed scale. A tripartition of the simplices, and canonical (co)chains and (co)cycles that are associated to the simplices will help us identify for which simplices we need to adapt their filtration values. These can be computed with a specific matrix reduction algorithm for persistent homology. The persistence diagram of the complex can help us guide the application of the hole operations. In the second part, we will present the Wrap complex as an alternative for shape reconstruction and will apply the hole operations on it.  16.3017.15 Mr Philip Smith (MIF, Liverpool)
Title. Skeletonization algorithms with theoretical guarantees for unorganized point clouds.
Abstract. We study the problem of approximating an unorganized cloud of points (in any Euclidean or metric space) by a 1dimensional graph or a skeleton. The following recent algorithms provide theoretical guarantees for an output skeleton: the 1dimensional Mapper, alphaReeb graphs and a Homologically Persistent Skeleton. All the three algorithms will be introduced on simple examples and then experimentally compared on the same synthetic and real data. The synthetic data are random point samples around planar graphs and sets of edge pixels obtained by a Canny edge detector on images from the Berkeley Segmentation Database BSD500. The criteria for comparison are the running time, topological types, geometric errors of reconstructed graphs.
 14.0014.45 Hubert Wagner (IST Austria)

 The 10th meeting was on 30th April 2018 at the University of Southampton.
 10:3011:20 Rachel Jeitziner (EPFL) TwoTier Mapper: a userindependent clustering method for global gene expression analysis based on topology
 11:3012:00 Mariam Pirashvili (Southampton) Improved understanding of aqueous solubility modeling through Topological Data Analysis
 14:0014:50 Ginestra Bianconi (QMUL) Emergent Hyperbolic Network Geometry and Frustrated Synchronization
 15:0015:50 Grzegorz Muszynski (Liverpool) Topological Analysis and Machine Learning for Detecting Atmospheric River Patterns in a Climate Model Output

 The 9th meeting was on 2nd February 2018 in the Queen Mary University of London.
 9.3010.30 (Scape 1.04) Gabor Elek (University of Lancaster) Topological graph limits
 11.0012.00 (Scape 1.04) Tim Evans (Imperial College London) Networks and Spacetime
 14.0015.00 (Scape 2.01) Ran Levi (Aberdeen) NeuroTopology: An interaction between topology and neuroscience
 15.3016.30 (Scape 2.01) Stephan Mescher (University of Leipzig) Topological complexity of aspherical spaces
 The network was initiated by Michael Farber (Queen Mary) with coorganisers at Aberdeen, Durham, Southampton.
 The first eight meetings were in 20152016 at Aberdeen, Durham, Queen Mary, Southampton.
Research topics of the Applied Geometry and Topology network
We run 23 halfday meetings per year with 34 talks on many topics of applied topology including (but not restricted to)
 Geometric Data Science parameterising moduli spaces of data objects up to practically important equivalence relatons.
 Topology of configuration spaces of particles and mechanisms of different types (including linkages) and their applications in robotics, molecular biology and materials chemistry.
 Topology of robot motion planning, complexity of algorithms for autonomous robot motion.
 Stochastic topology (random complexes, random manifolds, random groups etc).
 Applications of Topological Data Analysis to Computer Vision, Materials Science, Climate Science, Pattern Recognition and reconstruction of persistent topological structures in big and noisy data.
 Combinatorial and toric homotopy theory (simplicial complexes and polytopes, momentangle complexes, polyhedral products, etc).