Event Details

The purpose of the workshop is to discuss the latest methods of rigidity theory, geometric data science and artificial intelligence for real objects such as, but not limited to, solid 3D materials (periodic crystals or polymers) and molecular graphs including proteins, and more complex 3D genome structures.

Organised by Vitaliy Kurlin (Computer Science, Liverpool), Anthony Nixon (Mathematical Sciences, Lancaster) and Abbie Trewin (Chemistry, Lancaster) and jointly funded by Lancaster's Data Science Institute and Mathematics for AI and Real-world Systems project.

Agenda

Monday 9th June

1pm Coffee on arrival

2pm - Thérèse E. Malliavin and Martin Michael Müller, Université de Lorraine, CNRS, France.

Protein–membrane interactions with a twist

Within a framework of elasticity theory and geometry, the twister mechanism has been proposed some years ago for describing the interaction between a biofilament containing a twisted hydrophobic strip and a lipid membrane: this mechanism is capable of inducing deformations of the membrane, which can lead to its opening. The present work intends to extend this model to the Interactions between a membrane and protein regions conserving their folds using coarse-grained molecular dynamics simulations. The protein region is modelled as a cylinder stabilized by a tensegrity scheme, leading to an elasticity similar to that observed in real proteins. Recording molecular dynamics trajectories of this cylinder in the presence of a fluid lipid bilayer membrane allows investigation of the effect of the positions of the hydrophobic parts on the interaction with the membrane. The entire configuration space is explored by systematically varying the hydrophobic strip width, the twisting of the strip as well as the range of hydrophobic interactions between the cylinder and the membrane. Three different states are observed: no interaction between the cylinder and membrane, the cylinder in contact with the membrane surface and the cylinder inserted into the membrane with a variable tilt angle. The variations of the tilt angle are explained using a qualitative model based on the total hydrophobic moment of the cylinder. A deformation pattern of the membrane, previously predicted for the filament–membrane interaction by the twister model, is observed for the state when the cylinder is in contact with the membrane surface, which allows estimation of the applied torques.

3pm Coffee

3.30 - Michael Peach

Tuesday 10^th June

9.30 - Murad Banaji (University of Oxford, University of Lancaster)

Chemical reaction network theory: an overview of some recent results

A variety of models arising in the natural and social sciences have an underlying network structure. Individuals or populations interact with one another according to some rules; and these interactions lead in turn to changes in the state of the system. Our interest is in the structure-dynamics relationship in such models. We may want to know, for example, whether a given network structure permits or forbids multistationarity, oscillations, or certain bifurcations. We may also try to understand how subnetworks influence the dynamics, with an eye towards building networks with prescribed behaviours using simple building blocks. Often network models can be regarded, formally, as chemical reaction networks (CRNs), and CRN theory uses analysis, algebra, geometry and symbolic computation to explore the structure-dynamics relationship in such models. I will present an overview of some recent results in this area, focusing on two themes: the occurrence of bifurcations; and the influence of subnetworks on network dynamics.

10.30 Coffee

11am - Matthias Himmelman (TU Braunschweig)

PyRigi: A toolbox for the rigidity and flexibility of bar-joint frameworks

Despite their simplicity, bar-joint frameworks provide a powerful model for various geometric constraint systems and these structures find application in diverse real-world settings such as Computer Aided Design, Protein folding, polytope theory, sphere packing's and formation control. Their connection to physical systems highlights the importance of computational strategies for assessing rigidity and flexibility.

In an effort to unify the existing algorithmic approaches in rigidity theory, my collaborators—Matteo Gallet, Georg Grasegger and Jan Legerský—and I developed PyRigi, an intuitive and user-friendly Python package. In this presentation, I will provide an overview of the latest stable release. I will also introduce a new approach for efficiently and symbolically determining a framework’s second-order rigidity, as well as a homotopy continuation approach for numerically approximating deformation paths. Lastly, I will give an outlook on planned implementations and invite discussion on future directions beyond bar-joint rigidity.

12 – 2 Lunch

2pm - Abbie Trewin

3pm Coffee

3.30 - Andrea Fusiello (Universita' degli Studi di Udine)

The Viewing Graph: Rigidity Theory Applied to a Computer Vision Problem.

In this presentation, a collaborative study will be introduced that focuses on analysing the number of solutions to a polynomial system of equations derived from a problem in computer vision. This problem originates in the context of reconstructing geometric relationships from visual data, and it can be framed as a rigidity analysis of a specific type of graph known as the "viewing graph". The vertices of this graph represent perspective cameras, while the measurement function takes two 3x4 matrices as inputs — representing the projective transformations associated with the cameras — and outputs a 3x3 matrix that encapsulates the geometric constraints between the cameras.This approach provides insights into the theoretical aspects of the problem and has practical implications for computer vision tasks such as structure-from-motion and 3D scene reconstruction.

Wednesday 11^th June

9.30 - Jiayi Li (Max Planck Institute)

Geometry of Neural Networks with Algebraic Activations

Abstract: We consider neural networks with polynomial and rational activation functions. The choice of activation function in deep learning architectures is crucial for practical tasks and largely impacts the performance of a neural network. Leveraging tools from numerical algebraic geometry, we establish precise measures for the expressive power of neural networks with polynomial activation functions, by studying the image of the parametrization map from weights to functions, which forms an irreducible algebraic variety upon taking closure. In addition, we study the presence or absence of spurious valleys in the loss surface and contrast the topological properties of the loss landscape when activation coefficients are fixed versus trainable.

10.30 Coffee

11am - Ben Smith (Lancaster)

Discrete geometry of deep learning

A common choice of activation function in a deep neural network is ReLU, the function that sends x to max(x,0). The resulting function from such a deep network is a piecewise linear function, and hence is ripe for study via methods in discrete geometry. This includes classical methods of convex and polyhedral geometry, as well as more modern approaches using tropical geometry. In this talk, we will give an overview of how these discrete and geometric techniques have been used to analyse deep neural networks, and open avenues for future research.

12 – 2 Lunch

2pm - Catherine Mollart

3pm Coffee

3.30 - Lilja Metsalampi (Aalto)

Uniqueness of size-2 PSD factorizations using rigidity theory

A positive semidefinite (PSD) factorization of size-k of a nonnegative p x q matrix M is a collection of k x k symmetric PSD matrices A_1, ..., A_p, B_1, ..., B_q, such that the (i,j)-th entry of M is given by the trace product of A_i and B_j. The smallest k for which M admits a PSD factorization of size-k is called the PSD rank of M. A matrix is of minimal PSD rank, if it is a rank- k(k+1)/2 and PSD rank-k matrix. PSD factorizations originally appeared in semidefinite programming where PSD rank is related to the complexity of a semidefinite program over a convex set. Since then, applications of PSD factorizations have emerged in other fields also, most notably in quantum information theory. In this talk we consider the uniqueness of PSD factorizations. We study the uniqueness using notions from rigidity theory. We introduce s-infinitesimal motions of PSD factorizations and characterize 1- and 2-infinitesimally rigid size-2 factorizations. Lastly, we connect infinitesimal rigidity of size-2 PSD factorizations to uniqueness via global rigidity. This is a joint work with Kristen Dawson (SFSU), Serkan Hoşten (SFSU) and Kaie Kubjas (Aalto University).

Thursday 12^th June

9.30 - Jack Trainer

10.30 Coffee

11am - Vitaliy Kurlin (Liverpool)

The Principle of Molecular Rigidity

Understanding structure-property relationships for molecules requires a mathematical definition of a molecular structure. Any molecule keeps its functional properties under rigid motion (a composition of translations) within the same ambient environment.

However, flexible deformations affecting the rigid shape can also change molecular properties. Hence the strongest equivalence between molecules in practice is rigid motion, which is a composition of translations and rotations. Then a molecular structure can be mathematically defined as the class of all atomic sets that can be

exactly matched with each other by rigid motion. Since molecules often consist of indistinguishable atoms, the resulting problem is to design a complete invariant of unordered points under rigid motion with Lipschitz continuous distance metrics, all computable in polynomial time of the number of points, for a fixed dimension. It was a big embarrassment that, after the classification of triangles was known since ancient times, even the case of four unordered points in the plane had no better than a brute force solution involving exponentially many permutations of given points. The talk will present a recent extension of polynomial-time invariants [1], which satisfy the harder condition of realisability providing a continuous parametrisation of the moduli space of rigid clouds like a geographic-style map of Earth. The experiments on the world's largest

3D molecular databases QM9 (130K+ entries) and GEOM (37M+ entries) confirmed the Principle of Molecular Rigidity saying that any real molecule is uniquely determined by precise enough geometry of only atomic centres. [1] D.Widdowson, V.Kurlin. Recognizing rigid patterns of unlabeled point clouds by complete and continuous isometry invariants with no false negatives and no false positives. Proceedings of CVPR 2023 (Computer Vision and Pattern Recognition), p.1275-1284.

12 – 2 Lunch

2pm - Henry Moss (Lancaster)

Experimental Design in the Age of GenAI

Generative AI stands to revolutionize how experiments are conceived, orchestrated, and iterated. This talk outlines a general framework that can harness generative AI within experimental design loops, then examines the widely adopted approach of Latent Space Bayesian Optimization. Our proposed methodology diverges from conventional frameworks by decoupling the discriminative (surrogate) and generative models, rather than tightly integrating Gaussian Processes with Variational Autoencoders. Through independent training and a straightforward Bayesian update, we enable an efficient sampling strategy that identifies candidate structures with high scores.

3pm Coffee

3.30 - Olga Anosova (Liverpool)

Geometry of proteins in the PDB and AlphaFold databases

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 is the rigid shape of a protein chain represented by atomic positions in 3D space. Since different geometric structures often have distinct functional properties, it is important to continuously quantify differences in rigid shapes 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 talk will introduce 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 [1]. The new invariant has been used to detect thousands of (near-)duplicates in the Protein Data Bank [2], whose presence

inevitably skews machine learning predictions. The resulting invariant space allows low-dimensional maps with analytically defined coordinates that reveal substantial variability in the protein universe.

[1] Olga Anosova, Alexey Gorelov, Will Jeffcott, Ziqiu Jiang, Vitaliy Kurlin. A complete and bicontinuous invariant of protein backbones

under rigid motion. MATCH Comm. Math. Comp. Chemistry, v.94 (1), p.97-134, 2025.

[2] A.Wlodawer, Z.Dauter, P.Rubach, W.Minor, M.Jaskolski, W.Jeffcott, Z.Jiang, O.Anosova, V.Kurlin Duplicate entries in the Protein Data

Bank: how to detect and handle them. Acta Crystallographica D, v.81 (4), p.170-180, 2025.

Friday 13^th June

9.30 - Bernd Schulze (Lancaster)

Crystallographic local and global rigidity

Abstract: Rigidity Theory is concerned with the analysis of rigidity and flexibility of bar-joint frameworks and related discrete geometric constraint systems. This area has a rich history, tracing back to classical work by Euler, Cauchy, and Maxwell on the rigidity of polyhedra and skeletal structures. In the first part of this talk, I will introduce the theory of local and global rigidity for bar-joint frameworks, as well as related structures such as body-bar and body-hinge frameworks. I will provide an overview of key results, with particular emphasis on combinatorial characterisations of generic rigidity that are checkable in polynomial time. In the second part, I will discuss recent progress in extending these results to infinite periodic and crystallographic frameworks. These developments have growing applications in areas such as materials science, structural biology, and engineering.

10.30 Coffee

11am - Louis Theran

12pm - Daniel Widdowson (Liverpool)

Crystal Geomap: a visualisation & comparison tool for crystals.

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.

END

Confirmed speakers:

• Murad Banaji (Oxford)
• Matthias Himmelman (TU Braunschweig)
• Catherine Mollart (Birmingham)
• Therese Malliavin (Lorraine)
• Henry Moss (Lancaster)
• Lilja Metsalampi (Aalto)
• Michael Peach (Lancaster)
• Bernd Schulze (Lancaster)
• Ellena Sherrett (Lancaster)
• Jack Trainer (Lancaster)
• Andrea Fusiello (Universita' degli Studi di Udine)
• Jiayi Li (Max Planck Institute)
• Louis Theran (St Andrews)
• Daniel Widdowson (Liverpool)
• Olga Anosova (Liverpool)
• Vitaliy Kurlin (Liverpool)
• Abbie Trewin (Lancaster)
• Ben Smith (Lancaster)

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