Higher-order interactions arise in many physical, biological and social phenomena. There has been a lot of effort in recent years to study higher-order combinatorial structures such as hypergraphs or simplicial complexes from a geometric point of view, by developing notions of curvature, stochastic processes and optimal transport for such discrete spaces. In a distinct line of research, graphs and simplicial complexes are one of the main ingredients in the field of Topological Data Analysis, in which the topological properties of such combinatorial spaces are studied through a homological lens.
The aim of this workshop is to bring together both leading and upcoming researchers who use such approaches, with the goal of promoting collaborations across the different disciplines.
Confirmed Speakers:
Sophie Achard (CNRS)
Ulrich Bauer (TU Munich)
Anna Calissano (UCL)
Karel Devriendt (University of Oxford)
Daniela Egas Santander (MPI CBG)
Jürgen Jost (MPI MiS)
Steve Oudot (Inria-Saclay)
Michael Schaub (RWTH Aachen)
This workshop is part of an exchange programme between the CNRS (France) and the MPG (Germany).
We present topology aware signal processing tools [1] for the representation of simplicial complexes and cell complexes and signals defined on these. Specifically we discuss how harmonic flow embeddings that exploit topology offer a unified, interpretable framework for dynamic and static (edge-flow) data on complexes, and can be used for sparse representation tasks [2], outlier detection [3] and classification of trajectories [4,5].
References
[1] Schaub, M.T.; Zhu, Y.; Seby, J.-B.; Roddenberry, T.M. & Segarra, S. (2021), "Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond", Signal Processing., January, 2021. Vol. 187, pp. 108149.
[2] Hoppe, J. & Schaub, M.T. (2024), "Representing Edge Flows on Graphs via Sparse Cell Complexes", In Proceedings of the Second Learning on Graphs Conference., June, 2024. Vol. 231, pp. 1:1-1:22. PMLR.
[3] Frantzen, F. & Schaub, M.T. (2025), "HLSAD: Hodge Laplacian-based Simplicial Anomaly Detection", In Proceedings of the 31st ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2025)., August, 2025.
[4] Frantzen, F.; Seby, J.-B. & Schaub, M.T. (2021), "Outlier Detection for Trajectories via Flow-embeddings", In 2021 55th Asilomar Conference on Signals, Systems, and Computers., October, 2021. , pp. 1568-1572.
[5] Grande, V.P.; Hoppe, J.; Frantzen, F. & Schaub, M.T. (2024), "Topological Trajectory Classification and Landmark Inference on Simplicial Complexes", In 58th Annual Asilomar Conference on Signals, Systems, and Computers., October, 2024. , pp. 44-48.
Data may contain some unknown structure and may seem high dimensional. There exist various schemes for extracting dominant structures and efficiently representing them in 2D. A currently very popular scheme is UMAP. We clarify the underlying mathematics and introduce some new geometric ideas and on that basis develop an improved method.
References:
P.Joharinad, J.Jost: Mathematical principles of topological and geometric data analysis, Monograph, Math of Data, Springer, 2023
L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck: Data visualization with category theory and geometry, Monograph, Math of Data, Springer, 2025
L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck: IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations, Proc. AAAI Conf. Artificial Intelligence 39 (2025); arXiv:2407.17835, with code
L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck: Fuzzy simplicial sets and their application to geometric data analysis Applied Categorical Structures 33 (2025); arXiv:2406.11154
L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck: Merging Hazy Sets with m-Schemes: A Geometric Approach to Data Visualization, Adv.Theor.Math.Physics (2026)
A strong hypothesis in neuroscience is that many aspects of brain function are determined by the ‘’map of the brain’’ and that its computational power relies on its connectivity architecture. Impressive scientific and engineering advances in recent years generated a plethora of large brain networks of incredibly complex architectures.
A central feature of the architecture is its inherent directionality, which reflects the flow of information. Evidence shows that reciprocal connections and higher order motifs, such as directed cliques, emerge selectively rather than at random in biological neural networks. This raises fundamental questions in both mathematics and computational neuroscience. In this talk, we explore how such structure arises from the physicality of the neurons themselves and propose a framework to control and quantify the over or under representation of higher order motifs.
Topological Data Analysis (TDA) allows us to extract powerful topological, and higher-order information on the global shape of a data set or point cloud. Tools
like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, common machine learning applications like classification or applications in single-cell Biology require point-level information and features to be available. In our work, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry.
During the talk, we hope to illuminate how Topological Data Analysis can learn from ideas from Differential Geometry and Geometrical Machine Learning and vice-versa, and hope to paint a promising picture for future research joining the two areas.
Consider a finite point set in a Euclidean space colored by 0, ..., s. We can capture aspects of the mutual interaction of the s+1 colors by studying the growing space of points covered by the r-thickening of each of the colors. In other words: for every (s+1)-tuple of points, one per each color, consider the intersection of the disks centered in those points with radius r. We call such a shape an s-lune of radius r, and we study the growing union of all s-lunes. The lunar minimum spanning tree (LMST) is the minimum spanning tree of a filtered connectivity graph of the lunes: with sublevel sets the 1-skeletons of the nerves of the growing lunes.
We show that the expected length of LMST—equivalently, the 1-norm of the 0-persistence of the growing lunes—for colored point sets randomly drawn from a unit square asymptotically behaves as the square root of the cardinality times a constant, generalizing a classical result for Euclidean minimum spanning trees.
I will focus on the context for the definition and the result–its relation to the standard Euclidean minimum spanning trees, persistent homology and chromatic topological data analysis. The proof of the result relies on a reformulation using Delaunay mosaics, highlighting how methods motivated by making computations efficient feed back into proving interesting theoretical results.
Neural networks defined by polynomial or rational activations, as well as architectures with higher-order connectivity, give rise to algebraic varieties whose geometry encodes both the optimization landscape and the generalization behavior of learned representations. Using tools from numerical algebraic geometry—including ideals, algebraic stratification, and the resolution of singularities—we study the critical loci and degeneracies that shape learning dynamics. This approach reveals how phenomena such as symmetry, collapse, and instability correspond to singular structures within parameter space. The results provide a language for describing the hidden geometric of modern learning systems.
Given an unknown R^n-valued function f on a compact metric space X, can we estimate the persistent homology of f from a finite sample on X with known pairwise distances and function values? This question was answered more than a decade ago in the case n=1, assuming f is c-Lipschitz and X is a sufficiently regular geodesic metric space, and using filtered Rips complexes with fixed scale parameters for the approximation. Here we answer the question for arbitrary n, under similar conditions on X and f, and we investigate the choice of the scale parameters. We prove the statistical convergence of the persistent homology of a pair of filtered Rips complexes to the persistent homology of f, and we provide deviation bounds showing the quasi-minimax optimality of the estimator. Under more restrictive regularity conditions on X, we also prove the convergence of the persistent homology of a single filtered Rips complex to the persistent homology of f, and in fact we show that convergence happens already at the homotopy level, thus yielding a persistent version of Latschev's theorem on the homotopy type of Rips complexes of point samples on Riemannian manifolds.
This talk is based on two related manuscripts: one with Ethan Andre, Jingyi Li and David Loiseaux (arXiv:2412.04162); the other with Lukas Waas (in preparation).
In the scenario where multiple instances of networks with same nodes are available and nodes are attached to spatial features, it is worth combining both information in order to explain the role of the nodes. The explainability of node role in complex networks is very difficult, however crucial in different application scenarios such associal science, neuroscience, computer science. Many efforts have been made on the quantification of hubs revealing particular nodes in a network using a given structural property. Yet, for spatio-temporal networks, the identification of node role remains largely unexplored. In this talk, I will show limitations of classical methods on a real datasets coming from brain connectivity comparing healthy subjects to coma patients. Then, I will present recent work using equivalence relation of the nodal structural properties. Comparisons of graphs with same nodes set is evaluated with a new similarity score based on graph structural patterns. This score provides a nodal index to determine node role distinctiveness in a graph family. Finally, illustrations on different datasets concerning human brain functional connectivity will be described.
References:
S. Achard, C. Delon-Martin, P. E. Vértes, F. Renard, M. Schenck, F. Schneider, C. Heinrich, S. Kremer, and Edward T. Bullmore. (2012) Hubs of brain functional networks are radically reorganized in comatose patients. Proc Natl Acad Sci U.S.A., 109(50):20608-13, 2012.
Carboni, L., Dojat, M., & Achard, S. (2023). Nodal-statistics-based equivalence relation for graph collections. Physical Review E, 107(1), 014302.
Lauritzen, S. L., Rinaldo, A., & Sadeghi, K. (2017). On exchangeability in network models. arXiv preprint arXiv:1709.03885.
Barthelemy, M. Spatial networks. A complete introduction: from graph theory and statistical physics to real-world applications. Springer [2022] 437 pp.
Williams, M. J., & Musolesi, M. (2016). Spatio-temporal networks: reachability, centrality and robustness. Royal Society open science, 3(6), 160196.
The effective resistance is a function that measures the overall connectivity between pairs of vertices in a graph. While it originally served as a practical tool for electrical circuit design, the effective resistance has some striking mathematical features. In this talk, I will give a brief overview of two different perspectives on the effective resistance: a geometric perspective, centered around Fiedler's graph-simplex correspondence, and a combinatorial perspective, centered around Foster's theorem and Kirchhoff's matrix-tree theorem. In both cases, I will discuss a generalization of these classical results in a more modern language. Finally, I will briefly touch on some recent work on discrete curvature that sits at the intersection of these geometric and combinatorial theories.
Background material:
Biggs, 1997: Algebraic potential theory on graphs, in Bulletin of LMS
Fiedler, 2011: Matrices and graphs in geometry, Cambridge University Press
Lyons, 2014: Determinantal probability: Basic properties and conjectures, in Proc. ICM
Devriendt, 2022: Graph geometry from effective resistances, Oxford PhD thesis
Devriendt, 2025: Graphs with nonnegative resistance curvature, in Annals of Combinatorics
Persistent homology is a key tool in Topological Data Analysis, used to study the shape of data. Its use in practice is often limited because standard constructions like the Vietoris–Rips filtration lead to an explosive growth in the number of simplices. This scaling problem restricts applications to small datasets and low-dimensional homology.
In this talk we´ll tackle this exponential increase in simplicies by focusing on underlying geometric structures that generate these complexes: covers of the data.
We present a framework to compute persistent homology starting from a sequence of cover refinements. The presence of cover refinements induces simplicial contractions that curb the growth of the complex, greatly reducing the number of simplices. Importantly, the method produces the same topological invariants as traditional techniques—without approximation.
By shifting the focus to covers, this perspective makes persistent homology more scalable and also clarifies connections between classical results, such as the relationship between Čech and Vietoris–Rips persistence.
Many invariants of topological data analysis, in particular persistent homology, are constructed in a manner that passes through the world of homotopy theory, before finally producing an algebraic invariant. While this passage through homotopy theory comes with many advantages from a theoretical perspective, it also means that these invariants are often insensitive to features not detected by classical homotopy theory.
For example, classical persistent homology only has limited discriminative power when it comes to distinguishing singular data sets.
In this talk, I will explain how a more refined and recently developed version of homotopy theory – so-called stratified homotopy theory – can be used in TDA to study point clouds approximating singular spaces.
We develop a geometric framework for codimension 1 persistent homology by using Alexander duality to construct canonical, volume-minimizing cycle representatives in embedded filtered simplicial complexes. For a complex K, the connected components of the complement induce cycles for a homology basis in codimension 1 at each filtration value. Using the merge tree of the complement, we keep track of how these volume-optimal representatives evolve with the filtration of K, and equip each interval in the barcode with a sequence of canonical, volume-optimal representative cycles. If time permits, we also present an efficient algorithm for computing these sequences of representatives.
We apply geometric functionals to these representative cycles, such as path length, enclosed volume, or excess curvature. This way, we obtain a real-valued function for each interval, which captures geometric information about K. Deriving from this construction, we introduce generalized persistence landscapes, which specialize to standard persistence landscape with the constant-one functional. Generalized landscapes can distinguish point clouds with similar persistent homology but distinct shape, which we demonstrate by concrete examples.
Apparent pairs (also known as evident, shallow, close, steepness, Pareto, or minimal pairs) are a fundamental construction at the interface of persistent homology and discrete Morse theory. They play a key role in the context of algorithmic and computational topology. Besides their explicit use in efficient computation of persistent homology, I will show how they can be employed in the study of the geometry and topology of Gromov-hyperbolic spaces and their Rips complexes, and to from a bridge between Morse theoretic approaches to shape reconstruction from point clouds, specifically, relating lexicographically optimal cycles in the Delaunay filtration to Wrap complexes.
A spatial graph is a specific type of graph with spatial attributes associated with the nodes and the edges. It is a smart modelling choice for capturing the skeleton of a shape, a blood vessel network, a porous tissue, and many other data objects with intrinsically complex geometry. In this talk, we describe how spatial graphs can be analysed using a specific metric (the Fused Gromov–Wasserstein metric). We extend a testing procedure between distributions of spatial graphs, a depth measure to describe the distribution of spatial graphs, and a dimensionality reduction procedure based on preserving key topological features. We present this variety of methods on a dataset of cardiac fibrosis tissue and on a dataset of fungus mycelium networks.
References:
Vayer, T., Chapel, L., Flamary, R., Tavenard, R., & Courty, N. (2020). Fused Gromov-Wasserstein Distance for Structured Objects. Algorithms, 13(9), 212.
Dai, X., Lopez-Pintado, S. (2023). Tukey’s Depth for Object Data. Journal of the American Statistical Association, 118(543), 1760–1772.
Dubey, P., Chen, Y., & Müller, H. G. (2024). Metric statistics: Exploration and inference for random objects with distance profiles. The Annals of Statistics, 52(2), 757-792.
Hashemi, M., Gong, S., Ni, J., Fan, W., Prakash, B. A., & Jin, W. (2024). A comprehensive survey on graph reduction: sparsification, coarsening, and condensation. In Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, 8058-8066.
Calissano, A., Feragen, A., & Vantini, S. (2024). Populations of unlabelled networks: Graph space geometry and generalized geodesic principal components. Biometrika, 111(1), 147-170.
Graphical models and factor graphs are probabilistic models that incorporate prior knowledge of dependencies between variables; celebrated examples include hidden Markov models, and higher-order interactions are accounted for through hypergraphs. Computing the posterior distribution for a given collection of observations is called inference and is, in general, computationally very costly. In practice, one often resorts to variational inference, which consists in optimizing a weighted sum free energy over subcollections of variables, under the constraint that their probability distributions are compatible by marginalization. This compatibility condition defines the space of sections of specific presheaves, which we call projection presheaves. The General Belief Propagation algorithm is used to find the critical points of the weighted free energy. We will first explain how one can extend factor graphs to account for a broader class of relations between subcollections of variab!
les, by generalizing results from those specific presheaves to arbitrary presheaves over a poset. Given this broader framework, we show that the algorithm is functorial with respect to natural transformations. We then show that inference on minimal deformation retracts of hypergraphs seen as posets is sufficient for inference on the entire poset, and that for projection presheaves, a similar result holds when considering the undirected graph underlying the hypergraph. In collaboration with Léo Boitel.
Topological neural networks (TNNs) extend graph neural networks (GNNs) to simplicial complexes, cellular complexes, and other higher-order structures, to model topological features and higher-order interactions. As TNNs become more widely adopted, developing a better theoretical understanding of their properties becomes crucial. One phenomenon of interest is oversquashing, which is the compression of exponentially many features into fixed-width representations, which often degrades GNN performance on tasks. Even though oversquashing has been well studied in GNNs, it has remained largely unexamined for TNNs. In this talk, we present a first step toward a rigorous treatment of oversquashing in TNNs: We axiomatically model the computation graphs corresponding to TNNs as finite relational structures, and using this formulation we extend the results on oversquashing in GNNs to TNNs. In particular, we introduce "influence graphs" that model aggregate information flow in higher-order networks, and leverage these graphs to carry out higher-order sensitivity analysis, introduce new relevant higher-order discrete curvatures, establish bounds on the impact of local geometry and network depth, and quantify how hidden dimensions affect oversquashing. Lastly, we present a relational rewiring heuristic that adapts graph-rewiring techniques to higher-order networks, and demonstrably improves TNN performance in a manner consistent with graph rewiring. This talk is based on joint work with James Chapman, Marzieh Eidi, Karel Devriendt, and Guido Montúfar.
Representational similarity analysis (RSA) is widely used to analyze the alignment between humans and neural networks; however, conclusions based on this approach can be misleading without considering the underlying representational geometry.
Our work introduces a framework using Ollivier Ricci Curvature and Ricci Flow to analyze the fine-grained local structure of representations. This approach is agnostic to the source of the representational space, enabling a direct geometric comparison between human behavioral judgments and a model's vector embeddings. We apply it to compare human similarity judgments for 2D and 3D face stimuli with a baseline 2D-native network (VGG-Face) and a variant of it aligned to human behavior.
Our results suggest that geometry-aware analysis provides a more sensitive characterization of discrepancies and geometric dissimilarities in the underlying representations that remain only partially captured by RSA.
Notably, we reveal geometric inconsistencies in the alignment when moving from 2D to 3D viewing conditions.
This highlights how incorporating geometric information can expose alignment differences missed by traditional metrics, offering deeper insight into representational organization.
