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 (Inria-Grenoble)
Ulrich Bauer (TU Munich)
Anna Calissano (UCL)
Karel Devriendt (University of Oxford)
Heather Harrington (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)
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.
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.
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.
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.
