Estimating the persistent homology of Rn-valued functions using functional-geometric multifiltrations
Steve Oudot (INRIA Saclay - École Polytechnique)
This talk is about the following applied problem in topological data analysis, where the structure and stability of persistence modules plays a key role: given an unknown Rn-valued function f on a metric space X, how can we approximate the persistent homology of f from a finite sampling of 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 geometric complexes with fixed scale parameter for the approximation. Here we consider the question for arbitrary n and we answer it using a class of multiparameter filtrations called functional-geometric multifiltrations, which have gained attraction in recent years and have been studied for their stability properties. We prove the statistical convergence of the persistent homology of these multifiltrations, viewed as estimators of the persistent homology of the function, and we provide deviation bounds. Finally, we address questions pertaining to the computation of the estimator, the selection of its parameters, and the robustness to noise in the input function values and pairwise distances.
This is based on joint work with Ethan André, Jingyi Li, and David Loiseaux.
From Neurons to Networks: Understanding the Brain trough Algebraic Topology
Lida Kanari (University of Oxford)
Topological data analysis (TDA), and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In neuroscience, TDA methods have proven especially valuable, enabling analyses of individual cells to large-scale neuronal networks. A key example is the Topological Morphology Descriptor (TMD, [1]), which encodes the spatial structure of branching trees as persistence barcodes through a radial filtration. The TMD has been successfully applied to classify and cluster neurons and microglia, bridging the space of trees with the space of barcodes. In this talk, I will present recent results in the topological representation of brain cells, focusing on neurons. I will then demonstrate how algebraic topology provides insights into the relationships between single neurons and networks, bridging different computational scales.
A central question in neuroscience concerns the organizational principles that distinguish the human brain from other species. Our findings [2] suggest that human pyramidal cells, the most abundant cell type in the mammalian cortex, are remarkably more complex. These neurons form denser and more interconnected networks than those found in other species. This increased connectivity appears to stem from the complex branching patterns of their dendrites, the tree-like structures that receive signals from other neurons. Using topological data analysis, we propose a mechanistic understanding for the formation of these complex networks. This greater dendritic complexity, a unique characteristic of human neurons, may help explain the exceptional computational abilities and flexibility of the human brain.
[1] Kanari et al. 2018. A Topological representation of branching neuronal morphologies
[2] Kanari et al. 2025. Of mice and men: Dendritic architecture differentiates human from mouse neuronal networks
Barcode matchings, Wasserstein metrics, and applications
Andrea Guidolin (University of Southampton)
Matchings between persistence barcodes play a key role in persistence theory, providing tractable combinatorial ways to compare topological summaries of data. A theory of matchings induced by morphisms of persistence modules, called induced matchings, was first developed by Bauer and Lesnick. In this talk, we present an alternative notion of induced matching between persistence barcodes, derived from an algebraic decomposition theorem. We study the interaction between induced matchings and a notion of cost for morphisms based on p-norms of persistence modules, showing that matchings can be viewed as simplified morphisms with a reduced cost. This result has both theoretical and practical implications. On the theoretical side, induced matchings yield a concise proof of the algebraic Wasserstein stability of persistent homology. On the practical side, our results enable efficient computation of some stable vectorisations of persistence barcodes defined via Wasserstein metrics.
The talk is based on joint work with Jens Agerberg, Arthur Reidenbach, Isaac Ren and Martina Scolamiero.
Exploring the space of encodings
Karim Adiprasito (Jussieu Institute of Mathematics)
Given a task of modelling a geometric object, several ways come to mind. A variety, a simplicial complex, a pointcloud, and many more. I will not focus on a specific encoding. Instead, I will focus on surveying what is known about the space of different encodings, its connectivity, diameter and more. An example question addressed is: Given two triangulations of the same space, how can we transition from one to the other?
The algorithmic complexity of knots and surfaces in 3-manifolds
Mehdi Yazdi (King's College of London)
Abstract: Algorithmic questions about knots and surfaces in 3-manifolds, in particular in the Euclidean 3-space, go back at least to Dehn in early 20th century. Yet it is still unknown whether there is a polynomial time algorithm to decide if a knot diagram represents the unknot. In this talk I will give an overview of some of the main advances in the field, and then discuss the following two recent results:
The genus of a knot in a 3-manifold is the minimum genus of a compact orientable surface whose boundary is the given knot. In particular a knot is the unknot if and only if its genus is 0. In joint work with Marc Lackenby we showed that the problem of upper bound for the genus of a knot in a fixed closed orientable 3-manifold is in co-NP, answering a question of Agol, Hass, and Thurston from 2002. Previously this was known for the case of rational homology 3-spheres by the work of Lackenby.
In work in progress with Marc Lackenby and Eric Sedgwick we show that deciding if a given closed orientable 3-manifold contains an embedded closed incompressible surface of a given genus is NP-complete.
Higher-Order Functional Connectivity: Unveiling Individuality and Task Engagement Beyond Pairwise Interaction
Giovanni Petri (Northeastearn University of London)
I will present a topological framework for analyzing brain functional connectivity that moves beyond pairwise correlations to capture and describe higher-order interactions.
By computing instantaneous co-fluctuations at multiple orders and analyzing the resulting topological structures, we reveal local signatures—violating triangles and homological scaffolds—that encode information invisible to traditional network approaches.
Building on these topological descriptions, I will demonstrate how local higher-order features dramatically outperform standard methods for two critical applications: task decoding and functional brain fingerprinting.
While global topological measures show no advantage, local indicators enable superior discrimination between cognitive states and significantly enhance individual identification, particularly through interactions between unimodal sensory systems and transmodal networks.
These findings establish a direct link between the topological organization of brain activity and individual-specific functional patterns.
Finally, I will discuss how this framework connects topology and information theory, showing that local topological signatures can be interpreted as markers of synergistic information integration.
This correspondence suggests that the geometry of co-fluctuations captures how information is distributed, shared, and recombined across the brain, offering a principled route toward an information-topological theory of cognition.”
Counting bars in Sobolev spaces
Vukašin Stojisavljević (University of Oxford)
Classical Morrey-Sobolev inequality bounds the oscillation of a function by its Sobolev norms. Motivated by this result, we introduce methods from topological data analysis into the study of oscillations of functions. This idea leads to a Sobolev-type estimate on the number of sufficiently long bars in the associated barcode. As an application of this estimate, we generalize a number of classical theorems, such as Courant's nodal domain theorem and Bézout's theorem, to the setting of linear combinations of eigenfunctions of elliptic operators. The described approach to oscillations brings to light a bar-counting invariant which is of independent interest. The talk is based on a joint work with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.
Minimum Spanning Tree Regularization for Self-Supervised Learning
Julie Mordacq (INRIA Saclay - École Polytechnique)
Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties of effective representations: (i) avoiding dimensional collapse—where the learned features occupy only a low-dimensional subspace—and (ii) promoting uniformity in the induced distribution.
In this talk, I will present T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (or, equivalently, the total persistence in degree 0 of the Rips filtration) over the learned representations. I will discuss theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and enhances distribution uniformity on arbitrary compact Riemannian manifolds. Finally, I will show empirical results on synthetic data and standard SSL benchmarks, illustrating how T-REGS improves representation quality in practice.
Simplicial approximation, Delaunay triangulations, and the list homomorphism problem
Raphaël Tinarrage (IST Austria)
Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivisions produce poorly shaped simplices, and the star condition introduces many vertices. I will present ideas to address these difficulties, drawing on spherical Delaunay triangulations, economical simplicial mapping cones, locally equiconnected spaces, and connections with the list homomorphism problem.
Multi-parameter Module Approximation (MMA): An Interpretable Descriptor for Multiparameter Persistence
David Loiseaux (INRIA Saclay - École Polytechnique)
In Topological Data Analysis (TDA), the persistence barcode is a foundational and effective invariant for single-parameter persistent homology. However, analyzing complex, noisy, or multi-labeled data often necessitates multiparameter filtrations. While these filtrations capture richer, and more subtle features, the resulting multiparameter persistent homology (MPH) is notoriously complex, lacking a simple complete discrete descriptor like the barcode. This challenge severely limits its interpretability and thus its practical usage.
In this talk, I will introduce a descriptor for MPH, Multiparameter Module Approximation (MMA). This descriptor can be obtained by enforcing
interpretability of MPH, has some strong recovery properties and is efficiently computable.