The workshop focuses on recent advances on algorithms and data structures for computational geometry and topology, persistent homology and topological data analysis, geometric statistics, stochastic geometry, shape analysis and modeling, geodesic methods in Riemannian manifolds, Lie groups and metric spaces, geometric and topological aspects of machine learning, applications of geometry and topology in biology and medicine, geometry processing, discrete differential geometry.
Organizers
Speakers
Semi-plenary speakers
IST Austria
University of Goettingen
Invited speakers
CNRS
Tallahassee
Université Paris-Saclay
INRIA
University of Houston
INRIA
Purdue University
University of Warwick
Jan Jendrysiak
MPI of Molecular Biology and Genetics
University of Potsdam
SUNY Albany
Rutgers University
Lawrence Berkeley National Labs
ETH Zuerich
IRIT Toulouse
KTH Stockholm
WPI Vienna
Thursday, 9 July
![[HS 06 – 1st Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS06.png){kind=link}
14:00-15:00 semi-plenary talk
Stephan Huckeman (University of Goettingen)
Sample spaces modeling phylogenetic trees (such as BHV space or wald space) intrinsically carry a stratified structure featuring infinite negative curvatures: too many less resolved tree topologies meet at higher resolved tree topologies, preventing a manifold structure there. In result probability distributions of phylogenetic trees may feature stickiness of their Fréchet means: after a finite random sample size, the asymptotic distribution of sample means collapses. In the worst case, they collapse toward a single point, posing serious complications for asymptotic nonparametric statistics. In this talk we take a closer look at this phenomenon, exploring the various flavors of stickiness and their interdependence. For instance sample stickiness (above), perturbation and topological (e.g. total variation or Wasserstein) stickiness, as well as modulation and directional stickiness. The latter promise to be of value where current asymptotic nonparametric statistical method struggle or even fail.
This is joint work with Lars Lammers and Do Tran Van.
15:00-15:30
Blanche Buet (LMO/Université Paris Saclay)
We propose a natural framework for the study of surfaces and their different discretizations based on varifolds. Varifolds have been introduced by Almgren to carry out the study of minimal surfaces. Though mainly used in the context of rectifiable sets, they turn out to be well suited to the study of discrete type objects as well. While the structure of varifold is flexible enough to adapt to both regular and discrete objects, it allows to define variational notions of mean curvature and second fundamental form based on the divergence theorem.
Though flexible, varifolds require the knowledge of the dimension of the shape to be considered. We then embed all d-dimensional Grassmannians into symmetric positive semi definite matrices with trace 1, that we endow with a distance coinciding with the Riemannian one in each Grassmannians.
Building upon the aforementioned embedding of Grassmannians, we propose a generalization of varifolds, that we call flagfolds, in order to model multi-dimensional shapes. The notion of first variation extends to such flagfolds and we are investigating whether some form of Allard’s rectifiability theorem extends as well.
Collaborators: C. Labourie and X. Pennec.
15:30-16:00
Nicolas Charon (University of Houston)
This talk will introduce established extrinsic and intrinsic Riemannian shape analysis frameworks, that allow to tackle statistical and machine learning problems on geometric datasets, i.e. datasets in which observations are curves, surfaces, shape graphs… From there, I shall discuss how recent developments in deep learning could help solving such problems for larger scale datasets and, conversely, what shape analysis tools can bring in developing more robust machine learning pipelines.
16:00-16:30 Coffee Break
16:30-17:00
Antoine Commaret (INRIA)
We present a method for approximating the boundary area of a set X embedded in Euclidean space and satisfying mild assumptions, from an approximating set Y, with an error linear in the Hausdorff distance between the two sets. This rate is optimal among classes of sets containing piecewise-smooth objects. Our results rely on tools from persistent homology and geometric measure theory (notably the principal kinematic formula), giving rise to a framework we call Persistent Geometry.
Our approach also allows us to approximate more generally the so-called intrinsic volumes of X, namely global quantities related to curvature, of which the boundary area is a special case.
17:00-17:30
Patrick Schnider (ETH Zürich)
The famous Ham-Sandwich theorem states that any d point sets in d-dimensional Euclidean space can be simultaneously bisected by a single hyperplane. The alpha-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the alpha-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincaré-Miranda theorem and allows us to generalize the result to and beyond oriented matroids.
This is joint work with Michaela Borzechowski, Sebastian Haslebacher, Hung Hoang and Simon Weber.
17:00-18:30 Poster Session
Friday, 10 July
14:00-14:30
Karen Haberman (University of Warwick)
In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by n ≥ 2 distinct landmark points in R^d. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimisation problem which minimises a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold Q of n distinct landmark points in R^d can be endowed with a Riemannian metric g such that the above optimisation problem is equivalent to the geodesic boundary value problem for g on Q. Despite its importance for modelling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold (Q,g) is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterisation of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel. I will further discuss joint work with Stephen C. Preston and Stefan Sommer which, for any number of landmarks in R^d and again with respect to a radial kernel, provides a sharp criterion guaranteeing geodesic completeness or geodesic incompleteness, respectively, of (Q,g).
14:30-15:00
Tom Szwagier (IRIT Toulouse)
A flag is a sequence of nested linear subspaces of increasing dimension (or, equivalently, a sequence of mutually-orthogonal subspaces). The goal of this talk is to convince you that, although abstract at first glance, these objects can play an important role in statistics. A first fundamental contribution is the discovery of a new type of parsimony in covariance matrices, related to eigenvalue multiplicity. We argue that empirical eigenvalues whose relative distance is below a certain threshold should be equalized. This result has a significant statistical impact: it motivates a transition from principal component analysis to principal subspace analysis, with clear gains in interpretability. Several extensions are proposed, including a lasso-like relaxation on the eigenvalues and a Bayesian approximation of the marginal likelihood. Our methodology extends to Gaussian mixture models and many dimensionality reduction methods, demonstrating through various applications the relevance of flags in statistics.
15:00-15:30
Alice Barbara Tumpach (WPI Vienna)
We introduce Varifold Moments Invariants (VMI) as a unifying framework for many previously introduced Moment Invariants. These invariants are deeply related to other contour features that are invariant under translations and rotations, like Extended Gaussian Image, Elliptic Fourier Descriptors or Shape Distributions. The advantage of the varifold approach to moments consists in being able to combine the geometry of the region, its boundary, and the family of lines tangent to it, in order to create a substantial number of invariant features with high discriminating power and clear geometric meaning. By coupling our VMI feature extraction with the light feature classifiers Random Forest or Multi-Layer-Perceptron, we outperform state-of-the-art approaches based on contours, while decreasing drastically the computational cost to the point of allowing our algorithm to run on light devices. We tested our approach on classification tasks on a large number of widely-used datasets of various types (leaves, objects, cells) and achieved high accuracy with a low number of geometrically interpretable features.
15:30-16:00
Martin Bauer (Tallahassee)
We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case.
The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael–Simon–Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces.
16:00-16:30 Coffee Break
16:30-17:30 semi-plenary talk
Herbert Edelsbrunner (IST Austria)
A classic result on Euclidean minimum spanning trees (EMSTs) is the existence of an asymptotic constant, c, such that the expected length of the EMST of n points sampled uniformly at random in the unit square is c times the square root of n, in the limit when n goes to infinity. However, the value of c is not known. Prior to this work, the known bounds were 0.6008.
17:30-18:00
Facundo Mémoli (Rutgers University)
The Gromov–Hausdorff distance is a fundamental notion for comparing compact
metric spaces, but exact values are notoriously difficult to determine. Already
for finite metric spaces, its exact computation leads to NP-hard
combinatorial optimization problems, and there are relatively few
purpose-built algorithms for computing it directly. In practice, one often uses
heuristics, relaxations, or related distances for metric and metric-measure
comparison, such as the Gromov–Wasserstein distance. A guiding principle of
this project is that the Gromov–Hausdorff sphere problem is a model case for
this broader family of comparison distances: understanding it should provide
ground-truth benchmarks for algorithms and relaxations, while also highlighting
useful connections among metric geometry, topology, and symmetry.
In this talk I will give a status report on an ongoing project centered on the
sphere problem: determining the Gromov–Hausdorff distance between round spheres
with their standard geodesic metrics, $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$. I will review the current state of knowledge, including known exact values, the best available upper and lower bounds, the main methods behind them, and the principal remaining open cases.
The sphere problem is canonical but nontrivial: the spaces are simple, while the
optimal correspondences are far from obvious. Upper bounds come from explicit
constructions of correspondences between spheres, whereas lower bounds arise
from topological obstructions, including Borsuk–Ulam-type arguments,
Vietoris–Rips complexes, and equivariant topology. I will also discuss how an
equivariant Gromov–Hausdorff distance for group actions clarifies the role of
symmetry and provides sharper tools for the problem.
18:00-18:30
Hana Dal Poz Kourimska (Universität Potsdam)
Assumptions about the reach of a manifold are crucial for ensuring the correctness of many geometric and topological algorithms. However, these assumptions are often coupled with a strict smoothness requirement—typically at least C^2.
In recent work, we proved that any manifold with positive reach can be approximated arbitrarily well by a smooth manifold without significantly reducing the reach. This result implies that almost all theorems established for C^2 manifolds with a certain reach naturally extend to non-C^2 manifolds with the same reach—essentially for free!
In this talk, I hope to share my excitement about the power of the reach. I’ll start by clearly motivating what the reach is, before diving into our result and briefly sketching the proof.
Saturday, 11 July
14:00-14:30
Michael Lesnick (SUNY Albany)
It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation. We systematically study the algorithmic implications of this idea for diagrams indexed by a finite poset. We say an initial functor F: C→D with C small is minimal if the sets of objects and morphisms of C each have minimum cardinality, among the sources of all initial functors with target D. For Q a finite poset or Q an interval in ℕ^d (i.e., a convex, connected subposet), we describe all minimal initial functors F: P→Q and in particular, show that F is always a subposet inclusion. We give efficient algorithms to compute a choice of minimal initial functor. In the case that Q is an interval in ℕ^d, we give asymptotically optimal bounds on |P|, the number of relations in P (including identities), in terms of the number n of minima of Q: We show that |P|=Θ(n) for d\leq 3, and |P|=Θ(n^2) for d>3. We apply these results to give new bounds on the cost of computing lim G for a functor G: Q→ Vec valued in vector spaces. For Q connected, we also give new bounds on the cost of computing the generalized rank of G (i.e., the rank of the induced map lim G → colim G), which is of interest in topological data analysis.
14:30-15:00
Tamal Dey (Purdue University)
Recent advances in combinatorial dynamical systems, extending classical discrete Morse theory, have stimulated growing algorithmic interest in combinatorial vector fields. In this context, the authors of [1] recently introduced the Conley–Morse persistence barcode, a topological summary that captures the evolution of invariant sets (Conley indices) in a changing vector field through homological persistence. In this work, we present an efficient algorithm for computing this barcode. Our approach incrementally updates the matrix decomposition R=DU used in standard persistence algorithms via a collection of suitably defined atomic operations. The correctness of the algorithm follows from a new interpretation of the transition diagram introduced in [1].
[1] Conley–Morse Persistence Barcode: A Homological Signature of a Combinatorial Bifurcation, Tamal K. Dey, Michal Lipinsky, and Manuel Soriano-Trigueros, Foundations of Computational Mathematics (2026), to appear.
15:00-15:30
Matthieu Carrière (INRIA)
In this talk, I will introduce MMA (Multipersistence Module Approximation): an algorithm based on matching functions for computing instances of approximate decompositions of any multi-parameter persistence module, with some precision parameter delta. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Moreover, MMA is robust: when computed with so-called compatible matching functions, MMA produces approximate decompositions that preserve diagonal barcodes. Finally, I will present a range of applications where approximate decompositions produced by MMA can improve upon existing single-parameter TDA models. Joint work with: David Loiseaux, Andrew J. Blumberg.
15:30-16:00
Francesca Tombari (KTH Stockholm)
In persistence theory, a central question is how to compare data and the measurements derived from it. This is typically addressed by comparing the associated persistence modules or their invariants. In the multiparameter setting, several metrics have been introduced, among which the matching distance has received considerable attention, especially regarding its computational aspects.
In this talk, I will introduce the convex matching distance, an alternative metric to the matching distance. While the two are not directly comparable, the convex matching distance depends on fewer parameters, making it potentially more computationally effective. I will discuss its geometric properties, with a focus on the parameter values at which the distance is realised. These results yield both theoretical insight and practical consequences, as they significantly reduce the search space needed for computation. I will also present experimental results that highlight its discriminative power, comparing it with the matching distance.
This presentation is based on joint work with F. Conti, P. Frosini, U. Fugacci, E. Mosig García, N. Quercioli and S. Scaramuccia.
16:00-16:30 Coffee Break
16:30-17:00
Dominique Attali (CNRS, Gipsa-lab)
Given a finite set of points P that sample an unknown shape M, we aim to approximate M based solely on P. This problem, known as shape reconstruction, has been widely studied, both in computer graphics and computational geometry. In this talk, I will present some algorithms for shape reconstruction and state conditions under which those algorithms are topologically correct, that is, output a simplicial complex whose support is homeomorphic to M.
17:00-17:30
Dmitriy Morozov (Lawrence Berkeley National Labs)
Given a zigzag filtration, we want to find its barcode representatives, i.e., a compatible choice of bases for the homology groups that diagonalize the linear maps in the zigzag. To achieve this, we convert the input zigzag to a levelset zigzag of a real-valued function. This function generates a Mayer-Vietoris pyramid of spaces, which generates an infinite strip of homology groups. We call the origins of indecomposable (diamond) summands of this strip their apexes and give an algorithm to find representative cycles in these apexes from ordinary persistence computation. The resulting representatives map back to the levelset zigzag and thus yield barcode representatives for the input zigzag. Our algorithm for lifting a p-dimensional cycle from ordinary persistence to an apex representative takes O(p⋅m log m) time. From this we can recover zigzag representatives in time O(log m + C), where C is the size of the output.
Posters
![[PZ06 – 1st Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_1-stock.png){kind=link}
Olivier Bisson
(Université Côte d’Azur)
Considering an atlas that partitions the brain cortex into n regions, functional connectivity between regions can be represented by an n×n matrix, often called a connectome. Here we focus on functional connectomes derived from resting-state fMRI, where each entry is the Pearson correlation between the mean fMRI time series of a pair of regions. Recent studies suggest that resting-state connectivity is time-varying, reflecting a temporal succession of multiple functional networks’ activation. To capture these dynamics, we compute correlation matrices over overlapping sliding windows, yielding a time series of full-rank correlation matrices that tracks evolving functional brain connectivity. We study the smooth manifold of full-rank correlation matrices Cor+(n) from a computational-geometric viewpoint, equipped with log-Euclidean metrics, i.e., Riemannian metrics obtained by pulling back a fixed Euclidean inner product through a global diffeomorphism so that the chosen coordinates are flat. To justify and unify these constructions, we develop a self-contained theory of log-Euclidean Lie groups: any manifold diffeomorphic to a finite-dimensional vector space V inherits an abelian Lie group structure transported from V, and log-Euclidean metrics coincide with bi-invariant metrics. We prove that any two log-Euclidean metrics on manifolds of the same dimension are related by an explicit global Riemannian isometry. This yields concrete correspondences between the standard log-Euclidean SPD geometry on S+(n−1) and recent log-Euclidean geometries on Cor+(n), explaining why intrinsic constructions (geodesic interpolation, means, gradient flows) agree across parametrizations. From a principal-bundle viewpoint, we show that the quotient metric associated with the action of positive diagonal matrices on S+(n) coincides, when transported to Cor+(n), with the off-log metric. We then construct an explicit log-Euclidean metric on S+(n) for which the standard inclusion Cor+(n)↪S+(n) becomes an isometric, totally geodesic embedding. This yields explicit computational tools for geodesics and adapted first-order statistics. Finally, the nested isometric embeddings constructed here provide a simple way to compare SPD and full-rank correlation matrices of different dimensions by embedding them into a common ambient size while preserving all intrinsic distances.
Jake Cordes
(University at Albany)
Classifying the stratospheric polar vortex provides predictability for surface weather on extended-range timescales. However, providing a scientifically sound classification is challenging: all the definitions proposed in over 60 years of study depend on empirically chosen parameters and yield different results when one of them changes. Moreover, as they are based on static thresholds, it is not straightforward to use them to study the spatiotemporal evolution of the vortex. Here, we introduce SuPerPoV, a score system that computes displacement and split ratios of the polar vortex using tools from topological data analysis, thus providing a sound classification of the polar vortex. The scores are computed by adapting superlevel set persistence and comparing prominent features. Our definition is entirely threshold-free and implemented open source. The scores generally recovers previous definitions and are output for a user-defined number of days, thus showing the evolution of the event. SuPerPoV offers a paradigm shift in the study of the polar vortex, hopefully bringing a deeper understanding of the polar vortex and related extreme events, such as sudden stratospheric warmings.
Alejandro Estrada-Llesta
(University of Vienna)
Astronomical observations lie on the past light cone (LC), a null three-dimensional hypersurface $\mathcal{LC}$ in spacetime $\mathcal{M}$, while theoretical predictions are defined throughout four-dimensional spacetime.
To compare with data, these predictions must be restricted to $\mathcal{LC}$. Therefore, modeling and computing the geometrical features of $\mathcal{LC}$ is a key task to describe the data we have.
In this work, we present a novel general-relativistic, metric-agnostic framework that samples the LC by integrating null geodesics in their Hamiltonian form.
This yields a decomposition of $\mathcal{LC}$ into a family of two-dimensional cross-sections ${\mathcal{S}{\lambda}}{\lambda}$ obtained by slicing the LC with the geodesic parameter.
Simultaneously, along the geodesic curves, by parallel transport and projection, we compute a tetrad that retains tangent-space information.
That, for $q\in \mathcal{S}{\lambda}$, allows us to decompose vectors in $T_q\mathcal{M}$ into components along the null geodesic generator and its orthogonal space. With this structure and an intrinsic LC parameterization, we can compute the first and second fundamental forms of each $\mathcal{S}{\lambda}$ surface at \tit{well behaved} points of $\mathcal{LC}$.
We validate our approach and implementation applying it to spatially maximally symmetric spacetimes both in spherical and Cartesian coordinates, where the integration of the geodesics and the evaluation of the geometrical properties are non-trivial.
Additionally, we present convergence rates for geodesic trajectories and cross-section geometry, and constraint-violation metrics.
Finally, as a step towards observables, we discuss a tangent-space–aware interpolation that assembles discrete ray samples into a continuous LC approximation, designed to preserve and approximate LC geometry, aiming to facilitate light-cone–restricted comparisons of theory to data.
Joseph Jung
(George Mason University)
The microstructure of a polycrystalline material is characterized by the geometry, arrangement, and orientation of its grains, features that all strongly influence a material’s physical properties. Advances in imaging technology allow us to accurately view these structures as micrographs with visually distinct grain morphologies. To enable a more quantitative analysis of the morphology of a microstructure, we view the grain boundary network as an embedded graph. Using this abstraction, we introduce the Volume Exchange Metric (VEM): a distance between micrograph windows defined by the minimum amount of area swept out to transform one grain boundary network to another. This area is motivated by the physical energy dissipated for such a transformation, to which it is proportional. We apply the VEM by building off of previous work by Miley et al. where they represent a microstructure as a probability distribution of windows sampled from a micrograph. The advantage of their method is that it enables statistical comparison of material micrographs by their local geometries. To compare microstructures, we define a distance between window distributions by employing a modified Wasserstein distance with the VEM serving as a physically informed cost function. We empirically validate this approach on a dataset of five synthetically generated microstructures with distinct grain boundary configurations. Our results show that this method can reliably distinguish microstructures even with a relatively small number of sampled windows and can be used as a standard for downstream microstructure validation procedures.
Alexandros Keros
(University of Edinburgh)
Synthesising advanced nanomaterials relies on successfully navigating the rich space of nanoparticle configurations and phases. Understanding this phase structure can help us craft practical equilibration pathways that lead to particle arrangements with desired and novel properties. However, the size and high-dimensionality of the space of attainable configurations hinders reliable phase identification and navigation for all but the smallest of systems. We hereby propose a topology-driven visualisation and exploration tool for the equilibrated configuration space of colloidal nanoparticles. Our method offers paradigmatic evidence of the topological hypothesis, which relates phase transitions of a statistical mechanical system with topological variations in the structure of its energetic equipotential level sets, while also revealing structural polymorphy and configurational degeneracies. The structure of potential energy level sets is viewed through an annotated Mapper graph based on molecular dynamics simulation data, whose structure reflects salient topological features of the high-dimensional configuration space. Our approach exposes the rich multiparametric phase structure of colloidal systems, as well as colloidal particle configurations whose existence has been hypothesised by simpler models.
Minh Le
(Ho Chi Minh City Open University)
Magnitude homology has been developed through a sequence of contributions, initially grounded in combinatorial constructions and subsequently extended to a general categorical framework. However, most variants of magnitude homology are formulated in a static framework. In contrast, our approach considers dynamical paths that encode the temporal structure of a graph G relative to the time parameter t. In other words, each vertex in our path (i.e., a 0-path) encodes the entire structure at time t, represented through Mobius inversion in a poset P. Equivalently, we can view this path as a Markov chain with an initial distribution and a transition probability matrix, evolving through an iterative process. This framework generalizes multiplex Markov chains by introducing a joint product of two dynamical paths. Furthermore, a stability theorem is established by comparing the distance between the associated simplicial complexes and the k-depth WL distance between the corresponding Markov chains.
Michał Lipiński
(IST Austria)
A connection matrix is an algebraic invariant capturing information about connecting orbits in a dynamical system and can be seen as a generalization of a Morse complex. This project is motivated by two recurring questions in the study of connection matrices: whether it is feasible to find all possible connection matrices for a given dynamical system as well as how to classify those matrices.
Building upon the recent algorithms for computing connection matrices for combinatorial dynamical systems, a connection matrix can be obtained by finding the Morse complex corresponding to a Forman combinatorial vector field inscribed in the original multivector field. However, there exist connection matrices that cannot be obtained directly by applying the algorithm.
To investigate the source of this discrepancy we construct a topological space AP(V) of acyclic subpartitions of the original combinatorial dynamical system V. In particular, this space is inhabited by all the Forman vector fields leading to connection matrices that can be obtained with the algorithm. Subsequently, using the notion of continuation of Morse decompositions, we subdivide the space into smaller regions corresponding to different gradient dynamics. With this preparation, we study the transfer morphisms connecting two neighboring Forman vector fields and the corresponding connection matrices, and analyze how the topological constraints of AP(V) affect the composition of those morphisms.
The project is still at an early stage, but the observations give hope for a better understanding of the nature of connection matrices. One of the examples we explore was originally studied by Reineck (1990). It consists of an attracting periodic orbit surrounded by three repellers and three saddles. We observed that for a combinatorial model of that system the constructed space of acyclic partitions takes the form of a Möbius strip.
This project is a joint work with Thomas Wanne.
Juan Manuel López Medel
(ICMAT Madrid)
In this poster, I would like to present two discretization options for (almost-)Dirac structures using the notion of retractions and discretization maps on manifolds.
We focus on the preservation of the presymplectic structure induced by the Dirac structure, rather than on the preservation of the Courant bracket. That is why we use (almost-)Dirac structures.
The two methods also exhibit better numerical behaviour than non-symplectic methods such as RK2, as we will see in the point-vortex example.
Applications of these methods are very wide because of the versatility of Dirac structures. These structures are used for modelling complex physical systems, such as port-Hamiltonian systems, and they also generalize Poisson manifolds.
The second discretization method is the most interesting. Instead of applying a discretization directly, we apply the constraint algorithm to the system. After doing that, we adapt the discretization map to the final manifold of the algorithm, in order to preserve the resulting constraints.
For example, in a nonholonomic system, we apply this algorithm to preserve the constraints.
Additionally, we apply the proposed discretization techniques to construct numerical integrators for port-Hamiltonian systems, and we discuss how to combine the discretization procedure with the constraint algorithm associated with systems of implicit differential equations.
Shiwa Mardokh-Rouhani
(University Of Manchester)
Currently, models in computational mechanics and applied science are mainly built on continuum descriptions. Although successful, they assume smooth fields on an underlying geometric domain. In many modern applications, this assumption is restrictive. Complex materials, evolving networks, fractured media, biological tissues, and data-driven systems often exhibit discrete structure, changing connectivity, and multi-scale behaviour not naturally represented by continuum fields. Standard discretisation techniques typically approximate a continuum model rather than treating the discrete structure as fundamental, which can obscure key physical mechanisms related to topology, connectivity, and evolving structure.
Combinatorial Mesh Calculus (CMC) provides an alternative foundation for modelling physical systems directly on discrete structures. Instead of starting from a continuum formulation, CMC begins with a combinatorial description in terms of cells and their incidence relations. Physical quantities are represented as discrete differential forms, and governing relations arise from intrinsic algebraic operators defined purely by connectivity. Transport, conservation, and interaction laws are formulated variationally without reference to an embedding space or continuum limit.
Systems where connectivity influences behaviour, topology evolves, or multiple physical processes interact across scales are naturally treated within CMC. It supports rigorous modelling of mechanical assemblies, complex materials, and networked systems by separating geometry, topology, and transport in a consistent algebraic framework. It also enables integrating computational modelling with experimental data through scale-bridging calibration. CMC can estimate macroscopic parameters and predict interior structural changes, thanks to its topology-preserving, meshing-independent, intrinsically discrete formulation that ensures exact conservation.
CMC addresses multidimensional, multi-scale, multiphase, and coupled problems. Examples include contaminant transport in porous and fractured rock, mechanical processes such as cracking and fracturing where topology evolves, and coupled transport–mechanical processes during wetting–drying or freezing–thawing cycles, where crack networks evolve and permeability changes can be predicted from the evolving microstructure.
Roy Nehme
(Sorbonne Université)
This poster presents recent results of my paper Pruning Distance of Upset-Decomposable Persistence Modules.
In the one-parameter setting, pointwise finite-dimensional persistence modules decompose into interval modules, and the bottleneck distance agrees with the interleaving distance by the isometry theorem. In the multiparameter setting, this picture breaks down: indecomposables modules are not interval and the isometry theorem fails to hold. As a consequence, computable Bottleneck distance fails to retain the same stability properties, while the Interleaving distance remains stable but is NP-hard to compute.
The poster explains the role of upset decompositions, the motivation for pruning, and the main stability theorem showing that the pruning distance is controlled by the interleaving distance. It also presents the result of my paper: on the class of upset-decomposable persistence modules, the pruning distance is Lipschitz equivalent with respect to the interleaving, proved using a directed graph formalism.
Iason Papadopoulos
(University of Bremen)
Network realignment complexes, introduced by Kozlov in [1], model the space of ways to transform labeled spanning tree networks by local leaf slide operations. Vertices correspond to spanning trees, edges correspond to elementary realignments, and higher dimensional cubes encode commuting families of independent realignments. Kozlov proved, using discrete Morse theory, that, for the complete base graph, these complexes admit a deformation retraction onto a complete graph, and posed the problem of determining the diameter of the associated realignment graph.
In this project, we study network realignment complexes with a prescribed base graph, restricting the allowed spanning trees and realignments according to the edges of the base graph. We describe the resulting cubical complexes combinatorially and determine their exact homotopy type in the cases considered. The analysis extends Kozlov’s approach by adapting the discrete Morse matching to the constrained setting, where new phenomena such as split components and barrier vertices appear.
We also investigate the metric geometry of the 1-skeleton, the network realignment graph. We prove general upper bounds for its diameter using centroid based routing through star trees, and develop lower bound techniques based on invariants of leaf slides, including a maximal matching metric, which gives the current best lower bound. These bounds give new evidence toward the diameter problem and clarify which combinatorial features of tree realignments control long geodesics.
Together, these results extend the topological theory of network realignment complexes beyond the complete graph case and provide new tools for studying their associated reconfiguration metrics.
[1] Kozlov, D.N. Network realignment complexes. J Appl. and Comput. Topology 9, 8 (2025)
Donato Quiccione
(University of Plymouth)
Structural MRI is one of the main modalities for measuring brain atrophy, but most morphometric methods depend on template registration and voxel-wise statistics. These steps make brains comparable, yet they can also suppress subject-specific geometry and become unreliable when anatomy is altered by disease. We introduce a registration-free approach that uses persistent homology to construct coordinate-free shape descriptors of brain atrophy directly from tissue segmentations. The choice of filtration determines what aspect of brain geometry the homological cycles encode.
The approach consists of two complementary pipelines. The first targets tissue loss through EDT superlevel filtrations of parenchymal masks, summarizing slice-wise H₁ persistence into landscape L¹ curves that describe regional thinning along each anatomical axis. The second targets compensatory cerebrospinal fluid (CSF) expansion through α-complex filtrations of the CSF complement, extracting H₂ persistence diagrams that capture sulcal widening and ventricular enlargement. Together, these descriptors encode two geometric consequences of atrophy: retraction of brain tissue and expansion of surrounding cavities.
Synthetic erosion experiments confirm that both pipelines respond systematically to induced atrophy. We apply the method to structural MRI from ADNI, where the first pipeline separates Alzheimer’s disease from cognitively normal controls (ROC-AUC 0.895), and the second detects greater longitudinal progression in the disease group via bottleneck distance on H₂ diagrams.
These results show that persistent homology provides an interpretable and biologically grounded basis for studying how brain atrophy reshapes tissue and CSF geometry.
Marc Raffelsiefen
(University of Bremen)
In topological data analysis, point clouds are studied by investigating the topological properties of the multifiltered simplicial complexes built on them. Reducing the complexity of these complexes can significantly speed-up the computation of topological features, such as multiparameter persistent homology, which is otherwise often not feasible due to their size and dimension. Discrete Morse theory is considered to be an effective tool for this purpose. In contrast to ordinary discrete Morse theory, here the filtered homotopy type must be preserved. To achieve this, the notion of compatibility has been introduced in the literature, ensuring that the filtered chain homotopy type is preserved.
First, we develop a corresponding topological result. By the Patchwork theorem, the union of Morse matchings on the compatible sets produces a compatible matching. While compatible sets can be arbitrary convex subposets of simplicial complexes in general, for a simplicial complex filtered via a max-extension function, we prove that each compatible set can be naturally transformed into the face poset of a simplicial complex. Consequently, any state-of-the-art discrete Morse theory algorithm can be applied directly and in parallel to the compatible sets to compute compatible matchings.
In addition, given any Morse matching (not necessarily compatible) on a filtered simplicial complex, we give an upper bound for the interleaving distance between the reduced and the original complex. This leads to the concept of approximative discrete Morse theory- a new approach that allows for more cells to be matched while preserving most of the persistent homological information.
Bria Razanaparany
(TU Graz)
Degree bifiltration (Rips/\v{C}ech) have been used widely in topological data analysis. The degree bifiltration is known to be stable under small perturbations, however it is only known to be robust to outliers in certain sense. We study the following problem to strengthen the use of degree bifiltration in TDA.
Given a $d$-dimensional compact submanifold $M$ of Euclidean space. Let $X$ be a finite set of points drawn according to the mixture measure where many points are uniformly drawn from $M$ with high density and some some points are drawn uniformly outside $M$ with low density (known as outliers). We will discuss that if one sample enough points, there is a region in the plane of bifiltration where the Degree (Rips/ \v{C}ech) bifiltration on $X$ recovers the topology of $M$. This justify that a significant topological signal of the data is still recovered using the degree bifiltration even if the data is noisy and contains some outliers. This is a bifiltrated version of many previous successful results in topological inference.
Tom Szwagier (IRIT Toulouse)
Many machine learning methods look for low-dimensional representations of the data. The underlying subspace can be estimated by first choosing a dimension q and then optimizing a certain objective function over the space of q-dimensional subspaces: the Grassmannian. Trying different q yields in general non-nested subspaces, which raises an important issue of consistency between the data representations. In this work, we propose a simple and easily implementable principle to enforce nestedness in subspace learning methods. It consists in lifting Grassmannian optimization criteria to flag manifolds (the space of nested subspaces of increasing dimension) via nested projectors. We apply the flag trick to several classical machine learning methods and show that it successfully addresses the nestedness issue.
Raphaël Tinarrage
(IST Austria)
Simplicial approximation is a classical technique for producing a simplicial map homotopic to a given continuous map, enabling computational topology algorithms on real data. However, the standard implementation quickly becomes intractable: barycentric subdivision creates poorly shaped simplices and an exponential blow-up in the number of vertices.
This poster will present a Delaunay-based alternative. Starting from a well-spaced point sample of the domain, we construct the Delaunay triangulation and iteratively refine it by inserting Steiner points via simple local rules (e.g., edge midpoints, facet minicenters, or facet centroids), recomputing the Delaunay complex after each insertion. This procedure is proven to converge: simplex diameters decrease geometrically with each step. An explicit shrink factor quantifies the improvement, showing this refinement is more efficient than barycentric subdivision while introducing fewer vertices.
Efficient simplicial approximation enables a range of topological methods for data analysis. I will highlight two applications: (1) computing the Hopf invariant of a vector field from physical data to characterize hopfions (topological solitons), and (2) triangulating certain classifying spaces (real Grassmannians and complex projective spaces) to compute characteristic classes (Stiefel–Whitney and Chern classes) from data.
Hang Wang
(KU Leuven)
We address the problem of approximating manifold-valued functions from input-output samples, where inputs lie in a Euclidean vector space and outputs reside on a Riemannian manifold. This task is fundamental to computational geometry and its applications, including parametric model order reduction, geometric modeling, and shape analysis.
We propose the multiple tangent spaces model (MTSM), a novel framework that unifies and extends two prior techniques: single-tangent-space pullback approximation (STSM) and Riemannian moving least squares (RMLS). The core idea is to construct a weighted Fréchet mean of pullbacks to multiple tangent spaces centered at strategically chosen anchor points on the manifold. Each local model pulls the function back to a tangent space via the logarithmic map, approximates the resulting vector-valued map using classical approximation methods, and pushes the result forward via the exponential map. The weighted Fréchet mean then synthesizes these predictions into a global approximation.
MTSM addresses key limitations of existing approaches: it overcomes the locality restrictions of STSM, which relies on a single tangent space, while substantially reducing the online computational cost of RMLS, which requires evaluating many local models. These advantages are achieved by replacing dense clusters of output samples with compact local surrogate models during an offline stage. We present a complete algorithmic description of both offline and online stages, along with a theoretical error bound that quantifies the approximation quality in terms of anchor point selection, local approximation accuracy, and manifold curvature.
Numerical experiments on benchmark problems from DTI estimation demonstrate that MTSM achieves superior approximation accuracy compared to STSM and offers significant computational savings over RMLS in the online phase. This work contributes to the computational geometry community’s growing toolkit for efficient representation and manipulation of manifold-valued data.