@article{20867, abstract = {We discuss the embeddability of subspaces of the Gromov–Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension n(n−1)∕2. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both nonembeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean–Hausdorff distance.}, author = {Zava, Nicolò}, issn = {1472-2739}, journal = {Algebraic & Geometric Topology}, number = {8}, pages = {5153--5174}, publisher = {Mathematical Sciences Publishers}, title = {{Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces}}, doi = {10.2140/agt.2025.25.5153}, volume = {25}, year = {2025}, } @unpublished{21016, abstract = {Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times.}, author = {Bleile, Yossi and Fajstrup, Lisbeth and Heiss, Teresa and Svane, Anne Marie and Sørensen, Søren Strandskov}, booktitle = {arXiv}, title = {{Identifying cobordisms using kernel persistence}}, doi = {10.48550/arXiv.2505.17858}, year = {2025}, } @unpublished{21050, abstract = {In 1873, James C. Maxwell conjectured that the electric field generated by $n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to a conjecture of Gabrielov, Novikov and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find smaller bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.}, author = {Edelsbrunner, Herbert and Fillmore, Christopher D and Olivera, Gonçalo}, booktitle = {arXiv}, title = {{Counting equilibria of the electrostatic potential}}, doi = {10.48550/ARXIV.2501.05315}, year = {2025}, } @article{21253, abstract = {We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n - 1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.}, author = {Pach, János and Saghafian, Morteza and Schnider, Patrick}, issn = {0925-7721}, journal = {Computational Geometry}, publisher = {Elsevier}, title = {{Decomposition of geometric graphs into star-forests}}, doi = {10.1016/j.comgeo.2025.102186}, volume = {129}, year = {2025}, } @article{17149, abstract = {The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.}, author = {Edelsbrunner, Herbert and Nikitenko, Anton}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, pages = {490--499}, publisher = {Springer Nature}, title = {{Average and expected distortion of Voronoi paths and scapes}}, doi = {10.1007/s00454-024-00660-y}, volume = {73}, year = {2025}, } @article{18626, abstract = {The local angle property of the (order-1) Delaunay triangulations of a generic set in R2 asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.}, author = {Edelsbrunner, Herbert and Garber, Alexey and Saghafian, Morteza}, issn = {1090-2082}, journal = {Advances in Mathematics}, publisher = {Elsevier}, title = {{Order-2 Delaunay triangulations optimize angles}}, doi = {10.1016/j.aim.2024.110055}, volume = {461}, year = {2025}, } @phdthesis{18979, abstract = {Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA. The first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue. The aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor. We describe a construction of a chromatic alpha complex, which enables efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline. The second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory. In the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot.}, author = {Draganov, Ondrej}, issn = {2663-337X}, keywords = {topological data analysis, chromatic point set, alpha complex, persistent homology, six pack, sheaf, microlocal discrete Morse, injective resolution, collapse, knot, discrete Morse theory}, pages = {140}, publisher = {Institute of Science and Technology Austria}, title = {{Structures and computations in topological data analysis}}, doi = {10.15479/at:ista:18979}, year = {2025}, } @article{19937, abstract = {Simplets are elementary units within simplicial complexes and are fundamental for analyzing the structure of simplicial complexes. Previous efforts have mainly focused on accurately counting or approximating the number of simplets rather than studying their frequencies. However, analyzing simplet frequencies is more practical for large-scale simplicial complexes. This paper introduces the Simplet Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies in simplicial complexes. Additionally, we provide a bound on the sample complexity required to approximate the SFD vector using any uniform sampling-based algorithm accurately. We extend the definition of simplet frequency distribution to encompass simplices, allowing for the analysis of simplet frequencies within simplices of simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex. Furthermore, we present a bound on the sample complexity required for accurately approximating the SDV and SDC for a set of simplices using any uniform sampling-based algorithm. We also introduce algorithms for approximating SFD, geometric SFD, SDV, and SDC. We also validate the theoretical bounds with experiments on random simplicial complexes and demonstrate the practical application through a case study.}, author = {Mahini, Mohammad and Beigy, Hamid and Qadami, Salman and Saghafian, Morteza}, issn = {0020-0255}, journal = {Information Sciences}, number = {11}, publisher = {Elsevier}, title = {{Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality}}, doi = {10.1016/j.ins.2025.122425}, volume = {719}, year = {2025}, } @inproceedings{20005, abstract = {We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.}, author = {Edelsbrunner, Herbert and Garber, Alexey and Saghafian, Morteza}, booktitle = {41st International Symposium on Computational Geometry}, isbn = {9783959773706}, issn = {1868-8969}, location = {Kanazawa, Japan}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{On spheres with k points inside}}, doi = {10.4230/LIPIcs.SoCG.2025.43}, volume = {332}, year = {2025}, } @inproceedings{20006, abstract = {In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.}, author = {Ost, Lara and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert}, booktitle = {41st International Symposium on Computational Geometry}, isbn = {9783959773706}, issn = {1868-8969}, location = {Kanazawa, Japan}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Banana trees for the persistence in time series experimentally}}, doi = {10.4230/LIPIcs.SoCG.2025.71}, volume = {332}, year = {2025}, } @article{20260, abstract = {The medial axis of a set consists of the points in the ambient space without a unique closest point in the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a skeleton topologically equivalent to the original set. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities, various prunings of the medial axis have been proposed in the computational geometry community. Here, we examine one type of pruning, called burning. Because of the good experimental results it was hoped that the burning method of simplifying the medial axis would be stable. In this work, we show a simple example that dashes such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of a shape where the medial axis goes from collapsible to non-collapsible. More precisely, we consider the standard deformation retract from the closed ball to Bing’s house with two rooms, but stop just short of the point where Bing’s house becomes two dimensional. This way we obtain an isotopy from the 3-ball to a thickened version of Bing’s house. Under this isotopy, the medial axis goes from collapsible to non-collapsible. We stress that this isotopy can be made generic, in the sense of singularity theory, as developed by Arnol’d and Thom.}, author = {Chambers, Erin Wolf and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs}, issn = {2730-9657}, journal = {La Matematica}, pages = {811--828}, publisher = {Springer Nature}, title = {{Burning or collapsing the medial axis is unstable}}, doi = {10.1007/s44007-025-00170-0}, volume = {4}, year = {2025}, } @article{20293, abstract = {Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert}, issn = {1099-4300}, journal = {Entropy}, number = {8}, publisher = {MDPI}, title = {{Tight bounds between the Jensen–Shannon divergence and the minmax divergence}}, doi = {10.3390/e27080854}, volume = {27}, year = {2025}, } @article{20323, abstract = {We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities.}, author = {Brown, Adam and Draganov, Ondrej}, issn = {0022-4049}, journal = {Journal of Pure and Applied Algebra}, number = {10}, publisher = {Elsevier}, title = {{Discrete microlocal Morse theory}}, doi = {10.1016/j.jpaa.2025.108068}, volume = {229}, year = {2025}, } @article{20490, abstract = {We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips. }, author = {Edelsbrunner, Herbert and Garber, Alexey and Ghafari, Mohadese and Heiss, Teresa and Saghafian, Morteza}, issn = {0195-6698}, journal = {European Journal of Combinatorics}, publisher = {Elsevier}, title = {{Flips in two-dimensional hypertriangulations}}, doi = {10.1016/j.ejc.2025.104248}, volume = {132}, year = {2025}, } @article{20585, abstract = {Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {2639-8001}, journal = {Foundations of Data Science}, pages = {30--62}, publisher = {American Institute of Mathematical Sciences}, title = {{Chromatic alpha complexes}}, doi = {10.3934/fods.2025003}, volume = {8}, year = {2025}, } @article{20657, abstract = {The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². }, author = {Edelsbrunner, Herbert and Pach, János}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, publisher = {Springer Nature}, title = {{Maximum Betti numbers of Čech complexes}}, doi = {10.1007/s00454-025-00796-5}, year = {2025}, } @inproceedings{20658, abstract = {The medial axis of a smoothly embedded surface in R^3 consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.}, author = {Edelsbrunner, Herbert and Stephenson, Elizabeth R and Thoresen, Martin H}, booktitle = {4th International Joint Conference on Discrete Geometry and Mathematical Morphology}, isbn = {9783032095435}, issn = {1611-3349}, location = {Groningen, The Netherlands}, pages = {133--147}, publisher = {Springer Nature}, title = {{The mid-sphere cousin of the medial axis transform}}, doi = {10.1007/978-3-032-09544-2_10}, volume = {16296}, year = {2025}, } @inproceedings{20729, abstract = {Persistence modules (defined as a sequence of vector spaces and linear maps between them) are a key tool in topological data analysis. They are easy to interpret and fast to compute. However, when considering persistence maps (i.e. maps between persistence modules), these properties are lost. We propose a new invariant for persistence maps consisting of a partial matching such that: it is easy to interpret, it is more discriminative than the image of the persistence map, and can be calculated with cubical complexity.}, author = {Gonzalez-Diaz, Rocio and Soriano Trigueros, Manuel and Torras-Casas, Alvaro}, booktitle = {Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation}, isbn = {9798400720758}, location = {Guanajuato, Mexico}, pages = {188--196}, publisher = {Association for Computing Machinery}, title = {{Additive partial matchings for persistent homology}}, doi = {10.1145/3747199.3747561}, year = {2025}, } @article{13182, abstract = {We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining a collection of sorted lists together with its persistence diagram.}, author = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, pages = {1101--1119}, publisher = {Springer Nature}, title = {{Geometric characterization of the persistence of 1D maps}}, doi = {10.1007/s41468-023-00126-9}, volume = {8}, year = {2024}, } @article{14345, abstract = {For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)).}, author = {Edelsbrunner, Herbert and Garber, Alexey and Ghafari, Mohadese and Heiss, Teresa and Saghafian, Morteza}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {29--48}, publisher = {Springer Nature}, title = {{On angles in higher order Brillouin tessellations and related tilings in the plane}}, doi = {10.1007/s00454-023-00566-1}, volume = {72}, year = {2024}, }