@article{21931, 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 (nonoptimal) 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 Conjecture 1.8 by 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 lower 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 Oliveira, Goncalo}, issn = {1460-244X}, journal = {Proceedings of the London Mathematical Society}, number = {5}, publisher = {Wiley}, title = {{Counting equilibria of the electrostatic potential}}, doi = {10.1112/plms.70163}, volume = {132}, year = {2026}, } @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 = {2026}, } @article{22648, abstract = {Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems—a young but already well-established theory. We introduce the Conley–Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of the Conley–Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose.}, author = {Dey, Tamal K. and Lipiński, Michał and Soriano Trigueros, Manuel}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, keywords = {Multivector field, Conley index, Morse decomposition, Bifurcation, Continuation, Zigzag persistence, Persistence barcode, Gentle algebra}, publisher = {Springer}, title = {{Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations}}, doi = {10.1007/s10208-026-09766-6}, year = {2026}, } @article{20980, abstract = {Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix. These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures—such as attractors, repellers, and orbits—in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of Dey et al. and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.}, author = {Dey, Tamal K. and Haas, Andrew and Lipiński, Michał}, issn = {1536-0040}, journal = {SIAM Journal on Applied Dynamical Systems}, number = {1}, pages = {108--130}, publisher = {SIAM}, title = {{Computing a connection matrix and persistence efficiently from a morse decomposition}}, doi = {10.1137/25m1739406}, volume = {25}, year = {2026}, } @inproceedings{22299, abstract = {The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.}, author = {Edelsbrunner, Herbert and Lipiński, Michał and Mrozek, Marian and Soriano Trigueros, Manuel and Zimin, Fedor}, booktitle = {42nd International Symposium on Computational Geometry}, isbn = {9783959774185}, issn = {1868-8969}, keywords = {Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions, Theory of computation → Computational geometry}, location = {New Brunswick, NJ, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The depth poset under transpositions in the filter}}, doi = {10.4230/LIPICS.SOCG.2026.41}, volume = {367}, year = {2026}, } @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}, pages = {597--624}, publisher = {Springer Nature}, title = {{Maximum Betti numbers of Čech complexes}}, doi = {10.1007/s00454-025-00796-5}, volume = {75}, year = {2026}, } @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}, keywords = {Topological data analysis, Delaunay mosaic, alpha complex, chromatic sets, persistent homology, kernel/image/cokernel persistent homology, radius function, discrete Morse theory, exact sequences}, pages = {30--62}, publisher = {AIMS}, title = {{Chromatic alpha complexes}}, doi = {10.3934/fods.2025003}, volume = {8}, year = {2026}, } @article{23087, abstract = {Disordered materials such as glasses are promising candidates as solid-state electrolytes for batteries, but they tend to exhibit low ionic conductivity and current methods for investigating and predicting the ionic conductivity of electrolyte materials are inefficient. Here, we propose a framework that summarizes the topological properties of the electrolyte material’s atomic structure with a weighted graph based on periodic boundary conditions. We then compute a variation of the max-flow algorithm, a commonly used model in continuum models of transportation and liquid flow, on this summary and show that it is compatible with the simulated periodic atomic structures. Using molecular dynamics simulations of various glassy solid-state electrolyte families including both mobile lithium and sodium ions, we validate the approach by showing a positive correlation between the area-normalized maximum flow predicted from the static glass structures and the ionic conductivity as obtained from the simulations, especially in the case of high-conductivity sulfide glass electrolytes. We envision that the developed methodology, which only relies on static atomic structure, will accelerate the discovery of highly conductive sulfide glassy electrolytes for high-performance all-solid-state batteries.}, author = {Pegoraro, Matteo and Christensen, Rasmus and Sørensen, Søren S. and Lindbjerg, Anders R. and Du, Tao and Bleile, Yossi and Svane, Anne Marie and Biscio, Christophe A.N. and Fajstrup, Lisbeth and Smedskjaer, Morten M.}, issn = {2041-1723}, journal = {Nature Communications}, publisher = {Springer Nature}, title = {{Predicting ionic conductivity of sulfide glass solid-state electrolytes using max-flow methodology}}, doi = {10.1038/s41467-026-77363-w}, volume = {17}, year = {2026}, } @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}, } @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}, } @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}, } @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}, } @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{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}, }