@inproceedings{14888, abstract = {A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a probabilistic FPT-approach in the number of popular faces.}, author = {De Nooijer, Phoebe and Terziadis, Soeren and Weinberger, Alexandra and Masárová, Zuzana and Mchedlidze, Tamara and Löffler, Maarten and Rote, Günter}, booktitle = {31st International Symposium on Graph Drawing and Network Visualization}, isbn = {9783031492747}, issn = {1611-3349}, location = {Isola delle Femmine, Palermo, Italy}, pages = {18--33}, publisher = {Springer Nature}, title = {{Removing popular faces in curve arrangements}}, doi = {10.1007/978-3-031-49275-4_2}, volume = {14466}, year = {2024}, } @inproceedings{15012, 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}, booktitle = {31st International Symposium on Graph Drawing and Network Visualization}, isbn = {9783031492716}, issn = {1611-3349}, location = {Isola delle Femmine, Palermo, Italy}, pages = {339--346}, publisher = {Springer Nature}, title = {{Decomposition of geometric graphs into star-forests}}, doi = {10.1007/978-3-031-49272-3_23}, volume = {14465}, year = {2024}, } @unpublished{15091, abstract = {Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective.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}, booktitle = {arXiv}, title = {{Chromatic alpha complexes}}, doi = {10.48550/arXiv.2212.03128}, year = {2024}, } @inproceedings{15093, abstract = {We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.}, author = {Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Henzinger, Monika H and Ost, Lara}, booktitle = {Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}, editor = {Woodruff, David P.}, location = {Alexandria, VA, USA}, pages = {243 -- 295}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Dynamically maintaining the persistent homology of time series}}, doi = {10.1137/1.9781611977912.11}, year = {2024}, } @phdthesis{15094, abstract = {Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in discrete geometry that have captivated mathematicians for centuries, if not millennia. This thesis seeks to cast new light on these structures by illustrating specific instances where a topological perspective, specifically through discrete Morse theory and persistent homology, provides valuable insights. At first glance, the topology of these geometric objects might seem uneventful: point sets essentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which is a contractible space, and the topology of a network primarily involves the enumeration of connected components and cycles within the network. However, beneath this apparent simplicity, there lies an array of intriguing structures, a small subset of which will be uncovered in this thesis. Focused on three case studies, each addressing one of the mentioned objects, this work will showcase connections that intertwine topology with diverse fields such as combinatorial geometry, algorithms and data structures, and emerging applications like spatial biology. }, author = {Cultrera di Montesano, Sebastiano}, issn = {2663-337X}, pages = {108}, publisher = {Institute of Science and Technology Austria}, title = {{Persistence and Morse theory for discrete geometric structures}}, doi = {10.15479/at:ista:15094}, year = {2024}, } @article{15247, abstract = {Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant <2 such that every family of subsets of an n-element set that contains no odd-sunflower consists of at most n sets. We construct such families of size at least 1.5021n. We also characterize minimal odd-sunflowers of triples.}, author = {Frankl, Peter and Pach, János and Pálvölgyi, Dömötör}, issn = {1096-0899}, journal = {Journal of Combinatorial Theory, Series A}, number = {8}, publisher = {Elsevier}, title = {{Odd-sunflowers}}, doi = {10.1016/j.jcta.2024.105889}, volume = {206}, year = {2024}, } @article{15380, abstract = {The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.}, 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 = {557--578}, publisher = {Springer Nature}, title = {{Depth in arrangements: Dehn–Sommerville–Euler relations with applications}}, doi = {10.1007/s41468-024-00173-w}, volume = {8}, year = {2024}, } @inproceedings{17144, abstract = {We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.}, author = {Kourimska, Hana and Lieutier, André and Wintraecken, Mathijs}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, issn = {1868-8969}, location = {Athens, Greece}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms}}, doi = {10.4230/LIPIcs.SoCG.2024.69}, volume = {293}, year = {2024}, } @inproceedings{17145, abstract = {Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.}, author = {Rote, Günter and Rüber, Moritz and Saghafian, Morteza}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, issn = {1868-8969}, location = {Athens, Greece}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Grid peeling of parabolas}}, doi = {10.4230/LIPIcs.SoCG.2024.76}, volume = {293}, year = {2024}, } @inproceedings{17146, 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. In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry.}, author = {Edelsbrunner, Herbert and Pach, János}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, issn = {1868-8969}, location = {Athens, Greece}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Maximum Betti numbers of Čech complexes}}, doi = {10.4230/LIPIcs.SoCG.2024.53}, volume = {293}, year = {2024}, } @inproceedings{17170, abstract = {In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. - In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. - The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction. We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.}, author = {Attali, Dominique and Kourimska, Hana and Fillmore, Christopher D and Ghosh, Ishika and Lieutier, André and Stephenson, Elizabeth R and Wintraecken, Mathijs}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, issn = {1868-8969}, location = {Athens, Greece}, pages = {11:1--11:19}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds}}, doi = {10.4230/LIPIcs.SoCG.2024.11}, volume = {293}, year = {2024}, } @article{17190, abstract = {For a locally finite set, 𝐴⊆ℝ𝑑 , the 𝑘 th Brillouin zone of 𝑎∈𝐴 is the region of points 𝑥∈ℝ𝑑 for which ‖𝑥−𝑎‖ is the 𝑘 th smallest among the Euclidean distances between 𝑥 and the points in 𝐴 . If 𝐴 is a lattice, the 𝑘 th Brillouin zones of the points in 𝐴 are translates of each other, and together they tile space. Depending on the value of 𝑘 , they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in ℝ2 , and the convergence of the maximum volume of a chamber to zero for the integer lattice.}, author = {Edelsbrunner, Herbert and Garber, Alexey and Ghafaris, Mohadese and Heiss, Teresa and Saghafiant, Morteza and Wintraecken, Mathijs}, issn = {0895-4801}, journal = {SIAM Journal on Discrete Mathematics}, number = {2}, pages = {1784--1807}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Brillouin zones of integer lattices and their perturbations}}, doi = {10.1137/22M1489071}, volume = {38}, year = {2024}, } @inproceedings{18556, abstract = {Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization}, isbn = {9783959773430}, issn = {1868-8969}, location = {Vienna, Austria}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The Euclidean MST-ratio for bi-colored lattices}}, doi = {10.4230/LIPIcs.GD.2024.3}, volume = {320}, year = {2024}, } @article{18604, abstract = {A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a randomized FPT-time algorithm where the parameter is the number of popular faces.}, author = {De Nooijer, Phoebe and Terziadis, Soeren and Weinberger, Alexandra and Masárová, Zuzana and Mchedlidze, Tamara and Löffler, Maarten and Rote, Günter}, issn = {1526-1719}, journal = {Journal of Graph Algorithms and Applications}, number = {2}, pages = {47--82}, publisher = {Brown University}, title = {{Removing popular faces in curve arrangements}}, doi = {10.7155/jgaa.v28i2.2988}, volume = {28}, year = {2024}, } @phdthesis{18667, abstract = {Many chemical and physical properties of materials are determined by the material’s shape, for example the size of its pores and the width of its tunnels. This makes materials science a prime application area for geometrical and topological methods. Nevertheless many methods in topological data analysis have not been satisfyingly extended to the needs of materials science. This thesis provides new methods and new mathematical theorems targeted at those specific needs by answering four different research questions. While the motivation for each of the research questions arises from materials science, the methods are versatile and can be applied in different areas as well. The first research question is concerned with image data, for example a three-dimensional computed tomography (CT) scan of a material, like sand or stone. There are two commonly used topologies for digital images and depending on the application either of them might be required. However, software for computing the topological data analysis method persistence homology, usually supports only one of the two topologies. We answer the question how to compute persistent homology of an image with respect to one of the two topologies using software that is intended for the other topology. The second research question is concerned with image data as well, and asks how much of the topological information of an image is lost when the resolution is coarsened. As computer tomography scanners are more expensive the higher the resolution, it is an important question in materials science to know which resolution is enough to get satisfying persistent homology. We give theoretical bounds on the information loss based on different geometrical properties of the object to be scanned. In addition, we conduct experiments on sand and stone CT image data. The third research question is motivated by comparing crystalline materials efficiently. As the atoms within a crystal repeat periodically, crystalline materials are either modeled by unmanageable infinite periodic point sets, or by one of their fundamental domains, which is unstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with appropriate properties such that comparing the crystals can be eased by comparing the fingerprints instead. We define the density fingerprint and prove the necessary properties. The fourth research question is motivated by studying the hole-structure or connectedness, i.e. persistent homology or merge trees, of crystalline materials. A common way to deal with periodicity is to take a fundamental domain and identify opposite boundaries to form a torus. However, computing persistent homology or merge trees on that torus loses some of the information materials scientists are interested in and is additionally not stable under certain noise. We therefore decorate the merge tree stemming from the torus with additional information describing the density and growth rate of the periodic copies of a component within a growing spherical window. We prove all desired properties, like stability and efficient computability.}, author = {Heiss, Teresa}, isbn = {978-3-99078-052-7}, issn = {2663-337X}, keywords = {persistent homology, topological data analysis, periodic, crystalline materials, images, fingerprint}, pages = {111}, publisher = {Institute of Science and Technology Austria}, title = {{New methods for applying topological data analysis to materials science}}, doi = {10.15479/at:ista:18667}, year = {2024}, } @unpublished{18673, abstract = {Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes O((n+m)logn) time, in which n and m are the numbers of vertices and edges in the quotient complex, respectively. }, author = {Edelsbrunner, Herbert and Heiss, Teresa}, booktitle = {arXiv}, title = {{Merge trees of periodic filtrations}}, doi = {10.48550/arXiv.2408.16575}, year = {2024}, } @unpublished{18981, 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}, booktitle = {arXiv}, title = {{Discrete microlocal Morse theory}}, doi = {10.48550/arXiv.2209.14993}, year = {2024}, } @inproceedings{18998, abstract = {Word embeddings represent language vocabularies as clouds of d-dimensional points. We investigate how information is conveyed by the general shape of these clouds, instead of representing the semantic meaning of each token. Specifically, we use the notion of persistent homology from topological data analysis (TDA) to measure the distances between language pairs from the shape of their unlabeled embeddings. These distances quantify the degree of non-isometry of the embeddings. To distinguish whether these differences are random training errors or capture real information about the languages, we use the computed distance matrices to construct language phylogenetic trees over 81 Indo-European languages. Careful evaluation shows that our reconstructed trees exhibit strong and statistically-significant similarities to the reference.}, author = {Draganov, Ondrej and Skiena, Steven}, booktitle = {Findings of the Association for Computational Linguistics: EMNLP 2024}, location = {Miami, FL, United States}, pages = {12080--12099}, publisher = {Association for Computational Linguistics}, title = {{The shape of word embeddings: Quantifying non-isometry with topological data analysis}}, doi = {10.18653/v1/2024.findings-emnlp.705}, year = {2024}, } @unpublished{18999, abstract = {Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{Chromatic topological data analysis}}, doi = {10.48550/ARXIV.2406.04102}, year = {2024}, } @article{17891, abstract = {Abstract Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback–Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics.}, author = {Edelsbrunner, Herbert and Ölsböck, Katharina and Wagner, Hubert}, issn = {1099-4300}, journal = {Entropy}, number = {8}, publisher = {MDPI}, title = {{Understanding higher-order interactions in information space}}, doi = {10.3390/e26080637}, volume = {26}, year = {2024}, }