[{"status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"publisher":"Springer Nature","day":"01","type":"journal_article","has_accepted_license":"1","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"date_published":"2026-01-01T00:00:00Z","date_updated":"2026-01-05T13:21:56Z","year":"2026","oa_version":"Published Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"24-47","_id":"20456","month":"01","title":"On the size of chromatic Delaunay mosaics","article_processing_charge":"Yes (via OA deal)","PlanS_conform":"1","related_material":{"record":[{"id":"15090","status":"public","relation":"earlier_version"}]},"citation":{"ieee":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” <i>Discrete and Computational Geometry</i>, vol. 75. Springer Nature, pp. 24–47, 2026.","short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47.","mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>, vol. 75, Springer Nature, 2026, pp. 24–47, doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>.","ista":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 75, 24–47.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>.","apa":"Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>","ama":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. 2026;75:24-47. doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>"},"ddc":["510"],"arxiv":1,"article_type":"original","OA_type":"hybrid","doi":"10.1007/s00454-025-00778-7","OA_place":"publisher","acknowledgement":"The fourth author thanks Boris Aronov for insightful discussions on the size of the overlay of Voronoi tessellations. Open access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.","file":[{"content_type":"application/pdf","checksum":"0addb5c1b78142f9fb453bfa04695400","date_updated":"2026-01-05T13:21:20Z","date_created":"2026-01-05T13:21:20Z","file_id":"20952","file_name":"2026_DiscreteCompGeom_Biswas.pdf","creator":"dernst","file_size":570922,"relation":"main_file","access_level":"open_access","success":1}],"oa":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"publication":"Discrete and Computational Geometry","file_date_updated":"2026-01-05T13:21:20Z","scopus_import":"1","author":[{"first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890"},{"orcid":"0000-0001-6249-0832","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano"},{"orcid":"0000-0003-0464-3823","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","full_name":"Draganov, Ondrej","last_name":"Draganov"},{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"intvolume":"        75","isi":1,"ec_funded":1,"volume":75,"quality_controlled":"1","license":"https://creativecommons.org/licenses/by/4.0/","abstract":[{"text":"Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.","lang":"eng"}],"publication_status":"published","date_created":"2025-10-12T22:01:26Z","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2212.03121"],"isi":["001584166900001"]},"corr_author":"1","language":[{"iso":"eng"}]},{"citation":{"chicago":"Jabal Ameli, Afrouz, Faezeh Motiei, and Morteza Saghafian. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” In <i>20th International Conference and Workshops on Algorithms and Computation</i>, 16444:386–401. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">https://doi.org/10.1007/978-981-95-7127-7_26</a>.","ista":"Jabal Ameli A, Motiei F, Saghafian M. 2026. On the MST-ratio: Theoretical bounds and complexity of finding the maximum. 20th International Conference and Workshops on Algorithms and Computation. WALCOM: International Conference and Workshops on Algorithms and Computation, LNCS, vol. 16444, 386–401.","ama":"Jabal Ameli A, Motiei F, Saghafian M. On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In: <i>20th International Conference and Workshops on Algorithms and Computation</i>. Vol 16444. Springer Nature; 2026:386-401. doi:<a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">10.1007/978-981-95-7127-7_26</a>","apa":"Jabal Ameli, A., Motiei, F., &#38; Saghafian, M. (2026). On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In <i>20th International Conference and Workshops on Algorithms and Computation</i> (Vol. 16444, pp. 386–401). Perugia, Italy: Springer Nature. <a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">https://doi.org/10.1007/978-981-95-7127-7_26</a>","ieee":"A. Jabal Ameli, F. Motiei, and M. Saghafian, “On the MST-ratio: Theoretical bounds and complexity of finding the maximum,” in <i>20th International Conference and Workshops on Algorithms and Computation</i>, Perugia, Italy, 2026, vol. 16444, pp. 386–401.","mla":"Jabal Ameli, Afrouz, et al. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” <i>20th International Conference and Workshops on Algorithms and Computation</i>, vol. 16444, Springer Nature, 2026, pp. 386–401, doi:<a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">10.1007/978-981-95-7127-7_26</a>.","short":"A. Jabal Ameli, F. Motiei, M. Saghafian, in:, 20th International Conference and Workshops on Algorithms and Computation, Springer Nature, 2026, pp. 386–401."},"conference":{"end_date":"2026-03-06","location":"Perugia, Italy","start_date":"2026-03-04","name":"WALCOM: International Conference and Workshops on Algorithms and Computation"},"article_processing_charge":"No","month":"02","title":"On the MST-ratio: Theoretical bounds and complexity of finding the maximum","_id":"21410","alternative_title":["LNCS"],"OA_type":"green","doi":"10.1007/978-981-95-7127-7_26","OA_place":"repository","arxiv":1,"day":"14","type":"conference","publisher":"Springer Nature","status":"public","page":"386-401","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2026-03-09T10:25:41Z","oa_version":"Preprint","year":"2026","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"}],"date_published":"2026-02-14T00:00:00Z","abstract":[{"text":"Given a finite set of red and blue points in R^d, the MST-ratio is defined as the total length of the Euclidean minimum spanning trees of the red points and the blue points, divided by the length of the Euclidean minimum spanning tree of their union. The MST-ratio has recently gained attention due to its direct interpretation in topological models for studying point sets with applications in spatial biology. The maximum MST-ratio of a point set is the maximum MST-ratio over all proper colorings of its points by red and blue. We prove that finding the maximum MST-ratio of a given point set is NP-hard when the dimension is part of the input. Moreover, we present a quadratic-time 3-approximation algorithm for this problem. As part of the proof, we show that in any metric space, the maximum MST-ratio is smaller than 3. Furthermore, we study the average MST-ratio over all colorings of a set of n points. We show that this average is always at least n-2/n-1, and for n random points uniformly distributed in a d-dimensional unit cube, the average tends to (math formular) in expectation as n approaches infinity.","lang":"eng"}],"quality_controlled":"1","volume":16444,"language":[{"iso":"eng"}],"date_created":"2026-03-08T23:01:45Z","external_id":{"arxiv":["2409.11079"]},"department":[{"_id":"HeEd"}],"publication_status":"published","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2409.11079"}],"publication_identifier":{"eissn":["1611-3349"],"issn":["0302-9743"],"isbn":["9789819571260"]},"oa":1,"acknowledgement":"A. J. Ameli—Supported by the project COALESCE (ERC grant no. 853234).\r\nM. Saghafian—Partially supported by the European Research Council (ERC), grant no. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","intvolume":"     16444","ec_funded":1,"author":[{"last_name":"Jabal Ameli","full_name":"Jabal Ameli, Afrouz","first_name":"Afrouz"},{"first_name":"Faezeh","last_name":"Motiei","full_name":"Motiei, Faezeh"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"scopus_import":"1","publication":"20th International Conference and Workshops on Algorithms and Computation"},{"article_number":"110055","abstract":[{"lang":"eng","text":"The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n 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."}],"quality_controlled":"1","volume":461,"corr_author":"1","language":[{"iso":"eng"}],"date_created":"2024-12-08T23:01:54Z","external_id":{"isi":["001370682500001"],"arxiv":["2310.18238"]},"department":[{"_id":"HeEd"}],"publication_status":"published","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2310.18238"}],"publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"oa":1,"acknowledgement":"Work by the first and third authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation.","isi":1,"intvolume":"       461","ec_funded":1,"publication":"Advances in Mathematics","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Alexey","full_name":"Garber, Alexey","last_name":"Garber"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"scopus_import":"1","citation":{"mla":"Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>, vol. 461, 110055, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025.","ama":"Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. 2025;461. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055."},"article_processing_charge":"No","month":"02","title":"Order-2 Delaunay triangulations optimize angles","_id":"18626","OA_type":"green","article_type":"original","doi":"10.1016/j.aim.2024.110055","OA_place":"repository","arxiv":1,"day":"01","publisher":"Elsevier","type":"journal_article","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2025-04-15T07:16:53Z","year":"2025","oa_version":"Preprint","date_published":"2025-02-01T00:00:00Z","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}]},{"article_number":"122425","abstract":[{"lang":"eng","text":"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."}],"volume":719,"quality_controlled":"1","date_created":"2025-06-30T08:48:48Z","department":[{"_id":"HeEd"}],"external_id":{"isi":["001516170500002"]},"language":[{"iso":"eng"}],"corr_author":"1","publication_status":"published","acknowledgement":"The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which improved this paper.\r\nWork by the first and fourth authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35.","publication_identifier":{"issn":["0020-0255"]},"scopus_import":"1","author":[{"full_name":"Mahini, Mohammad","last_name":"Mahini","first_name":"Mohammad"},{"first_name":"Hamid","full_name":"Beigy, Hamid","last_name":"Beigy"},{"full_name":"Qadami, Salman","last_name":"Qadami","first_name":"Salman"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"publication":"Information Sciences","intvolume":"       719","isi":1,"ec_funded":1,"article_processing_charge":"No","citation":{"ieee":"M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality,” <i>Information Sciences</i>, vol. 719, no. 11. Elsevier, 2025.","mla":"Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>, vol. 719, no. 11, 122425, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).","ista":"Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. 719(11), 122425.","chicago":"Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>.","apa":"Mahini, M., Beigy, H., Qadami, S., &#38; Saghafian, M. (2025). Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>","ama":"Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. 2025;719(11). doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>"},"_id":"19937","month":"11","title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","article_type":"original","OA_type":"closed access","doi":"10.1016/j.ins.2025.122425","issue":"11","type":"journal_article","day":"01","publisher":"Elsevier","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2025-11-01T00:00:00Z","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"date_updated":"2025-12-30T09:05:32Z","oa_version":"None","year":"2025"},{"publication_status":"published","external_id":{"pmid":["40870326"],"isi":["001557476000001"]},"department":[{"_id":"HeEd"}],"date_created":"2025-09-07T22:01:33Z","language":[{"iso":"eng"}],"corr_author":"1","volume":27,"quality_controlled":"1","abstract":[{"text":"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.","lang":"eng"}],"article_number":"854","scopus_import":"1","author":[{"orcid":"0000-0002-2548-617X","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"id":"2E36B656-F248-11E8-B48F-1D18A9856A87","first_name":"Ziga","last_name":"Virk","full_name":"Virk, Ziga"},{"first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner","full_name":"Wagner, Hubert"}],"file_date_updated":"2025-09-08T07:55:48Z","publication":"Entropy","ec_funded":1,"isi":1,"intvolume":"        27","acknowledgement":"This research received partial funding from the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis of Neural Networks’. The APC was waived.","DOAJ_listed":"1","file":[{"access_level":"open_access","relation":"main_file","success":1,"content_type":"application/pdf","checksum":"65c5399c4015d9c8abb8c7a96f3d7836","date_updated":"2025-09-08T07:55:48Z","file_name":"2025_Entropy_Akopyan.pdf","creator":"dernst","date_created":"2025-09-08T07:55:48Z","file_id":"20309","file_size":379340}],"pmid":1,"oa":1,"publication_identifier":{"eissn":["1099-4300"]},"ddc":["500"],"OA_place":"publisher","doi":"10.3390/e27080854","article_type":"original","OA_type":"gold","_id":"20293","title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence","month":"08","article_processing_charge":"Yes","citation":{"ama":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. 2025;27(8). doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>","apa":"Akopyan, A., Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2025). Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. MDPI. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>","chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>. MDPI, 2025. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>.","ista":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.","mla":"Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>, vol. 27, no. 8, 854, MDPI, 2025, doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>.","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).","ieee":"A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between the Jensen–Shannon divergence and the minmax divergence,” <i>Entropy</i>, vol. 27, no. 8. MDPI, 2025."},"PlanS_conform":"1","has_accepted_license":"1","date_published":"2025-08-01T00:00:00Z","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"oa_version":"Published Version","year":"2025","date_updated":"2025-09-30T14:32:31Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"type":"journal_article","publisher":"MDPI","day":"01","issue":"8"},{"publication_identifier":{"issn":["0022-4049"]},"oa":1,"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35","file":[{"checksum":"39bcad462278c9322ef810af7db67f56","content_type":"application/pdf","date_updated":"2025-12-30T07:55:08Z","file_id":"20886","date_created":"2025-12-30T07:55:08Z","creator":"dernst","file_name":"2025_JourPureAppliedAlgebra_Brown.pdf","file_size":3090836,"relation":"main_file","access_level":"open_access","success":1}],"intvolume":"       229","ec_funded":1,"scopus_import":"1","author":[{"last_name":"Brown","full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam"},{"orcid":"0000-0003-0464-3823","first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","last_name":"Draganov"}],"file_date_updated":"2025-12-30T07:55:08Z","publication":"Journal of Pure and Applied Algebra","quality_controlled":"1","volume":229,"article_number":"108068","abstract":[{"text":"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.","lang":"eng"}],"publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2025-09-10T05:40:09Z","external_id":{"arxiv":["2209.14993"]},"department":[{"_id":"HeEd"}],"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","issue":"10","publisher":"Elsevier","day":"01","type":"journal_article","date_updated":"2025-12-30T07:55:21Z","year":"2025","oa_version":"Published Version","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"has_accepted_license":"1","date_published":"2025-10-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"10","title":"Discrete microlocal Morse theory","_id":"20323","related_material":{"record":[{"id":"18981","relation":"earlier_version","status":"public"}]},"PlanS_conform":"1","citation":{"ista":"Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure and Applied Algebra. 229(10), 108068.","chicago":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>.","apa":"Brown, A., &#38; Draganov, O. (2025). Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>","ama":"Brown A, Draganov O. Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. 2025;229(10). doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.","mla":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10, 108068, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>.","short":"A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025)."},"article_processing_charge":"Yes (via OA deal)","ddc":["510"],"arxiv":1,"OA_type":"hybrid","article_type":"original","doi":"10.1016/j.jpaa.2025.108068","OA_place":"publisher"},{"quality_controlled":"1","volume":132,"abstract":[{"lang":"eng","text":"We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n\r\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.\r\n"}],"article_number":"104248","publication_status":"epub_ahead","corr_author":"1","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2212.11380"],"isi":["001599061500002"]},"date_created":"2025-10-19T22:01:31Z","oa":1,"publication_identifier":{"issn":["0195-6698"]},"acknowledgement":"Work by all authors but the second is supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation . The second author thanks Jesús A. De Loera for useful discussions on flips and non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic graphs.","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2212.11380","open_access":"1"}],"ec_funded":1,"isi":1,"intvolume":"       132","publication":"European Journal of Combinatorics","scopus_import":"1","author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"last_name":"Garber","full_name":"Garber, Alexey","first_name":"Alexey"},{"last_name":"Ghafari","full_name":"Ghafari, Mohadese","first_name":"Mohadese"},{"full_name":"Heiss, Teresa","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa","orcid":"0000-0002-1780-2689"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"title":"Flips in two-dimensional hypertriangulations","month":"10","_id":"20490","citation":{"chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. 2025;132. doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2025). Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” <i>European Journal of Combinatorics</i>, vol. 132. Elsevier, 2025.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2025).","mla":"Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>, vol. 132, 104248, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>."},"article_processing_charge":"No","arxiv":1,"doi":"10.1016/j.ejc.2025.104248","OA_place":"repository","OA_type":"green","article_type":"original","status":"public","publisher":"Elsevier","day":"10","type":"journal_article","year":"2025","oa_version":"Preprint","date_updated":"2025-12-01T12:57:29Z","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"date_published":"2025-10-10T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"arxiv":1,"article_type":"original","OA_type":"green","OA_place":"repository","doi":"10.3934/fods.2025003","_id":"20585","month":"03","title":"Chromatic alpha complexes","article_processing_charge":"No","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"15091"}]},"citation":{"chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences, 2025. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2025;8:30-62. doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2025). Chromatic alpha complexes. <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>Foundations of Data Science</i>, vol. 8. American Institute of Mathematical Sciences, pp. 30–62, 2025.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2025) 30–62.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>, vol. 8, American Institute of Mathematical Sciences, 2025, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>."},"date_published":"2025-03-01T00:00:00Z","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"date_updated":"2025-11-04T12:25:47Z","year":"2025","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"30-62","status":"public","type":"journal_article","day":"01","publisher":"American Institute of Mathematical Sciences","publication_status":"epub_ahead","date_created":"2025-11-02T23:01:33Z","external_id":{"arxiv":["2212.03128"]},"department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"corr_author":"1","volume":8,"quality_controlled":"1","abstract":[{"text":"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.","lang":"eng"}],"author":[{"last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0003-0464-3823","last_name":"Draganov","full_name":"Draganov, Ondrej","first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"scopus_import":"1","publication":"Foundations of Data Science","intvolume":"         8","ec_funded":1,"acknowledgement":"This project has received funding from the European Research\r\nCouncil (ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.","publication_identifier":{"eissn":["2639-8001"]}},{"article_processing_charge":"Yes (via OA deal)","PlanS_conform":"1","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"17146"}]},"citation":{"ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>.","short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry (2025).","ista":"Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry.","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>.","apa":"Edelsbrunner, H., &#38; Pach, J. (2025). Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. 2025. doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>"},"_id":"20657","month":"11","title":"Maximum Betti numbers of Čech complexes","article_type":"original","OA_type":"hybrid","OA_place":"publisher","doi":"10.1007/s00454-025-00796-5","ddc":["510"],"arxiv":1,"day":"10","type":"journal_article","publisher":"Springer Nature","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2025-11-10T00:00:00Z","has_accepted_license":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"date_updated":"2025-12-01T15:19:21Z","year":"2025","oa_version":"Published Version","abstract":[{"lang":"eng","text":"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. "}],"quality_controlled":"1","date_created":"2025-11-19T09:44:58Z","department":[{"_id":"HeEd"}],"external_id":{"isi":["001610592600001"],"arxiv":["2310.14801"]},"corr_author":"1","language":[{"iso":"eng"}],"publication_status":"epub_ahead","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-025-00796-5"}],"acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János","full_name":"Pach, János","last_name":"Pach"}],"scopus_import":"1","publication":"Discrete & Computational Geometry","isi":1,"ec_funded":1},{"citation":{"chicago":"Bleile, Yossi, Lisbeth Fajstrup, Teresa Heiss, Anne Marie Svane, and Søren Strandskov Sørensen. “Identifying Cobordisms Using Kernel Persistence.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2505.17858\">https://doi.org/10.48550/arXiv.2505.17858</a>.","ista":"Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. arXiv, 2505.17858.","ama":"Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2505.17858\">10.48550/arXiv.2505.17858</a>","apa":"Bleile, Y., Fajstrup, L., Heiss, T., Svane, A. M., &#38; Sørensen, S. S. (n.d.). Identifying cobordisms using kernel persistence. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2505.17858\">https://doi.org/10.48550/arXiv.2505.17858</a>","ieee":"Y. Bleile, L. Fajstrup, T. Heiss, A. M. Svane, and S. S. Sørensen, “Identifying cobordisms using kernel persistence,” <i>arXiv</i>. .","mla":"Bleile, Yossi, et al. “Identifying Cobordisms Using Kernel Persistence.” <i>ArXiv</i>, 2505.17858, doi:<a href=\"https://doi.org/10.48550/arXiv.2505.17858\">10.48550/arXiv.2505.17858</a>.","short":"Y. Bleile, L. Fajstrup, T. Heiss, A.M. Svane, S.S. Sørensen, ArXiv (n.d.)."},"article_number":"2505.17858","article_processing_charge":"No","abstract":[{"lang":"eng","text":"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."}],"month":"05","title":"Identifying cobordisms using kernel persistence","_id":"21016","language":[{"iso":"eng"}],"doi":"10.48550/arXiv.2505.17858","OA_place":"repository","date_created":"2026-01-20T10:12:21Z","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2505.17858"]},"publication_status":"submitted","arxiv":1,"type":"preprint","day":"23","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2505.17858","open_access":"1"}],"oa":1,"acknowledgement":"Y. B. B. and L. F. were funded by the Independent Research Fund Denmark, grant\r\nnumber 1026-00037. T. H. was partially supported by the European Research Council\r\n(ERC) Horizon 2020, grant number 788183.","status":"public","ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"orcid":"0000-0002-4861-9174","last_name":"Bleile","full_name":"Bleile, Yossi","id":"920a7385-7995-11ef-9bfd-8c434cd8f3c2","first_name":"Yossi"},{"last_name":"Fajstrup","full_name":"Fajstrup, Lisbeth","first_name":"Lisbeth"},{"last_name":"Heiss","full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa","orcid":"0000-0002-1780-2689"},{"first_name":"Anne Marie","last_name":"Svane","full_name":"Svane, Anne Marie"},{"last_name":"Sørensen","full_name":"Sørensen, Søren Strandskov","first_name":"Søren Strandskov"}],"publication":"arXiv","date_updated":"2026-01-21T10:34:57Z","year":"2025","oa_version":"Preprint","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"}],"date_published":"2025-05-23T00:00:00Z"},{"pmid":1,"acknowledgement":"The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on computational experiments that motivated the work reported in this paper. The authors also thank Daniel Bonnema for proofreading and noticing an issue with the original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","file":[{"date_updated":"2025-04-23T07:31:32Z","content_type":"application/pdf","checksum":"ffb0c818222138f9f113f4bbea41e834","file_size":283443,"file_id":"19610","date_created":"2025-04-23T07:31:32Z","creator":"dernst","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","relation":"main_file","access_level":"open_access","success":1}],"oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"file_date_updated":"2025-04-23T07:31:32Z","scopus_import":"1","publication":"Discrete & Computational Geometry","author":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0659-3201"}],"intvolume":"        73","isi":1,"ec_funded":1,"abstract":[{"text":"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.","lang":"eng"}],"volume":73,"quality_controlled":"1","date_created":"2024-06-16T22:01:07Z","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2012.03350"],"pmid":["39974750"],"isi":["001238566200004"]},"language":[{"iso":"eng"}],"corr_author":"1","publication_status":"published","day":"01","publisher":"Springer Nature","type":"journal_article","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"490-499","date_published":"2025-03-01T00:00:00Z","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"has_accepted_license":"1","date_updated":"2026-02-16T12:18:50Z","year":"2025","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","citation":{"ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>.","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>","ama":"Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. 2025;73:490-499. doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>","ieee":"H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” <i>Discrete &#38; Computational Geometry</i>, vol. 73. Springer Nature, pp. 490–499, 2025.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>.","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499."},"_id":"17149","month":"03","title":"Average and expected distortion of Voronoi paths and scapes","article_type":"original","OA_type":"hybrid","doi":"10.1007/s00454-024-00660-y","OA_place":"publisher","ddc":["510"],"arxiv":1},{"ddc":["510"],"doi":"10.1007/s44007-025-00170-0","OA_place":"publisher","article_type":"original","OA_type":"hybrid","_id":"20260","title":"Burning or collapsing the medial axis is unstable","month":"12","article_processing_charge":"Yes (via OA deal)","citation":{"mla":"Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>, vol. 4, Springer Nature, 2025, pp. 811–28, doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>.","short":"E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica 4 (2025) 811–828.","ieee":"E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning or collapsing the medial axis is unstable,” <i>La Matematica</i>, vol. 4. Springer Nature, pp. 811–828, 2025.","apa":"Chambers, E. W., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2025). Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>","ama":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. 2025;4:811-828. doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>","ista":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica. 4, 811–828.","chicago":"Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>."},"PlanS_conform":"1","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"21021"}]},"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"},{"name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073"}],"has_accepted_license":"1","date_published":"2025-12-01T00:00:00Z","oa_version":"Published Version","year":"2025","date_updated":"2026-04-07T11:42:48Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"811-828","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"publisher":"Springer Nature","day":"01","type":"journal_article","publication_status":"published","department":[{"_id":"HeEd"}],"date_created":"2025-08-31T22:01:33Z","corr_author":"1","language":[{"iso":"eng"}],"volume":4,"quality_controlled":"1","abstract":[{"text":"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.","lang":"eng"}],"publication":"La Matematica","author":[{"full_name":"Chambers, Erin Wolf","last_name":"Chambers","first_name":"Erin Wolf"},{"last_name":"Fillmore","full_name":"Fillmore, Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D"},{"orcid":"0000-0002-6862-208X","full_name":"Stephenson, Elizabeth R","last_name":"Stephenson","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","first_name":"Elizabeth R"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"scopus_import":"1","file_date_updated":"2025-12-30T07:52:58Z","ec_funded":1,"intvolume":"         4","acknowledgement":"We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie Yan, and Tao Ju for sharing code to generate the examples. We further thank Abigail Thompson for discussion on the conjecture and James Damon for sharing his insight in singularity theory. We thank the reviewers for their detailed reviews, which helped to improve the exposition.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). Partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ and the European Research Council (ERC), grant no. 788183, ‘Alpha Shape Theory Extended’. The first author was supported in part by the National Science Foundation through grants DBI-1759807, CCF-1907612, and CCF-2444309. The fourth author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) M-3073, ANR grant StratMesh, ANR-24-CE48-1899, and the welcome package from IDEX of the Université Côte d’Azur, ANR-15-IDEX-01.","file":[{"file_name":"2025_LaMatematica_Chambers.pdf","creator":"dernst","file_id":"20885","date_created":"2025-12-30T07:52:58Z","file_size":2678640,"content_type":"application/pdf","checksum":"e2043259194bfcdf3d74c4da8a5a853f","date_updated":"2025-12-30T07:52:58Z","success":1,"access_level":"open_access","relation":"main_file"}],"oa":1,"publication_identifier":{"eissn":["2730-9657"]}},{"acknowledged_ssus":[{"_id":"ScienComp"}],"abstract":[{"text":"This thesis consists of three chapters, each corresponding to one publication. While each of these projects tackles a topic in a different area of research, they all share a common thread in the type of topological structure they handle - a partition of space into volumes separated by interfaces that meet in non-manifold junctions.\r\n\r\nIn Chapter 2, we study clusters of soap bubbles from a simulation perspective. In particular, we develop a surface-only algorithm that couples large scale motion and shape deformation of soap bubble clusters with the small scale evolution of the thin film's thickness, which is responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. We model film thickness as a reduced degree of freedom in the Navier-Stokes equations and from them derive three sets of equations governing normal and tangential motion of the soap film surface, as well as the evolution of the thin film thickness. We discretize these equations on a non-manifold triangle mesh, extending and adapting operators to handle complex topology. We also present an incompressible fluid solver for 2.5D films and an advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film.\r\n\r\nIn Chapter 3, we introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts mesh defects, such as overlaps, self-intersections, and inversions into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations, such as those presented in Chapter 2, but with an order of magnitude more interacting bubbles than what we could achieve before, and Boolean unions of non-manifold meshes consisting of millions of triangles.\r\n\r\nLastly, in Chapter 4, we utilize developments in the theory of random geometric complexes facilitated by observations from Discrete Morse theory. We survey the methods and results obtained with this new approach, and discuss some of its shortcomings. We use simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.","lang":"eng"}],"supervisor":[{"orcid":"0000-0001-6646-5546","last_name":"Wojtan","full_name":"Wojtan, Christopher J","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","first_name":"Christopher J"}],"publication_status":"published","degree_awarded":"PhD","corr_author":"1","language":[{"iso":"eng"}],"department":[{"_id":"ChWo"},{"_id":"GradSch"}],"date_created":"2025-04-29T09:39:34Z","oa":1,"publication_identifier":{"issn":["2663-337X"]},"acknowledgement":"The project in Chapter 2 has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No. 638176. The project in Chapter 3 was funded in part by the European Union (ERC-2021-COG 101045083 CoDiNA). The project in Chapter 4 has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It was also partially supported by the DFG Collaborative Research Center TRR 109, 'Discretization in Geometry and Dynamics', through grant no. I02979-N35 of the Austrian Science Fund (FWF). Thank you for providing funds to support my work.","file":[{"access_level":"closed","relation":"source_file","creator":"cchlebak","file_name":"Thesis_source_Heiss_Synak.zip","date_created":"2025-04-30T14:02:25Z","file_id":"19633","file_size":60670543,"checksum":"f00b519c27529daa0c3b2d4102b4fa7b","content_type":"application/x-zip-compressed","date_updated":"2025-04-30T14:02:25Z"},{"creator":"cchlebak","file_name":"Thesis_PDFA_Heiss_Synak.pdf","date_created":"2025-04-30T14:02:42Z","file_id":"19634","file_size":21319043,"content_type":"application/pdf","checksum":"6e40a2fd3b1b881af1385670854a682e","date_updated":"2025-04-30T15:49:16Z","access_level":"open_access","relation":"main_file"}],"ec_funded":1,"author":[{"last_name":"Synak","full_name":"Synak, Peter","id":"331776E2-F248-11E8-B48F-1D18A9856A87","first_name":"Peter"}],"file_date_updated":"2025-04-30T15:49:16Z","title":"Methods for fluid simulation, surface tracking, and statistics of non-manifold structures","month":"04","_id":"19630","citation":{"apa":"Synak, P. (2025). <i>Methods for fluid simulation, surface tracking, and statistics of non-manifold structures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-19630\">https://doi.org/10.15479/AT-ISTA-19630</a>","ama":"Synak P. Methods for fluid simulation, surface tracking, and statistics of non-manifold structures. 2025. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-19630\">10.15479/AT-ISTA-19630</a>","ista":"Synak P. 2025. Methods for fluid simulation, surface tracking, and statistics of non-manifold structures. Institute of Science and Technology Austria.","chicago":"Synak, Peter. “Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/AT-ISTA-19630\">https://doi.org/10.15479/AT-ISTA-19630</a>.","short":"P. Synak, Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures, Institute of Science and Technology Austria, 2025.","mla":"Synak, Peter. <i>Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-19630\">10.15479/AT-ISTA-19630</a>.","ieee":"P. Synak, “Methods for fluid simulation, surface tracking, and statistics of non-manifold structures,” Institute of Science and Technology Austria, 2025."},"related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"8135"},{"relation":"part_of_dissertation","status":"public","id":"17219"},{"id":"8384","status":"public","relation":"part_of_dissertation"}]},"article_processing_charge":"No","ddc":["519","006"],"OA_place":"publisher","doi":"10.15479/AT-ISTA-19630","alternative_title":["ISTA Thesis"],"status":"public","type":"dissertation","day":"29","publisher":"Institute of Science and Technology Austria","year":"2025","oa_version":"Published Version","date_updated":"2026-04-16T08:29:34Z","project":[{"_id":"2533E772-B435-11E9-9278-68D0E5697425","grant_number":"638176","name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales","call_identifier":"H2020"},{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083"},{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"_id":"2533E772-B435-11E9-9278-68D0E5697425","grant_number":"638176","name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales","call_identifier":"H2020"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"date_published":"2025-04-29T00:00:00Z","has_accepted_license":"1","page":"106","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd"},{"title":"On angles in higher order Brillouin tessellations and related tilings in the plane","month":"07","_id":"14345","citation":{"mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, vol. 72, Springer Nature, 2024, pp. 29–48, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete and Computational Geometry</i>, vol. 72. Springer Nature, pp. 29–48, 2024.","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2024;72:29-48. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>."},"article_processing_charge":"Yes (via OA deal)","arxiv":1,"ddc":["510"],"doi":"10.1007/s00454-023-00566-1","article_type":"original","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","publisher":"Springer Nature","day":"01","type":"journal_article","oa_version":"Published Version","year":"2024","date_updated":"2025-04-23T08:41:59Z","date_published":"2024-07-01T00:00:00Z","has_accepted_license":"1","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"page":"29-48","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","volume":72,"abstract":[{"lang":"eng","text":"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))."}],"publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","external_id":{"arxiv":["2204.01076"],"pmid":["39610762"],"isi":["001060727600004"]},"department":[{"_id":"HeEd"}],"date_created":"2023-09-17T22:01:10Z","oa":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"file":[{"access_level":"open_access","relation":"main_file","success":1,"date_updated":"2024-07-22T09:43:19Z","checksum":"b207b4e00f904e8ea8a30e24f0251f79","content_type":"application/pdf","file_size":892019,"file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","creator":"dernst","file_id":"17301","date_created":"2024-07-22T09:43:19Z"}],"acknowledgement":"Work by all authors but A. Garber is supported by the European Research Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation.","pmid":1,"ec_funded":1,"isi":1,"intvolume":"        72","publication":"Discrete and Computational Geometry","file_date_updated":"2024-07-22T09:43:19Z","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Alexey","full_name":"Garber, Alexey","last_name":"Garber"},{"first_name":"Mohadese","last_name":"Ghafari","full_name":"Ghafari, Mohadese"},{"orcid":"0000-0002-1780-2689","full_name":"Heiss, Teresa","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"scopus_import":"1"},{"year":"2024","oa_version":"Published Version","date_updated":"2025-04-15T07:16:58Z","has_accepted_license":"1","date_published":"2024-06-06T00:00:00Z","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"},{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships"},{"grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","day":"06","type":"conference","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","ddc":["000"],"doi":"10.4230/LIPIcs.SoCG.2024.87","alternative_title":["LIPIcs"],"title":"The ultimate frontier: An optimality construction for homotopy inference (media exposition)","month":"06","_id":"18097","citation":{"apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). The ultimate frontier: An optimality construction for homotopy inference (media exposition). In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>","ama":"Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality construction for homotopy inference (media exposition). In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. The ultimate frontier: An optimality construction for homotopy inference (media exposition). 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 87.","chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, Andre Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>.","mla":"Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","ieee":"D. Attali <i>et al.</i>, “The ultimate frontier: An optimality construction for homotopy inference (media exposition),” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293."},"article_processing_charge":"Yes","conference":{"start_date":"2024-06-11","name":"SoCG: Symposium on Computational Geometry","end_date":"2024-06-14","location":"Athens, Greece"},"ec_funded":1,"intvolume":"       293","author":[{"last_name":"Attali","full_name":"Attali, Dominique","first_name":"Dominique"},{"orcid":"0000-0001-7841-0091","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","last_name":"Kourimska","full_name":"Kourimska, Hana"},{"full_name":"Fillmore, Christopher D","last_name":"Fillmore","first_name":"Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"full_name":"Ghosh, Ishika","last_name":"Ghosh","first_name":"Ishika","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff"},{"first_name":"Andre","last_name":"Lieutier","full_name":"Lieutier, Andre"},{"first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","last_name":"Stephenson","full_name":"Stephenson, Elizabeth R","orcid":"0000-0002-6862-208X"},{"orcid":"0000-0002-7472-2220","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken"}],"file_date_updated":"2024-09-19T10:30:37Z","publication":"40th International Symposium on Computational Geometry","publication_identifier":{"isbn":["9783959773164"]},"oa":1,"file":[{"checksum":"9355c2e60b8ec285e1b22719c5b73f1a","content_type":"application/pdf","date_updated":"2024-09-19T10:30:37Z","date_created":"2024-09-19T10:30:37Z","file_id":"18098","creator":"dernst","file_name":"2024_LIPICs_Attali.pdf","file_size":3507177,"relation":"main_file","access_level":"open_access","success":1}],"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I02979-N35. Mathijs Wintraecken: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d’Azur.\r\nWe thank Jean-Daniel Boissonnat, Herbert Edelsbrunner, and Mariette Yvinec for discussion.","publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","department":[{"_id":"HeEd"}],"date_created":"2024-09-19T10:29:48Z","quality_controlled":"1","volume":293,"abstract":[{"lang":"eng","text":"In our companion paper \"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds\" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting."}],"article_number":"87"},{"publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773430"]},"oa":1,"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, \"Discretization in Geometry and Dynamics\", Austrian Science Fund (FWF), grant no. I 02979-N35.","file":[{"date_updated":"2024-11-18T07:49:25Z","content_type":"application/pdf","checksum":"5f9b35e115c3d375e99be78da9054cb4","file_size":908541,"date_created":"2024-11-18T07:49:25Z","file_id":"18560","file_name":"2024_LIPIcs_CultreradiMontesano.pdf","creator":"dernst","relation":"main_file","access_level":"open_access","success":1}],"isi":1,"intvolume":"       320","ec_funded":1,"scopus_import":"1","author":[{"first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej"},{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"publication":"32nd International Symposium on Graph Drawing and Network Visualization","file_date_updated":"2024-11-18T07:49:25Z","quality_controlled":"1","volume":320,"article_number":"3","abstract":[{"lang":"eng","text":"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."}],"publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2024-11-17T23:01:47Z","department":[{"_id":"HeEd"}],"external_id":{"isi":["001540278400001"],"arxiv":["2403.10204"]},"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","type":"conference","day":"28","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_updated":"2025-12-02T13:50:50Z","year":"2024","oa_version":"Published Version","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"}],"has_accepted_license":"1","date_published":"2024-10-28T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"10","title":"The Euclidean MST-ratio for bi-colored lattices","_id":"18556","citation":{"apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In <i>32nd International Symposium on Graph Drawing and Network Visualization</i> (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">https://doi.org/10.4230/LIPIcs.GD.2024.3</a>","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: <i>32nd International Symposium on Graph Drawing and Network Visualization</i>. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">10.4230/LIPIcs.GD.2024.3</a>","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2024. The Euclidean MST-ratio for bi-colored lattices. 32nd International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LIPIcs, vol. 320, 3.","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">https://doi.org/10.4230/LIPIcs.GD.2024.3</a>.","mla":"Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">10.4230/LIPIcs.GD.2024.3</a>.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, in:, 32nd International Symposium on Graph Drawing and Network Visualization, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vienna, Austria, 2024, vol. 320."},"conference":{"end_date":"2024-09-20","location":"Vienna, Austria","start_date":"2024-09-18","name":"GD: Graph Drawing and Network Visualization"},"article_processing_charge":"Yes","ddc":["510"],"arxiv":1,"OA_type":"gold","alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.GD.2024.3","OA_place":"publisher"},{"volume":8,"quality_controlled":"1","abstract":[{"lang":"eng","text":"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."}],"publication_status":"published","external_id":{"pmid":["39308789"]},"department":[{"_id":"HeEd"}],"date_created":"2024-05-12T22:01:03Z","language":[{"iso":"eng"}],"corr_author":"1","acknowledgement":"The authors thank Uli Wagner and Emo Welzl for comments on an earlier version of this paper, and for pointing out related work in the prior literature.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","file":[{"date_updated":"2025-04-23T08:01:36Z","content_type":"application/pdf","checksum":"0ee15c1493a6413cf356ab2f32c81a9e","file_size":522831,"file_id":"19612","date_created":"2025-04-23T08:01:36Z","file_name":"2024_JourApplCompTopo_BiswasRa.pdf","creator":"dernst","relation":"main_file","access_level":"open_access","success":1}],"pmid":1,"publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"oa":1,"publication":"Journal of Applied and Computational Topology","scopus_import":"1","author":[{"orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita","last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita"},{"full_name":"Cultrera Di Montesano, Sebastiano","last_name":"Cultrera Di Montesano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"file_date_updated":"2025-04-23T08:01:36Z","ec_funded":1,"intvolume":"         8","_id":"15380","title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","month":"09","article_processing_charge":"Yes (via OA deal)","citation":{"ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 8, 557–578.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. 2024;8:557-578. doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 557–578, 2024.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578.","mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 557–78, doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>."},"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"11658"}]},"ddc":["510"],"doi":"10.1007/s41468-024-00173-w","OA_place":"publisher","article_type":"original","OA_type":"hybrid","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"day":"01","type":"journal_article","publisher":"Springer Nature","has_accepted_license":"1","date_published":"2024-09-01T00:00:00Z","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"year":"2024","oa_version":"Published Version","date_updated":"2025-05-14T09:27:57Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"557-578"},{"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","day":"01","type":"conference","date_updated":"2025-04-15T07:16:58Z","oa_version":"Published Version","year":"2024","has_accepted_license":"1","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"},{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073","name":"Learning and triangulating manifolds via collapses"}],"date_published":"2024-06-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"06","title":"The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms","_id":"17144","citation":{"chicago":"Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>.","ista":"Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 69.","ama":"Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>","apa":"Kourimska, H., Lieutier, A., &#38; Wintraecken, M. (2024). The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>","ieee":"H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","short":"H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","mla":"Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>."},"conference":{"name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","end_date":"2024-06-14"},"article_processing_charge":"No","ddc":["510"],"arxiv":1,"alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.SoCG.2024.69","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"oa":1,"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nSupported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Cô d'Azur.\r\nWe are greatly indebted to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher Fillmore, and Elizabeth Stephenson for early discussions and all members of the Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape team (Inria) for the atmosphere in which this research was conducted.","file":[{"date_updated":"2024-06-17T08:33:40Z","content_type":"application/pdf","checksum":"b40ff456c19294adb5d9613fcfd751c6","file_size":1612558,"file_id":"17150","date_created":"2024-06-17T08:33:40Z","creator":"dernst","file_name":"2024_LIPICS_Kourimska.pdf","relation":"main_file","access_level":"open_access","success":1}],"intvolume":"       293","ec_funded":1,"author":[{"id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","last_name":"Kourimska","full_name":"Kourimska, Hana","orcid":"0000-0001-7841-0091"},{"full_name":"Lieutier, André","last_name":"Lieutier","first_name":"André"},{"last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","orcid":"0000-0002-7472-2220"}],"publication":"40th International Symposium on Computational Geometry","scopus_import":"1","file_date_updated":"2024-06-17T08:33:40Z","quality_controlled":"1","volume":293,"article_number":"69","abstract":[{"text":"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.","lang":"eng"}],"publication_status":"published","language":[{"iso":"eng"}],"date_created":"2024-06-16T22:01:06Z","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2212.01118"]}},{"publication":"40th International Symposium on Computational Geometry","file_date_updated":"2024-06-17T08:46:33Z","author":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","full_name":"Pach, János","last_name":"Pach"}],"scopus_import":"1","intvolume":"       293","ec_funded":1,"acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart for early discussions on the linked circles construction in three dimensions, and Gábor Tardos for helpful remarks and suggestions.","file":[{"date_updated":"2024-06-17T08:46:33Z","content_type":"application/pdf","checksum":"5442d44fb89d77477a87668d6e61aac9","file_size":766562,"date_created":"2024-06-17T08:46:33Z","file_id":"17152","creator":"dernst","file_name":"2024_LIPICS_Edelsbrunner.pdf","relation":"main_file","access_level":"open_access","success":1}],"oa":1,"publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"publication_status":"published","date_created":"2024-06-16T22:01:06Z","external_id":{"arxiv":["2310.14801"]},"department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"volume":293,"quality_controlled":"1","article_number":"53","abstract":[{"lang":"eng","text":"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."}],"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"}],"has_accepted_license":"1","date_published":"2024-06-01T00:00:00Z","date_updated":"2025-12-01T15:19:20Z","oa_version":"Published Version","year":"2024","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"day":"01","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","type":"conference","ddc":["510"],"arxiv":1,"alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.SoCG.2024.53","_id":"17146","month":"06","title":"Maximum Betti numbers of Čech complexes","conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11"},"article_processing_charge":"No","related_material":{"record":[{"id":"20657","status":"public","relation":"later_version"}]},"citation":{"short":"H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>.","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>","apa":"Edelsbrunner, H., &#38; Pach, J. (2024). Maximum Betti numbers of Čech complexes. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>.","ista":"Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 53."}},{"type":"conference","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","day":"06","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"11:1-11:19","has_accepted_license":"1","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"name":"Learning and triangulating manifolds via collapses","grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"date_published":"2024-06-06T00:00:00Z","date_updated":"2025-04-15T07:16:57Z","year":"2024","oa_version":"Published Version","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11","location":"Athens, Greece","end_date":"2024-06-14"},"article_processing_charge":"No","citation":{"apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>","ama":"Attali D, Kourimska H, Fillmore CD, et al. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 11:1-11:19.","chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” In <i>40th International Symposium on Computational Geometry</i>, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19.","mla":"Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>.","ieee":"D. Attali <i>et al.</i>, “Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293, p. 11:1-11:19."},"_id":"17170","month":"06","title":"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds","alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.SoCG.2024.11","ddc":["516"],"arxiv":1,"file":[{"date_updated":"2024-06-25T11:47:26Z","checksum":"6a2ddc8b51aa58f197a8b294750f1f8d","content_type":"application/pdf","file_size":20886142,"file_id":"17171","date_created":"2024-06-25T11:47:26Z","creator":"cfillmor","file_name":"LIPIcs.SoCG.2024.11.pdf","relation":"main_file","access_level":"open_access","success":1}],"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nWintraecken, Mathijs: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d'Azur.","oa":1,"publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773164"]},"file_date_updated":"2024-06-25T11:47:26Z","scopus_import":"1","author":[{"full_name":"Attali, Dominique","last_name":"Attali","first_name":"Dominique"},{"full_name":"Kourimska, Hana","last_name":"Kourimska","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","orcid":"0000-0001-7841-0091"},{"last_name":"Fillmore","full_name":"Fillmore, Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D"},{"first_name":"Ishika","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","full_name":"Ghosh, Ishika","last_name":"Ghosh"},{"full_name":"Lieutier, André","last_name":"Lieutier","first_name":"André"},{"orcid":"0000-0002-6862-208X","last_name":"Stephenson","full_name":"Stephenson, Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","first_name":"Elizabeth R"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220"}],"publication":"40th International Symposium on Computational Geometry","intvolume":"       293","ec_funded":1,"abstract":[{"lang":"eng","text":"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."}],"volume":293,"quality_controlled":"1","date_created":"2024-06-25T11:45:58Z","external_id":{"arxiv":["2206.10485"]},"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publication_status":"published"}]