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It is known that the edge set of every complete geometric graph on n vertices can be partitioned into O(n3∕2) crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set A of n randomly selected points, uniformly distributed in [0,1]2, with probability tending to 1 as n→∞, the edge set of Kn[A] can be covered by O(nlogn) crossing-free paths and by O(n√logn) crossing-free matchings. On the other hand, we construct n-element point sets such that covering the edge set of Kn[A] requires a quadratic number of monotone paths."}],"citation":{"ieee":"A. Dumitrescu, J. Pach, M. Saghafian, and A. Scott, “Covering complete geometric graphs by monotone paths,” <i>Combinatorics and Number Theory</i>, vol. 15, no. 1. Mathematical Sciences Publishers, pp. 73–82, 2026.","ama":"Dumitrescu A, Pach J, Saghafian M, Scott A. Covering complete geometric graphs by monotone paths. <i>Combinatorics and Number Theory</i>. 2026;15(1):73-82. doi:<a href=\"https://doi.org/10.2140/cnt.2026.15.73\">10.2140/cnt.2026.15.73</a>","apa":"Dumitrescu, A., Pach, J., Saghafian, M., &#38; Scott, A. (2026). Covering complete geometric graphs by monotone paths. <i>Combinatorics and Number Theory</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/cnt.2026.15.73\">https://doi.org/10.2140/cnt.2026.15.73</a>","chicago":"Dumitrescu, Adrian, János Pach, Morteza Saghafian, and Alex Scott. “Covering Complete Geometric Graphs by Monotone Paths.” <i>Combinatorics and Number Theory</i>. Mathematical Sciences Publishers, 2026. <a href=\"https://doi.org/10.2140/cnt.2026.15.73\">https://doi.org/10.2140/cnt.2026.15.73</a>.","mla":"Dumitrescu, Adrian, et al. “Covering Complete Geometric Graphs by Monotone Paths.” <i>Combinatorics and Number Theory</i>, vol. 15, no. 1, Mathematical Sciences Publishers, 2026, pp. 73–82, doi:<a href=\"https://doi.org/10.2140/cnt.2026.15.73\">10.2140/cnt.2026.15.73</a>.","short":"A. Dumitrescu, J. Pach, M. Saghafian, A. Scott, Combinatorics and Number Theory 15 (2026) 73–82.","ista":"Dumitrescu A, Pach J, Saghafian M, Scott A. 2026. Covering complete geometric graphs by monotone paths. Combinatorics and Number Theory. 15(1), 73–82."},"scopus_import":"1","oa_version":"Preprint","status":"public","type":"journal_article","publication_identifier":{"issn":["2996-2196"],"eissn":["2996-220X"]},"OA_type":"green","arxiv":1,"date_published":"2026-04-17T00:00:00Z","article_processing_charge":"No","acknowledgement":"Research partially supported by ERC Advanced Grant \"GeoScape\", no. 882971 and\r\nHungarian NKFIH grant no. K-131529. Work by the third author is supported by EPSRC grant\r\nEP/X013642/1. Work by the third author is 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.","title":"Covering complete geometric graphs by monotone paths","volume":15,"month":"04","date_updated":"2026-05-07T07:45:24Z","quality_controlled":"1","external_id":{"arxiv":["2507.10840"]}},{"corr_author":"1","type":"software","status":"public","day":"15","publisher":"Institute of Science and Technology Austria","has_accepted_license":"1","tmp":{"short":"MIT","legal_code_url":"https://opensource.org/licenses/MIT","name":"The MIT License"},"department":[{"_id":"HeEd"}],"user_id":"68b8ca59-c5b3-11ee-8790-cd641c68093d","_id":"21971","keyword":["quadratics","mathematics","dendrites","geometry","topology"],"citation":{"short":"Y. Bokor Bleile, E. Cortinovis, (2026).","ista":"Bokor Bleile Y, Cortinovis E. 2026. Quadrix, Institute of Science and Technology Austria, <a href=\"https://doi.org/10.15479/AT-ISTA-21971\">10.15479/AT-ISTA-21971</a>.","mla":"Bokor Bleile, Yossi, and Emanuele Cortinovis. <i>Quadrix</i>. 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This project provides efficient tools for representing dendritic trees, computing quadric error metrics, and visualizing eigenvalue distributions on hexagonal plots.\r\n\r\nThis library implements quadric-based geometric analysis of dendritic structures, commonly found in neuroscience applications. 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Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0003-0464-3823","first_name":"Ondrej","last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","first_name":"Morteza"}],"oa":1,"publication_status":"published","file":[{"file_size":570922,"date_updated":"2026-01-05T13:21:20Z","content_type":"application/pdf","relation":"main_file","creator":"dernst","checksum":"0addb5c1b78142f9fb453bfa04695400","access_level":"open_access","file_name":"2026_DiscreteCompGeom_Biswas.pdf","date_created":"2026-01-05T13:21:20Z","file_id":"20952","success":1}],"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape 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4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"has_accepted_license":"1","external_id":{"arxiv":["2212.03121"],"isi":["001584166900001"]},"license":"https://creativecommons.org/licenses/by/4.0/","quality_controlled":"1","date_updated":"2026-01-05T13:21:56Z","month":"01","volume":75,"PlanS_conform":"1","title":"On the size of chromatic Delaunay mosaics","date_published":"2026-01-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)","arxiv":1,"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.","isi":1,"OA_type":"hybrid","related_material":{"record":[{"status":"public","id":"15090","relation":"earlier_version"}]},"type":"journal_article","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"status":"public","scopus_import":"1","citation":{"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>","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.","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.","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>.","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>"},"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"}],"oa_version":"Published Version","publisher":"Springer Nature"},{"dataavailabilitystatement":"The code used for the computational experiments is available in https://github.com/Cimagroup/IBloFunMatch","language":[{"iso":"eng"}],"ddc":["500"],"author":[{"full_name":"Gonzalez-Diaz, Rocio","first_name":"Rocio","last_name":"Gonzalez-Diaz"},{"orcid":"0000-0003-2449-1433","last_name":"Soriano Trigueros","first_name":"Manuel","full_name":"Soriano Trigueros, Manuel","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8"},{"first_name":"Alvaro","last_name":"Torras-Casas","full_name":"Torras-Casas, Alvaro"}],"oa":1,"publication_status":"epub_ahead","doi":"10.1016/j.jsc.2026.102598","intvolume":"       138","mathsc":["55N31","16G20"],"year":"2026","date_created":"2026-07-13T09:43:38Z","article_type":"original","day":"23","researchdata_availability":"yes","corr_author":"1","OA_place":"publisher","publication":"Journal of Symbolic Computation","_id":"22291","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"has_accepted_license":"1","external_id":{"arxiv":["2006.11100"]},"date_updated":"2026-07-13T12:00:07Z","quality_controlled":"1","month":"06","volume":138,"PlanS_conform":"1","title":"Additive partial matchings induced by persistence maps","date_published":"2026-06-23T00:00:00Z","article_processing_charge":"No","arxiv":1,"acknowledgement":"This project was partially funded by MCIN/AEI and the NextGenerationEU/PRTR, under project TED2021-129438B-I00. The authors thank IMUS-Maria de Maeztu grant CEX2024-001517-M - Apoyo a Unidades de Excelencia María de Maeztu for supporting this research, funded by MICIU/AEI/ 10.13039/501100011033. The authors would also like to thank Lars M Salbu for fruitful discussions regarding the operators from Definition 4.1 and their relation with the order relations introduced in Definition 3.2.","OA_type":"hybrid","supplementarymaterial":"no","type":"journal_article","publication_identifier":{"issn":["0747-7171"],"eissn":["1095-855X"]},"status":"public","scopus_import":"1","abstract":[{"lang":"eng","text":"Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and fast-to-compute invariant known as the persistence diagram. However, this is no longer the case for maps between persistence modules (i.e. persistence maps). We propose a new invariant for persistence maps, consisting of a partial matching between the persistence diagrams of the domain and codomain modules. We show that this invariant is additive with respect to the direct sum decomposition of persistence maps, is more discriminative than the image module, and is computable in cubic time. Furthermore, we provide an implementation and demonstrate its efficiency by integrating it with edge collapse techniques for flag complexes (e.g., Vietoris–Rips complexes). As a key technical contribution, we describe how to induce a persistence map between two flag complexes that have been independently simplified via edge collapses, even when a direct simplicial map between them is no longer available."}],"keyword":["Persistence module","Persistence map","Persistent homology"],"article_number":"102598","citation":{"ama":"Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. Additive partial matchings induced by persistence maps. <i>Journal of Symbolic Computation</i>. 2026;138. doi:<a href=\"https://doi.org/10.1016/j.jsc.2026.102598\">10.1016/j.jsc.2026.102598</a>","ieee":"R. Gonzalez-Diaz, M. Soriano Trigueros, and A. Torras-Casas, “Additive partial matchings induced by persistence maps,” <i>Journal of Symbolic Computation</i>, vol. 138. Elsevier, 2026.","mla":"Gonzalez-Diaz, Rocio, et al. “Additive Partial Matchings Induced by Persistence Maps.” <i>Journal of Symbolic Computation</i>, vol. 138, 102598, Elsevier, 2026, doi:<a href=\"https://doi.org/10.1016/j.jsc.2026.102598\">10.1016/j.jsc.2026.102598</a>.","short":"R. Gonzalez-Diaz, M. Soriano Trigueros, A. Torras-Casas, Journal of Symbolic Computation 138 (2026).","ista":"Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. 2026. Additive partial matchings induced by persistence maps. Journal of Symbolic Computation. 138, 102598.","apa":"Gonzalez-Diaz, R., Soriano Trigueros, M., &#38; Torras-Casas, A. (2026). Additive partial matchings induced by persistence maps. <i>Journal of Symbolic Computation</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jsc.2026.102598\">https://doi.org/10.1016/j.jsc.2026.102598</a>","chicago":"Gonzalez-Diaz, Rocio, Manuel Soriano Trigueros, and Alvaro Torras-Casas. “Additive Partial Matchings Induced by Persistence Maps.” <i>Journal of Symbolic Computation</i>. Elsevier, 2026. <a href=\"https://doi.org/10.1016/j.jsc.2026.102598\">https://doi.org/10.1016/j.jsc.2026.102598</a>."},"oa_version":"Published Version","publisher":"Elsevier","das_tickbox":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1016/j.jsc.2026.102598"}]},{"publication_identifier":{"isbn":["9783959774185"],"eissn":["1868-8969"]},"type":"conference","status":"public","supplementarymaterial":"no","OA_type":"gold","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","das_tickbox":"0","oa_version":"Published Version","article_number":"41:1-41:18","keyword":["Algebraic topology","Lefschetz complexes","persistent homology","vines and vineyards","birth-death pairs","shallow pairs","relations","partial orders","transpositions","Theory of computation → Computational geometry"],"abstract":[{"text":"The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.","lang":"eng"}],"scopus_import":"1","citation":{"ieee":"H. Edelsbrunner, M. Lipiński, M. Mrozek, M. Soriano Trigueros, and F. Zimin, “The depth poset under transpositions in the filter,” in <i>42nd International Symposium on Computational Geometry</i>, New Brunswick, NJ, United States, 2026, vol. 367.","ama":"Edelsbrunner H, Lipiński M, Mrozek M, Soriano Trigueros M, Zimin F. The depth poset under transpositions in the filter. In: <i>42nd International Symposium on Computational Geometry</i>. Vol 367. Schloss Dagstuhl – Leibniz-Zentrum für Informatik; 2026. doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">10.4230/LIPICS.SOCG.2026.41</a>","chicago":"Edelsbrunner, Herbert, Michał Lipiński, Marian Mrozek, Manuel Soriano Trigueros, and Fedor Zimin. “The Depth Poset under Transpositions in the Filter.” In <i>42nd International Symposium on Computational Geometry</i>, Vol. 367. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">https://doi.org/10.4230/LIPICS.SOCG.2026.41</a>.","apa":"Edelsbrunner, H., Lipiński, M., Mrozek, M., Soriano Trigueros, M., &#38; Zimin, F. (2026). The depth poset under transpositions in the filter. In <i>42nd International Symposium on Computational Geometry</i> (Vol. 367). New Brunswick, NJ, United States: Schloss Dagstuhl – Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">https://doi.org/10.4230/LIPICS.SOCG.2026.41</a>","short":"H. Edelsbrunner, M. Lipiński, M. Mrozek, M. Soriano Trigueros, F. Zimin, in:, 42nd International Symposium on Computational Geometry, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026.","ista":"Edelsbrunner H, Lipiński M, Mrozek M, Soriano Trigueros M, Zimin F. 2026. The depth poset under transpositions in the filter. 42nd International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 367, 41:1-41:18.","mla":"Edelsbrunner, Herbert, et al. “The Depth Poset under Transpositions in the Filter.” <i>42nd International Symposium on Computational Geometry</i>, vol. 367, 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026, doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">10.4230/LIPICS.SOCG.2026.41</a>."},"month":"05","external_id":{"arxiv":["2511.21961"]},"quality_controlled":"1","date_updated":"2026-07-14T06:09:32Z","title":"The depth poset under transpositions in the filter","acknowledgement":"The authors thank Jakub Leśkiewicz and Bartosz Furmanek for discussions\r\nthat helped improve the paper. Herbert Edelsbrunner: DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35\r\nMichał Lipiński: European Union’s Horizon 2020 research and innovation programme under the\r\nMarie Skłodowska-Curie Grant Agreement No. 101034413\r\nMarian Mrozek: Polish National Science Center under Opus Grant 2019/35/B/ST1/00874 and Opus\r\nGrant 2025/57/B/ST1/00550","article_processing_charge":"Yes","date_published":"2026-05-27T00:00:00Z","arxiv":1,"volume":367,"OA_place":"publisher","corr_author":"1","publication":"42nd International Symposium on Computational Geometry","researchdata_availability":"no","day":"27","has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"department":[{"_id":"HeEd"},{"_id":"GradSch"}],"ec_funded":1,"conference":{"start_date":"2026-06-02","end_date":"2026-06-05","name":"SoCG: Symposium on Computational Geometry","location":"New Brunswick, NJ, United States"},"_id":"22299","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"file":[{"success":1,"file_id":"22329","date_created":"2026-07-14T06:08:05Z","file_name":"2026_LIPIcSSoCG_Edelsbrunner.pdf","access_level":"open_access","creator":"dernst","checksum":"9dfb96ee66985c724b499b0e5888dc8e","content_type":"application/pdf","date_updated":"2026-07-14T06:08:05Z","relation":"main_file","file_size":2902144}],"doi":"10.4230/LIPICS.SOCG.2026.41","oa":1,"publication_status":"published","intvolume":"       367","file_date_updated":"2026-07-14T06:08:05Z","author":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert"},{"orcid":"0000-0001-9789-9750","first_name":"Michał","last_name":"Lipiński","full_name":"Lipiński, Michał","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e"},{"orcid":"0000-0002-0619-6417","first_name":"Marian","last_name":"Mrozek","full_name":"Mrozek, Marian"},{"full_name":"Soriano Trigueros, Manuel","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","orcid":"0000-0003-2449-1433","last_name":"Soriano Trigueros","first_name":"Manuel"},{"last_name":"Zimin","first_name":"Fedor","id":"afd27eda-91c1-11f0-aad8-c6edbec24c04","full_name":"Zimin, Fedor"}],"language":[{"iso":"eng"}],"ddc":["500"],"date_created":"2026-07-13T09:56:38Z","alternative_title":["LIPIcs"],"year":"2026"},{"file":[{"file_id":"22328","success":1,"date_created":"2026-07-14T05:55:34Z","access_level":"open_access","file_name":"2026_ISPRS_EghbaliMardekheh.pdf","creator":"dernst","checksum":"7c088dd179cfee6074a350c95671dbe3","relation":"main_file","content_type":"application/pdf","date_updated":"2026-07-14T05:55:34Z","file_size":2697680}],"doi":"10.5194/isprs-annals-x-4-w8-2025-493-2026","oa":1,"publication_status":"published","author":[{"full_name":"Eghbali Mardekheh, Masoumeh","first_name":"Masoumeh","last_name":"Eghbali Mardekheh"},{"last_name":"Argany","first_name":"Meysam","full_name":"Argany, Meysam"},{"orcid":"0000-0001-6746-4174","last_name":"Karimipour","first_name":"Farid","full_name":"Karimipour, Farid","id":"2A2BCDC4-CF62-11E9-BE5E-3B1EE6697425"},{"last_name":"Sedghitabar","first_name":"Seyed Mohammad","full_name":"Sedghitabar, Seyed Mohammad"}],"language":[{"iso":"eng"}],"ddc":["500"],"file_date_updated":"2026-07-14T05:55:34Z","date_created":"2026-07-13T09:51:29Z","year":"2026","publication":"8th ISPRS Geospatial Conference","OA_place":"publisher","researchdata_availability":"no","day":"29","has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"page":"493-500","department":[{"_id":"HeEd"}],"issue":"-4/W8-2025","conference":{"location":"Tehran, Iran","start_date":"2025-12-15","end_date":"2025-12-17","name":"ISPRS: Conference on Photogrammetry, Remote Sensing and Spatial Information Sciences,"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"22298","month":"05","date_updated":"2026-07-14T05:56:30Z","quality_controlled":"1","acknowledgement":"We sincerely thank the Iranian Airports and Air Navigation\r\nCompany for sharing statistical data on flights from Iranian\r\nairports, separated by origin and destination, for this research.","article_processing_charge":"No","date_published":"2026-05-29T00:00:00Z","title":"Real time interactive web GIS based modelling of high traffic air corridors in Iran using origin destination matrix analysis","volume":"X","status":"public","publication_identifier":{"issn":["2194-9050"]},"type":"conference","OA_type":"gold","supplementarymaterial":"no","das_tickbox":"0","publisher":"Copernicus Publications","oa_version":"Published Version","scopus_import":"1","citation":{"apa":"Eghbali Mardekheh, M., Argany, M., Karimipour, F., &#38; Sedghitabar, S. M. (2026). Real time interactive web GIS based modelling of high traffic air corridors in Iran using origin destination matrix analysis. In <i>8th ISPRS Geospatial Conference</i> (Vol. X, pp. 493–500). Tehran, Iran: Copernicus Publications. <a href=\"https://doi.org/10.5194/isprs-annals-x-4-w8-2025-493-2026\">https://doi.org/10.5194/isprs-annals-x-4-w8-2025-493-2026</a>","chicago":"Eghbali Mardekheh, Masoumeh, Meysam Argany, Farid Karimipour, and Seyed Mohammad Sedghitabar. “Real Time Interactive Web GIS Based Modelling of High Traffic Air Corridors in Iran Using Origin Destination Matrix Analysis.” In <i>8th ISPRS Geospatial Conference</i>, X:493–500. Copernicus Publications, 2026. <a href=\"https://doi.org/10.5194/isprs-annals-x-4-w8-2025-493-2026\">https://doi.org/10.5194/isprs-annals-x-4-w8-2025-493-2026</a>.","mla":"Eghbali Mardekheh, Masoumeh, et al. “Real Time Interactive Web GIS Based Modelling of High Traffic Air Corridors in Iran Using Origin Destination Matrix Analysis.” <i>8th ISPRS Geospatial Conference</i>, vol. X, no. 4/W8-2025, Copernicus Publications, 2026, pp. 493–500, doi:<a href=\"https://doi.org/10.5194/isprs-annals-x-4-w8-2025-493-2026\">10.5194/isprs-annals-x-4-w8-2025-493-2026</a>.","short":"M. Eghbali Mardekheh, M. Argany, F. Karimipour, S.M. Sedghitabar, in:, 8th ISPRS Geospatial Conference, Copernicus Publications, 2026, pp. 493–500.","ista":"Eghbali Mardekheh M, Argany M, Karimipour F, Sedghitabar SM. 2026. Real time interactive web GIS based modelling of high traffic air corridors in Iran using origin destination matrix analysis. 8th ISPRS Geospatial Conference. ISPRS: Conference on Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. X, 493–500.","ieee":"M. Eghbali Mardekheh, M. Argany, F. Karimipour, and S. M. Sedghitabar, “Real time interactive web GIS based modelling of high traffic air corridors in Iran using origin destination matrix analysis,” in <i>8th ISPRS Geospatial Conference</i>, Tehran, Iran, 2026, vol. X, no. 4/W8-2025, pp. 493–500.","ama":"Eghbali Mardekheh M, Argany M, Karimipour F, Sedghitabar SM. Real time interactive web GIS based modelling of high traffic air corridors in Iran using origin destination matrix analysis. In: <i>8th ISPRS Geospatial Conference</i>. Vol X. Copernicus Publications; 2026:493-500. doi:<a href=\"https://doi.org/10.5194/isprs-annals-x-4-w8-2025-493-2026\">10.5194/isprs-annals-x-4-w8-2025-493-2026</a>"},"keyword":["Air Traffic simulation","Dynamic Air Traffic Corridors","Origin Destination matrix","Interactive Web GIS","Iran Aviation Network","Spatiotemporal Modelling"],"abstract":[{"lang":"eng","text":"Air traffic management is a critical component of aviation, aiming to balance safety, capacity and demand within controlled airspace. This research analyses Iran's domestic air transport network (2018-2021) by developing annual Origin-Destination (OD) impedance matrices based on flight frequency. The methodology quantifies network connectivity, revealing a highly centralized structure dominated by core corridors like Tehran-Mashhad, which carried over 10,500 flights in 2019. The COVID-19 pandemic caused a severe disruption in 2020, with traffic on major routes falling by nearly half, followed by a partial recovery in 2021. Analysis shows core hubs rebounded faster than peripheral airports, widening accessibility gaps. Through origin-destination matrix processing, the system identifies high density air routes and analyses historical trends in airspace utilization. The impedance matrix, validated against data (R²=0.87), successfully maps the intense service on hub links and the high impedance of sparse peripheral routes. Then an interactive Web GIS platform was implemented for spatiotemporal analysis of air traffic density."}]},{"publication_identifier":{"issn":["2663-337X"]},"type":"dissertation","status":"public","related_material":{"record":[{"relation":"part_of_dissertation","id":"20260","status":"public"},{"id":"21051","relation":"part_of_dissertation","status":"public"},{"relation":"part_of_dissertation","id":"21050","status":"public"}]},"publisher":"Institute of Science and Technology Austria","oa_version":"Published Version","abstract":[{"lang":"eng","text":"This thesis examines how geometry and topology intersect in the representation, transformation, and analysis of complex shapes. It considers how continuous manifolds relate to their discrete analogues, how topological structures evolve in persistence vineyards, and how tools from topological data analysis can illuminate problems in mathematical physics. Central to this exploration is the question of how structure, both geometric and topological, persists or changes under approximation, sampling, or deformation. The work develops new approaches to skeletal and grid-based representations of surfaces, reveals the full expressive capacity of persistence vineyards, and applies topological methods to the longstanding problem of equilibria in electrostatic fields. These threads braid together into a broader understanding of how topology and geometry inform one another across theory, computation, and application."}],"citation":{"ieee":"C. D. Fillmore, “Braiding geometry and topology to study shapes and data,” Institute of Science and Technology Austria, 2026.","ama":"Fillmore CD. Braiding geometry and topology to study shapes and data. 2026. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-21021\">10.15479/AT-ISTA-21021</a>","chicago":"Fillmore, Christopher D. “Braiding Geometry and Topology to Study Shapes and Data.” Institute of Science and Technology Austria, 2026. <a href=\"https://doi.org/10.15479/AT-ISTA-21021\">https://doi.org/10.15479/AT-ISTA-21021</a>.","apa":"Fillmore, C. D. (2026). <i>Braiding geometry and topology to study shapes and data</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-21021\">https://doi.org/10.15479/AT-ISTA-21021</a>","ista":"Fillmore CD. 2026. Braiding geometry and topology to study shapes and data. Institute of Science and Technology Austria.","short":"C.D. Fillmore, Braiding Geometry and Topology to Study Shapes and Data, Institute of Science and Technology Austria, 2026.","mla":"Fillmore, Christopher D. <i>Braiding Geometry and Topology to Study Shapes and Data</i>. Institute of Science and Technology Austria, 2026, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-21021\">10.15479/AT-ISTA-21021</a>."},"month":"01","date_updated":"2026-07-22T06:33:54Z","title":"Braiding geometry and topology to study shapes and data","acknowledgement":"The research presented in this thesis was funded by the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’.\r\n","date_published":"2026-01-21T00:00:00Z","article_processing_charge":"No","acknowledged_ssus":[{"_id":"M-Shop"},{"_id":"ScienComp"}],"OA_place":"publisher","corr_author":"1","day":"21","page":"122","supervisor":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","first_name":"Uli","last_name":"Wagner","orcid":"0000-0002-1494-0568"}],"has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"department":[{"_id":"GradSch"},{"_id":"HeEd"},{"_id":"UlWa"}],"_id":"21021","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","file":[{"file_size":55954297,"date_updated":"2026-01-30T11:40:09Z","content_type":"application/pdf","relation":"main_file","checksum":"4c0889130095c31d4e5088c5b8dfd607","creator":"cfillmor","file_name":"2025_Fillmore_Christopher_Thesis.pdf","access_level":"open_access","date_created":"2026-01-26T19:44:46Z","file_id":"21046"},{"access_level":"closed","file_name":"Thesis.zip","creator":"cfillmor","checksum":"d69afb71d82ab98f856886126ee7303a","file_id":"21047","date_created":"2026-01-26T19:46:20Z","content_type":"application/x-zip-compressed","relation":"source_file","date_updated":"2026-01-26T19:46:20Z","file_size":166080788}],"doi":"10.15479/AT-ISTA-21021","oa":1,"publication_status":"published","degree_awarded":"PhD","file_date_updated":"2026-01-30T11:40:09Z","author":[{"last_name":"Fillmore","first_name":"Christopher D","full_name":"Fillmore, Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"}],"language":[{"iso":"eng"}],"ddc":["514","516"],"date_created":"2026-01-20T21:38:40Z","alternative_title":["ISTA Thesis"],"year":"2026"},{"title":"Counting equilibria of the electrostatic potential","date_published":"2026-05-01T00:00:00Z","article_processing_charge":"No","arxiv":1,"volume":132,"month":"05","external_id":{"arxiv":["2501.05315"]},"quality_controlled":"1","date_updated":"2026-07-22T06:33:54Z","publisher":"Wiley","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2501.05315"}],"article_number":"e70163","citation":{"ama":"Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic potential. <i>Proceedings of the London Mathematical Society</i>. 2026;132(5). doi:<a href=\"https://doi.org/10.1112/plms.70163\">10.1112/plms.70163</a>","ieee":"H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of the electrostatic potential,” <i>Proceedings of the London Mathematical Society</i>, vol. 132, no. 5. Wiley, 2026.","ista":"Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. 132(5), e70163.","short":"H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical Society 132 (2026).","mla":"Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” <i>Proceedings of the London Mathematical Society</i>, vol. 132, no. 5, e70163, Wiley, 2026, doi:<a href=\"https://doi.org/10.1112/plms.70163\">10.1112/plms.70163</a>.","chicago":"Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting Equilibria of the Electrostatic Potential.” <i>Proceedings of the London Mathematical Society</i>. Wiley, 2026. <a href=\"https://doi.org/10.1112/plms.70163\">https://doi.org/10.1112/plms.70163</a>.","apa":"Edelsbrunner, H., Fillmore, C. D., &#38; Oliveira, G. (2026). Counting equilibria of the electrostatic potential. <i>Proceedings of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/plms.70163\">https://doi.org/10.1112/plms.70163</a>"},"scopus_import":"1","abstract":[{"text":"In 1873, James C. Maxwell conjectured that the electric field generated by n point charges in generic position has at most (n-1)^2 isolated zeroes. The first (nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov, and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.","lang":"eng"}],"oa_version":"Preprint","type":"journal_article","publication_identifier":{"eissn":["1460-244X"],"issn":["0024-6115"]},"status":"public","OA_type":"green","related_material":{"record":[{"status":"public","id":"21050","relation":"earlier_version"}]},"date_created":"2026-05-31T22:02:13Z","article_type":"original","year":"2026","oa":1,"publication_status":"published","doi":"10.1112/plms.70163","intvolume":"       132","language":[{"iso":"eng"}],"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Christopher D","last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","full_name":"Fillmore, Christopher D"},{"first_name":"Goncalo","last_name":"Oliveira","id":"58abbde8-f455-11eb-a497-98c8fd71b905","full_name":"Oliveira, Goncalo"}],"_id":"21931","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"5","department":[{"_id":"HeEd"},{"_id":"TaHa"}],"corr_author":"1","OA_place":"repository","publication":"Proceedings of the London Mathematical Society","day":"01"},{"oa_version":"Preprint","scopus_import":"1","citation":{"ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2026) 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, 2026, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>.","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, 2026. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). 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>","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2026;8:30-62. doi:<a href=\"https://doi.org/10.3934/fods.2025003\">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, 2026."},"keyword":["Topological data analysis","Delaunay mosaic","alpha complex","chromatic sets","persistent homology","kernel/image/cokernel persistent homology","radius function","discrete Morse theory","exact sequences"],"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"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2212.03128","open_access":"1"}],"publisher":"American Institute of Mathematical Sciences","OA_type":"green","related_material":{"record":[{"status":"public","id":"15091","relation":"earlier_version"}]},"status":"public","publication_identifier":{"eissn":["2639-8001"]},"type":"journal_article","volume":8,"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.","date_published":"2026-03-01T00:00:00Z","arxiv":1,"article_processing_charge":"No","title":"Chromatic alpha complexes","quality_controlled":"1","date_updated":"2026-07-23T12:03:56Z","external_id":{"arxiv":["2212.03128"]},"month":"03","department":[{"_id":"HeEd"}],"ec_funded":1,"_id":"20585","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","page":"30-62","day":"01","publication":"Foundations of Data Science","corr_author":"1","OA_place":"repository","year":"2026","article_type":"original","date_created":"2025-11-02T23:01:33Z","author":[{"orcid":"0000-0001-6249-0832","first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","first_name":"Ondrej","last_name":"Draganov","orcid":"0000-0003-0464-3823"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"},{"last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"ddc":["510"],"language":[{"iso":"eng"}],"mathsc":["62R40","55N31","68T09","57Q70"],"intvolume":"         8","doi":"10.3934/fods.2025003","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"publication_status":"published","oa":1},{"month":"02","date_updated":"2026-07-23T11:58:37Z","quality_controlled":"1","external_id":{"isi":["001599061500002"],"arxiv":["2212.11380"]},"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.","isi":1,"article_processing_charge":"No","arxiv":1,"date_published":"2026-02-01T00:00:00Z","title":"Flips in two-dimensional hypertriangulations","volume":132,"status":"public","publication_identifier":{"issn":["0195-6698"]},"type":"journal_article","OA_type":"green","das_tickbox":"0","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2212.11380","open_access":"1"}],"publisher":"Elsevier","oa_version":"Preprint","article_number":"104248","abstract":[{"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","lang":"eng"}],"citation":{"apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2026). 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>","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>. Elsevier, 2026. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>.","mla":"Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>, vol. 132, 104248, Elsevier, 2026, doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2026. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2026).","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, 2026.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. 2026;132. doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>"},"scopus_import":"1","intvolume":"       132","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","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"doi":"10.1016/j.ejc.2025.104248","oa":1,"publication_status":"published","author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Alexey","last_name":"Garber","full_name":"Garber, Alexey"},{"full_name":"Ghafari, Mohadese","last_name":"Ghafari","first_name":"Mohadese"},{"full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1780-2689","last_name":"Heiss","first_name":"Teresa"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"language":[{"iso":"eng"}],"article_type":"original","date_created":"2025-10-19T22:01:31Z","year":"2026","publication":"European Journal of Combinatorics","OA_place":"repository","corr_author":"1","day":"01","department":[{"_id":"HeEd"}],"ec_funded":1,"_id":"20490","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"date_updated":"2026-07-27T08:15:58Z","quality_controlled":"1","external_id":{"isi":["001610592600001"],"arxiv":["2310.14801"]},"month":"03","volume":75,"isi":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. Open access funding provided by Institute of Science and Technology (IST Austria).","article_processing_charge":"Yes (via OA deal)","arxiv":1,"date_published":"2026-03-01T00:00:00Z","PlanS_conform":"1","title":"Maximum Betti numbers of Čech complexes","OA_type":"hybrid","related_material":{"record":[{"id":"17146","relation":"earlier_version","status":"public"}]},"supplementarymaterial":"no","status":"public","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"type":"journal_article","oa_version":"Published Version","citation":{"chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2026. <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. (2026). 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>","short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry 75 (2026) 597–624.","ista":"Edelsbrunner H, Pach J. 2026. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry. 75, 597–624.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>, vol. 75, Springer Nature, 2026, pp. 597–624, doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>.","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” <i>Discrete &#38; Computational Geometry</i>, vol. 75. Springer Nature, pp. 597–624, 2026.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. 2026;75:597-624. doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>"},"scopus_import":"1","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. "}],"das_tickbox":"0","publisher":"Springer Nature","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Pach, János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach","first_name":"János"}],"language":[{"iso":"eng"}],"ddc":["510"],"file_date_updated":"2026-07-27T08:11:05Z","intvolume":"        75","file":[{"date_created":"2026-07-27T08:11:05Z","file_id":"22410","success":1,"creator":"dernst","checksum":"c17c014dbbf5be702195c737890251c8","access_level":"open_access","file_name":"2026_DiscreteCompGeom_Edelsbrunner.pdf","file_size":546483,"content_type":"application/pdf","relation":"main_file","date_updated":"2026-07-27T08:11:05Z"}],"project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"}],"doi":"10.1007/s00454-025-00796-5","oa":1,"publication_status":"published","year":"2026","article_type":"original","date_created":"2025-11-19T09:44:58Z","researchdata_availability":"no","day":"01","publication":"Discrete & Computational Geometry","OA_place":"publisher","corr_author":"1","ec_funded":1,"department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"20657","has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"page":"597-624"},{"type":"journal_article","publication_identifier":{"eissn":["1472-2739"],"issn":["1472-2747"]},"status":"public","OA_type":"diamond","publisher":"Mathematical Sciences Publishers","citation":{"ieee":"N. Zava, “Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces,” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8. Mathematical Sciences Publishers, pp. 5153–5174, 2025.","ama":"Zava N. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. <i>Algebraic &#38; Geometric Topology</i>. 2025;25(8):5153-5174. doi:<a href=\"https://doi.org/10.2140/agt.2025.25.5153\">10.2140/agt.2025.25.5153</a>","chicago":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers, 2025. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>.","apa":"Zava, N. (2025). Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>","ista":"Zava N. 2025. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. Algebraic &#38; Geometric Topology. 25(8), 5153–5174.","short":"N. Zava, Algebraic &#38; Geometric Topology 25 (2025) 5153–5174.","mla":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8, Mathematical Sciences Publishers, 2025, pp. 5153–74, doi:<a href=\"https://doi.org/10.2140/agt.2025.25.5153\">10.2140/agt.2025.25.5153</a>."},"abstract":[{"text":"We discuss the embeddability of subspaces of the Gromov–Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension n(n−1)∕2. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both nonembeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean–Hausdorff distance.","lang":"eng"}],"scopus_import":"1","oa_version":"Published Version","month":"11","external_id":{"arxiv":["2303.04730"]},"quality_controlled":"1","date_updated":"2026-01-05T12:19:09Z","PlanS_conform":"1","title":"Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces","date_published":"2025-11-20T00:00:00Z","article_processing_charge":"No","arxiv":1,"acknowledgement":"The author was supported by the FWF Grant, Project number I4245-N35. The author would like to thank Thomas Weighill for the helpful discussions around Theorem 3.10, and Takamitsu Yamauchi for bringing to my attention the fundamental reference [35]. Furthermore, the author\r\nis thankful for the detailed and helpful comments of the reviewer of this manuscript.","volume":25,"OA_place":"publisher","corr_author":"1","publication":"Algebraic & Geometric Topology","day":"20","page":"5153-5174","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"20867","department":[{"_id":"HeEd"}],"issue":"8","oa":1,"publication_status":"published","project":[{"name":"Algebraic Footprints of Geometric Features in Homology","call_identifier":"FWF","grant_number":"I04245","_id":"26AD5D90-B435-11E9-9278-68D0E5697425"}],"file":[{"file_size":574389,"date_updated":"2026-01-05T12:16:38Z","relation":"main_file","content_type":"application/pdf","date_created":"2026-01-05T12:16:38Z","file_id":"20943","success":1,"creator":"dernst","checksum":"1e05b4f17a44500ae1ae1e21bc636f6a","access_level":"open_access","file_name":"2025_AlgebraicGeomTopology_Zava.pdf"}],"doi":"10.2140/agt.2025.25.5153","intvolume":"        25","file_date_updated":"2026-01-05T12:16:38Z","language":[{"iso":"eng"}],"ddc":["500"],"author":[{"id":"c8b3499c-7a77-11eb-b046-aa368cbbf2ad","full_name":"Zava, Nicolò","first_name":"Nicolò","last_name":"Zava","orcid":"0000-0001-8686-1888"}],"date_created":"2025-12-29T12:09:09Z","article_type":"original","year":"2025"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"21253","department":[{"_id":"HeEd"}],"corr_author":"1","OA_place":"repository","publication":"Computational Geometry","day":"01","date_created":"2026-02-16T15:48:42Z","article_type":"original","year":"2025","oa":1,"publication_status":"published","doi":"10.1016/j.comgeo.2025.102186","intvolume":"       129","language":[{"iso":"eng"}],"author":[{"full_name":"Pach, János","first_name":"János","last_name":"Pach"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"},{"first_name":"Patrick","last_name":"Schnider","full_name":"Schnider, Patrick"}],"publisher":"Elsevier","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2306.13201"}],"article_number":"102186","citation":{"mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>, vol. 129, 102186, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>.","ista":"Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186.","short":"J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025).","apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2025). Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>.","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. 2025;129. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” <i>Computational Geometry</i>, vol. 129. Elsevier, 2025."},"abstract":[{"text":"We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n - 1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.","lang":"eng"}],"oa_version":"Preprint","type":"journal_article","publication_identifier":{"issn":["0925-7721"]},"status":"public","OA_type":"green","related_material":{"record":[{"status":"public","id":"15012","relation":"earlier_version"}]},"title":"Decomposition of geometric graphs into star-forests","article_processing_charge":"No","arxiv":1,"date_published":"2025-12-01T00:00:00Z","acknowledgement":"A preliminary version of this note has been published in the proceedings of the 31st International Symposium on Graph Drawing and Network Visualization, Palermo, 2023. The authors would like to thank the anonymous referees for their valuable comments.","volume":129,"month":"12","external_id":{"arxiv":["2306.13201"]},"date_updated":"2026-04-16T09:12:36Z","quality_controlled":"1"},{"title":"Average and expected distortion of Voronoi paths and scapes","arxiv":1,"date_published":"2025-03-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)","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.","isi":1,"volume":73,"month":"03","external_id":{"pmid":["39974750"],"arxiv":["2012.03350"],"isi":["001238566200004"]},"quality_controlled":"1","date_updated":"2026-02-16T12:18:50Z","publisher":"Springer Nature","scopus_import":"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"}],"citation":{"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>.","ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","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>","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>.","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."},"oa_version":"Published Version","type":"journal_article","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"status":"public","OA_type":"hybrid","date_created":"2024-06-16T22:01:07Z","article_type":"original","year":"2025","oa":1,"publication_status":"published","file":[{"relation":"main_file","date_updated":"2025-04-23T07:31:32Z","content_type":"application/pdf","file_size":283443,"file_id":"19610","success":1,"date_created":"2025-04-23T07:31:32Z","access_level":"open_access","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","creator":"dernst","checksum":"ffb0c818222138f9f113f4bbea41e834"}],"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_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","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"doi":"10.1007/s00454-024-00660-y","intvolume":"        73","file_date_updated":"2025-04-23T07:31:32Z","ddc":["510"],"language":[{"iso":"eng"}],"author":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert"},{"full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0659-3201","first_name":"Anton","last_name":"Nikitenko"}],"page":"490-499","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"17149","ec_funded":1,"department":[{"_id":"HeEd"}],"pmid":1,"OA_place":"publisher","corr_author":"1","publication":"Discrete & Computational Geometry","day":"01"},{"oa":1,"publication_status":"published","doi":"10.1016/j.aim.2024.110055","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"intvolume":"       461","language":[{"iso":"eng"}],"author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"date_created":"2024-12-08T23:01:54Z","article_type":"original","year":"2025","OA_place":"repository","corr_author":"1","publication":"Advances in Mathematics","day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"18626","ec_funded":1,"department":[{"_id":"HeEd"}],"month":"02","external_id":{"arxiv":["2310.18238"],"isi":["001370682500001"]},"quality_controlled":"1","date_updated":"2025-04-15T07:16:53Z","title":"Order-2 Delaunay triangulations optimize angles","arxiv":1,"date_published":"2025-02-01T00:00:00Z","article_processing_charge":"No","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,"volume":461,"type":"journal_article","publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"status":"public","OA_type":"green","publisher":"Elsevier","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2310.18238"}],"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."}],"scopus_import":"1","citation":{"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>.","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>","short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","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>.","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>"},"oa_version":"Preprint"},{"alternative_title":["ISTA Thesis"],"date_created":"2025-01-31T17:04:40Z","year":"2025","project":[{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"file":[{"file_id":"18983","date_created":"2025-01-31T16:58:30Z","file_name":"Thesis.zip","access_level":"closed","creator":"odragano","checksum":"af6567e5d35e5eb330b8925ae37f1998","relation":"source_file","date_updated":"2025-01-31T16:58:30Z","content_type":"application/zip","file_size":11899491},{"date_created":"2025-02-04T16:22:07Z","file_id":"19000","creator":"odragano","checksum":"c3fef68e35b9dc2020b2ca6006da6343","file_name":"Thesis.pdf","access_level":"open_access","file_size":8857514,"date_updated":"2025-02-04T16:22:07Z","relation":"main_file","content_type":"application/pdf"}],"doi":"10.15479/at:ista:18979","publication_status":"published","oa":1,"author":[{"orcid":"0000-0003-0464-3823","last_name":"Draganov","first_name":"Ondrej","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"}],"ddc":["514","004"],"language":[{"iso":"eng"}],"degree_awarded":"PhD","file_date_updated":"2025-02-04T16:22:07Z","supervisor":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert"}],"has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"page":"140","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"_id":"18979","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","corr_author":"1","OA_place":"publisher","day":"03","acknowledgement":"The research presented in this thesis was funded with the Wittgenstein Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.\r\n","article_processing_charge":"No","date_published":"2025-02-03T00:00:00Z","title":"Structures and computations in topological data analysis","month":"02","date_updated":"2026-04-07T11:47:30Z","publisher":"Institute of Science and Technology Austria","oa_version":"Published Version","keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"citation":{"ama":"Draganov O. Structures and computations in topological data analysis. 2025. doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>","ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025.","mla":"Draganov, Ondrej. <i>Structures and Computations in Topological Data Analysis</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>.","short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","ista":"Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria.","apa":"Draganov, O. (2025). <i>Structures and computations in topological data analysis</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>","chicago":"Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>."},"abstract":[{"text":"Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe describe a construction of a chromatic alpha complex, which enables  efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot.","lang":"eng"}],"status":"public","publication_identifier":{"issn":["2663-337X"]},"type":"dissertation","related_material":{"record":[{"status":"public","id":"15091","relation":"part_of_dissertation"},{"id":"18981","relation":"part_of_dissertation","status":"public"}]}},{"abstract":[{"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.","lang":"eng"}],"article_number":"122425","scopus_import":"1","citation":{"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>","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.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (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>.","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.","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>"},"oa_version":"None","publisher":"Elsevier","OA_type":"closed access","type":"journal_article","publication_identifier":{"issn":["0020-0255"]},"status":"public","volume":719,"title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","article_processing_charge":"No","date_published":"2025-11-01T00:00:00Z","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.","isi":1,"external_id":{"isi":["001516170500002"]},"date_updated":"2025-12-30T09:05:32Z","quality_controlled":"1","month":"11","_id":"19937","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"ec_funded":1,"issue":"11","day":"01","corr_author":"1","publication":"Information Sciences","year":"2025","date_created":"2025-06-30T08:48:48Z","article_type":"original","language":[{"iso":"eng"}],"author":[{"full_name":"Mahini, Mohammad","last_name":"Mahini","first_name":"Mohammad"},{"full_name":"Beigy, Hamid","first_name":"Hamid","last_name":"Beigy"},{"first_name":"Salman","last_name":"Qadami","full_name":"Qadami, Salman"},{"first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"publication_status":"published","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","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"doi":"10.1016/j.ins.2025.122425","intvolume":"       719"},{"OA_type":"gold","status":"public","publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"type":"conference","oa_version":"Published Version","citation":{"ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","ama":"Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43.","mla":"Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>."},"article_number":"43","abstract":[{"lang":"eng","text":"We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers."}],"scopus_import":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_updated":"2025-07-14T07:26:14Z","quality_controlled":"1","external_id":{"arxiv":["2410.21204"]},"month":"06","volume":332,"acknowledgement":"Herbert Edelsbrunner: partially supported by the Wittgenstein Prize, Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35","article_processing_charge":"Yes","arxiv":1,"date_published":"2025-06-20T00:00:00Z","title":"On spheres with k points inside","day":"20","publication":"41st International Symposium on Computational Geometry","corr_author":"1","OA_place":"publisher","department":[{"_id":"HeEd"}],"conference":{"location":"Kanazawa, Japan","end_date":"2025-06-27","name":"SoCG: Symposium on Computational Geometry","start_date":"2025-06-23"},"_id":"20005","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"author":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"},{"full_name":"Garber, Alexey","first_name":"Alexey","last_name":"Garber"},{"last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"language":[{"iso":"eng"}],"ddc":["510"],"file_date_updated":"2025-07-14T07:24:22Z","intvolume":"       332","file":[{"file_size":661893,"content_type":"application/pdf","relation":"main_file","date_updated":"2025-07-14T07:24:22Z","checksum":"b5313ed8575ea87913c71a6e3c7513c8","creator":"dernst","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","access_level":"open_access","date_created":"2025-07-14T07:24:22Z","success":1,"file_id":"20016"}],"doi":"10.4230/LIPIcs.SoCG.2025.43","project":[{"_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"}],"publication_status":"published","oa":1,"year":"2025","alternative_title":["LIPIcs"],"date_created":"2025-07-13T22:01:22Z"},{"month":"06","external_id":{"arxiv":["2405.17920"]},"quality_controlled":"1","date_updated":"2025-12-30T11:04:33Z","title":"Banana trees for the persistence in time series experimentally","date_published":"2025-06-20T00:00:00Z","arxiv":1,"article_processing_charge":"Yes","acknowledgement":"Lara Ost: Supported by the Vienna Graduate School on Computational Optimization\r\n(VGSCO), FWF project no. W1260-N35.\r\nSebastiano Cultrera di Montesano: Supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard.\r\nHerbert Edelsbrunner: Partially supported by the Wittgenstein Prize, FWF grant no. Z 342-N31,\r\nand by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.","volume":332,"type":"conference","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773706"]},"status":"public","related_material":{"link":[{"url":"https://github.com/laraost/BananaPersist","relation":"software"}]},"OA_type":"gold","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","citation":{"chicago":"Ost, Lara, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner. “Banana Trees for the Persistence in Time Series Experimentally.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>.","apa":"Ost, L., Cultrera di Montesano, S., &#38; Edelsbrunner, H. (2025). Banana trees for the persistence in time series experimentally. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>","ista":"Ost L, Cultrera di Montesano S, Edelsbrunner H. 2025. Banana trees for the persistence in time series experimentally. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 71.","short":"L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","mla":"Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 71, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>.","ieee":"L. Ost, S. Cultrera di Montesano, and H. Edelsbrunner, “Banana trees for the persistence in time series experimentally,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","ama":"Ost L, Cultrera di Montesano S, Edelsbrunner H. Banana trees for the persistence in time series experimentally. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>"},"abstract":[{"text":"In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.","lang":"eng"}],"article_number":"71","oa_version":"Published Version","publication_status":"published","oa":1,"doi":"10.4230/LIPIcs.SoCG.2025.71","project":[{"name":"Vienna Graduate School on Computational Optimization","_id":"9B9290DE-BA93-11EA-9121-9846C619BF3A","grant_number":"W1260-N35"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"file":[{"file_size":834623,"content_type":"application/pdf","relation":"main_file","date_updated":"2025-07-14T08:23:38Z","checksum":"3a4a7a707a56e0cfdf51428782dee55a","creator":"dernst","access_level":"open_access","file_name":"2025_LIPIcs.SoCG_Ost.pdf","date_created":"2025-07-14T08:23:38Z","file_id":"20017","success":1}],"intvolume":"       332","file_date_updated":"2025-07-14T08:23:38Z","ddc":["000"],"language":[{"iso":"eng"}],"author":[{"full_name":"Ost, Lara","first_name":"Lara","last_name":"Ost"},{"last_name":"Cultrera di Montesano","first_name":"Sebastiano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera di Montesano, Sebastiano"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833"}],"date_created":"2025-07-13T22:01:22Z","alternative_title":["LIPIcs"],"year":"2025","OA_place":"publisher","corr_author":"1","publication":"41st International Symposium on Computational Geometry","day":"20","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"has_accepted_license":"1","_id":"20006","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"conference":{"location":"Kanazawa, Japan","end_date":"2025-06-27","name":"SoCG: Symposium on Computational Geometry","start_date":"2025-06-23"}},{"publication_identifier":{"eissn":["2730-9657"]},"type":"journal_article","status":"public","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"21021"}]},"OA_type":"hybrid","publisher":"Springer Nature","oa_version":"Published Version","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"}],"citation":{"short":"E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica 4 (2025) 811–828.","ista":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica. 4, 811–828.","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>.","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>.","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>","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."},"scopus_import":"1","month":"12","date_updated":"2026-04-07T11:42:48Z","quality_controlled":"1","PlanS_conform":"1","title":"Burning or collapsing the medial axis is unstable","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.","article_processing_charge":"Yes (via OA deal)","date_published":"2025-12-01T00:00:00Z","volume":4,"corr_author":"1","OA_place":"publisher","publication":"La Matematica","day":"01","page":"811-828","has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"department":[{"_id":"HeEd"}],"ec_funded":1,"_id":"20260","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"file_size":2678640,"content_type":"application/pdf","date_updated":"2025-12-30T07:52:58Z","relation":"main_file","creator":"dernst","checksum":"e2043259194bfcdf3d74c4da8a5a853f","access_level":"open_access","file_name":"2025_LaMatematica_Chambers.pdf","date_created":"2025-12-30T07:52:58Z","file_id":"20885","success":1}],"doi":"10.1007/s44007-025-00170-0","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"},{"name":"Learning and triangulating manifolds via collapses","grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"publication_status":"published","oa":1,"intvolume":"         4","file_date_updated":"2025-12-30T07:52:58Z","author":[{"full_name":"Chambers, Erin Wolf","last_name":"Chambers","first_name":"Erin Wolf"},{"last_name":"Fillmore","first_name":"Christopher D","full_name":"Fillmore, Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"orcid":"0000-0002-6862-208X","last_name":"Stephenson","first_name":"Elizabeth R","full_name":"Stephenson, Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-7472-2220","last_name":"Wintraecken","first_name":"Mathijs","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"ddc":["510"],"language":[{"iso":"eng"}],"date_created":"2025-08-31T22:01:33Z","article_type":"original","year":"2025"}]