[{"ddc":["500"],"acknowledgement":"The authors would like to thank Michael Lesnick and Primoz Skraba for their helpful comments regarding sparse approximations of filtrations. We are also grateful to the anonymous referees for their careful reading and constructive suggestions. The three authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35, the U.S. National Science Foundation (NSF-DMS), grant no. 2005630, and a JSPS Grant-in-Aid for Transformative Research Areas (A) (22H05107, Y.H.), EPSRC Research Grant EP/Y008642/1.","file_date_updated":"2026-03-09T11:29:30Z","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"title":"Maximum persistent Betti numbers of Čech complexes","publication":"Journal of Applied and Computational Topology","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Kahle, Matthew","first_name":"Matthew","last_name":"Kahle"},{"full_name":"Kanazawa, Shu","first_name":"Shu","last_name":"Kanazawa"}],"project":[{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"fulldoi":"https://doi.org/10.1007/s41468-026-00233-3","file":[{"content_type":"application/pdf","date_updated":"2026-03-09T11:29:30Z","file_id":"21416","creator":"dernst","relation":"main_file","checksum":"0bf6dc430cafa40c08f260fe17d54595","success":1,"access_level":"open_access","file_size":323111,"date_created":"2026-03-09T11:29:30Z","file_name":"2026_JourAppliedCompTopology_Edelsbrunner.pdf"}],"OA_place":"publisher","publisher":"Springer Nature","volume":10,"year":"2026","citation":{"ieee":"H. Edelsbrunner, M. Kahle, and S. Kanazawa, “Maximum persistent Betti numbers of Čech complexes,” <i>Journal of Applied and Computational Topology</i>, vol. 10. Springer Nature, 2026.","chicago":"Edelsbrunner, Herbert, Matthew Kahle, and Shu Kanazawa. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>.","ista":"Edelsbrunner H, Kahle M, Kanazawa S. 2026. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 10, 5.","ama":"Edelsbrunner H, Kahle M, Kanazawa S. Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. 2026;10. doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>","mla":"Edelsbrunner, Herbert, et al. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>, vol. 10, 5, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>.","apa":"Edelsbrunner, H., Kahle, M., &#38; Kanazawa, S. (2026). Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>","short":"H. Edelsbrunner, M. Kahle, S. Kanazawa, Journal of Applied and Computational Topology 10 (2026)."},"department":[{"_id":"HeEd"}],"scopus_import":"1","external_id":{"arxiv":["2409.05241"]},"date_updated":"2026-03-09T11:31:29Z","doi":"10.1007/s41468-026-00233-3","arxiv":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"OA_type":"hybrid","date_created":"2026-03-08T23:01:45Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"        10","month":"03","quality_controlled":"1","PlanS_conform":"1","_id":"21407","day":"01","type":"journal_article","oa":1,"article_number":"5","article_type":"original","language":[{"iso":"eng"}],"oa_version":"Published Version","date_published":"2026-03-01T00:00:00Z","publication_status":"published","abstract":[{"text":"This note proves that only a linear number of holes in a Cech complex of n points in R^d\r\ncan persist over an interval of constant length. Specifically, for any fixed dimension p <\r\nd and fixed ε > 0, the number of p-dimensional holes in the ˇ Cech complex at radius 1\r\nthat persist to radius 1+ε is bounded above by a constant times n,where n is the number\r\nof points. The proof uses a packing argument supported by relating theCˇ ech complexes\r\nwith corresponding snap complexes over the cells in a partition of space. The argument\r\nis self-contained and elementary, relying on geometric and combinatorial constructions\r\nrather than on the existing theory of sparse approximations or interleavings. The bound\r\nalso applies to Alpha complexes and Vietoris–Rips complexes. While our result can be\r\ninferred from prior work on sparse filtrations, to our knowledge, no explicit statement\r\nor direct proof of this bound appears in the literature.","lang":"eng"}],"has_accepted_license":"1","status":"public","article_processing_charge":"Yes (in subscription journal)"},{"language":[{"iso":"eng"}],"article_type":"original","isi":1,"article_processing_charge":"Yes (via OA deal)","date_published":"2026-01-01T00:00:00Z","has_accepted_license":"1","publication_status":"published","abstract":[{"lang":"eng","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."}],"status":"public","oa_version":"Published Version","intvolume":"        75","month":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-10-12T22:01:26Z","OA_type":"hybrid","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"arxiv":1,"oa":1,"_id":"20456","type":"journal_article","day":"01","quality_controlled":"1","corr_author":"1","PlanS_conform":"1","year":"2026","citation":{"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>","short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47.","mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>, vol. 75, Springer Nature, 2026, pp. 24–47, doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>.","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.","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.","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>","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>."},"department":[{"_id":"HeEd"}],"volume":75,"publisher":"Springer Nature","OA_place":"publisher","ec_funded":1,"external_id":{"arxiv":["2212.03121"],"isi":["001584166900001"]},"doi":"10.1007/s00454-025-00778-7","date_updated":"2026-01-05T13:21:56Z","scopus_import":"1","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"related_material":{"record":[{"relation":"earlier_version","id":"15090","status":"public"}]},"file_date_updated":"2026-01-05T13:21:20Z","ddc":["510"],"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.","fulldoi":"https://doi.org/10.1007/s00454-025-00778-7","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"file":[{"access_level":"open_access","date_updated":"2026-01-05T13:21:20Z","creator":"dernst","relation":"main_file","file_id":"20952","content_type":"application/pdf","success":1,"checksum":"0addb5c1b78142f9fb453bfa04695400","file_name":"2026_DiscreteCompGeom_Biswas.pdf","date_created":"2026-01-05T13:21:20Z","file_size":570922}],"publication":"Discrete and Computational Geometry","author":[{"last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","first_name":"Ranita","orcid":"0000-0002-5372-7890"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"},{"last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0464-3823","first_name":"Ondrej","full_name":"Draganov, Ondrej"},{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"page":"24-47","title":"On the size of chromatic Delaunay mosaics"},{"oa_version":"Preprint","article_processing_charge":"No","isi":1,"abstract":[{"lang":"eng","text":"We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n\r\n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.\r\n"}],"status":"public","publication_status":"published","date_published":"2026-02-01T00:00:00Z","article_type":"original","language":[{"iso":"eng"}],"article_number":"104248","day":"01","_id":"20490","type":"journal_article","corr_author":"1","quality_controlled":"1","oa":1,"OA_type":"green","arxiv":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.11380"}],"month":"02","intvolume":"       132","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-10-19T22:01:31Z","doi":"10.1016/j.ejc.2025.104248","date_updated":"2026-07-23T11:58:37Z","external_id":{"isi":["001599061500002"],"arxiv":["2212.11380"]},"scopus_import":"1","ec_funded":1,"OA_place":"repository","department":[{"_id":"HeEd"}],"citation":{"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.","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>.","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>","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2026. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.","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>.","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>","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2026)."},"year":"2026","volume":132,"publisher":"Elsevier","title":"Flips in two-dimensional hypertriangulations","fulldoi":"https://doi.org/10.1016/j.ejc.2025.104248","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"publication":"European Journal of Combinatorics","author":[{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"full_name":"Garber, Alexey","first_name":"Alexey","last_name":"Garber"},{"full_name":"Ghafari, Mohadese","first_name":"Mohadese","last_name":"Ghafari"},{"id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss","orcid":"0000-0002-1780-2689","first_name":"Teresa","full_name":"Heiss, Teresa"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"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.","publication_identifier":{"issn":["0195-6698"]},"das_tickbox":"0"},{"OA_place":"publisher","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","volume":367,"year":"2026","citation":{"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.","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>","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>.","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.","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.","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>."},"department":[{"_id":"HeEd"},{"_id":"GradSch"}],"scopus_import":"1","external_id":{"arxiv":["2511.21961"]},"doi":"10.4230/LIPICS.SOCG.2026.41","date_updated":"2026-08-12T09:02:56Z","ec_funded":1,"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","ddc":["500"],"file_date_updated":"2026-07-14T06:08:05Z","das_tickbox":"0","publication_identifier":{"isbn":["9783959774185"],"eissn":["1868-8969"]},"title":"The depth poset under transpositions in the filter","publication":"42nd International Symposium on Computational Geometry","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Lipiński","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","full_name":"Lipiński, Michał","first_name":"Michał","orcid":"0000-0001-9789-9750"},{"last_name":"Mrozek","first_name":"Marian","orcid":"0000-0002-0619-6417","full_name":"Mrozek, Marian"},{"full_name":"Soriano Trigueros, Manuel","orcid":"0000-0003-2449-1433","first_name":"Manuel","last_name":"Soriano Trigueros","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8"},{"full_name":"Zimin, Fedor","first_name":"Fedor","last_name":"Zimin","id":"afd27eda-91c1-11f0-aad8-c6edbec24c04"}],"conference":{"location":"New Brunswick, NJ, United States","start_date":"2026-06-02","name":"SoCG: Symposium on Computational Geometry","end_date":"2026-06-05"},"fulldoi":"https://doi.org/10.4230/LIPICS.SOCG.2026.41","project":[{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020"}],"file":[{"checksum":"9dfb96ee66985c724b499b0e5888dc8e","success":1,"content_type":"application/pdf","date_updated":"2026-07-14T06:08:05Z","file_id":"22329","creator":"dernst","relation":"main_file","access_level":"open_access","file_size":2902144,"date_created":"2026-07-14T06:08:05Z","file_name":"2026_LIPIcSSoCG_Edelsbrunner.pdf"}],"article_number":"41:1-41:18","language":[{"iso":"eng"}],"alternative_title":["LIPIcs"],"oa_version":"Published Version","date_published":"2026-05-27T00:00:00Z","supplementarymaterial":"no","has_accepted_license":"1","abstract":[{"lang":"eng","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."}],"status":"public","publication_status":"published","article_processing_charge":"Yes","arxiv":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"OA_type":"gold","keyword":["Algebraic topology","Lefschetz complexes","persistent homology","vines and vineyards","birth-death pairs","shallow pairs","relations","partial orders","transpositions","Theory of computation → Computational geometry"],"researchdata_availability":"no","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-07-13T09:56:38Z","intvolume":"       367","month":"05","corr_author":"1","quality_controlled":"1","day":"27","_id":"22299","type":"conference","oa":1},{"doi":"10.1007/s00454-025-00796-5","date_updated":"2026-10-02T11:48:00Z","external_id":{"isi":["001610592600001"],"arxiv":["2310.14801"]},"scopus_import":"1","ec_funded":1,"OA_place":"publisher","department":[{"_id":"HeEd"}],"year":"2026","citation":{"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>.","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.","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.","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>.","ista":"Edelsbrunner H, Pach J. 2026. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry. 75, 597–624.","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>"},"publisher":"Springer Nature","volume":75,"title":"Maximum Betti numbers of Čech complexes","page":"597-624","file":[{"access_level":"open_access","content_type":"application/pdf","relation":"main_file","file_id":"22410","creator":"dernst","date_updated":"2026-07-27T08:11:05Z","checksum":"c17c014dbbf5be702195c737890251c8","success":1,"file_name":"2026_DiscreteCompGeom_Edelsbrunner.pdf","date_created":"2026-07-27T08:11:05Z","file_size":546483}],"fulldoi":"https://doi.org/10.1007/s00454-025-00796-5","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"publication":"Discrete & Computational Geometry","author":[{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"full_name":"Pach, János","first_name":"János","last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4"}],"file_date_updated":"2026-07-27T08:11:05Z","ddc":["510"],"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).","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"related_material":{"record":[{"relation":"earlier_version","id":"17146","status":"public"}]},"das_tickbox":"0","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","isi":1,"supplementarymaterial":"no","has_accepted_license":"1","status":"public","publication_status":"published","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. "}],"date_published":"2026-03-01T00:00:00Z","article_type":"original","language":[{"iso":"eng"}],"type":"journal_article","_id":"20657","day":"01","quality_controlled":"1","PlanS_conform":"1","corr_author":"1","oa":1,"OA_type":"hybrid","arxiv":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"month":"03","intvolume":"        75","date_created":"2025-11-19T09:44:58Z","researchdata_availability":"no","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"publication_identifier":{"eissn":["2639-8001"]},"related_material":{"record":[{"status":"public","id":"15091","relation":"earlier_version"}]},"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.","ddc":["510"],"fulldoi":"https://doi.org/10.3934/fods.2025003","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"author":[{"last_name":"Cultrera di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"},{"orcid":"0000-0003-0464-3823","first_name":"Ondrej","full_name":"Draganov, Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"publication":"Foundations of Data Science","title":"Chromatic alpha complexes","page":"30-62","department":[{"_id":"HeEd"}],"citation":{"short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2026) 30–62.","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). Chromatic alpha complexes. <i>Foundations of Data Science</i>. AIMS. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>, vol. 8, AIMS, 2026, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2026;8:30-62. 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>. AIMS, 2026. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>Foundations of Data Science</i>, vol. 8. AIMS, pp. 30–62, 2026."},"year":"2026","publisher":"AIMS","volume":8,"OA_place":"repository","ec_funded":1,"date_updated":"2026-10-02T11:56:12Z","doi":"10.3934/fods.2025003","external_id":{"arxiv":["2212.03128"]},"scopus_import":"1","month":"03","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.03128"}],"intvolume":"         8","mathsc":["62R40","55N31","68T09","57Q70"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-11-02T23:01:33Z","keyword":["Topological data analysis","Delaunay mosaic","alpha complex","chromatic sets","persistent homology","kernel/image/cokernel persistent homology","radius function","discrete Morse theory","exact sequences"],"OA_type":"green","arxiv":1,"oa":1,"type":"journal_article","_id":"20585","day":"01","quality_controlled":"1","corr_author":"1","article_type":"original","language":[{"iso":"eng"}],"article_processing_charge":"No","publication_status":"published","abstract":[{"lang":"eng","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."}],"has_accepted_license":"1","status":"public","date_published":"2026-03-01T00:00:00Z","oa_version":"Preprint"},{"date_updated":"2025-04-15T07:16:53Z","doi":"10.1016/j.aim.2024.110055","external_id":{"isi":["001370682500001"],"arxiv":["2310.18238"]},"scopus_import":"1","ec_funded":1,"OA_place":"repository","department":[{"_id":"HeEd"}],"year":"2025","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>.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","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>","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025.","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>.","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)."},"publisher":"Elsevier","volume":461,"title":"Order-2 Delaunay triangulations optimize angles","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"fulldoi":"https://doi.org/10.1016/j.aim.2024.110055","publication":"Advances in Mathematics","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"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.","publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"oa_version":"Preprint","article_processing_charge":"No","isi":1,"status":"public","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."}],"publication_status":"published","date_published":"2025-02-01T00:00:00Z","article_type":"original","language":[{"iso":"eng"}],"article_number":"110055","type":"journal_article","_id":"18626","day":"01","quality_controlled":"1","corr_author":"1","oa":1,"OA_type":"green","arxiv":1,"month":"02","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2310.18238","open_access":"1"}],"intvolume":"       461","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2024-12-08T23:01:54Z"},{"title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","fulldoi":"https://doi.org/10.1016/j.ins.2025.122425","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"publication":"Information Sciences","author":[{"first_name":"Mohammad","full_name":"Mahini, Mohammad","last_name":"Mahini"},{"full_name":"Beigy, Hamid","first_name":"Hamid","last_name":"Beigy"},{"full_name":"Qadami, Salman","first_name":"Salman","last_name":"Qadami"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza"}],"issue":"11","acknowledgement":"The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which improved this paper.\r\nWork by the first and fourth authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35.","publication_identifier":{"issn":["0020-0255"]},"external_id":{"isi":["001516170500002"]},"date_updated":"2025-12-30T09:05:32Z","doi":"10.1016/j.ins.2025.122425","scopus_import":"1","ec_funded":1,"year":"2025","citation":{"ieee":"M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality,” <i>Information Sciences</i>, vol. 719, no. 11. Elsevier, 2025.","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>.","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.","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>","mla":"Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>, vol. 719, no. 11, 122425, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).","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>"},"department":[{"_id":"HeEd"}],"volume":719,"publisher":"Elsevier","_id":"19937","type":"journal_article","day":"01","corr_author":"1","quality_controlled":"1","OA_type":"closed access","intvolume":"       719","month":"11","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-06-30T08:48:48Z","oa_version":"None","isi":1,"article_processing_charge":"No","date_published":"2025-11-01T00:00:00Z","status":"public","publication_status":"published","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_type":"original","language":[{"iso":"eng"}],"article_number":"122425"},{"OA_place":"publisher","department":[{"_id":"HeEd"}],"citation":{"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>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","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>","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>.","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>","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.","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."},"year":"2025","volume":332,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_updated":"2025-07-14T07:26:14Z","doi":"10.4230/LIPIcs.SoCG.2025.43","external_id":{"arxiv":["2410.21204"]},"scopus_import":"1","file_date_updated":"2025-07-14T07:24:22Z","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","ddc":["510"],"publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"title":"On spheres with k points inside","file":[{"file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","date_created":"2025-07-14T07:24:22Z","file_size":661893,"access_level":"open_access","content_type":"application/pdf","creator":"dernst","file_id":"20016","date_updated":"2025-07-14T07:24:22Z","relation":"main_file","checksum":"b5313ed8575ea87913c71a6e3c7513c8","success":1}],"project":[{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2025.43","conference":{"start_date":"2025-06-23","name":"SoCG: Symposium on Computational Geometry","end_date":"2025-06-27","location":"Kanazawa, Japan"},"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"last_name":"Garber","full_name":"Garber, Alexey","first_name":"Alexey"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"publication":"41st International Symposium on Computational Geometry","language":[{"iso":"eng"}],"article_number":"43","alternative_title":["LIPIcs"],"oa_version":"Published Version","article_processing_charge":"Yes","has_accepted_license":"1","publication_status":"published","status":"public","abstract":[{"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.","lang":"eng"}],"date_published":"2025-06-20T00:00:00Z","OA_type":"gold","arxiv":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"month":"06","intvolume":"       332","date_created":"2025-07-13T22:01:22Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"20","_id":"20005","type":"conference","quality_controlled":"1","corr_author":"1","oa":1},{"alternative_title":["LIPIcs"],"language":[{"iso":"eng"}],"article_number":"71","article_processing_charge":"Yes","publication_status":"published","has_accepted_license":"1","abstract":[{"lang":"eng","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."}],"status":"public","date_published":"2025-06-20T00:00:00Z","oa_version":"Published Version","month":"06","intvolume":"       332","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-07-13T22:01:22Z","OA_type":"gold","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"arxiv":1,"oa":1,"type":"conference","_id":"20006","day":"20","quality_controlled":"1","corr_author":"1","department":[{"_id":"HeEd"}],"year":"2025","citation":{"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>.","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>","short":"L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","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>.","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.","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>","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."},"volume":332,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","OA_place":"publisher","doi":"10.4230/LIPIcs.SoCG.2025.71","date_updated":"2025-12-30T11:04:33Z","external_id":{"arxiv":["2405.17920"]},"scopus_import":"1","publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"related_material":{"link":[{"relation":"software","url":"https://github.com/laraost/BananaPersist"}]},"file_date_updated":"2025-07-14T08:23:38Z","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.","ddc":["000"],"file":[{"access_level":"open_access","success":1,"checksum":"3a4a7a707a56e0cfdf51428782dee55a","file_id":"20017","date_updated":"2025-07-14T08:23:38Z","relation":"main_file","creator":"dernst","content_type":"application/pdf","file_name":"2025_LIPIcs.SoCG_Ost.pdf","date_created":"2025-07-14T08:23:38Z","file_size":834623}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2025.71","project":[{"grant_number":"W1260-N35","name":"Vienna Graduate School on Computational Optimization","_id":"9B9290DE-BA93-11EA-9121-9846C619BF3A"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"conference":{"location":"Kanazawa, Japan","start_date":"2025-06-23","name":"SoCG: Symposium on Computational Geometry","end_date":"2025-06-27"},"publication":"41st International Symposium on Computational Geometry","author":[{"last_name":"Ost","first_name":"Lara","full_name":"Ost, Lara"},{"full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano"},{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"}],"title":"Banana trees for the persistence in time series experimentally"},{"language":[{"iso":"eng"}],"article_type":"original","article_number":"854","isi":1,"article_processing_charge":"Yes","date_published":"2025-08-01T00:00:00Z","abstract":[{"lang":"eng","text":"Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence."}],"status":"public","publication_status":"published","has_accepted_license":"1","oa_version":"Published Version","intvolume":"        27","month":"08","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2025-09-07T22:01:33Z","OA_type":"gold","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"oa":1,"DOAJ_listed":"1","_id":"20293","day":"01","type":"journal_article","quality_controlled":"1","corr_author":"1","PlanS_conform":"1","citation":{"chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>. MDPI, 2025. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>.","ama":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. 2025;27(8). doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>","ista":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.","ieee":"A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between the Jensen–Shannon divergence and the minmax divergence,” <i>Entropy</i>, vol. 27, no. 8. MDPI, 2025.","mla":"Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>, vol. 27, no. 8, 854, MDPI, 2025, doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>.","apa":"Akopyan, A., Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2025). Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. MDPI. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025)."},"year":"2025","department":[{"_id":"HeEd"}],"volume":27,"publisher":"MDPI","OA_place":"publisher","ec_funded":1,"external_id":{"pmid":["40870326"],"isi":["001557476000001"]},"pmid":1,"date_updated":"2025-09-30T14:32:31Z","doi":"10.3390/e27080854","scopus_import":"1","publication_identifier":{"eissn":["1099-4300"]},"file_date_updated":"2025-09-08T07:55:48Z","issue":"8","acknowledgement":"This research received partial funding from the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis of Neural Networks’. The APC was waived.","ddc":["500"],"fulldoi":"https://doi.org/10.3390/e27080854","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"file":[{"access_level":"open_access","file_id":"20309","relation":"main_file","creator":"dernst","date_updated":"2025-09-08T07:55:48Z","content_type":"application/pdf","success":1,"checksum":"65c5399c4015d9c8abb8c7a96f3d7836","file_name":"2025_Entropy_Akopyan.pdf","date_created":"2025-09-08T07:55:48Z","file_size":379340}],"publication":"Entropy","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","first_name":"Arseniy"},{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"id":"2E36B656-F248-11E8-B48F-1D18A9856A87","last_name":"Virk","first_name":"Ziga","full_name":"Virk, Ziga"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner","full_name":"Wagner, Hubert","first_name":"Hubert"}],"title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence"},{"ec_funded":1,"doi":"10.15479/AT-ISTA-19630","date_updated":"2026-10-02T11:19:03Z","department":[{"_id":"ChWo"},{"_id":"GradSch"}],"citation":{"ista":"Synak P. 2025. Methods for fluid simulation, surface tracking, and statistics of non-manifold structures. Institute of Science and Technology Austria.","ama":"Synak P. Methods for fluid simulation, surface tracking, and statistics of non-manifold structures. 2025. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-19630\">10.15479/AT-ISTA-19630</a>","chicago":"Synak, Peter. “Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/AT-ISTA-19630\">https://doi.org/10.15479/AT-ISTA-19630</a>.","ieee":"P. Synak, “Methods for fluid simulation, surface tracking, and statistics of non-manifold structures,” Institute of Science and Technology Austria, 2025.","apa":"Synak, P. (2025). <i>Methods for fluid simulation, surface tracking, and statistics of non-manifold structures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-19630\">https://doi.org/10.15479/AT-ISTA-19630</a>","short":"P. Synak, Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures, Institute of Science and Technology Austria, 2025.","mla":"Synak, Peter. <i>Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-19630\">10.15479/AT-ISTA-19630</a>."},"year":"2025","degree_awarded":"PhD","publisher":"Institute of Science and Technology Austria","OA_place":"publisher","file":[{"file_name":"Thesis_source_Heiss_Synak.zip","date_created":"2025-04-30T14:02:25Z","file_size":60670543,"access_level":"closed","content_type":"application/x-zip-compressed","creator":"cchlebak","file_id":"19633","relation":"source_file","date_updated":"2025-04-30T14:02:25Z","checksum":"f00b519c27529daa0c3b2d4102b4fa7b"},{"file_size":21319043,"date_created":"2025-04-30T14:02:42Z","file_name":"Thesis_PDFA_Heiss_Synak.pdf","checksum":"6e40a2fd3b1b881af1385670854a682e","content_type":"application/pdf","date_updated":"2025-04-30T15:49:16Z","relation":"main_file","creator":"cchlebak","file_id":"19634","access_level":"open_access"}],"fulldoi":"https://doi.org/10.15479/AT-ISTA-19630","project":[{"grant_number":"638176","name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales","_id":"2533E772-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083","name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena"},{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"638176","name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales","_id":"2533E772-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"author":[{"last_name":"Synak","id":"331776E2-F248-11E8-B48F-1D18A9856A87","full_name":"Synak, Peter","first_name":"Peter"}],"title":"Methods for fluid simulation, surface tracking, and statistics of non-manifold structures","page":"106","publication_identifier":{"issn":["2663-337X"]},"related_material":{"record":[{"status":"public","id":"8135","relation":"part_of_dissertation"},{"id":"8384","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"17219"}]},"file_date_updated":"2025-04-30T15:49:16Z","ddc":["519","006"],"acknowledgement":"The project in Chapter 2 has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No. 638176. The project in Chapter 3 was funded in part by the European Union (ERC-2021-COG 101045083 CoDiNA). The project in Chapter 4 has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It was also partially supported by the DFG Collaborative Research Center TRR 109, 'Discretization in Geometry and Dynamics', through grant no. I02979-N35 of the Austrian Science Fund (FWF). Thank you for providing funds to support my work.","article_processing_charge":"No","acknowledged_ssus":[{"_id":"ScienComp"}],"has_accepted_license":"1","status":"public","publication_status":"published","abstract":[{"text":"This thesis consists of three chapters, each corresponding to one publication. While each of these projects tackles a topic in a different area of research, they all share a common thread in the type of topological structure they handle - a partition of space into volumes separated by interfaces that meet in non-manifold junctions.\r\n\r\nIn Chapter 2, we study clusters of soap bubbles from a simulation perspective. In particular, we develop a surface-only algorithm that couples large scale motion and shape deformation of soap bubble clusters with the small scale evolution of the thin film's thickness, which is responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. We model film thickness as a reduced degree of freedom in the Navier-Stokes equations and from them derive three sets of equations governing normal and tangential motion of the soap film surface, as well as the evolution of the thin film thickness. We discretize these equations on a non-manifold triangle mesh, extending and adapting operators to handle complex topology. We also present an incompressible fluid solver for 2.5D films and an advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film.\r\n\r\nIn Chapter 3, we introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts mesh defects, such as overlaps, self-intersections, and inversions into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations, such as those presented in Chapter 2, but with an order of magnitude more interacting bubbles than what we could achieve before, and Boolean unions of non-manifold meshes consisting of millions of triangles.\r\n\r\nLastly, in Chapter 4, we utilize developments in the theory of random geometric complexes facilitated by observations from Discrete Morse theory. We survey the methods and results obtained with this new approach, and discuss some of its shortcomings. We use simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.","lang":"eng"}],"date_published":"2025-04-29T00:00:00Z","oa_version":"Published Version","alternative_title":["ISTA Thesis"],"language":[{"iso":"eng"}],"oa":1,"_id":"19630","type":"dissertation","day":"29","supervisor":[{"full_name":"Wojtan, Christopher J","first_name":"Christopher J","orcid":"0000-0001-6646-5546","last_name":"Wojtan","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87"}],"corr_author":"1","month":"04","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_created":"2025-04-29T09:39:34Z"},{"isi":1,"article_processing_charge":"Yes (via OA deal)","date_published":"2025-03-01T00:00:00Z","status":"public","supplementarymaterial":"no","has_accepted_license":"1","abstract":[{"lang":"eng","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."}],"publication_status":"published","oa_version":"Published Version","article_type":"original","language":[{"iso":"eng"}],"oa":1,"day":"01","_id":"17149","type":"journal_article","corr_author":"1","quality_controlled":"1","intvolume":"        73","month":"03","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2024-06-16T22:01:07Z","researchdata_availability":"no","OA_type":"hybrid","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"arxiv":1,"ec_funded":1,"pmid":1,"external_id":{"isi":["001238566200004"],"arxiv":["2012.03350"],"pmid":["39974750"]},"doi":"10.1007/s00454-024-00660-y","date_updated":"2026-10-02T11:48:46Z","scopus_import":"1","year":"2025","citation":{"ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","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>","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>.","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.","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>","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","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>."},"department":[{"_id":"HeEd"}],"volume":73,"publisher":"Springer Nature","OA_place":"publisher","fulldoi":"https://doi.org/10.1007/s00454-024-00660-y","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"file":[{"access_level":"open_access","checksum":"ffb0c818222138f9f113f4bbea41e834","success":1,"content_type":"application/pdf","creator":"dernst","file_id":"19610","date_updated":"2025-04-23T07:31:32Z","relation":"main_file","date_created":"2025-04-23T07:31:32Z","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","file_size":283443}],"publication":"Discrete & Computational Geometry","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"full_name":"Nikitenko, Anton","first_name":"Anton","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko"}],"page":"490-499","title":"Average and expected distortion of Voronoi paths and scapes","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"das_tickbox":"0","file_date_updated":"2025-04-23T07:31:32Z","ddc":["510"],"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."},{"alternative_title":["ISTA Thesis"],"language":[{"iso":"eng"}],"article_processing_charge":"No","status":"public","publication_status":"published","has_accepted_license":"1","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"}],"date_published":"2025-02-03T00:00:00Z","oa_version":"Published Version","month":"02","date_created":"2025-01-31T17:04:40Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"oa":1,"_id":"18979","day":"03","type":"dissertation","supervisor":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"}],"corr_author":"1","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"year":"2025","citation":{"short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","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>","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>.","ista":"Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria.","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>","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>.","ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025."},"degree_awarded":"PhD","publisher":"Institute of Science and Technology Austria","OA_place":"publisher","date_updated":"2026-10-02T11:57:00Z","doi":"10.15479/at:ista:18979","publication_identifier":{"issn":["2663-337X"]},"related_material":{"record":[{"status":"public","id":"15091","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"18981","status":"public"}]},"file_date_updated":"2025-02-04T16:22:07Z","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","ddc":["514","004"],"file":[{"file_name":"Thesis.zip","date_created":"2025-01-31T16:58:30Z","file_size":11899491,"access_level":"closed","content_type":"application/zip","relation":"source_file","date_updated":"2025-01-31T16:58:30Z","file_id":"18983","creator":"odragano","checksum":"af6567e5d35e5eb330b8925ae37f1998"},{"date_created":"2025-02-04T16:22:07Z","file_name":"Thesis.pdf","file_size":8857514,"access_level":"open_access","checksum":"c3fef68e35b9dc2020b2ca6006da6343","content_type":"application/pdf","file_id":"19000","creator":"odragano","relation":"main_file","date_updated":"2025-02-04T16:22:07Z"}],"project":[{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"fulldoi":"https://doi.org/10.15479/at:ista:18979","author":[{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov","orcid":"0000-0003-0464-3823","first_name":"Ondrej","full_name":"Draganov, Ondrej"}],"title":"Structures and computations in topological data analysis","page":"140"},{"ec_funded":1,"external_id":{"arxiv":["2209.14993"]},"doi":"10.1016/j.jpaa.2025.108068","date_updated":"2026-10-02T11:57:00Z","scopus_import":"1","year":"2025","citation":{"mla":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10, 108068, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>.","apa":"Brown, A., &#38; Draganov, O. (2025). Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>","short":"A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).","chicago":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>.","ista":"Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure and Applied Algebra. 229(10), 108068.","ama":"Brown A, Draganov O. Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. 2025;229(10). doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025."},"department":[{"_id":"HeEd"}],"publisher":"Elsevier","volume":229,"OA_place":"publisher","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"fulldoi":"https://doi.org/10.1016/j.jpaa.2025.108068","file":[{"file_size":3090836,"file_name":"2025_JourPureAppliedAlgebra_Brown.pdf","date_created":"2025-12-30T07:55:08Z","checksum":"39bcad462278c9322ef810af7db67f56","success":1,"content_type":"application/pdf","relation":"main_file","date_updated":"2025-12-30T07:55:08Z","file_id":"20886","creator":"dernst","access_level":"open_access"}],"publication":"Journal of Pure and Applied Algebra","author":[{"full_name":"Brown, Adam","first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","last_name":"Brown"},{"full_name":"Draganov, Ondrej","first_name":"Ondrej","orcid":"0000-0003-0464-3823","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"}],"title":"Discrete microlocal Morse theory","publication_identifier":{"issn":["0022-4049"]},"related_material":{"record":[{"status":"public","id":"18981","relation":"earlier_version"}]},"issue":"10","file_date_updated":"2025-12-30T07:55:08Z","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35","ddc":["510"],"article_processing_charge":"Yes (via OA deal)","date_published":"2025-10-01T00:00:00Z","abstract":[{"text":"We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities.","lang":"eng"}],"publication_status":"published","status":"public","has_accepted_license":"1","oa_version":"Published Version","article_type":"original","language":[{"iso":"eng"}],"article_number":"108068","oa":1,"day":"01","_id":"20323","type":"journal_article","quality_controlled":"1","PlanS_conform":"1","corr_author":"1","intvolume":"       229","month":"10","date_created":"2025-09-10T05:40:09Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_type":"hybrid","arxiv":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"}},{"oa":1,"_id":"17170","type":"conference","day":"06","quality_controlled":"1","month":"06","intvolume":"       293","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","researchdata_availability":"no","date_created":"2024-06-25T11:45:58Z","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"arxiv":1,"article_processing_charge":"No","abstract":[{"text":"In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. - In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. - The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction. We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.","lang":"eng"}],"publication_status":"published","supplementarymaterial":"no","status":"public","has_accepted_license":"1","date_published":"2024-06-06T00:00:00Z","oa_version":"Published Version","alternative_title":["LIPIcs"],"language":[{"iso":"eng"}],"file":[{"file_size":20886142,"date_created":"2024-06-25T11:47:26Z","file_name":"LIPIcs.SoCG.2024.11.pdf","content_type":"application/pdf","creator":"cfillmor","date_updated":"2024-06-25T11:47:26Z","file_id":"17171","relation":"main_file","checksum":"6a2ddc8b51aa58f197a8b294750f1f8d","success":1,"access_level":"open_access"}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.11","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073","name":"Learning and triangulating manifolds via collapses"}],"conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11"},"publication":"40th International Symposium on Computational Geometry","author":[{"first_name":"Dominique","full_name":"Attali, Dominique","last_name":"Attali"},{"first_name":"Hana","orcid":"0000-0001-7841-0091","full_name":"Kourimska, Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","last_name":"Kourimska"},{"full_name":"Fillmore, Christopher D","first_name":"Christopher D","last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","last_name":"Ghosh","full_name":"Ghosh, Ishika","first_name":"Ishika"},{"first_name":"André","full_name":"Lieutier, André","last_name":"Lieutier"},{"last_name":"Stephenson","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","first_name":"Elizabeth R","orcid":"0000-0002-6862-208X","full_name":"Stephenson, Elizabeth R"},{"last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220","first_name":"Mathijs"}],"title":"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds","page":"11:1-11:19","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773164"]},"das_tickbox":"0","file_date_updated":"2024-06-25T11:47:26Z","ddc":["516"],"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nWintraecken, Mathijs: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d'Azur.","ec_funded":1,"doi":"10.4230/LIPIcs.SoCG.2024.11","date_updated":"2026-10-02T11:23:43Z","external_id":{"arxiv":["2206.10485"]},"scopus_import":"1","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"citation":{"chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” In <i>40th International Symposium on Computational Geometry</i>, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>.","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 11:1-11:19.","ama":"Attali D, Kourimska H, Fillmore CD, et al. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>","ieee":"D. Attali <i>et al.</i>, “Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293, p. 11:1-11:19.","mla":"Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19.","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>"},"year":"2024","volume":293,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik"},{"page":"29-48","title":"On angles in higher order Brillouin tessellations and related tilings in the plane","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"fulldoi":"https://doi.org/10.1007/s00454-023-00566-1","file":[{"access_level":"open_access","content_type":"application/pdf","creator":"dernst","date_updated":"2024-07-22T09:43:19Z","relation":"main_file","file_id":"17301","checksum":"b207b4e00f904e8ea8a30e24f0251f79","success":1,"file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","date_created":"2024-07-22T09:43:19Z","file_size":892019}],"publication":"Discrete and Computational Geometry","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"last_name":"Ghafari","full_name":"Ghafari, Mohadese","first_name":"Mohadese"},{"full_name":"Heiss, Teresa","first_name":"Teresa","orcid":"0000-0002-1780-2689","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"file_date_updated":"2024-07-22T09:43:19Z","acknowledgement":"Work by all authors but A. Garber is supported by the European Research Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation.","ddc":["510"],"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"das_tickbox":"0","pmid":1,"external_id":{"isi":["001060727600004"],"pmid":["39610762"],"arxiv":["2204.01076"]},"doi":"10.1007/s00454-023-00566-1","date_updated":"2026-10-02T11:45:50Z","scopus_import":"1","ec_funded":1,"citation":{"mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, vol. 72, Springer Nature, 2024, pp. 29–48, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete and Computational Geometry</i>, vol. 72. Springer Nature, pp. 29–48, 2024.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2024;72:29-48. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48."},"year":"2024","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","volume":72,"day":"01","_id":"14345","type":"journal_article","quality_controlled":"1","corr_author":"1","oa":1,"arxiv":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"intvolume":"        72","month":"07","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","researchdata_availability":"no","date_created":"2023-09-17T22:01:10Z","oa_version":"Published Version","isi":1,"article_processing_charge":"Yes (via OA deal)","date_published":"2024-07-01T00:00:00Z","abstract":[{"lang":"eng","text":"For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2  is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970))."}],"publication_status":"published","has_accepted_license":"1","supplementarymaterial":"yes","status":"public","article_type":"original","language":[{"iso":"eng"}]},{"_id":"17144","day":"01","type":"conference","quality_controlled":"1","oa":1,"arxiv":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"intvolume":"       293","month":"06","date_created":"2024-06-16T22:01:06Z","researchdata_availability":"no","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa_version":"Published Version","article_processing_charge":"No","date_published":"2024-06-01T00:00:00Z","status":"public","abstract":[{"text":"We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.","lang":"eng"}],"has_accepted_license":"1","publication_status":"published","supplementarymaterial":"no","language":[{"iso":"eng"}],"article_number":"69","alternative_title":["LIPIcs"],"title":"The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.69","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2024-06-14","location":"Athens, Greece"},"project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"grant_number":"M03073","name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"file":[{"checksum":"b40ff456c19294adb5d9613fcfd751c6","success":1,"content_type":"application/pdf","file_id":"17150","relation":"main_file","date_updated":"2024-06-17T08:33:40Z","creator":"dernst","access_level":"open_access","file_size":1612558,"date_created":"2024-06-17T08:33:40Z","file_name":"2024_LIPICS_Kourimska.pdf"}],"publication":"40th International Symposium on Computational Geometry","author":[{"id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","last_name":"Kourimska","first_name":"Hana","orcid":"0000-0001-7841-0091","full_name":"Kourimska, Hana"},{"last_name":"Lieutier","first_name":"André","full_name":"Lieutier, André"},{"full_name":"Wintraecken, Mathijs","first_name":"Mathijs","orcid":"0000-0002-7472-2220","last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"file_date_updated":"2024-06-17T08:33:40Z","ddc":["510"],"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nSupported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Cô d'Azur.\r\nWe are greatly indebted to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher Fillmore, and Elizabeth Stephenson for early discussions and all members of the Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape team (Inria) for the atmosphere in which this research was conducted.","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"das_tickbox":"0","external_id":{"arxiv":["2212.01118"]},"doi":"10.4230/LIPIcs.SoCG.2024.69","date_updated":"2026-10-02T11:46:39Z","scopus_import":"1","ec_funded":1,"citation":{"short":"H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","apa":"Kourimska, H., Lieutier, A., &#38; Wintraecken, M. (2024). The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>","mla":"Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>.","ieee":"H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","ama":"Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>","ista":"Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 69.","chicago":"Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>."},"year":"2024","department":[{"_id":"HeEd"}],"volume":293,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik"},{"article_number":"53","language":[{"iso":"eng"}],"alternative_title":["LIPIcs"],"oa_version":"Published Version","publication_status":"published","supplementarymaterial":"no","abstract":[{"text":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry.","lang":"eng"}],"status":"public","has_accepted_license":"1","date_published":"2024-06-01T00:00:00Z","article_processing_charge":"No","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","researchdata_availability":"no","date_created":"2024-06-16T22:01:06Z","month":"06","intvolume":"       293","quality_controlled":"1","_id":"17146","day":"01","type":"conference","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","volume":293,"department":[{"_id":"HeEd"}],"year":"2024","citation":{"chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>","ista":"Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 53.","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>.","short":"H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","apa":"Edelsbrunner, H., &#38; Pach, J. (2024). Maximum Betti numbers of Čech complexes. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>"},"scopus_import":"1","doi":"10.4230/LIPIcs.SoCG.2024.53","date_updated":"2026-10-02T11:48:01Z","external_id":{"arxiv":["2310.14801"]},"ec_funded":1,"ddc":["510"],"acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart for early discussions on the linked circles construction in three dimensions, and Gábor Tardos for helpful remarks and suggestions.","file_date_updated":"2024-06-17T08:46:33Z","related_material":{"record":[{"relation":"later_version","status":"public","id":"20657"}]},"das_tickbox":"0","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"title":"Maximum Betti numbers of Čech complexes","publication":"40th International Symposium on Computational Geometry","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","full_name":"Pach, János","first_name":"János"}],"file":[{"date_created":"2024-06-17T08:46:33Z","file_name":"2024_LIPICS_Edelsbrunner.pdf","file_size":766562,"access_level":"open_access","file_id":"17152","creator":"dernst","relation":"main_file","date_updated":"2024-06-17T08:46:33Z","content_type":"application/pdf","success":1,"checksum":"5442d44fb89d77477a87668d6e61aac9"}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.53","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11"}},{"year":"2024","citation":{"mla":"Edelsbrunner, Herbert, et al. “Brillouin Zones of Integer Lattices and Their Perturbations.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 38, no. 2, Society for Industrial and Applied Mathematics, 2024, pp. 1784–807, doi:<a href=\"https://doi.org/10.1137/22M1489071\">10.1137/22M1489071</a>.","short":"H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, M. Wintraecken, SIAM Journal on Discrete Mathematics 38 (2024) 1784–1807.","apa":"Edelsbrunner, H., Garber, A., Ghafaris, M., Heiss, T., Saghafiant, M., &#38; Wintraecken, M. (2024). Brillouin zones of integer lattices and their perturbations. <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/22M1489071\">https://doi.org/10.1137/22M1489071</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafaris, Teresa Heiss, Morteza Saghafiant, and Mathijs Wintraecken. “Brillouin Zones of Integer Lattices and Their Perturbations.” <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics, 2024. <a href=\"https://doi.org/10.1137/22M1489071\">https://doi.org/10.1137/22M1489071</a>.","ama":"Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M. Brillouin zones of integer lattices and their perturbations. <i>SIAM Journal on Discrete Mathematics</i>. 2024;38(2):1784-1807. doi:<a href=\"https://doi.org/10.1137/22M1489071\">10.1137/22M1489071</a>","ista":"Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M. 2024. Brillouin zones of integer lattices and their perturbations. SIAM Journal on Discrete Mathematics. 38(2), 1784–1807.","ieee":"H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, and M. Wintraecken, “Brillouin zones of integer lattices and their perturbations,” <i>SIAM Journal on Discrete Mathematics</i>, vol. 38, no. 2. Society for Industrial and Applied Mathematics, pp. 1784–1807, 2024."},"department":[{"_id":"HeEd"}],"publisher":"Society for Industrial and Applied Mathematics","volume":38,"external_id":{"arxiv":["2204.01077"],"isi":["001292728600001"]},"doi":"10.1137/22M1489071","date_updated":"2026-10-02T11:49:10Z","scopus_import":"1","ec_funded":1,"issue":"2","acknowledgement":"The second author is partially supported by the Alexander von Humboldt Foundation. The sixth author is supported by the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement 754411, and by Austrian Science Fund(FWF) grant M-3073. All other authors are supported by European Research Council (ERC) grant 788183, by the Wittgenstein Prize, by Austrian Science Fund (FWF) grant Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF) grant I 02979-N35.","publication_identifier":{"issn":["0895-4801"]},"das_tickbox":"0","page":"1784-1807","title":"Brillouin zones of integer lattices and their perturbations","project":[{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"},{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073","name":"Learning and triangulating manifolds via collapses"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"}],"fulldoi":"https://doi.org/10.1137/22M1489071","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Garber","full_name":"Garber, Alexey","first_name":"Alexey"},{"full_name":"Ghafaris, Mohadese","first_name":"Mohadese","last_name":"Ghafaris"},{"orcid":"0000-0002-1780-2689","first_name":"Teresa","full_name":"Heiss, Teresa","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Saghafiant, Morteza","first_name":"Morteza","last_name":"Saghafiant"},{"last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","first_name":"Mathijs","orcid":"0000-0002-7472-2220"}],"publication":"SIAM Journal on Discrete Mathematics","article_type":"original","language":[{"iso":"eng"}],"oa_version":"Preprint","isi":1,"article_processing_charge":"No","date_published":"2024-06-07T00:00:00Z","supplementarymaterial":"no","status":"public","publication_status":"published","abstract":[{"text":"For a locally finite set, 𝐴⊆ℝ𝑑\r\n, the 𝑘\r\nth Brillouin zone of 𝑎∈𝐴\r\n is the region of points 𝑥∈ℝ𝑑\r\n for which ‖𝑥−𝑎‖\r\n is the 𝑘\r\nth smallest among the Euclidean distances between 𝑥\r\n and the points in 𝐴\r\n. If 𝐴\r\n is a lattice, the 𝑘\r\nth Brillouin zones of the points in 𝐴\r\n are translates of each other, and together they tile space. Depending on the value of 𝑘\r\n, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in ℝ2\r\n, and the convergence of the maximum volume of a chamber to zero for the integer lattice.","lang":"eng"}],"arxiv":1,"intvolume":"        38","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2204.01077","open_access":"1"}],"month":"06","date_created":"2024-06-30T22:01:05Z","researchdata_availability":"no","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","day":"07","_id":"17190","type":"journal_article","corr_author":"1","quality_controlled":"1","oa":1}]