[{"oa_version":"Published Version","file":[{"file_id":"20885","file_name":"2025_LaMatematica_Chambers.pdf","date_created":"2025-12-30T07:52:58Z","relation":"main_file","date_updated":"2025-12-30T07:52:58Z","creator":"dernst","checksum":"e2043259194bfcdf3d74c4da8a5a853f","access_level":"open_access","content_type":"application/pdf","file_size":2678640,"success":1}],"OA_type":"hybrid","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"publication_identifier":{"eissn":["2730-9657"]},"OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","corr_author":"1","year":"2025","status":"public","publisher":"Springer Nature","ec_funded":1,"publication_status":"published","date_updated":"2026-04-07T11:42:48Z","date_published":"2025-12-01T00:00:00Z","article_type":"original","month":"12","PlanS_conform":"1","article_processing_charge":"Yes (via OA deal)","department":[{"_id":"HeEd"}],"oa":1,"ddc":["510"],"acknowledgement":"We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie Yan, and Tao Ju for sharing code to generate the examples. We further thank Abigail Thompson for discussion on the conjecture and James Damon for sharing his insight in singularity theory. We thank the reviewers for their detailed reviews, which helped to improve the exposition.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). Partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ and the European Research Council (ERC), grant no. 788183, ‘Alpha Shape Theory Extended’. The first author was supported in part by the National Science Foundation through grants DBI-1759807, CCF-1907612, and CCF-2444309. The fourth author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) M-3073, ANR grant StratMesh, ANR-24-CE48-1899, and the welcome package from IDEX of the Université Côte d’Azur, ANR-15-IDEX-01.","citation":{"ieee":"E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning or collapsing the medial axis is unstable,” <i>La Matematica</i>, vol. 4. Springer Nature, pp. 811–828, 2025.","mla":"Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>, vol. 4, Springer Nature, 2025, pp. 811–28, doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>.","ama":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. 2025;4:811-828. doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>","apa":"Chambers, E. W., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2025). Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>","ista":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica. 4, 811–828.","short":"E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica 4 (2025) 811–828.","chicago":"Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>."},"intvolume":"         4","type":"journal_article","publication":"La Matematica","_id":"20260","doi":"10.1007/s44007-025-00170-0","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","call_identifier":"H2020"},{"name":"Learning and triangulating manifolds via collapses","grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"quality_controlled":"1","page":"811-828","author":[{"last_name":"Chambers","first_name":"Erin Wolf","full_name":"Chambers, Erin Wolf"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","full_name":"Fillmore, Christopher D","last_name":"Fillmore"},{"id":"2D04F932-F248-11E8-B48F-1D18A9856A87","first_name":"Elizabeth R","full_name":"Stephenson, Elizabeth R","last_name":"Stephenson","orcid":"0000-0002-6862-208X"},{"first_name":"Mathijs","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","last_name":"Wintraecken"}],"scopus_import":"1","date_created":"2025-08-31T22:01:33Z","abstract":[{"text":"The medial axis of a set consists of the points in the ambient space without a unique closest point in the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a skeleton topologically equivalent to the original set. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities, various prunings of the medial axis have been proposed in the computational geometry community. Here, we examine one type of pruning, called burning. Because of the good experimental results it was hoped that the burning method of simplifying the medial axis would be stable. In this work, we show a simple example that dashes such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of a shape where the medial axis goes from collapsible to non-collapsible. More precisely, we consider the standard deformation retract from the closed ball to Bing’s house with two rooms, but stop just short of the point where Bing’s house becomes two dimensional. This way we obtain an isotopy from the 3-ball to a thickened version of Bing’s house. Under this isotopy, the medial axis goes from collapsible to non-collapsible. We stress that this isotopy can be made generic, in the sense of singularity theory, as developed by Arnol’d and Thom.","lang":"eng"}],"day":"01","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"dissertation_contains","id":"21021","status":"public"}]},"title":"Burning or collapsing the medial axis is unstable","volume":4,"file_date_updated":"2025-12-30T07:52:58Z"},{"doi":"10.3390/e27080854","_id":"20293","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"}],"quality_controlled":"1","author":[{"last_name":"Akopyan","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","first_name":"Arseniy"},{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"first_name":"Ziga","full_name":"Virk, Ziga","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","last_name":"Virk"},{"last_name":"Wagner","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","full_name":"Wagner, Hubert"}],"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.","citation":{"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>","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>","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>.","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.","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>.","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).","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."},"external_id":{"isi":["001557476000001"],"pmid":["40870326"]},"ddc":["500"],"intvolume":"        27","publication":"Entropy","type":"journal_article","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."}],"day":"01","date_created":"2025-09-07T22:01:33Z","DOAJ_listed":"1","title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence","language":[{"iso":"eng"}],"file_date_updated":"2025-09-08T07:55:48Z","volume":27,"pmid":1,"scopus_import":"1","isi":1,"corr_author":"1","has_accepted_license":"1","issue":"8","year":"2025","status":"public","file":[{"success":1,"access_level":"open_access","content_type":"application/pdf","checksum":"65c5399c4015d9c8abb8c7a96f3d7836","file_size":379340,"creator":"dernst","date_updated":"2025-09-08T07:55:48Z","relation":"main_file","date_created":"2025-09-08T07:55:48Z","file_name":"2025_Entropy_Akopyan.pdf","file_id":"20309"}],"oa_version":"Published Version","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"OA_type":"gold","OA_place":"publisher","publication_identifier":{"eissn":["1099-4300"]},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"Yes","department":[{"_id":"HeEd"}],"article_number":"854","oa":1,"publisher":"MDPI","ec_funded":1,"month":"08","article_type":"original","date_published":"2025-08-01T00:00:00Z","date_updated":"2025-09-30T14:32:31Z","publication_status":"published","PlanS_conform":"1"},{"OA_place":"publisher","publication_identifier":{"issn":["0022-4049"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"file_id":"20886","relation":"main_file","file_name":"2025_JourPureAppliedAlgebra_Brown.pdf","date_created":"2025-12-30T07:55:08Z","date_updated":"2025-12-30T07:55:08Z","success":1,"creator":"dernst","access_level":"open_access","content_type":"application/pdf","file_size":3090836,"checksum":"39bcad462278c9322ef810af7db67f56"}],"oa_version":"Published Version","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"OA_type":"hybrid","status":"public","year":"2025","issue":"10","has_accepted_license":"1","corr_author":"1","month":"10","date_published":"2025-10-01T00:00:00Z","article_type":"original","date_updated":"2025-12-30T07:55:21Z","publication_status":"published","PlanS_conform":"1","publisher":"Elsevier","ec_funded":1,"oa":1,"article_processing_charge":"Yes (via OA deal)","article_number":"108068","department":[{"_id":"HeEd"}],"publication":"Journal of Pure and Applied Algebra","type":"journal_article","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","citation":{"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>","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>","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>.","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.","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."},"external_id":{"arxiv":["2209.14993"]},"ddc":["510"],"arxiv":1,"intvolume":"       229","quality_controlled":"1","author":[{"last_name":"Brown","id":"70B7FDF6-608D-11E9-9333-8535E6697425","full_name":"Brown, Adam","first_name":"Adam"},{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","first_name":"Ondrej","last_name":"Draganov","orcid":"0000-0003-0464-3823"}],"doi":"10.1016/j.jpaa.2025.108068","_id":"20323","project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"scopus_import":"1","title":"Discrete microlocal Morse theory","language":[{"iso":"eng"}],"related_material":{"record":[{"id":"18981","relation":"earlier_version","status":"public"}]},"file_date_updated":"2025-12-30T07:55:08Z","volume":229,"abstract":[{"lang":"eng","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."}],"day":"01","date_created":"2025-09-10T05:40:09Z"},{"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.11380"}],"title":"Flips in two-dimensional hypertriangulations","volume":132,"date_created":"2025-10-19T22:01:31Z","day":"10","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"}],"scopus_import":"1","isi":1,"quality_controlled":"1","author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"first_name":"Alexey","full_name":"Garber, Alexey","last_name":"Garber"},{"last_name":"Ghafari","full_name":"Ghafari, Mohadese","first_name":"Mohadese"},{"id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa","full_name":"Heiss, Teresa","last_name":"Heiss","orcid":"0000-0002-1780-2689"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"_id":"20490","doi":"10.1016/j.ejc.2025.104248","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","publication":"European Journal of Combinatorics","external_id":{"arxiv":["2212.11380"],"isi":["001599061500002"]},"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.","citation":{"ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. 2025;132. doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2025). Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>","mla":"Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>, vol. 132, 104248, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>.","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” <i>European Journal of Combinatorics</i>, vol. 132. Elsevier, 2025.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2025).","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248."},"intvolume":"       132","arxiv":1,"oa":1,"article_processing_charge":"No","article_number":"104248","department":[{"_id":"HeEd"}],"date_updated":"2025-12-01T12:57:29Z","publication_status":"epub_ahead","article_type":"original","date_published":"2025-10-10T00:00:00Z","month":"10","publisher":"Elsevier","ec_funded":1,"year":"2025","status":"public","corr_author":"1","publication_identifier":{"issn":["0195-6698"]},"OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","OA_type":"green"},{"department":[{"_id":"HeEd"}],"article_processing_charge":"No","date_published":"2025-03-01T00:00:00Z","article_type":"original","month":"03","publication_status":"epub_ahead","date_updated":"2025-11-04T12:25:47Z","ec_funded":1,"publisher":"American Institute of Mathematical Sciences","year":"2025","status":"public","corr_author":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"repository","publication_identifier":{"eissn":["2639-8001"]},"OA_type":"green","oa_version":"Preprint","volume":8,"title":"Chromatic alpha complexes","language":[{"iso":"eng"}],"related_material":{"record":[{"id":"15091","relation":"earlier_version","status":"public"}]},"day":"01","abstract":[{"text":"Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.","lang":"eng"}],"date_created":"2025-11-02T23:01:33Z","scopus_import":"1","author":[{"last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","full_name":"Cultrera di Montesano, Sebastiano"},{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","first_name":"Ondrej","last_name":"Draganov","orcid":"0000-0003-0464-3823"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"page":"30-62","quality_controlled":"1","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"doi":"10.3934/fods.2025003","_id":"20585","type":"journal_article","publication":"Foundations of Data Science","arxiv":1,"intvolume":"         8","citation":{"short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2025) 30–62.","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences, 2025. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2025;8:30-62. doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2025). Chromatic alpha complexes. <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>Foundations of Data Science</i>, vol. 8. American Institute of Mathematical Sciences, pp. 30–62, 2025.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>, vol. 8, American Institute of Mathematical Sciences, 2025, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>."},"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.","external_id":{"arxiv":["2212.03128"]}},{"type":"journal_article","publication":"Discrete & Computational Geometry","arxiv":1,"external_id":{"arxiv":["2310.14801"],"isi":["001610592600001"]},"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.","citation":{"ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. 2025. doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>","apa":"Edelsbrunner, H., &#38; Pach, J. (2025). Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>.","short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry (2025).","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>.","ista":"Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry."},"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"last_name":"Pach","first_name":"János","full_name":"Pach, János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4"}],"quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"_id":"20657","doi":"10.1007/s00454-025-00796-5","isi":1,"scopus_import":"1","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"earlier_version","id":"17146","status":"public"}]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-025-00796-5"}],"title":"Maximum Betti numbers of Čech complexes","date_created":"2025-11-19T09:44:58Z","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. "}],"day":"10","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"OA_place":"publisher","OA_type":"hybrid","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"oa_version":"Published Version","status":"public","year":"2025","has_accepted_license":"1","corr_author":"1","PlanS_conform":"1","publication_status":"epub_ahead","date_updated":"2025-12-01T15:19:21Z","month":"11","article_type":"original","date_published":"2025-11-10T00:00:00Z","ec_funded":1,"publisher":"Springer Nature","oa":1,"department":[{"_id":"HeEd"}],"article_processing_charge":"Yes (via OA deal)"},{"oa_version":"Preprint","alternative_title":["LNCS"],"OA_type":"green","publication_identifier":{"eissn":["1611-3349"],"isbn":["9783032095435"],"issn":["0302-9743"]},"OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","year":"2025","publisher":"Springer Nature","publication_status":"published","date_updated":"2025-11-24T10:05:11Z","date_published":"2025-11-01T00:00:00Z","month":"11","article_processing_charge":"No","conference":{"name":"DGMM: Discrete Geometry and Mathematical Morphology","start_date":"2025-11-03","location":"Groningen, The Netherlands","end_date":"2025-11-06"},"department":[{"_id":"HeEd"}],"oa":1,"external_id":{"arxiv":["2504.14743"]},"citation":{"ieee":"H. Edelsbrunner, E. R. Stephenson, and M. H. Thoresen, “The mid-sphere cousin of the medial axis transform,” in <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>, Groningen, The Netherlands, 2025, vol. 16296, pp. 133–147.","mla":"Edelsbrunner, Herbert, et al. “The Mid-Sphere Cousin of the Medial Axis Transform.” <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>, vol. 16296, Springer Nature, 2025, pp. 133–47, doi:<a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">10.1007/978-3-032-09544-2_10</a>.","apa":"Edelsbrunner, H., Stephenson, E. R., &#38; Thoresen, M. H. (2025). The mid-sphere cousin of the medial axis transform. In <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i> (Vol. 16296, pp. 133–147). Groningen, The Netherlands: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">https://doi.org/10.1007/978-3-032-09544-2_10</a>","ama":"Edelsbrunner H, Stephenson ER, Thoresen MH. The mid-sphere cousin of the medial axis transform. In: <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>. Vol 16296. Springer Nature; 2025:133-147. doi:<a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">10.1007/978-3-032-09544-2_10</a>","ista":"Edelsbrunner H, Stephenson ER, Thoresen MH. 2025. The mid-sphere cousin of the medial axis transform. 4th International Joint Conference on Discrete Geometry and Mathematical Morphology. DGMM: Discrete Geometry and Mathematical Morphology, LNCS, vol. 16296, 133–147.","chicago":"Edelsbrunner, Herbert, Elizabeth R Stephenson, and Martin H Thoresen. “The Mid-Sphere Cousin of the Medial Axis Transform.” In <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>, 16296:133–47. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">https://doi.org/10.1007/978-3-032-09544-2_10</a>.","short":"H. Edelsbrunner, E.R. Stephenson, M.H. Thoresen, in:, 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, Springer Nature, 2025, pp. 133–147."},"intvolume":"     16296","arxiv":1,"type":"conference","publication":"4th International Joint Conference on Discrete Geometry and Mathematical Morphology","_id":"20658","doi":"10.1007/978-3-032-09544-2_10","quality_controlled":"1","page":"133-147","author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"last_name":"Stephenson","orcid":"0000-0002-6862-208X","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","first_name":"Elizabeth R","full_name":"Stephenson, Elizabeth R"},{"full_name":"Thoresen, Martin H","first_name":"Martin H","id":"47CB1472-F248-11E8-B48F-1D18A9856A87","last_name":"Thoresen"}],"scopus_import":"1","date_created":"2025-11-23T23:01:37Z","day":"01","abstract":[{"text":"The medial axis of a smoothly embedded surface in R^3 consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.","lang":"eng"}],"language":[{"iso":"eng"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2504.14743","open_access":"1"}],"title":"The mid-sphere cousin of the medial axis transform","volume":16296},{"oa":1,"article_processing_charge":"Yes (in subscription journal)","department":[{"_id":"HeEd"}],"conference":{"name":"ISSAC: International Symposium on Symbolic and Algebraic Computation","start_date":"2025-07-28","location":"Guanajuato, Mexico","end_date":"2025-08-01"},"publication_status":"published","date_updated":"2025-12-09T13:46:42Z","date_published":"2025-11-10T00:00:00Z","month":"11","publisher":"Association for Computing Machinery","status":"public","year":"2025","has_accepted_license":"1","corr_author":"1","publication_identifier":{"isbn":["9798400720758"]},"OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","file":[{"date_updated":"2025-12-09T13:43:17Z","checksum":"1c299cca165a20e2518afe4fda63dbf1","file_size":761617,"access_level":"open_access","content_type":"application/pdf","creator":"dernst","success":1,"file_id":"20751","date_created":"2025-12-09T13:43:17Z","file_name":"2025_ISSAC_GonzalezDiaz.pdf","relation":"main_file"}],"OA_type":"hybrid","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"language":[{"iso":"eng"}],"title":"Additive partial matchings for persistent homology","file_date_updated":"2025-12-09T13:43:17Z","date_created":"2025-12-07T23:02:01Z","abstract":[{"text":"Persistence modules (defined as a sequence of vector spaces and linear maps between them) are a key tool in topological data analysis. They are easy to interpret and fast to compute. However, when considering persistence maps (i.e. maps between persistence modules), these properties are lost. We propose a new invariant for persistence maps consisting of a partial matching such that: it is easy to interpret, it is more discriminative than the image of the persistence map, and can be calculated with cubical complexity.","lang":"eng"}],"day":"10","scopus_import":"1","quality_controlled":"1","author":[{"last_name":"Gonzalez-Diaz","first_name":"Rocio","full_name":"Gonzalez-Diaz, Rocio"},{"orcid":"0000-0003-2449-1433","last_name":"Soriano Trigueros","first_name":"Manuel","full_name":"Soriano Trigueros, Manuel","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8"},{"last_name":"Torras-Casas","full_name":"Torras-Casas, Alvaro","first_name":"Alvaro"}],"page":"188-196","_id":"20729","doi":"10.1145/3747199.3747561","publication":"Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation","type":"conference","ddc":["510"],"acknowledgement":"Álvaro Torras-Casas contract is funded by the French Agence Nationale de la Recherche through the project reference ANR-22-CPJ1-0047-01. Rocio Gonzalez-Diaz is partially funded by the European Union under grant agreement no. 101070028-2 (REXASI-PRO).","citation":{"chicago":"Gonzalez-Diaz, Rocio, Manuel Soriano Trigueros, and Alvaro Torras-Casas. “Additive Partial Matchings for Persistent Homology.” In <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>, 188–96. Association for Computing Machinery, 2025. <a href=\"https://doi.org/10.1145/3747199.3747561\">https://doi.org/10.1145/3747199.3747561</a>.","short":"R. Gonzalez-Diaz, M. Soriano Trigueros, A. Torras-Casas, in:, Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation, Association for Computing Machinery, 2025, pp. 188–196.","ista":"Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. 2025. Additive partial matchings for persistent homology. Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation. ISSAC: International Symposium on Symbolic and Algebraic Computation, 188–196.","apa":"Gonzalez-Diaz, R., Soriano Trigueros, M., &#38; Torras-Casas, A. (2025). Additive partial matchings for persistent homology. In <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i> (pp. 188–196). Guanajuato, Mexico: Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3747199.3747561\">https://doi.org/10.1145/3747199.3747561</a>","ama":"Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. Additive partial matchings for persistent homology. In: <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>. Association for Computing Machinery; 2025:188-196. doi:<a href=\"https://doi.org/10.1145/3747199.3747561\">10.1145/3747199.3747561</a>","ieee":"R. Gonzalez-Diaz, M. Soriano Trigueros, and A. Torras-Casas, “Additive partial matchings for persistent homology,” in <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>, Guanajuato, Mexico, 2025, pp. 188–196.","mla":"Gonzalez-Diaz, Rocio, et al. “Additive Partial Matchings for Persistent Homology.” <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>, Association for Computing Machinery, 2025, pp. 188–96, doi:<a href=\"https://doi.org/10.1145/3747199.3747561\">10.1145/3747199.3747561</a>."}},{"has_accepted_license":"1","corr_author":"1","status":"public","year":"2024","file":[{"creator":"dernst","checksum":"d493df5088c222b88d9ca46b623ad0ee","content_type":"application/pdf","file_size":476896,"access_level":"open_access","success":1,"date_updated":"2025-01-09T07:39:41Z","file_name":"2024_JourApplCompTopo_Biswas.pdf","date_created":"2025-01-09T07:39:41Z","relation":"main_file","file_id":"18783"}],"oa_version":"Published Version","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"OA_type":"hybrid","OA_place":"publisher","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"Yes (via OA deal)","department":[{"_id":"HeEd"}],"oa":1,"publisher":"Springer Nature","ec_funded":1,"article_type":"original","date_published":"2024-10-01T00:00:00Z","month":"10","publication_status":"published","date_updated":"2026-04-07T12:58:47Z","doi":"10.1007/s41468-023-00126-9","_id":"13182","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"name":"Persistent Homology, Algorithms and Stochastic Geometry","grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342"}],"quality_controlled":"1","page":"1101-1119","author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","full_name":"Biswas, Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890"},{"first_name":"Sebastiano","full_name":"Cultrera Di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","last_name":"Cultrera Di Montesano"},{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"citation":{"apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. 2024;8:1101-1119. doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 1101–19, doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 1101–1119, 2024.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 1101–1119.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 8, 1101–1119."},"acknowledgement":"Open access funding provided by Austrian Science Fund (FWF). 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. The authors of this paper thank anonymous reviewers for their constructive criticism and Monika Henzinger for detailed comments on an earlier version of this paper.","ddc":["000"],"external_id":{"pmid":["39678706"]},"intvolume":"         8","publication":"Journal of Applied and Computational Topology","type":"journal_article","abstract":[{"text":"We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram.","lang":"eng"}],"day":"01","date_created":"2023-07-02T22:00:44Z","title":"Geometric characterization of the persistence of 1D maps","language":[{"iso":"eng"}],"related_material":{"record":[{"id":"15094","relation":"dissertation_contains","status":"public"}]},"file_date_updated":"2025-01-09T07:39:41Z","volume":8,"pmid":1,"scopus_import":"1"},{"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"last_name":"Ghafari","first_name":"Mohadese","full_name":"Ghafari, Mohadese"},{"first_name":"Teresa","full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1780-2689","last_name":"Heiss"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"page":"29-48","quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","call_identifier":"FWF","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"}],"doi":"10.1007/s00454-023-00566-1","_id":"14345","type":"journal_article","publication":"Discrete and Computational Geometry","arxiv":1,"intvolume":"        72","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.","citation":{"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.","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>.","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>","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>","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.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>."},"ddc":["510"],"external_id":{"isi":["001060727600004"],"arxiv":["2204.01076"],"pmid":["39610762"]},"file_date_updated":"2024-07-22T09:43:19Z","volume":72,"title":"On angles in higher order Brillouin tessellations and related tilings in the plane","language":[{"iso":"eng"}],"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))."}],"day":"01","date_created":"2023-09-17T22:01:10Z","isi":1,"scopus_import":"1","pmid":1,"year":"2024","status":"public","corr_author":"1","has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"file":[{"checksum":"b207b4e00f904e8ea8a30e24f0251f79","file_size":892019,"access_level":"open_access","content_type":"application/pdf","creator":"dernst","success":1,"date_updated":"2024-07-22T09:43:19Z","date_created":"2024-07-22T09:43:19Z","file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","relation":"main_file","file_id":"17301"}],"oa_version":"Published Version","oa":1,"department":[{"_id":"HeEd"}],"article_processing_charge":"Yes (via OA deal)","month":"07","date_published":"2024-07-01T00:00:00Z","article_type":"original","publication_status":"published","date_updated":"2025-04-23T08:41:59Z","ec_funded":1,"publisher":"Springer Nature"},{"date_created":"2024-01-28T23:01:43Z","day":"06","abstract":[{"text":"A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a probabilistic FPT-approach in the number of popular faces.","lang":"eng"}],"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2202.12175"}],"title":"Removing popular faces in curve arrangements","volume":14466,"scopus_import":"1","isi":1,"_id":"14888","doi":"10.1007/978-3-031-49275-4_2","quality_controlled":"1","page":"18-33","author":[{"first_name":"Phoebe","full_name":"De Nooijer, Phoebe","last_name":"De Nooijer"},{"last_name":"Terziadis","full_name":"Terziadis, Soeren","first_name":"Soeren"},{"last_name":"Weinberger","full_name":"Weinberger, Alexandra","first_name":"Alexandra"},{"orcid":"0000-0002-6660-1322","last_name":"Masárová","first_name":"Zuzana","full_name":"Masárová, Zuzana","id":"45CFE238-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Mchedlidze","first_name":"Tamara","full_name":"Mchedlidze, Tamara"},{"full_name":"Löffler, Maarten","first_name":"Maarten","last_name":"Löffler"},{"first_name":"Günter","full_name":"Rote, Günter","last_name":"Rote"}],"external_id":{"arxiv":["2202.12175"],"isi":["001207942000002"]},"acknowledgement":"This work was initiated at the 16th European Research Week on Geometric Graphs in Strobl in 2019. A.W. is supported by the Austrian Science Fund (FWF): W1230. S.T. has been funded by the Vienna Science and Technology Fund (WWTF) [10.47379/ICT19035]. A preliminary version of this work has been presented at the 38th European Workshop on Computational Geometry (EuroCG 2022) in Perugia [9]. A full version of this paper, which includes appendices but is otherwise identical, is available as a technical report [10].","citation":{"short":"P. De Nooijer, S. Terziadis, A. Weinberger, Z. Masárová, T. Mchedlidze, M. Löffler, G. Rote, in:, 31st International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 18–33.","chicago":"De Nooijer, Phoebe, Soeren Terziadis, Alexandra Weinberger, Zuzana Masárová, Tamara Mchedlidze, Maarten Löffler, and Günter Rote. “Removing Popular Faces in Curve Arrangements.” In <i>31st International Symposium on Graph Drawing and Network Visualization</i>, 14466:18–33. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/978-3-031-49275-4_2\">https://doi.org/10.1007/978-3-031-49275-4_2</a>.","ista":"De Nooijer P, Terziadis S, Weinberger A, Masárová Z, Mchedlidze T, Löffler M, Rote G. 2024. Removing popular faces in curve arrangements. 31st International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LNCS, vol. 14466, 18–33.","ama":"De Nooijer P, Terziadis S, Weinberger A, et al. Removing popular faces in curve arrangements. In: <i>31st International Symposium on Graph Drawing and Network Visualization</i>. Vol 14466. Springer Nature; 2024:18-33. doi:<a href=\"https://doi.org/10.1007/978-3-031-49275-4_2\">10.1007/978-3-031-49275-4_2</a>","apa":"De Nooijer, P., Terziadis, S., Weinberger, A., Masárová, Z., Mchedlidze, T., Löffler, M., &#38; Rote, G. (2024). Removing popular faces in curve arrangements. In <i>31st International Symposium on Graph Drawing and Network Visualization</i> (Vol. 14466, pp. 18–33). Isola delle Femmine, Palermo, Italy: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-031-49275-4_2\">https://doi.org/10.1007/978-3-031-49275-4_2</a>","mla":"De Nooijer, Phoebe, et al. “Removing Popular Faces in Curve Arrangements.” <i>31st International Symposium on Graph Drawing and Network Visualization</i>, vol. 14466, Springer Nature, 2024, pp. 18–33, doi:<a href=\"https://doi.org/10.1007/978-3-031-49275-4_2\">10.1007/978-3-031-49275-4_2</a>.","ieee":"P. De Nooijer <i>et al.</i>, “Removing popular faces in curve arrangements,” in <i>31st International Symposium on Graph Drawing and Network Visualization</i>, Isola delle Femmine, Palermo, Italy, 2024, vol. 14466, pp. 18–33."},"intvolume":"     14466","arxiv":1,"type":"conference","publication":"31st International Symposium on Graph Drawing and Network Visualization","article_processing_charge":"No","conference":{"end_date":"2023-09-22","location":"Isola delle Femmine, Palermo, Italy","start_date":"2023-09-20","name":"GD: Graph Drawing and Network Visualization"},"department":[{"_id":"UlWa"},{"_id":"HeEd"}],"oa":1,"publisher":"Springer Nature","publication_status":"published","date_updated":"2025-09-04T11:52:35Z","month":"01","date_published":"2024-01-06T00:00:00Z","year":"2024","status":"public","oa_version":"Preprint","alternative_title":["LNCS"],"publication_identifier":{"eissn":["1611-3349"],"isbn":["9783031492747"],"issn":["0302-9743"]},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345"},{"ec_funded":1,"publisher":"Springer Nature","date_published":"2024-01-01T00:00:00Z","month":"01","publication_status":"published","date_updated":"2026-04-16T09:12:37Z","department":[{"_id":"HeEd"}],"conference":{"name":"GD: Graph Drawing and Network Visualization","start_date":"2023-09-20","location":"Isola delle Femmine, Palermo, Italy","end_date":"2023-09-22"},"article_processing_charge":"No","oa":1,"alternative_title":["LNCS"],"oa_version":"Preprint","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publication_identifier":{"eissn":["1611-3349"],"eisbn":["9783031492723"],"issn":["0302-9743"],"isbn":["9783031492716"]},"year":"2024","status":"public","isi":1,"scopus_import":"1","day":"01","abstract":[{"lang":"eng","text":"We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n-1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs."}],"date_created":"2024-02-18T23:01:03Z","volume":14465,"title":"Decomposition of geometric graphs into star-forests","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.13201","open_access":"1"}],"language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","relation":"later_version","id":"21253"}]},"arxiv":1,"intvolume":"     14465","acknowledgement":"János Pach’s Research partially supported by European Research Council (ERC), grant “GeoScape” No. 882971 and by the Hungarian Science Foundation (NKFIH), grant K-131529. Work by Morteza Saghafian is partially supported by the European Research Council (ERC), grant No. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31.","citation":{"ista":"Pach J, Saghafian M, Schnider P. 2024. Decomposition of geometric graphs into star-forests. 31st International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LNCS, vol. 14465, 339–346.","short":"J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” In <i>31st International Symposium on Graph Drawing and Network Visualization</i>, 14465:339–46. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>.","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” in <i>31st International Symposium on Graph Drawing and Network Visualization</i>, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp. 339–346.","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>31st International Symposium on Graph Drawing and Network Visualization</i>, vol. 14465, Springer Nature, 2024, pp. 339–46, doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>.","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. In: <i>31st International Symposium on Graph Drawing and Network Visualization</i>. Vol 14465. Springer Nature; 2024:339-346. doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>","apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2024). Decomposition of geometric graphs into star-forests. In <i>31st International Symposium on Graph Drawing and Network Visualization</i> (Vol. 14465, pp. 339–346). Isola delle Femmine, Palermo, Italy: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>"},"external_id":{"isi":["001207939600023"],"arxiv":["2306.13201"]},"publication":"31st International Symposium on Graph Drawing and Network Visualization","type":"conference","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"doi":"10.1007/978-3-031-49272-3_23","_id":"15012","page":"339-346","author":[{"first_name":"János","full_name":"Pach, János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian"},{"last_name":"Schnider","first_name":"Patrick","full_name":"Schnider, Patrick"}],"quality_controlled":"1"},{"article_number":"2212.03128","department":[{"_id":"HeEd"}],"date_created":"2024-03-08T10:13:59Z","abstract":[{"text":"Motivated by applications in the medical sciences, we study finite chromatic\r\nsets in Euclidean space from a topological perspective. Based on the persistent\r\nhomology for images, kernels and cokernels, we design provably stable\r\nhomological quantifiers that describe the geometric micro- and macro-structure\r\nof how the color classes mingle. These can be efficiently computed using\r\nchromatic variants of Delaunay and alpha complexes, and code that does these\r\ncomputations is provided.","lang":"eng"}],"day":"07","article_processing_charge":"No","related_material":{"record":[{"status":"public","id":"20585","relation":"later_version"},{"relation":"dissertation_contains","id":"18979","status":"public"},{"id":"15094","relation":"dissertation_contains","status":"public"}]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2212.03128"}],"oa":1,"language":[{"iso":"eng"}],"title":"Chromatic alpha complexes","publication_status":"draft","date_updated":"2026-04-07T12:58:47Z","month":"02","date_published":"2024-02-07T00:00:00Z","corr_author":"1","_id":"15091","doi":"10.48550/arXiv.2212.03128","author":[{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0003-0464-3823","last_name":"Draganov","first_name":"Ondrej","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"status":"public","year":"2024","arxiv":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"oa_version":"Preprint","external_id":{"arxiv":["2212.03128"]},"citation":{"short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2212.03128\">https://doi.org/10.48550/arXiv.2212.03128</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv, 2212.03128.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2212.03128\">10.48550/arXiv.2212.03128</a>","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Chromatic alpha complexes. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2212.03128\">https://doi.org/10.48550/arXiv.2212.03128</a>","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>arXiv</i>. .","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>ArXiv</i>, 2212.03128, doi:<a href=\"https://doi.org/10.48550/arXiv.2212.03128\">10.48550/arXiv.2212.03128</a>."},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","publication":"arXiv","type":"preprint","OA_place":"repository"},{"publisher":"Society for Industrial and Applied Mathematics","ec_funded":1,"month":"01","date_published":"2024-01-04T00:00:00Z","publication_status":"published","date_updated":"2026-04-07T12:58:47Z","article_processing_charge":"No","department":[{"_id":"HeEd"},{"_id":"MoHe"}],"conference":{"name":"SODA: Symposium on Discrete Algorithms","start_date":"2024-01-07","location":"Alexandria, VA, USA","end_date":"2024-01-10"},"oa":1,"oa_version":"Preprint","publication_identifier":{"eisbn":["9781611977912"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","editor":[{"last_name":"Woodruff","full_name":"Woodruff, David P.","first_name":"David P."}],"corr_author":"1","year":"2024","status":"public","scopus_import":"1","day":"04","abstract":[{"lang":"eng","text":"We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right."}],"date_created":"2024-03-08T10:27:39Z","title":"Dynamically maintaining the persistent homology of time series","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2311.01115"}],"language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"15094"}]},"acknowledgement":"The  first  and  second  authors  are  funded  by  the  European  Research  Council  under  the European Union’s Horizon 2020 research and innovation programme, ERC grant no. 788183,“Alpha Shape Theory Extended (Alpha)”, by the Wittgenstein Prize, FWF grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.The third author received funding by the European Research Council under the European Union’s Horizon 2020research  and  innovation  programme,  ERC  grant  no.  101019564,  “The  Design  of  Modern  Fully  Dynamic  DataStructures (MoDynStruct)”, and by the Austrian Science Fund through the Wittgenstein Prize with FWF grant no. Z 422-N, and also by FWF grant no. I 5982-N, and by FWF grant no. P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024.  The fourth author is funded by the Vienna Graduate School on Computational Optimization, FWF project no. W1260-N35.","citation":{"apa":"Cultrera di Montesano, S., Edelsbrunner, H., Henzinger, M., &#38; Ost, L. (2024). Dynamically maintaining the persistent homology of time series. In D. P. Woodruff (Ed.), <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)</i> (pp. 243–295). Alexandria, VA, USA: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611977912.11\">https://doi.org/10.1137/1.9781611977912.11</a>","ama":"Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. Dynamically maintaining the persistent homology of time series. In: Woodruff DP, ed. <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)</i>. Society for Industrial and Applied Mathematics; 2024:243-295. doi:<a href=\"https://doi.org/10.1137/1.9781611977912.11\">10.1137/1.9781611977912.11</a>","mla":"Cultrera di Montesano, Sebastiano, et al. “Dynamically Maintaining the Persistent Homology of Time Series.” <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)</i>, edited by David P. Woodruff, Society for Industrial and Applied Mathematics, 2024, pp. 243–95, doi:<a href=\"https://doi.org/10.1137/1.9781611977912.11\">10.1137/1.9781611977912.11</a>.","ieee":"S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, and L. Ost, “Dynamically maintaining the persistent homology of time series,” in <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)</i>, Alexandria, VA, USA, 2024, pp. 243–295.","chicago":"Cultrera di Montesano, Sebastiano, Herbert Edelsbrunner, Monika Henzinger, and Lara Ost. “Dynamically Maintaining the Persistent Homology of Time Series.” In <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)</i>, edited by David P. Woodruff, 243–95. Society for Industrial and Applied Mathematics, 2024. <a href=\"https://doi.org/10.1137/1.9781611977912.11\">https://doi.org/10.1137/1.9781611977912.11</a>.","short":"S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, L. Ost, in:, D.P. Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Society for Industrial and Applied Mathematics, 2024, pp. 243–295.","ista":"Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. 2024. Dynamically maintaining the persistent homology of time series. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SODA: Symposium on Discrete Algorithms, 243–295."},"external_id":{"arxiv":["2311.01115"]},"arxiv":1,"publication":"Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)","type":"conference","doi":"10.1137/1.9781611977912.11","_id":"15093","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"_id":"bd9ca328-d553-11ed-ba76-dc4f890cfe62","name":"The design and evaluation of modern fully dynamic data structures","grant_number":"101019564","call_identifier":"H2020"},{"grant_number":"Z00422","name":"Efficient algorithms","_id":"34def286-11ca-11ed-8bc3-da5948e1613c"},{"name":"Fast Algorithms for a Reactive Network Layer","grant_number":"P33775","_id":"bd9e3a2e-d553-11ed-ba76-8aa684ce17fe"}],"quality_controlled":"1","page":"243 - 295","author":[{"full_name":"Cultrera di Montesano, Sebastiano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","last_name":"Cultrera di Montesano"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Monika H","full_name":"Henzinger, Monika H","id":"540c9bbd-f2de-11ec-812d-d04a5be85630","orcid":"0000-0002-5008-6530","last_name":"Henzinger"},{"last_name":"Ost","first_name":"Lara","full_name":"Ost, Lara"}]},{"type":"dissertation","ddc":["514","500","516"],"citation":{"mla":"Cultrera di Montesano, Sebastiano. <i>Persistence and Morse Theory for Discrete Geometric Structures</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:15094\">10.15479/at:ista:15094</a>.","ieee":"S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric structures,” Institute of Science and Technology Austria, 2024.","ama":"Cultrera di Montesano S. Persistence and Morse theory for discrete geometric structures. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:15094\">10.15479/at:ista:15094</a>","apa":"Cultrera di Montesano, S. (2024). <i>Persistence and Morse theory for discrete geometric structures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:15094\">https://doi.org/10.15479/at:ista:15094</a>","ista":"Cultrera di Montesano S. 2024. Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria.","short":"S. Cultrera di Montesano, Persistence and Morse Theory for Discrete Geometric Structures, Institute of Science and Technology Austria, 2024.","chicago":"Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete Geometric Structures.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:15094\">https://doi.org/10.15479/at:ista:15094</a>."},"author":[{"first_name":"Sebastiano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","last_name":"Cultrera di Montesano"}],"page":"108","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"grant_number":"I4887","name":"Persistent Homology, Algorithms and Stochastic Geometry","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"_id":"15094","doi":"10.15479/at:ista:15094","supervisor":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"}],"file_date_updated":"2024-03-14T14:14:35Z","language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","id":"15091","relation":"part_of_dissertation"},{"status":"public","id":"11660","relation":"part_of_dissertation"},{"id":"15090","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"15093","relation":"part_of_dissertation"},{"id":"13182","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"11658"}]},"title":"Persistence and Morse theory for discrete geometric structures","date_created":"2024-03-08T15:28:10Z","abstract":[{"lang":"eng","text":"Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in\r\ndiscrete geometry that have captivated mathematicians for centuries, if not millennia. This\r\nthesis seeks to cast new light on these structures by illustrating specific instances where a\r\ntopological perspective, specifically through discrete Morse theory and persistent homology,\r\nprovides valuable insights.\r\n\r\nAt first glance, the topology of these geometric objects might seem uneventful: point sets\r\nessentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which\r\nis a contractible space, and the topology of a network primarily involves the enumeration\r\nof connected components and cycles within the network. However, beneath this apparent\r\nsimplicity, there lies an array of intriguing structures, a small subset of which will be uncovered\r\nin this thesis.\r\n\r\nFocused on three case studies, each addressing one of the mentioned objects, this work\r\nwill showcase connections that intertwine topology with diverse fields such as combinatorial\r\ngeometry, algorithms and data structures, and emerging applications like spatial biology.\r\n\r\n"}],"day":"08","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publication_identifier":{"issn":["2663-337X"]},"OA_place":"publisher","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)","image":"/images/cc_by_nc_sa.png","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)"},"oa_version":"Published Version","alternative_title":["ISTA Thesis"],"file":[{"file_id":"15112","date_created":"2024-03-14T08:55:07Z","file_name":"Thesis Sebastiano.pdf","relation":"main_file","date_updated":"2024-03-14T08:55:07Z","checksum":"1e468bfa42a7dcf04d89f4dadc621c87","content_type":"application/pdf","file_size":4106872,"access_level":"open_access","creator":"scultrer","success":1},{"date_updated":"2024-03-14T14:14:35Z","file_size":4746234,"checksum":"bcbd213490f5a7e68855a092bbce93f1","content_type":"application/zip","access_level":"closed","creator":"scultrer","file_id":"15113","relation":"source_file","date_created":"2024-03-14T08:56:24Z","file_name":"Thesis (1).zip"}],"year":"2024","status":"public","degree_awarded":"PhD","corr_author":"1","has_accepted_license":"1","date_updated":"2026-04-07T12:58:48Z","publication_status":"published","month":"03","date_published":"2024-03-08T00:00:00Z","ec_funded":1,"publisher":"Institute of Science and Technology Austria","oa":1,"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"article_processing_charge":"No"},{"_id":"15247","doi":"10.1016/j.jcta.2024.105889","author":[{"last_name":"Frankl","full_name":"Frankl, Peter","first_name":"Peter"},{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János","full_name":"Pach, János","last_name":"Pach"},{"last_name":"Pálvölgyi","first_name":"Dömötör","full_name":"Pálvölgyi, Dömötör"}],"quality_controlled":"1","intvolume":"       206","arxiv":1,"external_id":{"isi":["001217739200001"],"arxiv":["2310.16701"]},"ddc":["510"],"citation":{"ieee":"P. Frankl, J. Pach, and D. Pálvölgyi, “Odd-sunflowers,” <i>Journal of Combinatorial Theory, Series A</i>, vol. 206, no. 8. Elsevier, 2024.","mla":"Frankl, Peter, et al. “Odd-Sunflowers.” <i>Journal of Combinatorial Theory, Series A</i>, vol. 206, no. 8, 105889, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">10.1016/j.jcta.2024.105889</a>.","apa":"Frankl, P., Pach, J., &#38; Pálvölgyi, D. (2024). Odd-sunflowers. <i>Journal of Combinatorial Theory, Series A</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">https://doi.org/10.1016/j.jcta.2024.105889</a>","ama":"Frankl P, Pach J, Pálvölgyi D. Odd-sunflowers. <i>Journal of Combinatorial Theory, Series A</i>. 2024;206(8). doi:<a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">10.1016/j.jcta.2024.105889</a>","ista":"Frankl P, Pach J, Pálvölgyi D. 2024. Odd-sunflowers. Journal of Combinatorial Theory, Series A. 206(8), 105889.","chicago":"Frankl, Peter, János Pach, and Dömötör Pálvölgyi. “Odd-Sunflowers.” <i>Journal of Combinatorial Theory, Series A</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">https://doi.org/10.1016/j.jcta.2024.105889</a>.","short":"P. Frankl, J. Pach, D. Pálvölgyi, Journal of Combinatorial Theory, Series A 206 (2024)."},"acknowledgement":"We are grateful to Balázs Keszegh, and to the members of the Miklós Schweitzer Competition committee of 2022 for valuable discussions, and Shira Zerbib for pointing out several important mathematical typos.","type":"journal_article","publication":"Journal of Combinatorial Theory, Series A","date_created":"2024-03-31T22:01:11Z","day":"01","abstract":[{"lang":"eng","text":"Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant <2 such that every family of subsets of an n-element set that contains no odd-sunflower consists of at most n sets. We construct such families of size at least 1.5021n. We also characterize minimal odd-sunflowers of triples."}],"volume":206,"file_date_updated":"2025-01-09T08:37:20Z","language":[{"iso":"eng"}],"title":"Odd-sunflowers","isi":1,"scopus_import":"1","corr_author":"1","has_accepted_license":"1","issue":"8","year":"2024","status":"public","OA_type":"hybrid","tmp":{"short":"CC BY-NC (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","image":"/images/cc_by_nc.png","name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)"},"oa_version":"Published Version","file":[{"creator":"dernst","checksum":"ffc29d65e712849f0d31009271e06a63","content_type":"application/pdf","file_size":366029,"access_level":"open_access","success":1,"date_updated":"2025-01-09T08:37:20Z","file_name":"2024_JourCombiTheoryA_Frankl.pdf","date_created":"2025-01-09T08:37:20Z","relation":"main_file","file_id":"18791"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_identifier":{"issn":["0097-3165"],"eissn":["1096-0899"]},"OA_place":"publisher","article_number":"105889","department":[{"_id":"HeEd"}],"article_processing_charge":"No","oa":1,"publisher":"Elsevier","date_updated":"2025-09-04T13:20:39Z","publication_status":"published","date_published":"2024-08-01T00:00:00Z","article_type":"original","month":"08"},{"year":"2024","status":"public","corr_author":"1","has_accepted_license":"1","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","file":[{"success":1,"checksum":"0ee15c1493a6413cf356ab2f32c81a9e","access_level":"open_access","file_size":522831,"content_type":"application/pdf","creator":"dernst","date_updated":"2025-04-23T08:01:36Z","relation":"main_file","date_created":"2025-04-23T08:01:36Z","file_name":"2024_JourApplCompTopo_BiswasRa.pdf","file_id":"19612"}],"OA_type":"hybrid","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"oa":1,"article_processing_charge":"Yes (via OA deal)","department":[{"_id":"HeEd"}],"publication_status":"published","date_updated":"2025-05-14T09:27:57Z","article_type":"original","date_published":"2024-09-01T00:00:00Z","month":"09","publisher":"Springer Nature","ec_funded":1,"quality_controlled":"1","page":"557-578","author":[{"first_name":"Ranita","full_name":"Biswas, Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890","last_name":"Biswas"},{"first_name":"Sebastiano","full_name":"Cultrera Di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","last_name":"Cultrera Di Montesano"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"_id":"15380","doi":"10.1007/s41468-024-00173-w","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"publication":"Journal of Applied and Computational Topology","type":"journal_article","ddc":["510"],"external_id":{"pmid":["39308789"]},"citation":{"short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 8, 557–578.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. 2024;8:557-578. doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 557–578, 2024.","mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 557–78, doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>."},"acknowledgement":"The authors thank Uli Wagner and Emo Welzl for comments on an earlier version of this paper, and for pointing out related work in the prior literature.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","intvolume":"         8","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"earlier_version","id":"11658","status":"public"}]},"title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","volume":8,"file_date_updated":"2025-04-23T08:01:36Z","date_created":"2024-05-12T22:01:03Z","abstract":[{"lang":"eng","text":"The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements."}],"day":"01","scopus_import":"1","pmid":1},{"has_accepted_license":"1","year":"2024","status":"public","oa_version":"Published Version","alternative_title":["LIPIcs"],"file":[{"file_name":"2024_LIPICS_Kourimska.pdf","date_created":"2024-06-17T08:33:40Z","relation":"main_file","file_id":"17150","creator":"dernst","access_level":"open_access","content_type":"application/pdf","checksum":"b40ff456c19294adb5d9613fcfd751c6","file_size":1612558,"success":1,"date_updated":"2024-06-17T08:33:40Z"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773164"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","conference":{"end_date":"2024-06-14","location":"Athens, Greece","name":"SoCG: Symposium on Computational Geometry"},"article_number":"69","department":[{"_id":"HeEd"}],"oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","ec_funded":1,"publication_status":"published","date_updated":"2025-04-15T07:16:58Z","date_published":"2024-06-01T00:00:00Z","month":"06","_id":"17144","doi":"10.4230/LIPIcs.SoCG.2024.69","project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"},{"grant_number":"754411","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073","name":"Learning and triangulating manifolds via collapses"}],"quality_controlled":"1","author":[{"last_name":"Kourimska","orcid":"0000-0001-7841-0091","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","full_name":"Kourimska, Hana"},{"full_name":"Lieutier, André","first_name":"André","last_name":"Lieutier"},{"last_name":"Wintraecken","orcid":"0000-0002-7472-2220","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","full_name":"Wintraecken, Mathijs"}],"external_id":{"arxiv":["2212.01118"]},"ddc":["510"],"citation":{"chicago":"Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>.","short":"H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","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.","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>","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>","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.","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>."},"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.","intvolume":"       293","arxiv":1,"type":"conference","publication":"40th International Symposium on Computational Geometry","date_created":"2024-06-16T22:01:06Z","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"}],"day":"01","language":[{"iso":"eng"}],"title":"The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms","volume":293,"file_date_updated":"2024-06-17T08:33:40Z","scopus_import":"1"},{"oa":1,"article_processing_charge":"No","department":[{"_id":"HeEd"}],"article_number":"76","conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11"},"date_updated":"2024-06-17T08:41:56Z","publication_status":"published","month":"06","date_published":"2024-06-01T00:00:00Z","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","year":"2024","status":"public","has_accepted_license":"1","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","file":[{"creator":"dernst","content_type":"application/pdf","file_size":1430896,"checksum":"fbad1de06383a6b7e8a1cb3e8c7205ce","access_level":"open_access","success":1,"date_updated":"2024-06-17T08:40:04Z","file_name":"2024_LIPICS_Rote.pdf","date_created":"2024-06-17T08:40:04Z","relation":"main_file","file_id":"17151"}],"alternative_title":["LIPIcs"],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"language":[{"iso":"eng"}],"title":"Grid peeling of parabolas","volume":293,"file_date_updated":"2024-06-17T08:40:04Z","date_created":"2024-06-16T22:01:06Z","day":"01","abstract":[{"lang":"eng","text":"Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes."}],"scopus_import":"1","quality_controlled":"1","author":[{"last_name":"Rote","first_name":"Günter","full_name":"Rote, Günter"},{"first_name":"Moritz","full_name":"Rüber, Moritz","last_name":"Rüber"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"_id":"17145","doi":"10.4230/LIPIcs.SoCG.2024.76","publication":"40th International Symposium on Computational Geometry","type":"conference","ddc":["510"],"external_id":{"arxiv":["2402.15787"]},"citation":{"apa":"Rote, G., Rüber, M., &#38; Saghafian, M. (2024). Grid peeling of parabolas. 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.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>","ama":"Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. 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.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>","mla":"Rote, Günter, et al. “Grid Peeling of Parabolas.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>.","ieee":"G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","chicago":"Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.” 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.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>.","short":"G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","ista":"Rote G, Rüber M, Saghafian M. 2024. Grid peeling of parabolas. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 76."},"acknowledgement":"Part of this work was done while G.R. enjoyed the hospitality of the Institute of Science and Technology Austria (ISTA) as a visiting professor during his sabbatical in the winter semester 2022/23.","intvolume":"       293","arxiv":1},{"file_date_updated":"2024-06-17T08:46:33Z","volume":293,"title":"Maximum Betti numbers of Čech complexes","related_material":{"record":[{"status":"public","id":"20657","relation":"later_version"}]},"language":[{"iso":"eng"}],"day":"01","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"}],"date_created":"2024-06-16T22:01:06Z","scopus_import":"1","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","full_name":"Pach, János","first_name":"János"}],"quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"doi":"10.4230/LIPIcs.SoCG.2024.53","_id":"17146","type":"conference","publication":"40th International Symposium on Computational Geometry","arxiv":1,"intvolume":"       293","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>.","short":"H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","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.","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>","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>","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>."},"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.","external_id":{"arxiv":["2310.14801"]},"ddc":["510"],"oa":1,"department":[{"_id":"HeEd"}],"article_number":"53","conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11"},"article_processing_charge":"No","month":"06","date_published":"2024-06-01T00:00:00Z","date_updated":"2025-12-01T15:19:20Z","publication_status":"published","ec_funded":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","status":"public","year":"2024","has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"file":[{"file_id":"17152","date_created":"2024-06-17T08:46:33Z","file_name":"2024_LIPICS_Edelsbrunner.pdf","relation":"main_file","date_updated":"2024-06-17T08:46:33Z","content_type":"application/pdf","file_size":766562,"access_level":"open_access","checksum":"5442d44fb89d77477a87668d6e61aac9","creator":"dernst","success":1}],"alternative_title":["LIPIcs"],"oa_version":"Published Version"}]