[{"publisher":"Springer Nature","department":[{"_id":"HeEd"}],"page":"597-624","oa":1,"file":[{"date_updated":"2026-07-27T08:11:05Z","content_type":"application/pdf","access_level":"open_access","file_id":"22410","relation":"main_file","creator":"dernst","success":1,"checksum":"c17c014dbbf5be702195c737890251c8","file_size":546483,"date_created":"2026-07-27T08:11:05Z","file_name":"2026_DiscreteCompGeom_Edelsbrunner.pdf"}],"OA_place":"publisher","publication":"Discrete & Computational Geometry","status":"public","das_tickbox":"0","ddc":["510"],"external_id":{"arxiv":["2310.14801"],"isi":["001610592600001"]},"quality_controlled":"1","article_processing_charge":"Yes (via OA deal)","_id":"20657","OA_type":"hybrid","isi":1,"acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31. Open access funding provided by Institute of Science and Technology (IST Austria).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"day":"01","date_created":"2025-11-19T09:44:58Z","publication_status":"published","intvolume":"        75","file_date_updated":"2026-07-27T08:11:05Z","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"}],"has_accepted_license":"1","corr_author":"1","volume":75,"date_published":"2026-03-01T00:00:00Z","PlanS_conform":"1","ec_funded":1,"researchdata_availability":"no","article_type":"original","language":[{"iso":"eng"}],"scopus_import":"1","month":"03","doi":"10.1007/s00454-025-00796-5","author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"},{"last_name":"Pach","first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","full_name":"Pach, János"}],"oa_version":"Published Version","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. ","lang":"eng"}],"citation":{"short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry 75 (2026) 597–624.","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” <i>Discrete &#38; Computational Geometry</i>, vol. 75. Springer Nature, pp. 597–624, 2026.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>, vol. 75, Springer Nature, 2026, pp. 597–624, doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>.","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>.","ista":"Edelsbrunner H, Pach J. 2026. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry. 75, 597–624.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. 2026;75:597-624. doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>","apa":"Edelsbrunner, H., &#38; Pach, J. (2026). Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>"},"supplementarymaterial":"no","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"type":"journal_article","arxiv":1,"date_updated":"2026-07-27T08:15:58Z","year":"2026","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"17146"}]},"title":"Maximum Betti numbers of Čech complexes"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"url":"https://doi.org/10.1007/s10208-026-09766-6","open_access":"1"}],"date_created":"2026-08-05T06:11:30Z","day":"04","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"publication_status":"epub_ahead","keyword":["Multivector field","Conley index","Morse decomposition","Bifurcation","Continuation","Zigzag persistence","Persistence barcode","Gentle algebra"],"oa":1,"publisher":"Springer","department":[{"_id":"HeEd"}],"publication":"Foundations of Computational Mathematics","status":"public","OA_place":"publisher","ddc":["500"],"external_id":{"arxiv":["2504.17105"]},"quality_controlled":"1","das_tickbox":"0","acknowledgement":"M.L. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. T.D. acknowledges the support of NSF funds CCF-2437030 and DMS-2301360. The authors would like to thank the anonymous reviewers for their careful reading of the paper. Their feedback significantly improved the quality of the article. T.D. and M.L. would like to acknowledge many thought-provoking discussions with Marian Mrozek on combinatorial dynamical systems and their continuations. M.S.T. would like to thank Álvaro Sánchez for insightful discussions about representation theory. Open access funding provided by Institute of Science and Technology (IST Austria).","_id":"22648","OA_type":"hybrid","article_processing_charge":"Yes (via OA deal)","oa_version":"Published Version","author":[{"full_name":"Dey, Tamal K.","first_name":"Tamal K.","last_name":"Dey"},{"last_name":"Lipiński","first_name":"Michał","orcid":"0000-0001-9789-9750","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","full_name":"Lipiński, Michał"},{"id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","full_name":"Soriano Trigueros, Manuel","last_name":"Soriano Trigueros","first_name":"Manuel","orcid":"0000-0003-2449-1433"}],"supplementarymaterial":"yes","abstract":[{"lang":"eng","text":"Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems—a young but already well-established theory. We introduce the Conley–Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of the Conley–Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose."}],"citation":{"ieee":"T. K. Dey, M. Lipiński, and M. Soriano Trigueros, “Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations,” <i>Foundations of Computational Mathematics</i>. Springer, 2026.","mla":"Dey, Tamal K., et al. “Conley-Morse Persistence Barcode: A Homological Signature of Combinatorial Bifurcations.” <i>Foundations of Computational Mathematics</i>, Springer, 2026, doi:<a href=\"https://doi.org/10.1007/s10208-026-09766-6\">10.1007/s10208-026-09766-6</a>.","ista":"Dey TK, Lipiński M, Soriano Trigueros M. 2026. Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations. Foundations of Computational Mathematics.","chicago":"Dey, Tamal K., Michał Lipiński, and Manuel Soriano Trigueros. “Conley-Morse Persistence Barcode: A Homological Signature of Combinatorial Bifurcations.” <i>Foundations of Computational Mathematics</i>. Springer, 2026. <a href=\"https://doi.org/10.1007/s10208-026-09766-6\">https://doi.org/10.1007/s10208-026-09766-6</a>.","short":"T.K. Dey, M. Lipiński, M. Soriano Trigueros, Foundations of Computational Mathematics (2026).","apa":"Dey, T. K., Lipiński, M., &#38; Soriano Trigueros, M. (2026). Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-026-09766-6\">https://doi.org/10.1007/s10208-026-09766-6</a>","ama":"Dey TK, Lipiński M, Soriano Trigueros M. Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations. <i>Foundations of Computational Mathematics</i>. 2026. doi:<a href=\"https://doi.org/10.1007/s10208-026-09766-6\">10.1007/s10208-026-09766-6</a>"},"date_updated":"2026-08-11T06:13:33Z","publication_identifier":{"eissn":["1615-3383"],"issn":["1615-3375"]},"type":"journal_article","arxiv":1,"title":"Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations","year":"2026","date_published":"2026-08-04T00:00:00Z","PlanS_conform":"1","ec_funded":1,"project":[{"call_identifier":"H2020","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program"}],"has_accepted_license":"1","corr_author":"1","language":[{"iso":"eng"}],"researchdata_availability":"no","article_type":"original","scopus_import":"1","doi":"10.1007/s10208-026-09766-6","month":"08"},{"publication_status":"published","intvolume":"        25","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"issue":"8","day":"20","date_created":"2025-12-29T12:09:09Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"20867","OA_type":"diamond","article_processing_charge":"No","acknowledgement":"The author was supported by the FWF Grant, Project number I4245-N35. The author would like to thank Thomas Weighill for the helpful discussions around Theorem 3.10, and Takamitsu Yamauchi for bringing to my attention the fundamental reference [35]. Furthermore, the author\r\nis thankful for the detailed and helpful comments of the reviewer of this manuscript.","external_id":{"arxiv":["2303.04730"]},"quality_controlled":"1","ddc":["500"],"OA_place":"publisher","status":"public","publication":"Algebraic & Geometric Topology","page":"5153-5174","department":[{"_id":"HeEd"}],"publisher":"Mathematical Sciences Publishers","file":[{"date_created":"2026-01-05T12:16:38Z","file_size":574389,"checksum":"1e05b4f17a44500ae1ae1e21bc636f6a","file_name":"2025_AlgebraicGeomTopology_Zava.pdf","success":1,"creator":"dernst","relation":"main_file","file_id":"20943","content_type":"application/pdf","date_updated":"2026-01-05T12:16:38Z","access_level":"open_access"}],"oa":1,"year":"2025","title":"Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces","type":"journal_article","arxiv":1,"publication_identifier":{"issn":["1472-2747"],"eissn":["1472-2739"]},"date_updated":"2026-01-05T12:19:09Z","citation":{"mla":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8, Mathematical Sciences Publishers, 2025, pp. 5153–74, doi:<a href=\"https://doi.org/10.2140/agt.2025.25.5153\">10.2140/agt.2025.25.5153</a>.","ieee":"N. Zava, “Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces,” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8. Mathematical Sciences Publishers, pp. 5153–5174, 2025.","chicago":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers, 2025. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>.","ista":"Zava N. 2025. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. Algebraic &#38; Geometric Topology. 25(8), 5153–5174.","short":"N. Zava, Algebraic &#38; Geometric Topology 25 (2025) 5153–5174.","apa":"Zava, N. (2025). Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>","ama":"Zava N. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. <i>Algebraic &#38; Geometric Topology</i>. 2025;25(8):5153-5174. doi:<a href=\"https://doi.org/10.2140/agt.2025.25.5153\">10.2140/agt.2025.25.5153</a>"},"abstract":[{"text":"We discuss the embeddability of subspaces of the Gromov–Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension n(n−1)∕2. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both nonembeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean–Hausdorff distance.","lang":"eng"}],"author":[{"orcid":"0000-0001-8686-1888","first_name":"Nicolò","last_name":"Zava","full_name":"Zava, Nicolò","id":"c8b3499c-7a77-11eb-b046-aa368cbbf2ad"}],"oa_version":"Published Version","doi":"10.2140/agt.2025.25.5153","month":"11","scopus_import":"1","article_type":"original","language":[{"iso":"eng"}],"corr_author":"1","has_accepted_license":"1","file_date_updated":"2026-01-05T12:16:38Z","project":[{"_id":"26AD5D90-B435-11E9-9278-68D0E5697425","grant_number":"I04245","name":"Algebraic Footprints of Geometric Features in Homology","call_identifier":"FWF"}],"PlanS_conform":"1","volume":25,"date_published":"2025-11-20T00:00:00Z"},{"day":"01","date_created":"2026-02-16T15:48:42Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2306.13201"}],"intvolume":"       129","publication_status":"published","publication":"Computational Geometry","status":"public","OA_place":"repository","oa":1,"publisher":"Elsevier","department":[{"_id":"HeEd"}],"acknowledgement":"A preliminary version of this note has been published in the proceedings of the 31st International Symposium on Graph Drawing and Network Visualization, Palermo, 2023. The authors would like to thank the anonymous referees for their valuable comments.","OA_type":"green","_id":"21253","article_processing_charge":"No","quality_controlled":"1","external_id":{"arxiv":["2306.13201"]},"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."}],"article_number":"102186","citation":{"apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2025). Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. 2025;129. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>, vol. 129, 102186, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>.","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” <i>Computational Geometry</i>, vol. 129. Elsevier, 2025.","ista":"Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186.","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>.","short":"J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025)."},"oa_version":"Preprint","author":[{"last_name":"Pach","first_name":"János","full_name":"Pach, János"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian"},{"last_name":"Schnider","first_name":"Patrick","full_name":"Schnider, Patrick"}],"related_material":{"record":[{"id":"15012","relation":"earlier_version","status":"public"}]},"title":"Decomposition of geometric graphs into star-forests","year":"2025","date_updated":"2026-04-16T09:12:36Z","publication_identifier":{"issn":["0925-7721"]},"arxiv":1,"type":"journal_article","language":[{"iso":"eng"}],"article_type":"original","date_published":"2025-12-01T00:00:00Z","volume":129,"corr_author":"1","doi":"10.1016/j.comgeo.2025.102186","month":"12"},{"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"type":"journal_article","arxiv":1,"date_updated":"2026-02-16T12:18:50Z","year":"2025","title":"Average and expected distortion of Voronoi paths and scapes","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Nikitenko","first_name":"Anton","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton"}],"oa_version":"Published Version","abstract":[{"lang":"eng","text":"The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes."}],"citation":{"ama":"Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. 2025;73:490-499. doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>.","ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","ieee":"H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” <i>Discrete &#38; Computational Geometry</i>, vol. 73. Springer Nature, pp. 490–499, 2025.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>."},"scopus_import":"1","month":"03","doi":"10.1007/s00454-024-00660-y","file_date_updated":"2025-04-23T07:31:32Z","has_accepted_license":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes"}],"corr_author":"1","volume":73,"date_published":"2025-03-01T00:00:00Z","ec_funded":1,"article_type":"original","language":[{"iso":"eng"}],"publication_status":"published","intvolume":"        73","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"pmid":1,"date_created":"2024-06-16T22:01:07Z","day":"01","ddc":["510"],"external_id":{"isi":["001238566200004"],"pmid":["39974750"],"arxiv":["2012.03350"]},"quality_controlled":"1","_id":"17149","OA_type":"hybrid","article_processing_charge":"Yes (via OA deal)","isi":1,"acknowledgement":"The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on computational experiments that motivated the work reported in this paper. The authors also thank Daniel Bonnema for proofreading and noticing an issue with the original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","publisher":"Springer Nature","page":"490-499","department":[{"_id":"HeEd"}],"file":[{"creator":"dernst","file_id":"19610","relation":"main_file","access_level":"open_access","date_updated":"2025-04-23T07:31:32Z","content_type":"application/pdf","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","date_created":"2025-04-23T07:31:32Z","file_size":283443,"checksum":"ffb0c818222138f9f113f4bbea41e834","success":1}],"oa":1,"OA_place":"publisher","publication":"Discrete & Computational Geometry","status":"public"},{"scopus_import":"1","doi":"10.1016/j.aim.2024.110055","month":"02","corr_author":"1","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"ec_funded":1,"date_published":"2025-02-01T00:00:00Z","volume":461,"article_type":"original","language":[{"iso":"eng"}],"type":"journal_article","arxiv":1,"publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"date_updated":"2025-04-15T07:16:53Z","year":"2025","title":"Order-2 Delaunay triangulations optimize angles","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"oa_version":"Preprint","citation":{"short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>.","mla":"Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>, vol. 461, 110055, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>.","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025.","ama":"Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. 2025;461. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>"},"article_number":"110055","abstract":[{"text":"The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.","lang":"eng"}],"quality_controlled":"1","external_id":{"isi":["001370682500001"],"arxiv":["2310.18238"]},"isi":1,"article_processing_charge":"No","_id":"18626","OA_type":"green","acknowledgement":"Work by the first and third authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation.","department":[{"_id":"HeEd"}],"publisher":"Elsevier","oa":1,"OA_place":"repository","status":"public","publication":"Advances in Mathematics","publication_status":"published","intvolume":"       461","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2310.18238","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2024-12-08T23:01:54Z","day":"01"},{"date_updated":"2026-04-07T11:47:30Z","type":"dissertation","publication_identifier":{"issn":["2663-337X"]},"title":"Structures and computations in topological data analysis","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"15091"},{"relation":"part_of_dissertation","status":"public","id":"18981"}]},"year":"2025","oa_version":"Published Version","author":[{"last_name":"Draganov","first_name":"Ondrej","orcid":"0000-0003-0464-3823","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej"}],"citation":{"ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025.","mla":"Draganov, Ondrej. <i>Structures and Computations in Topological Data Analysis</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>.","chicago":"Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>.","ista":"Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria.","short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","apa":"Draganov, O. (2025). <i>Structures and computations in topological data analysis</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>","ama":"Draganov O. Structures and computations in topological data analysis. 2025. doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>"},"abstract":[{"text":"Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe describe a construction of a chromatic alpha complex, which enables  efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot.","lang":"eng"}],"doi":"10.15479/at:ista:18979","month":"02","date_published":"2025-02-03T00:00:00Z","corr_author":"1","file_date_updated":"2025-02-04T16:22:07Z","has_accepted_license":"1","project":[{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"}],"language":[{"iso":"eng"}],"publication_status":"published","keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"degree_awarded":"PhD","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_created":"2025-01-31T17:04:40Z","day":"03","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"ddc":["514","004"],"acknowledgement":"The research presented in this thesis was funded with the Wittgenstein Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.\r\n","_id":"18979","article_processing_charge":"No","file":[{"access_level":"closed","content_type":"application/zip","date_updated":"2025-01-31T16:58:30Z","relation":"source_file","file_id":"18983","creator":"odragano","file_name":"Thesis.zip","checksum":"af6567e5d35e5eb330b8925ae37f1998","date_created":"2025-01-31T16:58:30Z","file_size":11899491},{"file_name":"Thesis.pdf","checksum":"c3fef68e35b9dc2020b2ca6006da6343","file_size":8857514,"date_created":"2025-02-04T16:22:07Z","access_level":"open_access","date_updated":"2025-02-04T16:22:07Z","content_type":"application/pdf","file_id":"19000","relation":"main_file","creator":"odragano"}],"oa":1,"supervisor":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833"}],"page":"140","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publisher":"Institute of Science and Technology Austria","status":"public","alternative_title":["ISTA Thesis"],"OA_place":"publisher"},{"publication_identifier":{"issn":["0020-0255"]},"type":"journal_article","date_updated":"2025-12-30T09:05:32Z","year":"2025","title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","author":[{"full_name":"Mahini, Mohammad","last_name":"Mahini","first_name":"Mohammad"},{"first_name":"Hamid","last_name":"Beigy","full_name":"Beigy, Hamid"},{"full_name":"Qadami, Salman","first_name":"Salman","last_name":"Qadami"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian"}],"oa_version":"None","abstract":[{"lang":"eng","text":"Simplets are elementary units within simplicial complexes and are fundamental for analyzing the structure of simplicial complexes. Previous efforts have mainly focused on accurately counting or approximating the number of simplets rather than studying their frequencies. However, analyzing simplet frequencies is more practical for large-scale simplicial complexes. This paper introduces the Simplet Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies in simplicial complexes. Additionally, we provide a bound on the sample complexity required to approximate the SFD vector using any uniform sampling-based algorithm accurately. We extend the definition of simplet frequency distribution to encompass simplices, allowing for the analysis of simplet frequencies within simplices of simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex. Furthermore, we present a bound on the sample complexity required for accurately approximating the SDV and SDC for a set of simplices using any uniform sampling-based algorithm. We also introduce algorithms for approximating SFD, geometric SFD, SDV, and SDC. We also validate the theoretical bounds with experiments on random simplicial complexes and demonstrate the practical application through a case study."}],"citation":{"chicago":"Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>.","ista":"Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. 719(11), 122425.","ieee":"M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality,” <i>Information Sciences</i>, vol. 719, no. 11. Elsevier, 2025.","mla":"Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>, vol. 719, no. 11, 122425, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).","apa":"Mahini, M., Beigy, H., Qadami, S., &#38; Saghafian, M. (2025). Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>","ama":"Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. 2025;719(11). doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>"},"article_number":"122425","scopus_import":"1","month":"11","doi":"10.1016/j.ins.2025.122425","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"corr_author":"1","date_published":"2025-11-01T00:00:00Z","volume":719,"ec_funded":1,"article_type":"original","language":[{"iso":"eng"}],"publication_status":"published","intvolume":"       719","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"11","date_created":"2025-06-30T08:48:48Z","day":"01","quality_controlled":"1","external_id":{"isi":["001516170500002"]},"article_processing_charge":"No","_id":"19937","OA_type":"closed access","isi":1,"acknowledgement":"The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which improved this paper.\r\nWork by the first and fourth authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35.","publisher":"Elsevier","department":[{"_id":"HeEd"}],"publication":"Information Sciences","status":"public"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-07-13T22:01:22Z","day":"20","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"intvolume":"       332","publication_status":"published","oa":1,"file":[{"success":1,"checksum":"b5313ed8575ea87913c71a6e3c7513c8","file_size":661893,"date_created":"2025-07-14T07:24:22Z","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","date_updated":"2025-07-14T07:24:22Z","content_type":"application/pdf","access_level":"open_access","file_id":"20016","relation":"main_file","creator":"dernst"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","department":[{"_id":"HeEd"}],"publication":"41st International Symposium on Computational Geometry","status":"public","alternative_title":["LIPIcs"],"OA_place":"publisher","ddc":["510"],"quality_controlled":"1","external_id":{"arxiv":["2410.21204"]},"acknowledgement":"Herbert Edelsbrunner: partially supported by the Wittgenstein Prize, Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35","article_processing_charge":"Yes","_id":"20005","OA_type":"gold","oa_version":"Published Version","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2025-06-23","location":"Kanazawa, Japan","end_date":"2025-06-27"},"author":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian"}],"abstract":[{"lang":"eng","text":"We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers."}],"article_number":"43","citation":{"ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","mla":"Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43.","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>","ama":"Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>"},"date_updated":"2025-07-14T07:26:14Z","publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"type":"conference","arxiv":1,"title":"On spheres with k points inside","year":"2025","date_published":"2025-06-20T00:00:00Z","volume":332,"has_accepted_license":"1","file_date_updated":"2025-07-14T07:24:22Z","project":[{"name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"corr_author":"1","language":[{"iso":"eng"}],"scopus_import":"1","month":"06","doi":"10.4230/LIPIcs.SoCG.2025.43"},{"day":"20","date_created":"2025-07-13T22:01:22Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"       332","publication_status":"published","alternative_title":["LIPIcs"],"status":"public","publication":"41st International Symposium on Computational Geometry","OA_place":"publisher","oa":1,"file":[{"success":1,"file_size":834623,"date_created":"2025-07-14T08:23:38Z","checksum":"3a4a7a707a56e0cfdf51428782dee55a","file_name":"2025_LIPIcs.SoCG_Ost.pdf","date_updated":"2025-07-14T08:23:38Z","content_type":"application/pdf","access_level":"open_access","creator":"dernst","file_id":"20017","relation":"main_file"}],"department":[{"_id":"HeEd"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","acknowledgement":"Lara Ost: Supported by the Vienna Graduate School on Computational Optimization\r\n(VGSCO), FWF project no. W1260-N35.\r\nSebastiano Cultrera di Montesano: Supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard.\r\nHerbert Edelsbrunner: Partially supported by the Wittgenstein Prize, FWF grant no. Z 342-N31,\r\nand by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.","_id":"20006","article_processing_charge":"Yes","OA_type":"gold","external_id":{"arxiv":["2405.17920"]},"quality_controlled":"1","ddc":["000"],"article_number":"71","citation":{"ista":"Ost L, Cultrera di Montesano S, Edelsbrunner H. 2025. Banana trees for the persistence in time series experimentally. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 71.","chicago":"Ost, Lara, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner. “Banana Trees for the Persistence in Time Series Experimentally.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>.","ieee":"L. Ost, S. Cultrera di Montesano, and H. Edelsbrunner, “Banana trees for the persistence in time series experimentally,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","mla":"Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 71, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>.","short":"L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","apa":"Ost, L., Cultrera di Montesano, S., &#38; Edelsbrunner, H. (2025). Banana trees for the persistence in time series experimentally. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>","ama":"Ost L, Cultrera di Montesano S, Edelsbrunner H. Banana trees for the persistence in time series experimentally. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>"},"abstract":[{"lang":"eng","text":"In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation."}],"oa_version":"Published Version","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2025-06-27","location":"Kanazawa, Japan","start_date":"2025-06-23"},"author":[{"full_name":"Ost, Lara","last_name":"Ost","first_name":"Lara"},{"last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera di Montesano, Sebastiano"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"}],"title":"Banana trees for the persistence in time series experimentally","related_material":{"link":[{"relation":"software","url":"https://github.com/laraost/BananaPersist"}]},"year":"2025","date_updated":"2025-12-30T11:04:33Z","arxiv":1,"type":"conference","publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"language":[{"iso":"eng"}],"date_published":"2025-06-20T00:00:00Z","volume":332,"corr_author":"1","project":[{"name":"Vienna Graduate School on Computational Optimization","grant_number":"W1260-N35","_id":"9B9290DE-BA93-11EA-9121-9846C619BF3A"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science"}],"has_accepted_license":"1","file_date_updated":"2025-07-14T08:23:38Z","month":"06","doi":"10.4230/LIPIcs.SoCG.2025.71","scopus_import":"1"},{"intvolume":"         4","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"01","date_created":"2025-08-31T22:01:33Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"ddc":["510"],"quality_controlled":"1","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.","_id":"20260","OA_type":"hybrid","article_processing_charge":"Yes (via OA deal)","file":[{"access_level":"open_access","date_updated":"2025-12-30T07:52:58Z","content_type":"application/pdf","creator":"dernst","file_id":"20885","relation":"main_file","success":1,"file_name":"2025_LaMatematica_Chambers.pdf","date_created":"2025-12-30T07:52:58Z","file_size":2678640,"checksum":"e2043259194bfcdf3d74c4da8a5a853f"}],"oa":1,"publisher":"Springer Nature","department":[{"_id":"HeEd"}],"page":"811-828","publication":"La Matematica","status":"public","OA_place":"publisher","date_updated":"2026-04-07T11:42:48Z","publication_identifier":{"eissn":["2730-9657"]},"type":"journal_article","related_material":{"record":[{"id":"21021","relation":"dissertation_contains","status":"public"}]},"title":"Burning or collapsing the medial axis is unstable","year":"2025","oa_version":"Published Version","author":[{"first_name":"Erin Wolf","last_name":"Chambers","full_name":"Chambers, Erin Wolf"},{"full_name":"Fillmore, Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","last_name":"Fillmore"},{"id":"2D04F932-F248-11E8-B48F-1D18A9856A87","full_name":"Stephenson, Elizabeth R","last_name":"Stephenson","first_name":"Elizabeth R","orcid":"0000-0002-6862-208X"},{"last_name":"Wintraecken","orcid":"0000-0002-7472-2220","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs"}],"abstract":[{"lang":"eng","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."}],"citation":{"apa":"Chambers, E. W., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2025). Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>","ama":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. 2025;4:811-828. doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>","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>.","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.","ista":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica. 4, 811–828.","chicago":"Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>.","short":"E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica 4 (2025) 811–828."},"scopus_import":"1","doi":"10.1007/s44007-025-00170-0","month":"12","date_published":"2025-12-01T00:00:00Z","volume":4,"PlanS_conform":"1","ec_funded":1,"has_accepted_license":"1","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"},{"name":"Learning and triangulating manifolds via collapses","grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"file_date_updated":"2025-12-30T07:52:58Z","corr_author":"1","language":[{"iso":"eng"}],"article_type":"original"},{"publication_identifier":{"eissn":["1099-4300"]},"type":"journal_article","date_updated":"2025-09-30T14:32:31Z","year":"2025","title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence","author":[{"first_name":"Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan","full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"full_name":"Virk, Ziga","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","first_name":"Ziga","last_name":"Virk"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Hubert","last_name":"Wagner","first_name":"Hubert"}],"oa_version":"Published Version","abstract":[{"text":"Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence.","lang":"eng"}],"article_number":"854","citation":{"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.","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>.","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.","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).","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>"},"scopus_import":"1","doi":"10.3390/e27080854","month":"08","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"file_date_updated":"2025-09-08T07:55:48Z","has_accepted_license":"1","corr_author":"1","date_published":"2025-08-01T00:00:00Z","volume":27,"ec_funded":1,"PlanS_conform":"1","article_type":"original","language":[{"iso":"eng"}],"publication_status":"published","DOAJ_listed":"1","intvolume":"        27","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"issue":"8","pmid":1,"date_created":"2025-09-07T22:01:33Z","day":"01","ddc":["500"],"external_id":{"isi":["001557476000001"],"pmid":["40870326"]},"quality_controlled":"1","OA_type":"gold","_id":"20293","article_processing_charge":"Yes","isi":1,"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.","publisher":"MDPI","department":[{"_id":"HeEd"}],"file":[{"file_name":"2025_Entropy_Akopyan.pdf","date_created":"2025-09-08T07:55:48Z","file_size":379340,"checksum":"65c5399c4015d9c8abb8c7a96f3d7836","success":1,"creator":"dernst","file_id":"20309","relation":"main_file","access_level":"open_access","content_type":"application/pdf","date_updated":"2025-09-08T07:55:48Z"}],"oa":1,"OA_place":"publisher","publication":"Entropy","status":"public"},{"article_type":"original","language":[{"iso":"eng"}],"corr_author":"1","has_accepted_license":"1","file_date_updated":"2025-12-30T07:55:08Z","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"PlanS_conform":"1","ec_funded":1,"volume":229,"date_published":"2025-10-01T00:00:00Z","doi":"10.1016/j.jpaa.2025.108068","month":"10","scopus_import":"1","citation":{"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.","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.","mla":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10, 108068, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>.","short":"A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).","apa":"Brown, A., &#38; Draganov, O. (2025). Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>","ama":"Brown A, Draganov O. Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. 2025;229(10). doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>"},"article_number":"108068","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."}],"author":[{"id":"70B7FDF6-608D-11E9-9333-8535E6697425","full_name":"Brown, Adam","last_name":"Brown","first_name":"Adam"},{"orcid":"0000-0003-0464-3823","first_name":"Ondrej","last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"}],"oa_version":"Published Version","year":"2025","title":"Discrete microlocal Morse theory","related_material":{"record":[{"id":"18981","relation":"earlier_version","status":"public"}]},"arxiv":1,"type":"journal_article","publication_identifier":{"issn":["0022-4049"]},"date_updated":"2025-12-30T07:55:21Z","OA_place":"publisher","status":"public","publication":"Journal of Pure and Applied Algebra","department":[{"_id":"HeEd"}],"publisher":"Elsevier","file":[{"access_level":"open_access","content_type":"application/pdf","date_updated":"2025-12-30T07:55:08Z","creator":"dernst","relation":"main_file","file_id":"20886","success":1,"file_name":"2025_JourPureAppliedAlgebra_Brown.pdf","date_created":"2025-12-30T07:55:08Z","file_size":3090836,"checksum":"39bcad462278c9322ef810af7db67f56"}],"oa":1,"OA_type":"hybrid","_id":"20323","article_processing_charge":"Yes (via OA deal)","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","quality_controlled":"1","external_id":{"arxiv":["2209.14993"]},"ddc":["510"],"issue":"10","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"day":"01","date_created":"2025-09-10T05:40:09Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","intvolume":"       229"},{"publisher":"Springer Nature","page":"133-147","department":[{"_id":"HeEd"}],"oa":1,"OA_place":"repository","publication":"4th International Joint Conference on Discrete Geometry and Mathematical Morphology","alternative_title":["LNCS"],"status":"public","quality_controlled":"1","external_id":{"arxiv":["2504.14743"]},"OA_type":"green","_id":"20658","article_processing_charge":"No","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2504.14743"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-11-23T23:01:37Z","day":"01","publication_status":"published","intvolume":"     16296","volume":16296,"date_published":"2025-11-01T00:00:00Z","language":[{"iso":"eng"}],"scopus_import":"1","month":"11","doi":"10.1007/978-3-032-09544-2_10","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Stephenson","orcid":"0000-0002-6862-208X","first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","full_name":"Stephenson, Elizabeth R"},{"last_name":"Thoresen","first_name":"Martin H","id":"47CB1472-F248-11E8-B48F-1D18A9856A87","full_name":"Thoresen, Martin H"}],"oa_version":"Preprint","conference":{"end_date":"2025-11-06","start_date":"2025-11-03","location":"Groningen, The Netherlands","name":"DGMM: Discrete Geometry and Mathematical Morphology"},"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"}],"citation":{"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>","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>.","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.","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>.","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.","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."},"publication_identifier":{"issn":["0302-9743"],"isbn":["9783032095435"],"eissn":["1611-3349"]},"arxiv":1,"type":"conference","date_updated":"2025-11-24T10:05:11Z","year":"2025","title":"The mid-sphere cousin of the medial axis transform"},{"date_published":"2025-11-10T00:00:00Z","file_date_updated":"2025-12-09T13:43:17Z","has_accepted_license":"1","corr_author":"1","language":[{"iso":"eng"}],"scopus_import":"1","month":"11","doi":"10.1145/3747199.3747561","conference":{"name":"ISSAC: International Symposium on Symbolic and Algebraic Computation","start_date":"2025-07-28","location":"Guanajuato, Mexico","end_date":"2025-08-01"},"oa_version":"Published Version","author":[{"first_name":"Rocio","last_name":"Gonzalez-Diaz","full_name":"Gonzalez-Diaz, Rocio"},{"last_name":"Soriano Trigueros","first_name":"Manuel","orcid":"0000-0003-2449-1433","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","full_name":"Soriano Trigueros, Manuel"},{"full_name":"Torras-Casas, Alvaro","first_name":"Alvaro","last_name":"Torras-Casas"}],"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"}],"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>.","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.","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>.","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.","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.","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>"},"date_updated":"2025-12-09T13:46:42Z","publication_identifier":{"isbn":["9798400720758"]},"type":"conference","title":"Additive partial matchings for persistent homology","year":"2025","oa":1,"file":[{"success":1,"file_name":"2025_ISSAC_GonzalezDiaz.pdf","checksum":"1c299cca165a20e2518afe4fda63dbf1","date_created":"2025-12-09T13:43:17Z","file_size":761617,"access_level":"open_access","date_updated":"2025-12-09T13:43:17Z","content_type":"application/pdf","file_id":"20751","relation":"main_file","creator":"dernst"}],"publisher":"Association for Computing Machinery","page":"188-196","department":[{"_id":"HeEd"}],"publication":"Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation","status":"public","OA_place":"publisher","ddc":["510"],"quality_controlled":"1","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).","_id":"20729","article_processing_charge":"Yes (in subscription journal)","OA_type":"hybrid","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"10","date_created":"2025-12-07T23:02:01Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"publication_status":"published"},{"publication":"arXiv","status":"public","OA_place":"repository","oa":1,"department":[{"_id":"HeEd"}],"acknowledgement":"Y. B. B. and L. F. were funded by the Independent Research Fund Denmark, grant\r\nnumber 1026-00037. T. H. was partially supported by the European Research Council\r\n(ERC) Horizon 2020, grant number 788183.","OA_type":"green","_id":"21016","article_processing_charge":"No","external_id":{"arxiv":["2505.17858"]},"das_tickbox":"1","day":"23","date_created":"2026-01-20T10:12:21Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2505.17858","open_access":"1"}],"publication_status":"submitted","language":[{"iso":"eng"}],"date_published":"2025-05-23T00:00:00Z","ec_funded":1,"project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"}],"month":"05","doi":"10.48550/arXiv.2505.17858","abstract":[{"text":"Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times.","lang":"eng"}],"article_number":"2505.17858","citation":{"ista":"Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. arXiv, 2505.17858.","chicago":"Bokor Bleile, Yossi, Lisbeth Fajstrup, Teresa Heiss, Anne Marie Svane, and Søren Strandskov Sørensen. “Identifying Cobordisms Using Kernel Persistence.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2505.17858\">https://doi.org/10.48550/arXiv.2505.17858</a>.","mla":"Bokor Bleile, Yossi, et al. “Identifying Cobordisms Using Kernel Persistence.” <i>ArXiv</i>, 2505.17858, doi:<a href=\"https://doi.org/10.48550/arXiv.2505.17858\">10.48550/arXiv.2505.17858</a>.","ieee":"Y. Bokor Bleile, L. Fajstrup, T. Heiss, A. M. Svane, and S. S. Sørensen, “Identifying cobordisms using kernel persistence,” <i>arXiv</i>. .","short":"Y. Bokor Bleile, L. Fajstrup, T. Heiss, A.M. Svane, S.S. Sørensen, ArXiv (n.d.).","apa":"Bokor Bleile, Y., Fajstrup, L., Heiss, T., Svane, A. M., &#38; Sørensen, S. S. (n.d.). Identifying cobordisms using kernel persistence. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2505.17858\">https://doi.org/10.48550/arXiv.2505.17858</a>","ama":"Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2505.17858\">10.48550/arXiv.2505.17858</a>"},"oa_version":"Preprint","author":[{"last_name":"Bleile","orcid":"0000-0002-4861-9174","first_name":"Yossi","id":"920a7385-7995-11ef-9bfd-8c434cd8f3c2","full_name":"Bleile, Yossi"},{"last_name":"Fajstrup","first_name":"Lisbeth","full_name":"Fajstrup, Lisbeth"},{"first_name":"Teresa","orcid":"0000-0002-1780-2689","last_name":"Heiss","full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Svane, Anne Marie","first_name":"Anne Marie","last_name":"Svane"},{"full_name":"Sørensen, Søren Strandskov","first_name":"Søren Strandskov","last_name":"Sørensen"}],"title":"Identifying cobordisms using kernel persistence","year":"2025","date_updated":"2026-07-22T06:33:07Z","type":"preprint","arxiv":1},{"article_number":"2501.05315","citation":{"short":"H. Edelsbrunner, C.D. Fillmore, G. Olivera, ArXiv (n.d.).","ieee":"H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the electrostatic potential,” <i>arXiv</i>. .","mla":"Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” <i>ArXiv</i>, 2501.05315, doi:<a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">10.48550/ARXIV.2501.05315</a>.","chicago":"Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting Equilibria of the Electrostatic Potential.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">https://doi.org/10.48550/ARXIV.2501.05315</a>.","ista":"Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv, 2501.05315.","ama":"Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">10.48550/ARXIV.2501.05315</a>","apa":"Edelsbrunner, H., Fillmore, C. D., &#38; Olivera, G. (n.d.). Counting equilibria of the electrostatic potential. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">https://doi.org/10.48550/ARXIV.2501.05315</a>"},"abstract":[{"lang":"eng","text":"In 1873, James C. Maxwell conjectured that the electric field generated by $n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov and Shapiro, who also posed two additional interesting conjectures.\r\n In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to a conjecture of Gabrielov, Novikov and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges.\r\n Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find smaller bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day."}],"oa_version":"Preprint","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Fillmore, Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","last_name":"Fillmore"},{"last_name":"Olivera","first_name":"Gonçalo","full_name":"Olivera, Gonçalo"}],"title":"Counting equilibria of the electrostatic potential","related_material":{"record":[{"id":"21021","status":"public","relation":"dissertation_contains"},{"status":"public","relation":"later_version","id":"21931"}]},"year":"2025","date_updated":"2026-07-22T06:33:55Z","type":"preprint","arxiv":1,"language":[{"iso":"eng"}],"date_published":"2025-03-20T00:00:00Z","corr_author":"1","doi":"10.48550/ARXIV.2501.05315","month":"03","day":"20","date_created":"2026-01-27T14:29:27Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2501.05315","open_access":"1"}],"publication_status":"draft","status":"public","publication":"arXiv","OA_place":"repository","oa":1,"department":[{"_id":"HeEd"}],"OA_type":"green","_id":"21050","article_processing_charge":"No","external_id":{"arxiv":["2501.05315"]},"das_tickbox":"1"},{"ddc":["000"],"quality_controlled":"1","external_id":{"pmid":["39678706"]},"OA_type":"hybrid","_id":"13182","article_processing_charge":"Yes (via OA deal)","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.","publisher":"Springer Nature","department":[{"_id":"HeEd"}],"page":"1101-1119","oa":1,"file":[{"file_id":"18783","relation":"main_file","creator":"dernst","access_level":"open_access","content_type":"application/pdf","date_updated":"2025-01-09T07:39:41Z","file_name":"2024_JourApplCompTopo_Biswas.pdf","checksum":"d493df5088c222b88d9ca46b623ad0ee","date_created":"2025-01-09T07:39:41Z","file_size":476896,"success":1}],"OA_place":"publisher","publication":"Journal of Applied and Computational Topology","status":"public","publication_status":"published","intvolume":"         8","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"pmid":1,"day":"01","date_created":"2023-07-02T22:00:44Z","scopus_import":"1","doi":"10.1007/s41468-023-00126-9","month":"10","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887","name":"Persistent Homology, Algorithms and Stochastic Geometry"},{"call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"}],"file_date_updated":"2025-01-09T07:39:41Z","has_accepted_license":"1","corr_author":"1","volume":8,"date_published":"2024-10-01T00:00:00Z","ec_funded":1,"article_type":"original","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"type":"journal_article","date_updated":"2026-04-07T12:58:47Z","year":"2024","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"15094"}]},"title":"Geometric characterization of the persistence of 1D maps","author":[{"last_name":"Biswas","orcid":"0000-0002-5372-7890","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita"},{"last_name":"Cultrera Di Montesano","first_name":"Sebastiano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera Di Montesano, Sebastiano"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian"}],"oa_version":"Published Version","abstract":[{"lang":"eng","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."}],"citation":{"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.","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>.","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.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 1101–1119.","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>"}},{"arxiv":1,"type":"journal_article","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"date_updated":"2025-04-23T08:41:59Z","year":"2024","title":"On angles in higher order Brillouin tessellations and related tilings in the plane","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"first_name":"Alexey","last_name":"Garber","full_name":"Garber, Alexey"},{"last_name":"Ghafari","first_name":"Mohadese","full_name":"Ghafari, Mohadese"},{"full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa","orcid":"0000-0002-1780-2689","last_name":"Heiss"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian"}],"oa_version":"Published Version","citation":{"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>","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>.","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>.","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."},"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))."}],"scopus_import":"1","month":"07","doi":"10.1007/s00454-023-00566-1","corr_author":"1","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF"}],"file_date_updated":"2024-07-22T09:43:19Z","has_accepted_license":"1","ec_funded":1,"volume":72,"date_published":"2024-07-01T00:00:00Z","article_type":"original","language":[{"iso":"eng"}],"publication_status":"published","intvolume":"        72","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"date_created":"2023-09-17T22:01:10Z","day":"01","pmid":1,"external_id":{"isi":["001060727600004"],"pmid":["39610762"],"arxiv":["2204.01076"]},"quality_controlled":"1","ddc":["510"],"isi":1,"_id":"14345","article_processing_charge":"Yes (via OA deal)","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.","department":[{"_id":"HeEd"}],"page":"29-48","publisher":"Springer Nature","oa":1,"file":[{"file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","date_created":"2024-07-22T09:43:19Z","file_size":892019,"checksum":"b207b4e00f904e8ea8a30e24f0251f79","success":1,"creator":"dernst","relation":"main_file","file_id":"17301","access_level":"open_access","date_updated":"2024-07-22T09:43:19Z","content_type":"application/pdf"}],"status":"public","publication":"Discrete and Computational Geometry"},{"day":"06","date_created":"2024-01-28T23:01:43Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2202.12175","open_access":"1"}],"intvolume":"     14466","publication_status":"published","alternative_title":["LNCS"],"status":"public","publication":"31st International Symposium on Graph Drawing and Network Visualization","oa":1,"department":[{"_id":"UlWa"},{"_id":"HeEd"}],"page":"18-33","publisher":"Springer Nature","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].","isi":1,"article_processing_charge":"No","_id":"14888","external_id":{"isi":["001207942000002"],"arxiv":["2202.12175"]},"quality_controlled":"1","citation":{"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.","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.","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.","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>","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>"},"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"}],"conference":{"name":"GD: Graph Drawing and Network Visualization","end_date":"2023-09-22","start_date":"2023-09-20","location":"Isola delle Femmine, Palermo, Italy"},"oa_version":"Preprint","author":[{"full_name":"De Nooijer, Phoebe","first_name":"Phoebe","last_name":"De Nooijer"},{"full_name":"Terziadis, Soeren","last_name":"Terziadis","first_name":"Soeren"},{"last_name":"Weinberger","first_name":"Alexandra","full_name":"Weinberger, Alexandra"},{"first_name":"Zuzana","orcid":"0000-0002-6660-1322","last_name":"Masárová","full_name":"Masárová, Zuzana","id":"45CFE238-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Tamara","last_name":"Mchedlidze","full_name":"Mchedlidze, Tamara"},{"full_name":"Löffler, Maarten","last_name":"Löffler","first_name":"Maarten"},{"first_name":"Günter","last_name":"Rote","full_name":"Rote, Günter"}],"title":"Removing popular faces in curve arrangements","year":"2024","date_updated":"2025-09-04T11:52:35Z","type":"conference","arxiv":1,"publication_identifier":{"eissn":["1611-3349"],"isbn":["9783031492747"],"issn":["0302-9743"]},"language":[{"iso":"eng"}],"volume":14466,"date_published":"2024-01-06T00:00:00Z","doi":"10.1007/978-3-031-49275-4_2","month":"01","scopus_import":"1"}]