[{"month":"05","doi":"10.1112/plms.70163","OA_place":"repository","issue":"5","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","day":"01","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2501.05315"}],"quality_controlled":"1","status":"public","department":[{"_id":"HeEd"},{"_id":"TaHa"}],"oa_version":"Preprint","date_published":"2026-05-01T00:00:00Z","external_id":{"arxiv":["2501.05315"]},"oa":1,"OA_type":"green","intvolume":"       132","corr_author":"1","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"first_name":"Christopher D","full_name":"Fillmore, Christopher D","last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"id":"58abbde8-f455-11eb-a497-98c8fd71b905","first_name":"Goncalo","full_name":"Oliveira, Goncalo","last_name":"Oliveira"}],"arxiv":1,"title":"Counting equilibria of the electrostatic potential","publisher":"Wiley","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2026","date_updated":"2026-07-22T06:33:54Z","_id":"21931","citation":{"ama":"Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic potential. <i>Proceedings of the London Mathematical Society</i>. 2026;132(5). doi:<a href=\"https://doi.org/10.1112/plms.70163\">10.1112/plms.70163</a>","mla":"Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” <i>Proceedings of the London Mathematical Society</i>, vol. 132, no. 5, e70163, Wiley, 2026, doi:<a href=\"https://doi.org/10.1112/plms.70163\">10.1112/plms.70163</a>.","ista":"Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. 132(5), e70163.","ieee":"H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of the electrostatic potential,” <i>Proceedings of the London Mathematical Society</i>, vol. 132, no. 5. Wiley, 2026.","short":"H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical Society 132 (2026).","apa":"Edelsbrunner, H., Fillmore, C. D., &#38; Oliveira, G. (2026). Counting equilibria of the electrostatic potential. <i>Proceedings of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/plms.70163\">https://doi.org/10.1112/plms.70163</a>","chicago":"Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting Equilibria of the Electrostatic Potential.” <i>Proceedings of the London Mathematical Society</i>. Wiley, 2026. <a href=\"https://doi.org/10.1112/plms.70163\">https://doi.org/10.1112/plms.70163</a>."},"publication_identifier":{"issn":["0024-6115"],"eissn":["1460-244X"]},"abstract":[{"text":"In 1873, James C. Maxwell conjectured that the electric field generated by n point charges in generic position has at most (n-1)^2 isolated zeroes. The first (nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov, and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.","lang":"eng"}],"article_number":"e70163","publication_status":"published","article_type":"original","publication":"Proceedings of the London Mathematical Society","volume":132,"date_created":"2026-05-31T22:02:13Z","type":"journal_article","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"21050"}]}},{"date_published":"2026-02-01T00:00:00Z","external_id":{"arxiv":["2212.11380"],"isi":["001599061500002"]},"project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"oa":1,"OA_type":"green","quality_controlled":"1","status":"public","department":[{"_id":"HeEd"}],"oa_version":"Preprint","corr_author":"1","acknowledgement":"Work by all authors but the second is supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation . The second author thanks Jesús A. De Loera for useful discussions on flips and non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic graphs.","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"full_name":"Ghafari, Mohadese","last_name":"Ghafari","first_name":"Mohadese"},{"first_name":"Teresa","orcid":"0000-0002-1780-2689","last_name":"Heiss","full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"arxiv":1,"intvolume":"       132","isi":1,"das_tickbox":"0","doi":"10.1016/j.ejc.2025.104248","month":"02","scopus_import":"1","article_processing_charge":"No","language":[{"iso":"eng"}],"day":"01","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2212.11380","open_access":"1"}],"OA_place":"repository","publication":"European Journal of Combinatorics","volume":132,"abstract":[{"lang":"eng","text":"We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n\r\n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.\r\n"}],"article_number":"104248","publication_status":"published","article_type":"original","date_created":"2025-10-19T22:01:31Z","type":"journal_article","year":"2026","title":"Flips in two-dimensional hypertriangulations","ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Elsevier","citation":{"ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. 2026;132. doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2026. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.","mla":"Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>, vol. 132, 104248, Elsevier, 2026, doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>.","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” <i>European Journal of Combinatorics</i>, vol. 132. Elsevier, 2026.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2026).","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2026). Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>. Elsevier, 2026. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>."},"publication_identifier":{"issn":["0195-6698"]},"date_updated":"2026-07-23T11:58:37Z","_id":"20490"},{"type":"journal_article","date_created":"2025-11-19T09:44:58Z","related_material":{"record":[{"relation":"earlier_version","id":"17146","status":"public"}]},"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"creator":"dernst","date_updated":"2026-07-27T08:11:05Z","date_created":"2026-07-27T08:11:05Z","content_type":"application/pdf","access_level":"open_access","success":1,"relation":"main_file","file_size":546483,"file_name":"2026_DiscreteCompGeom_Edelsbrunner.pdf","checksum":"c17c014dbbf5be702195c737890251c8","file_id":"22410"}],"ddc":["510"],"publication_status":"published","article_type":"original","abstract":[{"lang":"eng","text":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². "}],"volume":75,"publication":"Discrete & Computational Geometry","_id":"20657","date_updated":"2026-07-27T08:15:58Z","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"citation":{"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>","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>.","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>.","ista":"Edelsbrunner H, Pach J. 2026. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry. 75, 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.","short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry 75 (2026) 597–624."},"title":"Maximum Betti numbers of Čech complexes","ec_funded":1,"publisher":"Springer Nature","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2026","license":"https://creativecommons.org/licenses/by/4.0/","intvolume":"        75","isi":1,"page":"597-624","arxiv":1,"corr_author":"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).","supplementarymaterial":"no","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János","last_name":"Pach","full_name":"Pach, János"}],"quality_controlled":"1","status":"public","oa_version":"Published Version","department":[{"_id":"HeEd"}],"OA_type":"hybrid","oa":1,"date_published":"2026-03-01T00:00:00Z","external_id":{"arxiv":["2310.14801"],"isi":["001610592600001"]},"project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342"}],"OA_place":"publisher","has_accepted_license":"1","day":"01","researchdata_availability":"no","scopus_import":"1","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"month":"03","file_date_updated":"2026-07-27T08:11:05Z","PlanS_conform":"1","doi":"10.1007/s00454-025-00796-5","das_tickbox":"0"},{"has_accepted_license":"1","OA_place":"publisher","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"researchdata_availability":"no","scopus_import":"1","main_file_link":[{"url":"https://doi.org/10.1007/s10208-026-09766-6","open_access":"1"}],"day":"04","PlanS_conform":"1","month":"08","das_tickbox":"0","doi":"10.1007/s10208-026-09766-6","author":[{"last_name":"Dey","full_name":"Dey, Tamal K.","first_name":"Tamal K."},{"id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","last_name":"Lipiński","full_name":"Lipiński, Michał","orcid":"0000-0001-9789-9750","first_name":"Michał"},{"orcid":"0000-0003-2449-1433","full_name":"Soriano Trigueros, Manuel","last_name":"Soriano Trigueros","first_name":"Manuel","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8"}],"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).","supplementarymaterial":"yes","corr_author":"1","arxiv":1,"department":[{"_id":"HeEd"}],"oa_version":"Published Version","quality_controlled":"1","status":"public","project":[{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"external_id":{"arxiv":["2504.17105"]},"date_published":"2026-08-04T00:00:00Z","OA_type":"hybrid","oa":1,"date_updated":"2026-08-11T06:13:33Z","_id":"22648","citation":{"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>","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>.","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>","short":"T.K. Dey, M. Lipiński, M. Soriano Trigueros, Foundations of Computational Mathematics (2026).","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."},"publication_identifier":{"eissn":["1615-3383"],"issn":["1615-3375"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Springer","ec_funded":1,"title":"Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations","year":"2026","date_created":"2026-08-05T06:11:30Z","type":"journal_article","ddc":["500"],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"abstract":[{"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.","lang":"eng"}],"publication_status":"epub_ahead","article_type":"original","publication":"Foundations of Computational Mathematics","keyword":["Multivector field","Conley index","Morse decomposition","Bifurcation","Continuation","Zigzag persistence","Persistence barcode","Gentle algebra"]},{"arxiv":1,"page":"30-62","corr_author":"1","acknowledgement":"This project has received funding from the European Research\r\nCouncil (ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.","author":[{"first_name":"Sebastiano","orcid":"0000-0001-6249-0832","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Draganov, Ondrej","last_name":"Draganov","orcid":"0000-0003-0464-3823","first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"intvolume":"         8","oa":1,"OA_type":"green","date_published":"2026-03-01T00:00:00Z","external_id":{"arxiv":["2212.03128"]},"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"}],"status":"public","quality_controlled":"1","department":[{"_id":"HeEd"}],"oa_version":"Preprint","day":"01","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.03128"}],"scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","OA_place":"repository","has_accepted_license":"1","doi":"10.3934/fods.2025003","month":"03","related_material":{"record":[{"relation":"earlier_version","id":"15091","status":"public"}]},"ddc":["510"],"type":"journal_article","date_created":"2025-11-02T23:01:33Z","volume":8,"keyword":["Topological data analysis","Delaunay mosaic","alpha complex","chromatic sets","persistent homology","kernel/image/cokernel persistent homology","radius function","discrete Morse theory","exact sequences"],"publication":"Foundations of Data Science","article_type":"original","publication_status":"published","abstract":[{"lang":"eng","text":"Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided."}],"publication_identifier":{"eissn":["2639-8001"]},"citation":{"apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). Chromatic alpha complexes. <i>Foundations of Data Science</i>. AIMS. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>. AIMS, 2026. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>, vol. 8, AIMS, 2026, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>Foundations of Data Science</i>, vol. 8. AIMS, pp. 30–62, 2026.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2026) 30–62.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2026;8:30-62. doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>"},"_id":"20585","date_updated":"2026-08-12T06:19:48Z","year":"2026","mathsc":["62R40","55N31","68T09","57Q70"],"ec_funded":1,"title":"Chromatic alpha complexes","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"AIMS"},{"arxiv":1,"page":"108-130","acknowledgement":"This research was supported by NSF grants DMS-2301360 and CCF-2437030 as well as from the European Union's Horizon 2020 research and innovation programme under Marie Sk\\lodowska-Curie grant 101034413.\r\n","author":[{"first_name":"Tamal K.","last_name":"Dey","full_name":"Dey, Tamal K."},{"full_name":"Haas, Andrew","last_name":"Haas","first_name":"Andrew"},{"orcid":"0000-0001-9789-9750","full_name":"Lipiński, Michał","last_name":"Lipiński","first_name":"Michał","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e"}],"intvolume":"        25","OA_type":"green","oa":1,"date_published":"2026-01-01T00:00:00Z","external_id":{"arxiv":["2502.19369"]},"project":[{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"quality_controlled":"1","status":"public","oa_version":"Preprint","department":[{"_id":"HeEd"}],"day":"01","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2502.19369","open_access":"1"}],"scopus_import":"1","article_processing_charge":"No","language":[{"iso":"eng"}],"OA_place":"repository","issue":"1","doi":"10.1137/25m1739406","month":"01","ddc":["510"],"type":"journal_article","date_created":"2026-01-12T11:17:06Z","volume":25,"publication":"SIAM Journal on Applied Dynamical Systems","publication_status":"published","article_type":"original","abstract":[{"text":"Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix. These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures—such as attractors, repellers, and orbits—in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of Dey et al. and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.","lang":"eng"}],"publication_identifier":{"issn":["1536-0040"]},"citation":{"ama":"Dey TK, Haas A, Lipiński M. Computing a connection matrix and persistence efficiently from a morse decomposition. <i>SIAM Journal on Applied Dynamical Systems</i>. 2026;25(1):108-130. doi:<a href=\"https://doi.org/10.1137/25m1739406\">10.1137/25m1739406</a>","ista":"Dey TK, Haas A, Lipiński M. 2026. Computing a connection matrix and persistence efficiently from a morse decomposition. SIAM Journal on Applied Dynamical Systems. 25(1), 108–130.","mla":"Dey, Tamal K., et al. “Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition.” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 25, no. 1, SIAM, 2026, pp. 108–30, doi:<a href=\"https://doi.org/10.1137/25m1739406\">10.1137/25m1739406</a>.","short":"T.K. Dey, A. Haas, M. Lipiński, SIAM Journal on Applied Dynamical Systems 25 (2026) 108–130.","ieee":"T. K. Dey, A. Haas, and M. Lipiński, “Computing a connection matrix and persistence efficiently from a morse decomposition,” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 25, no. 1. SIAM, pp. 108–130, 2026.","apa":"Dey, T. K., Haas, A., &#38; Lipiński, M. (2026). Computing a connection matrix and persistence efficiently from a morse decomposition. <i>SIAM Journal on Applied Dynamical Systems</i>. SIAM. <a href=\"https://doi.org/10.1137/25m1739406\">https://doi.org/10.1137/25m1739406</a>","chicago":"Dey, Tamal K., Andrew Haas, and Michał Lipiński. “Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition.” <i>SIAM Journal on Applied Dynamical Systems</i>. SIAM, 2026. <a href=\"https://doi.org/10.1137/25m1739406\">https://doi.org/10.1137/25m1739406</a>."},"_id":"20980","date_updated":"2026-08-12T06:23:46Z","year":"2026","ec_funded":1,"title":"Computing a connection matrix and persistence efficiently from a morse decomposition","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"SIAM"},{"citation":{"ama":"Edelsbrunner H, Lipiński M, Mrozek M, Soriano Trigueros M, Zimin F. The depth poset under transpositions in the filter. In: <i>42nd International Symposium on Computational Geometry</i>. Vol 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2026. doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">10.4230/LIPICS.SOCG.2026.41</a>","ieee":"H. Edelsbrunner, M. Lipiński, M. Mrozek, M. Soriano Trigueros, and F. Zimin, “The depth poset under transpositions in the filter,” in <i>42nd International Symposium on Computational Geometry</i>, New Brunswick, NJ, United States, 2026, vol. 367.","short":"H. Edelsbrunner, M. Lipiński, M. Mrozek, M. Soriano Trigueros, F. Zimin, in:, 42nd International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026.","ista":"Edelsbrunner H, Lipiński M, Mrozek M, Soriano Trigueros M, Zimin F. 2026. The depth poset under transpositions in the filter. 42nd International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 367, 41:1-41:18.","mla":"Edelsbrunner, Herbert, et al. “The Depth Poset under Transpositions in the Filter.” <i>42nd International Symposium on Computational Geometry</i>, vol. 367, 41:1-41:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026, doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">10.4230/LIPICS.SOCG.2026.41</a>.","chicago":"Edelsbrunner, Herbert, Michał Lipiński, Marian Mrozek, Manuel Soriano Trigueros, and Fedor Zimin. “The Depth Poset under Transpositions in the Filter.” In <i>42nd International Symposium on Computational Geometry</i>, Vol. 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">https://doi.org/10.4230/LIPICS.SOCG.2026.41</a>.","apa":"Edelsbrunner, H., Lipiński, M., Mrozek, M., Soriano Trigueros, M., &#38; Zimin, F. (2026). The depth poset under transpositions in the filter. In <i>42nd International Symposium on Computational Geometry</i> (Vol. 367). New Brunswick, NJ, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">https://doi.org/10.4230/LIPICS.SOCG.2026.41</a>"},"publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959774185"]},"date_updated":"2026-08-12T09:02:56Z","_id":"22299","year":"2026","title":"The depth poset under transpositions in the filter","ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ddc":["500"],"file":[{"access_level":"open_access","success":1,"relation":"main_file","file_size":2902144,"file_name":"2026_LIPIcSSoCG_Edelsbrunner.pdf","checksum":"9dfb96ee66985c724b499b0e5888dc8e","file_id":"22329","creator":"dernst","date_updated":"2026-07-14T06:08:05Z","date_created":"2026-07-14T06:08:05Z","content_type":"application/pdf"}],"date_created":"2026-07-13T09:56:38Z","type":"conference","keyword":["Algebraic topology","Lefschetz complexes","persistent homology","vines and vineyards","birth-death pairs","shallow pairs","relations","partial orders","transpositions","Theory of computation → Computational geometry"],"publication":"42nd International Symposium on Computational Geometry","volume":367,"abstract":[{"lang":"eng","text":"The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies."}],"article_number":"41:1-41:18","publication_status":"published","scopus_import":"1","researchdata_availability":"no","language":[{"iso":"eng"}],"article_processing_charge":"Yes","day":"27","has_accepted_license":"1","OA_place":"publisher","das_tickbox":"0","doi":"10.4230/LIPICS.SOCG.2026.41","file_date_updated":"2026-07-14T06:08:05Z","month":"05","acknowledgement":"The authors thank Jakub Leśkiewicz and Bartosz Furmanek for discussions\r\nthat helped improve the paper. Herbert Edelsbrunner: DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35\r\nMichał Lipiński: European Union’s Horizon 2020 research and innovation programme under the\r\nMarie Skłodowska-Curie Grant Agreement No. 101034413\r\nMarian Mrozek: Polish National Science Center under Opus Grant 2019/35/B/ST1/00874 and Opus\r\nGrant 2025/57/B/ST1/00550","supplementarymaterial":"no","corr_author":"1","author":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","first_name":"Michał","orcid":"0000-0001-9789-9750","last_name":"Lipiński","full_name":"Lipiński, Michał"},{"first_name":"Marian","orcid":"0000-0002-0619-6417","last_name":"Mrozek","full_name":"Mrozek, Marian"},{"id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","first_name":"Manuel","orcid":"0000-0003-2449-1433","full_name":"Soriano Trigueros, Manuel","last_name":"Soriano Trigueros"},{"last_name":"Zimin","full_name":"Zimin, Fedor","first_name":"Fedor","id":"afd27eda-91c1-11f0-aad8-c6edbec24c04"}],"arxiv":1,"alternative_title":["LIPIcs"],"intvolume":"       367","date_published":"2026-05-27T00:00:00Z","external_id":{"arxiv":["2511.21961"]},"project":[{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"},{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"OA_type":"gold","oa":1,"conference":{"start_date":"2026-06-02","location":"New Brunswick, NJ, United States","name":"SoCG: Symposium on Computational Geometry","end_date":"2026-06-05"},"status":"public","quality_controlled":"1","oa_version":"Published Version","department":[{"_id":"HeEd"},{"_id":"GradSch"}]},{"publication":"Algebraic & Geometric Topology","volume":25,"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"}],"publication_status":"published","article_type":"original","file":[{"file_id":"20943","file_name":"2025_AlgebraicGeomTopology_Zava.pdf","checksum":"1e05b4f17a44500ae1ae1e21bc636f6a","success":1,"relation":"main_file","file_size":574389,"access_level":"open_access","content_type":"application/pdf","date_updated":"2026-01-05T12:16:38Z","date_created":"2026-01-05T12:16:38Z","creator":"dernst"}],"ddc":["500"],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"date_created":"2025-12-29T12:09:09Z","type":"journal_article","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Mathematical Sciences Publishers","title":"Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces","citation":{"ieee":"N. Zava, “Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces,” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8. Mathematical Sciences Publishers, pp. 5153–5174, 2025.","short":"N. Zava, Algebraic &#38; Geometric Topology 25 (2025) 5153–5174.","mla":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>, vol. 25, no. 8, Mathematical Sciences Publishers, 2025, pp. 5153–74, doi:<a href=\"https://doi.org/10.2140/agt.2025.25.5153\">10.2140/agt.2025.25.5153</a>.","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.","chicago":"Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers, 2025. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>.","apa":"Zava, N. (2025). Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. <i>Algebraic &#38; Geometric Topology</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/agt.2025.25.5153\">https://doi.org/10.2140/agt.2025.25.5153</a>","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>"},"publication_identifier":{"issn":["1472-2747"],"eissn":["1472-2739"]},"date_updated":"2026-01-05T12:19:09Z","_id":"20867","external_id":{"arxiv":["2303.04730"]},"project":[{"grant_number":"I04245","call_identifier":"FWF","name":"Algebraic Footprints of Geometric Features in Homology","_id":"26AD5D90-B435-11E9-9278-68D0E5697425"}],"date_published":"2025-11-20T00:00:00Z","oa":1,"OA_type":"diamond","department":[{"_id":"HeEd"}],"oa_version":"Published Version","quality_controlled":"1","status":"public","author":[{"id":"c8b3499c-7a77-11eb-b046-aa368cbbf2ad","full_name":"Zava, Nicolò","last_name":"Zava","orcid":"0000-0001-8686-1888","first_name":"Nicolò"}],"corr_author":"1","acknowledgement":"The author was supported by the FWF Grant, Project number I4245-N35. The author would like to thank Thomas Weighill for the helpful discussions around Theorem 3.10, and Takamitsu Yamauchi for bringing to my attention the fundamental reference [35]. Furthermore, the author\r\nis thankful for the detailed and helpful comments of the reviewer of this manuscript.","arxiv":1,"page":"5153-5174","intvolume":"        25","doi":"10.2140/agt.2025.25.5153","file_date_updated":"2026-01-05T12:16:38Z","PlanS_conform":"1","month":"11","language":[{"iso":"eng"}],"article_processing_charge":"No","scopus_import":"1","day":"20","has_accepted_license":"1","issue":"8","OA_place":"publisher"},{"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"15012"}]},"date_created":"2026-02-16T15:48:42Z","type":"journal_article","publication":"Computational Geometry","volume":129,"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."}],"publication_status":"published","article_type":"original","article_number":"102186","citation":{"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>.","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>","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” <i>Computational Geometry</i>, vol. 129. Elsevier, 2025.","short":"J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025).","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>, vol. 129, 102186, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>.","ista":"Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186.","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>"},"publication_identifier":{"issn":["0925-7721"]},"date_updated":"2026-04-16T09:12:36Z","_id":"21253","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Elsevier","title":"Decomposition of geometric graphs into star-forests","author":[{"first_name":"János","full_name":"Pach, János","last_name":"Pach"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"},{"first_name":"Patrick","full_name":"Schnider, Patrick","last_name":"Schnider"}],"corr_author":"1","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.","arxiv":1,"intvolume":"       129","external_id":{"arxiv":["2306.13201"]},"date_published":"2025-12-01T00:00:00Z","OA_type":"green","oa":1,"oa_version":"Preprint","department":[{"_id":"HeEd"}],"quality_controlled":"1","status":"public","language":[{"iso":"eng"}],"article_processing_charge":"No","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2306.13201"}],"day":"01","OA_place":"repository","doi":"10.1016/j.comgeo.2025.102186","month":"12"},{"intvolume":"        73","isi":1,"page":"490-499","arxiv":1,"corr_author":"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.","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","orcid":"0000-0002-0659-3201","full_name":"Nikitenko, Anton","last_name":"Nikitenko"}],"quality_controlled":"1","status":"public","oa_version":"Published Version","department":[{"_id":"HeEd"}],"oa":1,"OA_type":"hybrid","date_published":"2025-03-01T00:00:00Z","external_id":{"pmid":["39974750"],"isi":["001238566200004"],"arxiv":["2012.03350"]},"project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"OA_place":"publisher","has_accepted_license":"1","day":"01","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","month":"03","file_date_updated":"2025-04-23T07:31:32Z","doi":"10.1007/s00454-024-00660-y","type":"journal_article","date_created":"2024-06-16T22:01:07Z","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ddc":["510"],"file":[{"date_updated":"2025-04-23T07:31:32Z","date_created":"2025-04-23T07:31:32Z","content_type":"application/pdf","creator":"dernst","checksum":"ffb0c818222138f9f113f4bbea41e834","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","file_id":"19610","access_level":"open_access","file_size":283443,"success":1,"relation":"main_file"}],"publication_status":"published","article_type":"original","abstract":[{"text":"The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.","lang":"eng"}],"volume":73,"publication":"Discrete & Computational Geometry","_id":"17149","pmid":1,"date_updated":"2026-02-16T12:18:50Z","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"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>","ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>.","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.","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>."},"ec_funded":1,"title":"Average and expected distortion of Voronoi paths and scapes","publisher":"Springer Nature","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025"},{"publication":"Advances in Mathematics","volume":461,"abstract":[{"lang":"eng","text":"The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well."}],"article_number":"110055","publication_status":"published","article_type":"original","date_created":"2024-12-08T23:01:54Z","type":"journal_article","year":"2025","ec_funded":1,"title":"Order-2 Delaunay triangulations optimize angles","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Elsevier","citation":{"ama":"Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. 2025;461. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","mla":"Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>, vol. 461, 110055, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025."},"publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"date_updated":"2025-04-15T07:16:53Z","_id":"18626","date_published":"2025-02-01T00:00:00Z","external_id":{"arxiv":["2310.18238"],"isi":["001370682500001"]},"project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"}],"OA_type":"green","oa":1,"status":"public","quality_controlled":"1","department":[{"_id":"HeEd"}],"oa_version":"Preprint","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.","corr_author":"1","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"arxiv":1,"intvolume":"       461","isi":1,"doi":"10.1016/j.aim.2024.110055","month":"02","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","day":"01","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2310.18238","open_access":"1"}],"OA_place":"repository"},{"corr_author":"1","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","author":[{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov","full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823","first_name":"Ondrej"}],"page":"140","alternative_title":["ISTA Thesis"],"date_published":"2025-02-03T00:00:00Z","project":[{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342"}],"oa":1,"status":"public","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"oa_version":"Published Version","article_processing_charge":"No","language":[{"iso":"eng"}],"day":"03","has_accepted_license":"1","OA_place":"publisher","doi":"10.15479/at:ista:18979","file_date_updated":"2025-02-04T16:22:07Z","month":"02","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"related_material":{"record":[{"status":"public","id":"15091","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"18981"}]},"file":[{"creator":"odragano","content_type":"application/zip","date_created":"2025-01-31T16:58:30Z","date_updated":"2025-01-31T16:58:30Z","relation":"source_file","file_size":11899491,"access_level":"closed","file_id":"18983","file_name":"Thesis.zip","checksum":"af6567e5d35e5eb330b8925ae37f1998"},{"content_type":"application/pdf","date_updated":"2025-02-04T16:22:07Z","date_created":"2025-02-04T16:22:07Z","creator":"odragano","file_id":"19000","file_name":"Thesis.pdf","checksum":"c3fef68e35b9dc2020b2ca6006da6343","relation":"main_file","file_size":8857514,"access_level":"open_access"}],"ddc":["514","004"],"date_created":"2025-01-31T17:04:40Z","type":"dissertation","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","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"}],"publication_status":"published","citation":{"ama":"Draganov O. Structures and computations in topological data analysis. 2025. doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>","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>.","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>","ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025.","short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","ista":"Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria.","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>."},"publication_identifier":{"issn":["2663-337X"]},"date_updated":"2026-04-07T11:47:30Z","_id":"18979","supervisor":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"}],"year":"2025","title":"Structures and computations in topological data analysis","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publisher":"Institute of Science and Technology Austria"},{"intvolume":"       719","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.","corr_author":"1","author":[{"first_name":"Mohammad","full_name":"Mahini, Mohammad","last_name":"Mahini"},{"full_name":"Beigy, Hamid","last_name":"Beigy","first_name":"Hamid"},{"full_name":"Qadami, Salman","last_name":"Qadami","first_name":"Salman"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"status":"public","quality_controlled":"1","oa_version":"None","department":[{"_id":"HeEd"}],"OA_type":"closed access","date_published":"2025-11-01T00:00:00Z","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"external_id":{"isi":["001516170500002"]},"issue":"11","day":"01","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","month":"11","doi":"10.1016/j.ins.2025.122425","type":"journal_article","date_created":"2025-06-30T08:48:48Z","article_number":"122425","publication_status":"published","article_type":"original","abstract":[{"text":"Simplets are elementary units within simplicial complexes and are fundamental for analyzing the structure of simplicial complexes. Previous efforts have mainly focused on accurately counting or approximating the number of simplets rather than studying their frequencies. However, analyzing simplet frequencies is more practical for large-scale simplicial complexes. This paper introduces the Simplet Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies in simplicial complexes. Additionally, we provide a bound on the sample complexity required to approximate the SFD vector using any uniform sampling-based algorithm accurately. We extend the definition of simplet frequency distribution to encompass simplices, allowing for the analysis of simplet frequencies within simplices of simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex. Furthermore, we present a bound on the sample complexity required for accurately approximating the SDV and SDC for a set of simplices using any uniform sampling-based algorithm. We also introduce algorithms for approximating SFD, geometric SFD, SDV, and SDC. We also validate the theoretical bounds with experiments on random simplicial complexes and demonstrate the practical application through a case study.","lang":"eng"}],"volume":719,"publication":"Information Sciences","_id":"19937","date_updated":"2025-12-30T09:05:32Z","publication_identifier":{"issn":["0020-0255"]},"citation":{"chicago":"Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>.","apa":"Mahini, M., Beigy, H., Qadami, S., &#38; Saghafian, M. (2025). Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>","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.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).","ista":"Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. 719(11), 122425.","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>.","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>"},"title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","ec_funded":1,"publisher":"Elsevier","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","title":"On spheres with k points inside","year":"2025","_id":"20005","date_updated":"2025-07-14T07:26:14Z","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773706"]},"citation":{"chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","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.","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>"},"publication_status":"published","article_number":"43","abstract":[{"text":"We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.","lang":"eng"}],"volume":332,"publication":"41st International Symposium on Computational Geometry","type":"conference","date_created":"2025-07-13T22:01:22Z","ddc":["510"],"file":[{"content_type":"application/pdf","date_updated":"2025-07-14T07:24:22Z","date_created":"2025-07-14T07:24:22Z","creator":"dernst","file_id":"20016","checksum":"b5313ed8575ea87913c71a6e3c7513c8","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","file_size":661893,"relation":"main_file","success":1,"access_level":"open_access"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"month":"06","file_date_updated":"2025-07-14T07:24:22Z","doi":"10.4230/LIPIcs.SoCG.2025.43","OA_place":"publisher","has_accepted_license":"1","day":"20","language":[{"iso":"eng"}],"article_processing_charge":"Yes","scopus_import":"1","oa_version":"Published Version","department":[{"_id":"HeEd"}],"quality_controlled":"1","status":"public","conference":{"end_date":"2025-06-27","name":"SoCG: Symposium on Computational Geometry","location":"Kanazawa, Japan","start_date":"2025-06-23"},"OA_type":"gold","oa":1,"project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"}],"external_id":{"arxiv":["2410.21204"]},"date_published":"2025-06-20T00:00:00Z","intvolume":"       332","alternative_title":["LIPIcs"],"arxiv":1,"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"corr_author":"1","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"},{"external_id":{"arxiv":["2405.17920"]},"project":[{"grant_number":"W1260-N35","name":"Vienna Graduate School on Computational Optimization","_id":"9B9290DE-BA93-11EA-9121-9846C619BF3A"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF"}],"date_published":"2025-06-20T00:00:00Z","oa":1,"OA_type":"gold","department":[{"_id":"HeEd"}],"oa_version":"Published Version","conference":{"end_date":"2025-06-27","name":"SoCG: Symposium on Computational Geometry","location":"Kanazawa, Japan","start_date":"2025-06-23"},"quality_controlled":"1","status":"public","author":[{"first_name":"Lara","last_name":"Ost","full_name":"Ost, Lara"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"}],"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.","corr_author":"1","arxiv":1,"alternative_title":["LIPIcs"],"intvolume":"       332","doi":"10.4230/LIPIcs.SoCG.2025.71","file_date_updated":"2025-07-14T08:23:38Z","month":"06","article_processing_charge":"Yes","language":[{"iso":"eng"}],"scopus_import":"1","day":"20","has_accepted_license":"1","OA_place":"publisher","publication":"41st International Symposium on Computational Geometry","volume":332,"abstract":[{"text":"In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.","lang":"eng"}],"publication_status":"published","article_number":"71","ddc":["000"],"file":[{"creator":"dernst","date_created":"2025-07-14T08:23:38Z","date_updated":"2025-07-14T08:23:38Z","content_type":"application/pdf","access_level":"open_access","relation":"main_file","success":1,"file_size":834623,"file_name":"2025_LIPIcs.SoCG_Ost.pdf","checksum":"3a4a7a707a56e0cfdf51428782dee55a","file_id":"20017"}],"related_material":{"link":[{"url":"https://github.com/laraost/BananaPersist","relation":"software"}]},"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"date_created":"2025-07-13T22:01:22Z","type":"conference","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","title":"Banana trees for the persistence in time series experimentally","citation":{"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.","short":"L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","mla":"Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 71, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>.","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>.","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>"},"publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"date_updated":"2025-12-30T11:04:33Z","_id":"20006"},{"author":[{"full_name":"Chambers, Erin Wolf","last_name":"Chambers","first_name":"Erin Wolf"},{"full_name":"Fillmore, Christopher D","last_name":"Fillmore","first_name":"Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"orcid":"0000-0002-6862-208X","last_name":"Stephenson","full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220","first_name":"Mathijs"}],"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.","corr_author":"1","page":"811-828","intvolume":"         4","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses","grant_number":"M03073"}],"date_published":"2025-12-01T00:00:00Z","OA_type":"hybrid","oa":1,"department":[{"_id":"HeEd"}],"oa_version":"Published Version","status":"public","quality_controlled":"1","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"scopus_import":"1","day":"01","has_accepted_license":"1","OA_place":"publisher","doi":"10.1007/s44007-025-00170-0","PlanS_conform":"1","file_date_updated":"2025-12-30T07:52:58Z","month":"12","ddc":["510"],"file":[{"date_created":"2025-12-30T07:52:58Z","date_updated":"2025-12-30T07:52:58Z","content_type":"application/pdf","creator":"dernst","checksum":"e2043259194bfcdf3d74c4da8a5a853f","file_name":"2025_LaMatematica_Chambers.pdf","file_id":"20885","access_level":"open_access","file_size":2678640,"relation":"main_file","success":1}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"21021"}]},"date_created":"2025-08-31T22:01:33Z","type":"journal_article","publication":"La Matematica","volume":4,"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."}],"article_type":"original","publication_status":"published","citation":{"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>","chicago":"Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>.","apa":"Chambers, E. W., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2025). Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>","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.","short":"E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica 4 (2025) 811–828.","mla":"Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>, vol. 4, Springer Nature, 2025, pp. 811–28, doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>.","ista":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica. 4, 811–828."},"publication_identifier":{"eissn":["2730-9657"]},"date_updated":"2026-04-07T11:42:48Z","_id":"20260","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Springer Nature","ec_funded":1,"title":"Burning or collapsing the medial axis is unstable"},{"doi":"10.3390/e27080854","month":"08","PlanS_conform":"1","file_date_updated":"2025-09-08T07:55:48Z","day":"01","article_processing_charge":"Yes","language":[{"iso":"eng"}],"scopus_import":"1","issue":"8","OA_place":"publisher","has_accepted_license":"1","OA_type":"gold","oa":1,"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"external_id":{"isi":["001557476000001"],"pmid":["40870326"]},"date_published":"2025-08-01T00:00:00Z","department":[{"_id":"HeEd"}],"oa_version":"Published Version","quality_controlled":"1","status":"public","author":[{"first_name":"Arseniy","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan","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"},{"id":"2E36B656-F248-11E8-B48F-1D18A9856A87","full_name":"Virk, Ziga","last_name":"Virk","first_name":"Ziga"},{"first_name":"Hubert","full_name":"Wagner, Hubert","last_name":"Wagner","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"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.","corr_author":"1","isi":1,"intvolume":"        27","year":"2025","DOAJ_listed":"1","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publisher":"MDPI","title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence","ec_funded":1,"publication_identifier":{"eissn":["1099-4300"]},"citation":{"ama":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. 2025;27(8). doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>","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>.","apa":"Akopyan, A., Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2025). Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. MDPI. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).","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.","ista":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.","mla":"Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>, vol. 27, no. 8, 854, MDPI, 2025, doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>."},"_id":"20293","pmid":1,"date_updated":"2025-09-30T14:32:31Z","volume":27,"publication":"Entropy","publication_status":"published","article_type":"original","article_number":"854","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"}],"file":[{"file_name":"2025_Entropy_Akopyan.pdf","checksum":"65c5399c4015d9c8abb8c7a96f3d7836","file_id":"20309","access_level":"open_access","success":1,"relation":"main_file","file_size":379340,"date_updated":"2025-09-08T07:55:48Z","date_created":"2025-09-08T07:55:48Z","content_type":"application/pdf","creator":"dernst"}],"ddc":["500"],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"type":"journal_article","date_created":"2025-09-07T22:01:33Z"},{"day":"01","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","scopus_import":"1","issue":"10","OA_place":"publisher","has_accepted_license":"1","doi":"10.1016/j.jpaa.2025.108068","month":"10","file_date_updated":"2025-12-30T07:55:08Z","PlanS_conform":"1","arxiv":1,"author":[{"last_name":"Brown","full_name":"Brown, Adam","first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425"},{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","last_name":"Draganov","orcid":"0000-0003-0464-3823","first_name":"Ondrej"}],"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","corr_author":"1","intvolume":"       229","oa":1,"OA_type":"hybrid","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"external_id":{"arxiv":["2209.14993"]},"date_published":"2025-10-01T00:00:00Z","oa_version":"Published Version","department":[{"_id":"HeEd"}],"status":"public","quality_controlled":"1","publication_identifier":{"issn":["0022-4049"]},"citation":{"ama":"Brown A, Draganov O. Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. 2025;229(10). doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>","chicago":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>.","apa":"Brown, A., &#38; Draganov, O. (2025). Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.","short":"A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).","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>.","ista":"Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure and Applied Algebra. 229(10), 108068."},"_id":"20323","date_updated":"2025-12-30T07:55:21Z","year":"2025","publisher":"Elsevier","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ec_funded":1,"title":"Discrete microlocal Morse theory","ddc":["510"],"file":[{"creator":"dernst","content_type":"application/pdf","date_updated":"2025-12-30T07:55:08Z","date_created":"2025-12-30T07:55:08Z","relation":"main_file","success":1,"file_size":3090836,"access_level":"open_access","file_id":"20886","file_name":"2025_JourPureAppliedAlgebra_Brown.pdf","checksum":"39bcad462278c9322ef810af7db67f56"}],"related_material":{"record":[{"status":"public","id":"18981","relation":"earlier_version"}]},"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"type":"journal_article","date_created":"2025-09-10T05:40:09Z","volume":229,"publication":"Journal of Pure and Applied Algebra","article_type":"original","publication_status":"published","article_number":"108068","abstract":[{"text":"We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities.","lang":"eng"}]},{"month":"11","doi":"10.1007/978-3-032-09544-2_10","OA_place":"repository","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"No","day":"01","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2504.14743","open_access":"1"}],"conference":{"end_date":"2025-11-06","name":"DGMM: Discrete Geometry and Mathematical Morphology","location":"Groningen, The Netherlands","start_date":"2025-11-03"},"quality_controlled":"1","status":"public","oa_version":"Preprint","department":[{"_id":"HeEd"}],"date_published":"2025-11-01T00:00:00Z","external_id":{"arxiv":["2504.14743"]},"OA_type":"green","oa":1,"alternative_title":["LNCS"],"intvolume":"     16296","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert"},{"id":"2D04F932-F248-11E8-B48F-1D18A9856A87","first_name":"Elizabeth R","orcid":"0000-0002-6862-208X","last_name":"Stephenson","full_name":"Stephenson, Elizabeth R"},{"full_name":"Thoresen, Martin H","last_name":"Thoresen","first_name":"Martin H","id":"47CB1472-F248-11E8-B48F-1D18A9856A87"}],"arxiv":1,"page":"133-147","title":"The mid-sphere cousin of the medial axis transform","publisher":"Springer Nature","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","date_updated":"2025-11-24T10:05:11Z","_id":"20658","citation":{"ieee":"H. Edelsbrunner, E. R. Stephenson, and M. H. Thoresen, “The mid-sphere cousin of the medial axis transform,” in <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>, Groningen, The Netherlands, 2025, vol. 16296, pp. 133–147.","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.","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>.","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>.","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>"},"publication_identifier":{"isbn":["9783032095435"],"eissn":["1611-3349"],"issn":["0302-9743"]},"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"}],"publication_status":"published","publication":"4th International Joint Conference on Discrete Geometry and Mathematical Morphology","volume":16296,"date_created":"2025-11-23T23:01:37Z","type":"conference"},{"date_updated":"2025-12-09T13:46:42Z","_id":"20729","citation":{"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>","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>.","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>"},"publication_identifier":{"isbn":["9798400720758"]},"title":"Additive partial matchings for persistent homology","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Association for Computing Machinery","year":"2025","date_created":"2025-12-07T23:02:01Z","type":"conference","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"creator":"dernst","date_created":"2025-12-09T13:43:17Z","date_updated":"2025-12-09T13:43:17Z","content_type":"application/pdf","access_level":"open_access","relation":"main_file","success":1,"file_size":761617,"file_name":"2025_ISSAC_GonzalezDiaz.pdf","checksum":"1c299cca165a20e2518afe4fda63dbf1","file_id":"20751"}],"ddc":["510"],"abstract":[{"lang":"eng","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."}],"publication_status":"published","publication":"Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation","has_accepted_license":"1","OA_place":"publisher","scopus_import":"1","language":[{"iso":"eng"}],"article_processing_charge":"Yes (in subscription journal)","day":"10","file_date_updated":"2025-12-09T13:43:17Z","month":"11","doi":"10.1145/3747199.3747561","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).","corr_author":"1","author":[{"full_name":"Gonzalez-Diaz, Rocio","last_name":"Gonzalez-Diaz","first_name":"Rocio"},{"first_name":"Manuel","last_name":"Soriano Trigueros","full_name":"Soriano Trigueros, Manuel","orcid":"0000-0003-2449-1433","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8"},{"full_name":"Torras-Casas, Alvaro","last_name":"Torras-Casas","first_name":"Alvaro"}],"page":"188-196","status":"public","conference":{"end_date":"2025-08-01","name":"ISSAC: International Symposium on Symbolic and Algebraic Computation","location":"Guanajuato, Mexico","start_date":"2025-07-28"},"quality_controlled":"1","department":[{"_id":"HeEd"}],"oa_version":"Published Version","date_published":"2025-11-10T00:00:00Z","OA_type":"hybrid","oa":1}]