[{"article_type":"original","article_number":"5","scopus_import":"1","day":"01","doi":"10.1007/s41468-026-00233-3","year":"2026","project":[{"call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"author":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"},{"first_name":"Matthew","last_name":"Kahle","full_name":"Kahle, Matthew"},{"first_name":"Shu","last_name":"Kanazawa","full_name":"Kanazawa, Shu"}],"title":"Maximum persistent Betti numbers of Čech complexes","month":"03","PlanS_conform":"1","ddc":["500"],"type":"journal_article","file":[{"file_name":"2026_JourAppliedCompTopology_Edelsbrunner.pdf","content_type":"application/pdf","date_created":"2026-03-09T11:29:30Z","success":1,"checksum":"0bf6dc430cafa40c08f260fe17d54595","file_size":323111,"file_id":"21416","relation":"main_file","access_level":"open_access","creator":"dernst","date_updated":"2026-03-09T11:29:30Z"}],"oa":1,"date_created":"2026-03-08T23:01:45Z","abstract":[{"text":"This note proves that only a linear number of holes in a Cech complex of n points in R^d\r\ncan persist over an interval of constant length. Specifically, for any fixed dimension p <\r\nd and fixed ε > 0, the number of p-dimensional holes in the ˇ Cech complex at radius 1\r\nthat persist to radius 1+ε is bounded above by a constant times n,where n is the number\r\nof points. The proof uses a packing argument supported by relating theCˇ ech complexes\r\nwith corresponding snap complexes over the cells in a partition of space. The argument\r\nis self-contained and elementary, relying on geometric and combinatorial constructions\r\nrather than on the existing theory of sparse approximations or interleavings. The bound\r\nalso applies to Alpha complexes and Vietoris–Rips complexes. While our result can be\r\ninferred from prior work on sparse filtrations, to our knowledge, no explicit statement\r\nor direct proof of this bound appears in the literature.","lang":"eng"}],"status":"public","arxiv":1,"quality_controlled":"1","has_accepted_license":"1","OA_type":"hybrid","OA_place":"publisher","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"citation":{"chicago":"Edelsbrunner, Herbert, Matthew Kahle, and Shu Kanazawa. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>.","ieee":"H. Edelsbrunner, M. Kahle, and S. Kanazawa, “Maximum persistent Betti numbers of Čech complexes,” <i>Journal of Applied and Computational Topology</i>, vol. 10. Springer Nature, 2026.","apa":"Edelsbrunner, H., Kahle, M., &#38; Kanazawa, S. (2026). Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>","ama":"Edelsbrunner H, Kahle M, Kanazawa S. Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. 2026;10. doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>","mla":"Edelsbrunner, Herbert, et al. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>, vol. 10, 5, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>.","ista":"Edelsbrunner H, Kahle M, Kanazawa S. 2026. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 10, 5.","short":"H. Edelsbrunner, M. Kahle, S. Kanazawa, Journal of Applied and Computational Topology 10 (2026)."},"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"external_id":{"arxiv":["2409.05241"]},"_id":"21407","article_processing_charge":"Yes (in subscription journal)","date_updated":"2026-03-09T11:31:29Z","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"acknowledgement":"The authors would like to thank Michael Lesnick and Primoz Skraba for their helpful comments regarding sparse approximations of filtrations. We are also grateful to the anonymous referees for their careful reading and constructive suggestions. The three authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35, the U.S. National Science Foundation (NSF-DMS), grant no. 2005630, and a JSPS Grant-in-Aid for Transformative Research Areas (A) (22H05107, Y.H.), EPSRC Research Grant EP/Y008642/1.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"        10","file_date_updated":"2026-03-09T11:29:30Z","publication":"Journal of Applied and Computational Topology","date_published":"2026-03-01T00:00:00Z","volume":10,"oa_version":"Published Version","publisher":"Springer Nature","publication_status":"published"},{"article_type":"original","doi":"10.1007/s41468-024-00173-w","year":"2024","author":[{"orcid":"0000-0002-5372-7890","last_name":"Biswas","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita"},{"last_name":"Cultrera Di Montesano","first_name":"Sebastiano","orcid":"0000-0001-6249-0832","full_name":"Cultrera Di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"},{"last_name":"Saghafian","first_name":"Morteza","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes"}],"page":"557-578","scopus_import":"1","day":"01","month":"09","title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","ddc":["510"],"related_material":{"record":[{"id":"11658","status":"public","relation":"earlier_version"}]},"file":[{"checksum":"0ee15c1493a6413cf356ab2f32c81a9e","file_size":522831,"file_id":"19612","relation":"main_file","creator":"dernst","access_level":"open_access","date_updated":"2025-04-23T08:01:36Z","file_name":"2024_JourApplCompTopo_BiswasRa.pdf","date_created":"2025-04-23T08:01:36Z","content_type":"application/pdf","success":1}],"oa":1,"type":"journal_article","abstract":[{"text":"The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.","lang":"eng"}],"date_created":"2024-05-12T22:01:03Z","quality_controlled":"1","status":"public","corr_author":"1","has_accepted_license":"1","OA_place":"publisher","OA_type":"hybrid","language":[{"iso":"eng"}],"external_id":{"pmid":["39308789"]},"pmid":1,"department":[{"_id":"HeEd"}],"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"citation":{"ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 557–578, 2024.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 8, 557–578.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578.","mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 557–78, doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. 2024;8:557-578. doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>"},"article_processing_charge":"Yes (via OA deal)","date_updated":"2025-05-14T09:27:57Z","_id":"15380","acknowledgement":"The authors thank Uli Wagner and Emo Welzl for comments on an earlier version of this paper, and for pointing out related work in the prior literature.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"publication":"Journal of Applied and Computational Topology","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"         8","file_date_updated":"2025-04-23T08:01:36Z","oa_version":"Published Version","ec_funded":1,"date_published":"2024-09-01T00:00:00Z","volume":8,"publisher":"Springer Nature","publication_status":"published"},{"language":[{"iso":"eng"}],"OA_type":"hybrid","OA_place":"publisher","has_accepted_license":"1","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"citation":{"chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 1101–1119, 2024.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 1101–19, doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. 2024;8:1101-1119. doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 8, 1101–1119.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 1101–1119."},"department":[{"_id":"HeEd"}],"pmid":1,"external_id":{"pmid":["39678706"]},"_id":"13182","date_updated":"2026-04-07T12:58:47Z","article_processing_charge":"Yes (via OA deal)","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"acknowledgement":"Open access funding provided by Austrian Science Fund (FWF). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of this paper thank anonymous reviewers for their constructive criticism and Monika Henzinger for detailed comments on an earlier version of this paper.","file_date_updated":"2025-01-09T07:39:41Z","intvolume":"         8","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"Journal of Applied and Computational Topology","date_published":"2024-10-01T00:00:00Z","volume":8,"ec_funded":1,"oa_version":"Published Version","publication_status":"published","publisher":"Springer Nature","article_type":"original","day":"01","page":"1101-1119","scopus_import":"1","author":[{"orcid":"0000-0002-5372-7890","first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera Di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","last_name":"Cultrera Di Montesano","first_name":"Sebastiano"},{"first_name":"Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian"}],"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Persistent Homology, Algorithms and Stochastic Geometry"},{"grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"}],"doi":"10.1007/s41468-023-00126-9","year":"2024","title":"Geometric characterization of the persistence of 1D maps","month":"10","related_material":{"record":[{"id":"15094","relation":"dissertation_contains","status":"public"}]},"ddc":["000"],"type":"journal_article","file":[{"success":1,"file_name":"2024_JourApplCompTopo_Biswas.pdf","content_type":"application/pdf","date_created":"2025-01-09T07:39:41Z","access_level":"open_access","creator":"dernst","date_updated":"2025-01-09T07:39:41Z","relation":"main_file","file_id":"18783","checksum":"d493df5088c222b88d9ca46b623ad0ee","file_size":476896}],"oa":1,"date_created":"2023-07-02T22:00:44Z","abstract":[{"text":"We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram.","lang":"eng"}],"status":"public","quality_controlled":"1","corr_author":"1"},{"status":"public","quality_controlled":"1","corr_author":"1","type":"journal_article","oa":1,"date_created":"2023-03-26T22:01:08Z","main_file_link":[{"url":"https://inserm.hal.science/INRIA-SACLAY/hal-04083524v1","open_access":"1"}],"abstract":[{"text":"Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In this paper we introduce a slight variant of Kleinjohann’s and Bangert’s extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality.","lang":"eng"}],"title":"The reach of subsets of manifolds","month":"09","article_type":"original","scopus_import":"1","page":"619-641","day":"01","doi":"10.1007/s41468-023-00116-x","year":"2023","project":[{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"name":"Learning and triangulating manifolds via collapses","grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"author":[{"last_name":"Boissonnat","first_name":"Jean Daniel","full_name":"Boissonnat, Jean Daniel"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","first_name":"Mathijs","last_name":"Wintraecken","orcid":"0000-0002-7472-2220"}],"date_published":"2023-09-01T00:00:00Z","volume":7,"oa_version":"Submitted Version","ec_funded":1,"publisher":"Springer Nature","publication_status":"published","acknowledgement":"We thank Eddie Aamari, David Cohen-Steiner, Isa Costantini, Fred Chazal, Ramsay Dyer, André Lieutier, and Alef Sterk for discussion and Pierre Pansu for encouragement. We further acknowledge the anonymous reviewers whose comments helped improve the exposition.\r\nThe research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions). The first author is further supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002. The second author is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411 and the Austrian science fund (FWF) M-3073.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"         7","publication":"Journal of Applied and Computational Topology","_id":"12763","article_processing_charge":"No","date_updated":"2025-04-14T07:44:01Z","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"citation":{"chicago":"Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets of Manifolds.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s41468-023-00116-x\">https://doi.org/10.1007/s41468-023-00116-x</a>.","ieee":"J. D. Boissonnat and M. Wintraecken, “The reach of subsets of manifolds,” <i>Journal of Applied and Computational Topology</i>, vol. 7. Springer Nature, pp. 619–641, 2023.","apa":"Boissonnat, J. D., &#38; Wintraecken, M. (2023). The reach of subsets of manifolds. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-023-00116-x\">https://doi.org/10.1007/s41468-023-00116-x</a>","ama":"Boissonnat JD, Wintraecken M. The reach of subsets of manifolds. <i>Journal of Applied and Computational Topology</i>. 2023;7:619-641. doi:<a href=\"https://doi.org/10.1007/s41468-023-00116-x\">10.1007/s41468-023-00116-x</a>","mla":"Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets of Manifolds.” <i>Journal of Applied and Computational Topology</i>, vol. 7, Springer Nature, 2023, pp. 619–41, doi:<a href=\"https://doi.org/10.1007/s41468-023-00116-x\">10.1007/s41468-023-00116-x</a>.","ista":"Boissonnat JD, Wintraecken M. 2023. The reach of subsets of manifolds. Journal of Applied and Computational Topology. 7, 619–641.","short":"J.D. Boissonnat, M. Wintraecken, Journal of Applied and Computational Topology 7 (2023) 619–641."},"publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]}},{"title":"Probabilistic convergence and stability of random mapper graphs","month":"03","ddc":["510"],"article_type":"original","scopus_import":"1","page":"99-140","day":"01","year":"2021","doi":"10.1007/s41468-020-00063-x","author":[{"full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam","last_name":"Brown"},{"first_name":"Omer","last_name":"Bobrowski","full_name":"Bobrowski, Omer"},{"first_name":"Elizabeth","last_name":"Munch","full_name":"Munch, Elizabeth"},{"last_name":"Wang","first_name":"Bei","full_name":"Wang, Bei"}],"project":[{"call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships"}],"status":"public","quality_controlled":"1","arxiv":1,"type":"journal_article","file":[{"success":1,"date_created":"2021-02-11T14:43:59Z","content_type":"application/pdf","file_name":"2020_JourApplCompTopology_Brown.pdf","file_id":"9112","file_size":2090265,"checksum":"3f02e9d47c428484733da0f588a3c069","date_updated":"2021-02-11T14:43:59Z","creator":"dernst","access_level":"open_access","relation":"main_file"}],"oa":1,"date_created":"2021-02-11T14:41:02Z","abstract":[{"text":"We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.","lang":"eng"}],"_id":"9111","article_processing_charge":"Yes (via OA deal)","date_updated":"2025-04-14T07:43:51Z","issue":"1","has_accepted_license":"1","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"citation":{"ieee":"A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence and stability of random mapper graphs,” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.","chicago":"Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>.","ista":"Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 5(1), 99–140.","short":"A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational Topology 5 (2021) 99–140.","mla":"Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1, Springer Nature, 2021, pp. 99–140, doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>.","ama":"Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. 2021;5(1):99-140. doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>","apa":"Brown, A., Bobrowski, O., Munch, E., &#38; Wang, B. (2021). Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>"},"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"external_id":{"arxiv":["1909.03488"]},"date_published":"2021-03-01T00:00:00Z","volume":5,"oa_version":"Published Version","ec_funded":1,"publisher":"Springer Nature","publication_status":"published","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"acknowledgement":"AB was supported in part by the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No. 754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation, Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like to thank the Institute for Mathematics and its Applications for hosting a workshop titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen Access funding provided by Institute of Science and Technology (IST Austria).","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","file_date_updated":"2021-02-11T14:43:59Z","intvolume":"         5","publication":"Journal of Applied and Computational Topology"},{"title":"Čech-Delaunay gradient flow and homology inference for self-maps","month":"12","ddc":["500"],"article_type":"original","day":"01","scopus_import":"1","page":"455-480","author":[{"last_name":"Bauer","first_name":"U.","full_name":"Bauer, U."},{"last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"},{"first_name":"Grzegorz","last_name":"Jablonski","orcid":"0000-0002-3536-9866","id":"4483EF78-F248-11E8-B48F-1D18A9856A87","full_name":"Jablonski, Grzegorz"},{"last_name":"Mrozek","first_name":"M.","full_name":"Mrozek, M."}],"doi":"10.1007/s41468-020-00058-8","year":"2020","status":"public","quality_controlled":"1","type":"journal_article","oa":1,"file":[{"date_created":"2024-03-04T10:52:42Z","content_type":"application/pdf","file_name":"2020_JourApplCompTopology_Bauer.pdf","success":1,"relation":"main_file","date_updated":"2024-03-04T10:52:42Z","access_level":"open_access","creator":"dernst","file_size":851190,"checksum":"eed1168b6e66cd55272c19bb7fca8a1c","file_id":"15065"}],"date_created":"2024-03-04T10:47:49Z","abstract":[{"text":"We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.","lang":"eng"}],"_id":"15064","date_updated":"2024-03-04T10:54:04Z","article_processing_charge":"Yes (via OA deal)","issue":"4","language":[{"iso":"eng"}],"has_accepted_license":"1","citation":{"mla":"Bauer, U., et al. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” <i>Journal of Applied and Computational Topology</i>, vol. 4, no. 4, Springer Nature, 2020, pp. 455–80, doi:<a href=\"https://doi.org/10.1007/s41468-020-00058-8\">10.1007/s41468-020-00058-8</a>.","ama":"Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. Čech-Delaunay gradient flow and homology inference for self-maps. <i>Journal of Applied and Computational Topology</i>. 2020;4(4):455-480. doi:<a href=\"https://doi.org/10.1007/s41468-020-00058-8\">10.1007/s41468-020-00058-8</a>","ista":"Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. 2020. Čech-Delaunay gradient flow and homology inference for self-maps. Journal of Applied and Computational Topology. 4(4), 455–480.","short":"U. Bauer, H. Edelsbrunner, G. Jablonski, M. Mrozek, Journal of Applied and Computational Topology 4 (2020) 455–480.","apa":"Bauer, U., Edelsbrunner, H., Jablonski, G., &#38; Mrozek, M. (2020). Čech-Delaunay gradient flow and homology inference for self-maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-020-00058-8\">https://doi.org/10.1007/s41468-020-00058-8</a>","chicago":"Bauer, U., Herbert Edelsbrunner, Grzegorz Jablonski, and M. Mrozek. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s41468-020-00058-8\">https://doi.org/10.1007/s41468-020-00058-8</a>.","ieee":"U. Bauer, H. Edelsbrunner, G. Jablonski, and M. Mrozek, “Čech-Delaunay gradient flow and homology inference for self-maps,” <i>Journal of Applied and Computational Topology</i>, vol. 4, no. 4. Springer Nature, pp. 455–480, 2020."},"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"department":[{"_id":"HeEd"}],"volume":4,"date_published":"2020-12-01T00:00:00Z","oa_version":"Published Version","publication_status":"published","publisher":"Springer Nature","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"acknowledgement":"This research has been supported by the DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”, by Polish MNiSzW Grant No. 2621/7.PR/12/2013/2, by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453 and Grant No. DEC-2013/09/N/ST6/02995. Open Access funding provided by Projekt DEAL.","intvolume":"         4","file_date_updated":"2024-03-04T10:52:42Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"Journal of Applied and Computational Topology"},{"abstract":[{"lang":"eng","text":"In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound on the metric distortion between the distance measured in the ambient Euclidean space and the shortest path distance measured in the set. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the reach and the distance between the two points."}],"date_created":"2019-07-24T08:37:29Z","file":[{"relation":"main_file","date_updated":"2020-07-14T12:47:36Z","access_level":"open_access","creator":"dernst","file_size":2215157,"checksum":"a5b244db9f751221409cf09c97ee0935","file_id":"6741","date_created":"2019-07-31T08:09:56Z","content_type":"application/pdf","file_name":"2019_JournAppliedComputTopol_Boissonnat.pdf"}],"oa":1,"type":"journal_article","corr_author":"1","quality_controlled":"1","status":"public","year":"2019","doi":"10.1007/s41468-019-00029-8","author":[{"full_name":"Boissonnat, Jean-Daniel","first_name":"Jean-Daniel","last_name":"Boissonnat"},{"last_name":"Lieutier","first_name":"André","full_name":"Lieutier, André"},{"orcid":"0000-0002-7472-2220","last_name":"Wintraecken","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs"}],"project":[{"call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"scopus_import":"1","page":"29–58","day":"01","article_type":"original","ddc":["000"],"month":"06","title":"The reach, metric distortion, geodesic convexity and the variation of tangent spaces","publication":"Journal of Applied and Computational Topology","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file_date_updated":"2020-07-14T12:47:36Z","intvolume":"         3","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"publisher":"Springer Nature","publication_status":"published","oa_version":"Published Version","ec_funded":1,"date_published":"2019-06-01T00:00:00Z","volume":3,"department":[{"_id":"HeEd"}],"citation":{"ieee":"J.-D. Boissonnat, A. Lieutier, and M. Wintraecken, “The reach, metric distortion, geodesic convexity and the variation of tangent spaces,” <i>Journal of Applied and Computational Topology</i>, vol. 3, no. 1–2. Springer Nature, pp. 29–58, 2019.","chicago":"Boissonnat, Jean-Daniel, André Lieutier, and Mathijs Wintraecken. “The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2019. <a href=\"https://doi.org/10.1007/s41468-019-00029-8\">https://doi.org/10.1007/s41468-019-00029-8</a>.","apa":"Boissonnat, J.-D., Lieutier, A., &#38; Wintraecken, M. (2019). The reach, metric distortion, geodesic convexity and the variation of tangent spaces. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-019-00029-8\">https://doi.org/10.1007/s41468-019-00029-8</a>","short":"J.-D. Boissonnat, A. Lieutier, M. Wintraecken, Journal of Applied and Computational Topology 3 (2019) 29–58.","ista":"Boissonnat J-D, Lieutier A, Wintraecken M. 2019. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Journal of Applied and Computational Topology. 3(1–2), 29–58.","ama":"Boissonnat J-D, Lieutier A, Wintraecken M. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. <i>Journal of Applied and Computational Topology</i>. 2019;3(1-2):29–58. doi:<a href=\"https://doi.org/10.1007/s41468-019-00029-8\">10.1007/s41468-019-00029-8</a>","mla":"Boissonnat, Jean-Daniel, et al. “The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces.” <i>Journal of Applied and Computational Topology</i>, vol. 3, no. 1–2, Springer Nature, 2019, pp. 29–58, doi:<a href=\"https://doi.org/10.1007/s41468-019-00029-8\">10.1007/s41468-019-00029-8</a>."},"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"has_accepted_license":"1","language":[{"iso":"eng"}],"issue":"1-2","article_processing_charge":"Yes (via OA deal)","date_updated":"2025-04-14T07:44:06Z","_id":"6671"},{"article_type":"original","year":"2018","doi":"10.1007/s41468-018-0021-5","author":[{"full_name":"Filakovský, Marek","id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87","last_name":"Filakovský","first_name":"Marek"},{"full_name":"Franek, Peter","id":"473294AE-F248-11E8-B48F-1D18A9856A87","last_name":"Franek","first_name":"Peter","orcid":"0000-0001-8878-8397"},{"full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568","last_name":"Wagner","first_name":"Uli"},{"first_name":"Stephan Y","last_name":"Zhechev","full_name":"Zhechev, Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87"}],"project":[{"name":"Robust Invariants of Nonlinear Systems","grant_number":"M01980","call_identifier":"FWF","_id":"25F8B9BC-B435-11E9-9278-68D0E5697425"},{"name":"FWF Open Access Fund","_id":"3AC91DDA-15DF-11EA-824D-93A3E7B544D1","call_identifier":"FWF"}],"scopus_import":"1","page":"177-231","day":"01","month":"12","title":"Computing simplicial representatives of homotopy group elements","ddc":["514"],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"6681"}]},"file":[{"file_size":1056278,"checksum":"cf9e7fcd2a113dd4828774fc75cdb7e8","file_id":"6775","relation":"main_file","date_updated":"2020-07-14T12:47:40Z","creator":"dernst","access_level":"open_access","content_type":"application/pdf","date_created":"2019-08-08T06:55:21Z","file_name":"2018_JourAppliedComputTopology_Filakovsky.pdf"}],"oa":1,"type":"journal_article","abstract":[{"lang":"eng","text":"A central problem of algebraic topology is to understand the homotopy groups  𝜋𝑑(𝑋)  of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group  𝜋1(𝑋)  of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with   𝜋1(𝑋)  trivial), compute the higher homotopy group   𝜋𝑑(𝑋)  for any given   𝑑≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of   𝜋𝑑(𝑋) ,   𝑑≥2  as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of   𝜋𝑑(𝑋) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere   𝑆𝑑  to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes   𝜋𝑑(𝑋)  and represents its elements as simplicial maps from a suitable triangulation of the d-sphere   𝑆𝑑  to X. For fixed d, the algorithm runs in time exponential in   size(𝑋) , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed   𝑑≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of   𝜋𝑑(𝑋) , the size of the triangulation of   𝑆𝑑  on which the map is defined, is exponential in size(𝑋) ."}],"date_created":"2019-08-08T06:47:40Z","quality_controlled":"1","status":"public","corr_author":"1","has_accepted_license":"1","language":[{"iso":"eng"}],"department":[{"_id":"UlWa"}],"citation":{"apa":"Filakovský, M., Franek, P., Wagner, U., &#38; Zhechev, S. Y. (2018). Computing simplicial representatives of homotopy group elements. <i>Journal of Applied and Computational Topology</i>. Springer. <a href=\"https://doi.org/10.1007/s41468-018-0021-5\">https://doi.org/10.1007/s41468-018-0021-5</a>","short":"M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and Computational Topology 2 (2018) 177–231.","ista":"Filakovský M, Franek P, Wagner U, Zhechev SY. 2018. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2(3–4), 177–231.","mla":"Filakovský, Marek, et al. “Computing Simplicial Representatives of Homotopy Group Elements.” <i>Journal of Applied and Computational Topology</i>, vol. 2, no. 3–4, Springer, 2018, pp. 177–231, doi:<a href=\"https://doi.org/10.1007/s41468-018-0021-5\">10.1007/s41468-018-0021-5</a>.","ama":"Filakovský M, Franek P, Wagner U, Zhechev SY. Computing simplicial representatives of homotopy group elements. <i>Journal of Applied and Computational Topology</i>. 2018;2(3-4):177-231. doi:<a href=\"https://doi.org/10.1007/s41468-018-0021-5\">10.1007/s41468-018-0021-5</a>","ieee":"M. Filakovský, P. Franek, U. Wagner, and S. Y. Zhechev, “Computing simplicial representatives of homotopy group elements,” <i>Journal of Applied and Computational Topology</i>, vol. 2, no. 3–4. Springer, pp. 177–231, 2018.","chicago":"Filakovský, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. “Computing Simplicial Representatives of Homotopy Group Elements.” <i>Journal of Applied and Computational Topology</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s41468-018-0021-5\">https://doi.org/10.1007/s41468-018-0021-5</a>."},"publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"date_updated":"2026-04-08T13:56:01Z","_id":"6774","issue":"3-4","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"publication":"Journal of Applied and Computational Topology","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file_date_updated":"2020-07-14T12:47:40Z","intvolume":"         2","oa_version":"Published Version","date_published":"2018-12-01T00:00:00Z","volume":2,"publisher":"Springer","publication_status":"published"}]