[{"oa":1,"file":[{"success":1,"date_updated":"2026-06-22T07:39:21Z","file_size":2052749,"creator":"dernst","content_type":"application/pdf","file_name":"2026_LIPIcSSoCG_Leskiewicz.pdf","file_id":"22110","access_level":"open_access","date_created":"2026-06-22T07:39:21Z","checksum":"3be91c06fdf716c8735b6af64a09a921","relation":"main_file"}],"date_created":"2026-06-14T22:01:43Z","author":[{"first_name":"Jakub","full_name":"Leśkiewicz, Jakub","last_name":"Leśkiewicz"},{"first_name":"Bartosz","last_name":"Furmanek","full_name":"Furmanek, Bartosz"},{"id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","orcid":"0000-0001-9789-9750","first_name":"Michał","last_name":"Lipiński","full_name":"Lipiński, Michał"},{"first_name":"Dmitriy","last_name":"Morozov","full_name":"Morozov, Dmitriy"}],"tmp":{"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","short":"CC BY (4.0)"},"scopus_import":"1","project":[{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"acknowledgement":"Jakub Leśkiewicz wants to thank his supervisor, Prof. Marian Mrozek, forscientific guidance, patience, and opportunity to delay the rest of his duties while writing this work.\r\nThe author also extends thanks to his entire family, to Zuzanna Świątek, and to Mikołaj Kardyś,\r\nBEng, MSc, for providing meals during the most intensive periods of work. Jakub Leśkiewicz: The research was partially funded by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874 and Opus Grant 2025/57/B/ST1/00550. Bartosz Furmanek: The research was partially funded by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874 and Opus Grant 2025/57/B/ST1/00550. Michał Lipiński: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. \r\nDmitriy Morozov: This work was supported in part by the U.S. Department of Energy, Office\r\nof Science, Office of Advanced Scientific Computing Research, under Contract No. DE-AC02-\r\n05CH11231.","status":"public","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2026.72","date_published":"2026-05-27T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"05","has_accepted_license":"1","article_processing_charge":"No","keyword":["persistent homology","topological simplification","depth posets"],"OA_place":"publisher","quality_controlled":"1","OA_type":"gold","oa_version":"Published Version","arxiv":1,"department":[{"_id":"HeEd"}],"article_number":"72:1-72:17","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959774185"]},"das_tickbox":"0","abstract":[{"text":"Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c⋅n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.","lang":"eng"}],"citation":{"mla":"Leśkiewicz, Jakub, et al. “Topological Simplification Guided by Forbidden Regions.” <i>42nd International Symposium on Computational Geometry</i>, vol. 367, 72:1-72:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2026.72\">10.4230/LIPIcs.SoCG.2026.72</a>.","apa":"Leśkiewicz, J., Furmanek, B., Lipiński, M., &#38; Morozov, D. (2026). Topological simplification guided by forbidden regions. 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.2026.72\">https://doi.org/10.4230/LIPIcs.SoCG.2026.72</a>","short":"J. Leśkiewicz, B. Furmanek, M. Lipiński, D. Morozov, in:, 42nd International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026.","ama":"Leśkiewicz J, Furmanek B, Lipiński M, Morozov D. Topological simplification guided by forbidden regions. 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.2026.72\">10.4230/LIPIcs.SoCG.2026.72</a>","ista":"Leśkiewicz J, Furmanek B, Lipiński M, Morozov D. 2026. Topological simplification guided by forbidden regions. 42nd International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 367, 72:1-72:17.","ieee":"J. Leśkiewicz, B. Furmanek, M. Lipiński, and D. Morozov, “Topological simplification guided by forbidden regions,” in <i>42nd International Symposium on Computational Geometry</i>, New Brunswick, NJ, United States, 2026, vol. 367.","chicago":"Leśkiewicz, Jakub, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov. “Topological Simplification Guided by Forbidden Regions.” 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.2026.72\">https://doi.org/10.4230/LIPIcs.SoCG.2026.72</a>."},"intvolume":"       367","doi":"10.4230/LIPIcs.SoCG.2026.72","conference":{"end_date":"2026-06-05","start_date":"2026-06-02","location":"New Brunswick, NJ, United States","name":"SoCG: Symposium on Computational Geometry"},"date_updated":"2026-06-22T07:45:36Z","_id":"22002","year":"2026","file_date_updated":"2026-06-22T07:39:21Z","language":[{"iso":"eng"}],"publication":"42nd International Symposium on Computational Geometry","title":"Topological simplification guided by forbidden regions","corr_author":"1","external_id":{"arxiv":["2603.16416"]},"volume":367,"ec_funded":1,"alternative_title":["LIPIcs"],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","ddc":["500"],"type":"conference","day":"27","publication_status":"published"},{"month":"08","main_file_link":[{"url":"https://doi.org/10.1007/s10208-026-09766-6","open_access":"1"}],"has_accepted_license":"1","keyword":["Multivector field","Conley index","Morse decomposition","Bifurcation","Continuation","Zigzag persistence","Persistence barcode","Gentle algebra"],"article_processing_charge":"Yes (via OA deal)","supplementarymaterial":"yes","article_type":"original","OA_type":"hybrid","quality_controlled":"1","OA_place":"publisher","oa_version":"Published Version","department":[{"_id":"HeEd"}],"arxiv":1,"date_created":"2026-08-05T06:11:30Z","oa":1,"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).","author":[{"full_name":"Dey, Tamal K.","last_name":"Dey","first_name":"Tamal K."},{"first_name":"Michał","full_name":"Lipiński, Michał","last_name":"Lipiński","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","orcid":"0000-0001-9789-9750"},{"orcid":"0000-0003-2449-1433","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","full_name":"Soriano Trigueros, Manuel","last_name":"Soriano Trigueros","first_name":"Manuel"}],"scopus_import":"1","tmp":{"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","short":"CC BY (4.0)"},"project":[{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2026-08-04T00:00:00Z","fulldoi":"https://doi.org/10.1007/s10208-026-09766-6","status":"public","publication":"Foundations of Computational Mathematics","corr_author":"1","title":"Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations","year":"2026","_id":"22648","PlanS_conform":"1","language":[{"iso":"eng"}],"publisher":"Springer","external_id":{"arxiv":["2504.17105"]},"ec_funded":1,"type":"journal_article","ddc":["500"],"publication_status":"epub_ahead","day":"04","das_tickbox":"0","publication_identifier":{"issn":["1615-3375"],"eissn":["1615-3383"]},"abstract":[{"lang":"eng","text":"Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems—a young but already well-established theory. We introduce the Conley–Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of the Conley–Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose."}],"citation":{"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>.","ista":"Dey TK, Lipiński M, Soriano Trigueros M. 2026. Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations. Foundations of Computational Mathematics.","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.","short":"T.K. Dey, M. Lipiński, M. Soriano Trigueros, Foundations of Computational Mathematics (2026).","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>","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>","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>."},"date_updated":"2026-08-11T06:13:33Z","researchdata_availability":"no","doi":"10.1007/s10208-026-09766-6"},{"ddc":["510"],"type":"journal_article","day":"01","publication_status":"published","_id":"20980","year":"2026","language":[{"iso":"eng"}],"page":"108-130","publication":"SIAM Journal on Applied Dynamical Systems","title":"Computing a connection matrix and persistence efficiently from a morse decomposition","external_id":{"arxiv":["2502.19369"]},"volume":25,"ec_funded":1,"publisher":"SIAM","citation":{"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>","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>.","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.","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.","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>.","short":"T.K. Dey, A. Haas, M. Lipiński, SIAM Journal on Applied Dynamical Systems 25 (2026) 108–130.","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>"},"intvolume":"        25","doi":"10.1137/25m1739406","date_updated":"2026-08-12T06:23:46Z","issue":"1","publication_identifier":{"issn":["1536-0040"]},"abstract":[{"lang":"eng","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."}],"article_processing_charge":"No","quality_controlled":"1","OA_place":"repository","OA_type":"green","oa_version":"Preprint","department":[{"_id":"HeEd"}],"arxiv":1,"article_type":"original","month":"01","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2502.19369"}],"author":[{"first_name":"Tamal K.","last_name":"Dey","full_name":"Dey, Tamal K."},{"first_name":"Andrew","full_name":"Haas, Andrew","last_name":"Haas"},{"full_name":"Lipiński, Michał","last_name":"Lipiński","first_name":"Michał","orcid":"0000-0001-9789-9750","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e"}],"scopus_import":"1","project":[{"name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"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","status":"public","fulldoi":"https://doi.org/10.1137/25m1739406","date_published":"2026-01-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"date_created":"2026-01-12T11:17:06Z"},{"tmp":{"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","short":"CC BY (4.0)"},"scopus_import":"1","project":[{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"},{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"author":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert"},{"first_name":"Michał","full_name":"Lipiński, Michał","last_name":"Lipiński","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","orcid":"0000-0001-9789-9750"},{"first_name":"Marian","last_name":"Mrozek","full_name":"Mrozek, Marian","orcid":"0000-0002-0619-6417"},{"first_name":"Manuel","full_name":"Soriano Trigueros, Manuel","last_name":"Soriano Trigueros","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","orcid":"0000-0003-2449-1433"},{"id":"afd27eda-91c1-11f0-aad8-c6edbec24c04","first_name":"Fedor","last_name":"Zimin","full_name":"Zimin, Fedor"}],"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","fulldoi":"https://doi.org/10.4230/LIPICS.SOCG.2026.41","status":"public","date_published":"2026-05-27T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"file":[{"content_type":"application/pdf","creator":"dernst","file_size":2902144,"date_updated":"2026-07-14T06:08:05Z","success":1,"checksum":"9dfb96ee66985c724b499b0e5888dc8e","relation":"main_file","date_created":"2026-07-14T06:08:05Z","access_level":"open_access","file_id":"22329","file_name":"2026_LIPIcSSoCG_Edelsbrunner.pdf"}],"date_created":"2026-07-13T09:56:38Z","supplementarymaterial":"no","article_processing_charge":"Yes","keyword":["Algebraic topology","Lefschetz complexes","persistent homology","vines and vineyards","birth-death pairs","shallow pairs","relations","partial orders","transpositions","Theory of computation → Computational geometry"],"arxiv":1,"oa_version":"Published Version","department":[{"_id":"HeEd"},{"_id":"GradSch"}],"quality_controlled":"1","OA_place":"publisher","OA_type":"gold","month":"05","has_accepted_license":"1","intvolume":"       367","citation":{"apa":"Edelsbrunner, H., Lipiński, M., Mrozek, M., Soriano Trigueros, M., &#38; Zimin, F. (2026). The depth poset under transpositions in the filter. In <i>42nd International Symposium on Computational Geometry</i> (Vol. 367). New Brunswick, NJ, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">https://doi.org/10.4230/LIPICS.SOCG.2026.41</a>","mla":"Edelsbrunner, Herbert, et al. “The Depth Poset under Transpositions in the Filter.” <i>42nd International Symposium on Computational Geometry</i>, vol. 367, 41:1-41:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026, doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">10.4230/LIPICS.SOCG.2026.41</a>.","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.","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>.","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.","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>","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."},"researchdata_availability":"no","conference":{"location":"New Brunswick, NJ, United States","name":"SoCG: Symposium on Computational Geometry","end_date":"2026-06-05","start_date":"2026-06-02"},"doi":"10.4230/LIPICS.SOCG.2026.41","date_updated":"2026-08-12T09:02:56Z","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959774185"]},"article_number":"41:1-41:18","das_tickbox":"0","abstract":[{"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.","lang":"eng"}],"ddc":["500"],"type":"conference","day":"27","publication_status":"published","language":[{"iso":"eng"}],"file_date_updated":"2026-07-14T06:08:05Z","_id":"22299","year":"2026","corr_author":"1","title":"The depth poset under transpositions in the filter","publication":"42nd International Symposium on Computational Geometry","ec_funded":1,"volume":367,"external_id":{"arxiv":["2511.21961"]},"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","alternative_title":["LIPIcs"]},{"publisher":"Springer Nature","external_id":{"arxiv":["2312.08013"],"isi":["001356000500005"]},"volume":24,"ec_funded":1,"publication":"Qualitative Theory of Dynamical Systems","title":"Morse predecomposition of an invariant set","corr_author":"1","year":"2025","_id":"18580","file_date_updated":"2024-11-28T06:52:38Z","language":[{"iso":"eng"}],"publication_status":"published","day":"01","type":"journal_article","ddc":["514","510"],"abstract":[{"lang":"eng","text":"Motivated by the study of recurrent orbits and dynamics within a Morse set of a Morse decomposition we introduce the concept of Morse predecomposition of an isolated invariant set within the setting of both combinatorial and classical dynamical systems. While Morse decomposition summarizes solely the gradient part of a dynamical system, the developed generalization extends to the recurrent component as well. In particular, a chain recurrent set, which is indecomposable in terms of Morse decomposition, can be represented more finely in the Morse predecomposition framework. This generalization is achieved by forgoing the poset structure inherent to Morse decomposition and relaxing the notion of connection between Morse sets (elements of Morse decomposition) in favor of what we term ’links’. We prove that a Morse decomposition is a special case of Morse predecomposition indexed by a poset. Additionally, we show how a Morse predecomposition may be condensed back to retrieve a Morse decomposition."}],"article_number":"5","publication_identifier":{"issn":["1575-5460"],"eissn":["1662-3592"]},"issue":"1","date_updated":"2025-04-14T07:54:56Z","doi":"10.1007/s12346-024-01144-3","citation":{"ama":"Lipiński M, Mischaikow K, Mrozek M. Morse predecomposition of an invariant set. <i>Qualitative Theory of Dynamical Systems</i>. 2025;24(1). doi:<a href=\"https://doi.org/10.1007/s12346-024-01144-3\">10.1007/s12346-024-01144-3</a>","short":"M. Lipiński, K. Mischaikow, M. Mrozek, Qualitative Theory of Dynamical Systems 24 (2025).","ieee":"M. Lipiński, K. Mischaikow, and M. Mrozek, “Morse predecomposition of an invariant set,” <i>Qualitative Theory of Dynamical Systems</i>, vol. 24, no. 1. Springer Nature, 2025.","chicago":"Lipiński, Michał, Konstantin Mischaikow, and Marian Mrozek. “Morse Predecomposition of an Invariant Set.” <i>Qualitative Theory of Dynamical Systems</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s12346-024-01144-3\">https://doi.org/10.1007/s12346-024-01144-3</a>.","ista":"Lipiński M, Mischaikow K, Mrozek M. 2025. Morse predecomposition of an invariant set. Qualitative Theory of Dynamical Systems. 24(1), 5.","mla":"Lipiński, Michał, et al. “Morse Predecomposition of an Invariant Set.” <i>Qualitative Theory of Dynamical Systems</i>, vol. 24, no. 1, 5, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s12346-024-01144-3\">10.1007/s12346-024-01144-3</a>.","apa":"Lipiński, M., Mischaikow, K., &#38; Mrozek, M. (2025). Morse predecomposition of an invariant set. <i>Qualitative Theory of Dynamical Systems</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s12346-024-01144-3\">https://doi.org/10.1007/s12346-024-01144-3</a>"},"intvolume":"        24","has_accepted_license":"1","month":"02","article_type":"original","OA_place":"publisher","OA_type":"hybrid","quality_controlled":"1","oa_version":"Published Version","arxiv":1,"department":[{"_id":"UlWa"}],"isi":1,"article_processing_charge":"Yes (via OA deal)","date_created":"2024-11-24T23:01:47Z","oa":1,"file":[{"date_updated":"2024-11-28T06:52:38Z","success":1,"file_size":1483668,"creator":"mlipinsk","content_type":"application/pdf","file_id":"18595","file_name":"2025_predecomposition.pdf","date_created":"2024-11-28T06:52:38Z","access_level":"open_access","relation":"main_file","checksum":"73309a57cc798d696caa57b6aa1467d8"}],"date_published":"2025-02-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","fulldoi":"https://doi.org/10.1007/s12346-024-01144-3","status":"public","acknowledgement":"M.L. acknowledge support by the Dioscuri program initiated by the Max Planck Society, jointly managed with the National Science Centre (Poland), and mutually funded by the Polish Ministry of Science and Higher Education and the German Federal Ministry of Education and Research. M.L. also acknowledges that this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. Research of M.M. is partially supported by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874. The work of K.M. was partially supported by the National Science Foundation under awards DMS-1839294 and HDR TRIPODS award CCF-1934924, DARPA contract HR0011-16-2-0033, National Institutes of Health award R01 GM126555, Air Force Office of Scientific Research under award numbers FA9550-23-1-0011, AWD00010853-MOD002 and MURI FA9550-23-1-0400. K.M. was also supported by a grant from the Simons Foundation. Open access funding provided by Institute of Science and Technology (IST Austria). ","author":[{"first_name":"Michał","last_name":"Lipiński","full_name":"Lipiński, Michał","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","orcid":"0000-0001-9789-9750"},{"first_name":"Konstantin","full_name":"Mischaikow, Konstantin","last_name":"Mischaikow"},{"first_name":"Marian","full_name":"Mrozek, Marian","last_name":"Mrozek"}],"tmp":{"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","short":"CC BY (4.0)"},"scopus_import":"1","project":[{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}]}]