[{"supervisor":[{"full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning"}],"type":"dissertation","publisher":"Institute of Science and Technology Austria","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","language":[{"iso":"eng"}],"related_material":{"record":[{"id":"18173","relation":"part_of_dissertation","status":"public"},{"id":"18295","relation":"part_of_dissertation","status":"public"},{"id":"18294","relation":"part_of_dissertation","status":"public"},{"id":"18293","relation":"part_of_dissertation","status":"public"}]},"page":"195","month":"09","ddc":["512"],"_id":"18132","oa_version":"Published Version","degree_awarded":"PhD","file_date_updated":"2024-09-25T14:08:57Z","status":"public","date_published":"2024-09-23T00:00:00Z","has_accepted_license":"1","das_tickbox":"0","fulldoi":"https://doi.org/10.15479/at:ista:18132","department":[{"_id":"GradSch"},{"_id":"TiBr"}],"OA_place":"publisher","corr_author":"1","license":"https://creativecommons.org/licenses/by-nc/4.0/","publication_identifier":{"issn":["2663-337X"]},"doi":"10.15479/at:ista:18132","file":[{"file_size":5382106,"checksum":"2f8cf5cefdab108b1979caa8146cae9a","file_name":"PhDthesis (3).zip","date_updated":"2024-09-23T18:49:22Z","creator":"jglas","access_level":"closed","file_id":"18133","date_created":"2024-09-23T18:49:22Z","content_type":"application/x-zip-compressed","relation":"source_file"},{"file_name":"example-phd.pdf","checksum":"08bb6f14c42b47ff25882a2ce3ea0d8a","file_size":2380127,"file_id":"18140","access_level":"open_access","date_updated":"2024-09-25T14:08:57Z","creator":"jglas","content_type":"application/pdf","date_created":"2024-09-25T14:08:57Z","relation":"main_file","success":1}],"abstract":[{"text":"In this thesis, we are dealing with both arithmetic and geometric problems coming from the\r\nstudy of rational points with a particular focus on function fields over finite fields:\r\n(1) Using the circle method we produce upper bounds for the number of rational points of\r\nbounded height on diagonal cubic surfaces and fourfolds over Fq(t). This is based on\r\njoint work with Leonhard Hochfilzer.\r\n(2) We study rational points on smooth complete intersections X defined by cubic and\r\nquadratic hypersurfaces over Fq(t). We refine the Farey dissection of the “unit square”\r\ndeveloped by Vishe [202] and use the circle method with a Kloosterman refinement to\r\nestablish an asymptotic formula for the number of rational points of bounded height on\r\nX when dim(X) ≥ 23. Under the same hypotheses, we also verify weak approximation.\r\n(3) In joint work with Hochfilzer, we obtain upper bounds for the number of rational points of\r\nbounded height on del Pezzo surfaces of low degree over any global field. Our approach\r\nis to take hyperplane sections, which reduces the problem to uniform estimates for the\r\nnumber of rational points on curves.\r\n(4) We develop a version of the circle method capable of counting Fq-points on jet schemes\r\nof moduli spaces of rational curves on hypersurfaces. Combining this with a spreading\r\nout argument and a result of Mustaţă [150], this allows us to show that these moduli\r\nspaces only have canonical singularities under suitable assumptions on the degree and the\r\ndimension.\r\nIn addition, we give an overview of guiding questions and conjectures in the field of rational\r\npoints and explain the basic mechanism underlying the circle method.\r\n","lang":"eng"}],"author":[{"full_name":"Glas, Jakob","first_name":"Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","last_name":"Glas"}],"doi_confirm":"1","title":"Counting rational points over function fields","date_updated":"2026-10-02T11:44:29Z","alternative_title":["ISTA Thesis"],"day":"23","citation":{"ama":"Glas J. Counting rational points over function fields. 2024. doi:10.15479/at:ista:18132","chicago":"Glas, Jakob. “Counting Rational Points over Function Fields.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18132.","short":"J. Glas, Counting Rational Points over Function Fields, Institute of Science and Technology Austria, 2024.","mla":"Glas, Jakob. Counting Rational Points over Function Fields. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18132.","ieee":"J. Glas, “Counting rational points over function fields,” Institute of Science and Technology Austria, 2024.","apa":"Glas, J. (2024). Counting rational points over function fields. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18132","ista":"Glas J. 2024. Counting rational points over function fields. Institute of Science and Technology Austria."},"project":[{"grant_number":"P36278","name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"}],"researchdata_availability":"no","supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)","image":"/images/cc_by_nc.png","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","short":"CC BY-NC (4.0)"},"oa":1,"article_processing_charge":"No","year":"2024","date_created":"2024-09-23T18:58:08Z"},{"month":"10","ddc":["510"],"oa_version":"Published Version","_id":"18173","scopus_import":"1","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","language":[{"iso":"eng"}],"publication":"Journal of the London Mathematical Society","related_material":{"record":[{"status":"public","id":"18132","relation":"dissertation_contains"}]},"type":"journal_article","publisher":"London Mathematical Society","quality_controlled":"1","corr_author":"1","file":[{"access_level":"open_access","date_updated":"2024-10-07T08:51:01Z","creator":"dernst","file_id":"18181","checksum":"11ebf690363151026ce81f91f2220855","file_size":579601,"file_name":"2024_JLondonMathSoc_Glas.pdf","success":1,"relation":"main_file","date_created":"2024-10-07T08:51:01Z","content_type":"application/pdf"}],"doi":"10.1112/jlms.12991","publication_identifier":{"issn":["0024-6107"],"eissn":["1469-7750"]},"fulldoi":"https://doi.org/10.1112/jlms.12991","department":[{"_id":"TiBr"}],"date_published":"2024-10-01T00:00:00Z","has_accepted_license":"1","das_tickbox":"0","file_date_updated":"2024-10-07T08:51:01Z","issue":"4","status":"public","researchdata_availability":"no","arxiv":1,"article_number":"e12991","day":"01","volume":110,"intvolume":" 110","citation":{"ista":"Glas J. 2024. Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t). Journal of the London Mathematical Society. 110(4), e12991.","apa":"Glas, J. (2024). Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t). Journal of the London Mathematical Society. London Mathematical Society. https://doi.org/10.1112/jlms.12991","ieee":"J. Glas, “Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t),” Journal of the London Mathematical Society, vol. 110, no. 4. London Mathematical Society, 2024.","short":"J. Glas, Journal of the London Mathematical Society 110 (2024).","mla":"Glas, Jakob. “Rational Points on Complete Intersections of Cubic and Quadric Hypersurfaces over Fq(T).” Journal of the London Mathematical Society, vol. 110, no. 4, e12991, London Mathematical Society, 2024, doi:10.1112/jlms.12991.","chicago":"Glas, Jakob. “Rational Points on Complete Intersections of Cubic and Quadric Hypersurfaces over Fq(T).” Journal of the London Mathematical Society. London Mathematical Society, 2024. https://doi.org/10.1112/jlms.12991.","ama":"Glas J. Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t). Journal of the London Mathematical Society. 2024;110(4). doi:10.1112/jlms.12991"},"project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory","grant_number":"P36278"}],"title":"Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t)","date_updated":"2026-10-02T11:44:29Z","author":[{"id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","last_name":"Glas","first_name":"Jakob","full_name":"Glas, Jakob"}],"abstract":[{"lang":"eng","text":"Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t), provided char (Fq)>3. Under the same hypotheses, we also verify weak approximation."}],"article_processing_charge":"Yes (via OA deal)","date_created":"2024-10-06T22:01:11Z","year":"2024","acknowledgement":"The author would like to thank his supervisor Tim Browning for suggesting this project and many helpful conversations and Pankaj Vishe for useful comments. Moreover, he is grateful to Dante Bonolis and Julian Lyczak for sharing their expertise in exponential sums and geometry. While working on this paper, the author was supported by FWF grant (DOI 10.55776/P36278).","external_id":{"arxiv":["2306.02718"]},"oa":1,"article_type":"original","supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"}},{"title":"Canonical singularities on moduli spaces of rational curves via the circle method","date_updated":"2026-10-02T11:44:29Z","abstract":[{"lang":"eng","text":"By developing a suitable version of the circle method, we show that the space of degree e rational curves on a smooth hypersurface of degree d has only canonical singularities provided its dimension is sufficiently large with respect to e and d."}],"author":[{"full_name":"Glas, Jakob","id":"d6423cba-dc74-11ea-a0a7-ee61689ff5fb","last_name":"Glas","first_name":"Jakob"}],"arxiv":1,"researchdata_availability":"no","day":"26","citation":{"ieee":"J. Glas, “Canonical singularities on moduli spaces of rational curves via the circle method,” arXiv. .","apa":"Glas, J. (n.d.). Canonical singularities on moduli spaces of rational curves via the circle method. arXiv. https://doi.org/10.48550/arXiv.2405.16648","ista":"Glas J. Canonical singularities on moduli spaces of rational curves via the circle method. arXiv, 10.48550/arXiv.2405.16648.","chicago":"Glas, Jakob. “Canonical Singularities on Moduli Spaces of Rational Curves via the Circle Method.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2405.16648.","mla":"Glas, Jakob. “Canonical Singularities on Moduli Spaces of Rational Curves via the Circle Method.” ArXiv, doi:10.48550/arXiv.2405.16648.","short":"J. Glas, ArXiv (n.d.).","ama":"Glas J. Canonical singularities on moduli spaces of rational curves via the circle method. arXiv. doi:10.48550/arXiv.2405.16648"},"project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"}],"oa":1,"external_id":{"arxiv":["2405.16648"]},"supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"article_processing_charge":"No","year":"2024","date_created":"2024-10-10T13:15:43Z","language":[{"iso":"eng"}],"publication":"arXiv","publication_status":"submitted","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","related_material":{"record":[{"id":"19013","relation":"later_version","status":"public"},{"relation":"dissertation_contains","id":"18132","status":"public"}]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2405.16648"}],"type":"preprint","month":"05","_id":"18295","oa_version":"Preprint","das_tickbox":"0","date_published":"2024-05-26T00:00:00Z","status":"public","corr_author":"1","doi":"10.48550/arXiv.2405.16648","fulldoi":"https://doi.org/10.48550/arXiv.2405.16648","OA_place":"repository","department":[{"_id":"TiBr"}]},{"month":"06","ddc":["510"],"_id":"17144","scopus_import":"1","oa_version":"Published Version","publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"publication":"40th International Symposium on Computational Geometry","conference":{"name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","end_date":"2024-06-14"},"type":"conference","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","doi":"10.4230/LIPIcs.SoCG.2024.69","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773164"]},"file":[{"content_type":"application/pdf","date_created":"2024-06-17T08:33:40Z","relation":"main_file","success":1,"file_name":"2024_LIPICS_Kourimska.pdf","file_size":1612558,"checksum":"b40ff456c19294adb5d9613fcfd751c6","file_id":"17150","access_level":"open_access","date_updated":"2024-06-17T08:33:40Z","creator":"dernst"}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.69","ec_funded":1,"department":[{"_id":"HeEd"}],"date_published":"2024-06-01T00:00:00Z","has_accepted_license":"1","das_tickbox":"0","status":"public","file_date_updated":"2024-06-17T08:33:40Z","arxiv":1,"researchdata_availability":"no","article_number":"69","alternative_title":["LIPIcs"],"day":"01","volume":293,"citation":{"ama":"Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.69","chicago":"Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.69.","mla":"Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” 40th International Symposium on Computational Geometry, vol. 293, 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.69.","short":"H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","ieee":"H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms,” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293.","apa":"Kourimska, H., Lieutier, A., & Wintraecken, M. (2024). The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.69","ista":"Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 69."},"project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"},{"grant_number":"M03073","name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"intvolume":" 293","title":"The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms","date_updated":"2026-10-02T11:46:39Z","author":[{"orcid":"0000-0001-7841-0091","full_name":"Kourimska, Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","last_name":"Kourimska","first_name":"Hana"},{"first_name":"André","last_name":"Lieutier","full_name":"Lieutier, André"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220"}],"abstract":[{"lang":"eng","text":"We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms."}],"article_processing_charge":"No","year":"2024","date_created":"2024-06-16T22:01:06Z","acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nSupported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Cô d'Azur.\r\nWe are greatly indebted to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher Fillmore, and Elizabeth Stephenson for early discussions and all members of the Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape team (Inria) for the atmosphere in which this research was conducted.","external_id":{"arxiv":["2212.01118"]},"oa":1,"supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"}},{"quality_controlled":"1","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773164"]},"file":[{"file_name":"2024_LIPICS_Edelsbrunner.pdf","file_size":766562,"checksum":"5442d44fb89d77477a87668d6e61aac9","file_id":"17152","date_updated":"2024-06-17T08:46:33Z","creator":"dernst","access_level":"open_access","content_type":"application/pdf","date_created":"2024-06-17T08:46:33Z","relation":"main_file","success":1}],"doi":"10.4230/LIPIcs.SoCG.2024.53","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.53","department":[{"_id":"HeEd"}],"ec_funded":1,"date_published":"2024-06-01T00:00:00Z","has_accepted_license":"1","das_tickbox":"0","status":"public","file_date_updated":"2024-06-17T08:46:33Z","month":"06","ddc":["510"],"scopus_import":"1","_id":"17146","oa_version":"Published Version","publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication":"40th International Symposium on Computational Geometry","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"later_version","id":"20657","status":"public"}]},"conference":{"end_date":"2024-06-14","start_date":"2024-06-11","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece"},"type":"conference","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","article_processing_charge":"No","date_created":"2024-06-16T22:01:06Z","year":"2024","acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart for early discussions on the linked circles construction in three dimensions, and Gábor Tardos for helpful remarks and suggestions.","external_id":{"arxiv":["2310.14801"]},"oa":1,"supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"arxiv":1,"researchdata_availability":"no","article_number":"53","alternative_title":["LIPIcs"],"day":"01","volume":293,"citation":{"mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” 40th International Symposium on Computational Geometry, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.53.","short":"H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.53.","ista":"Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 53.","apa":"Edelsbrunner, H., & Pach, J. (2024). Maximum Betti numbers of Čech complexes. In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.53","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.53"},"intvolume":" 293","project":[{"call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"title":"Maximum Betti numbers of Čech complexes","date_updated":"2026-10-02T11:48:01Z","author":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"full_name":"Pach, János","first_name":"János","last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4"}],"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. In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry."}]},{"quality_controlled":"1","doi":"10.4230/LIPIcs.SoCG.2024.76","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773164"]},"file":[{"file_id":"17151","access_level":"open_access","creator":"dernst","date_updated":"2024-06-17T08:40:04Z","file_name":"2024_LIPICS_Rote.pdf","file_size":1430896,"checksum":"fbad1de06383a6b7e8a1cb3e8c7205ce","relation":"main_file","success":1,"content_type":"application/pdf","date_created":"2024-06-17T08:40:04Z"}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.76","department":[{"_id":"HeEd"}],"das_tickbox":"0","has_accepted_license":"1","date_published":"2024-06-01T00:00:00Z","file_date_updated":"2024-06-17T08:40:04Z","status":"public","month":"06","ddc":["510"],"scopus_import":"1","_id":"17145","oa_version":"Published Version","language":[{"iso":"eng"}],"publication":"40th International Symposium on Computational Geometry","publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","conference":{"end_date":"2024-06-14","start_date":"2024-06-11","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece"},"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","type":"conference","article_processing_charge":"No","date_created":"2024-06-16T22:01:06Z","year":"2024","acknowledgement":"Part of this work was done while G.R. enjoyed the hospitality of the Institute of Science and Technology Austria (ISTA) as a visiting professor during his sabbatical in the winter semester 2022/23.","oa":1,"external_id":{"arxiv":["2402.15787"]},"supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"researchdata_availability":"no","arxiv":1,"article_number":"76","day":"01","alternative_title":["LIPIcs"],"intvolume":" 293","citation":{"ama":"Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.76","apa":"Rote, G., Rüber, M., & Saghafian, M. (2024). Grid peeling of parabolas. In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.76","ieee":"G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293.","ista":"Rote G, Rüber M, Saghafian M. 2024. Grid peeling of parabolas. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 76.","chicago":"Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.76.","short":"G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","mla":"Rote, Günter, et al. “Grid Peeling of Parabolas.” 40th International Symposium on Computational Geometry, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.76."},"volume":293,"title":"Grid peeling of parabolas","date_updated":"2026-10-02T11:47:18Z","abstract":[{"text":"Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.","lang":"eng"}],"author":[{"first_name":"Günter","last_name":"Rote","full_name":"Rote, Günter"},{"last_name":"Rüber","first_name":"Moritz","full_name":"Rüber, Moritz"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}]},{"fulldoi":"https://doi.org/10.3390/e26080637","department":[{"_id":"HeEd"}],"quality_controlled":"1","file":[{"file_size":8025139,"checksum":"624a9e2c5b49d6c38b88b0f675467ba3","file_name":"2024_Entropy_Edelsbrunner.pdf","date_updated":"2024-09-09T09:01:12Z","creator":"dernst","access_level":"open_access","file_id":"17948","date_created":"2024-09-09T09:01:12Z","content_type":"application/pdf","success":1,"relation":"main_file"}],"publication_identifier":{"eissn":["1099-4300"]},"doi":"10.3390/e26080637","pmid":1,"status":"public","file_date_updated":"2024-09-09T09:01:12Z","issue":"8","date_published":"2024-08-01T00:00:00Z","das_tickbox":"1","has_accepted_license":"1","ddc":["510"],"month":"08","scopus_import":"1","_id":"17891","oa_version":"Published Version","isi":1,"dataavailabilitystatement":"The software and data used for this study is available at: https://git.ista.ac.at/katharina.oelsboeck/wrap_2_3-public/ (accessed on 19 July 2024).","type":"journal_article","publisher":"MDPI","publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication":"Entropy","language":[{"iso":"eng"}],"related_material":{"link":[{"relation":"software","url":"https://git.ista.ac.at/katharina.oelsboeck/wrap_2_3-public/"}]},"article_processing_charge":"Yes","date_created":"2024-09-08T22:01:11Z","year":"2024","article_type":"original","supplementarymaterial":"yes","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"acknowledgement":"We thank Anton Nikitenko for first observing that the Wrap complex can be characterized as stated in Claim (ii) of the Wrap Complex Lemma, and Ondrej Draganov for correcting a critical mistake in one of our formulas in Section 2.","external_id":{"pmid":["39202107"],"isi":["001305543500001"]},"oa":1,"day":"01","volume":26,"intvolume":" 26","citation":{"short":"H. Edelsbrunner, K. Ölsböck, H. Wagner, Entropy 26 (2024).","mla":"Edelsbrunner, Herbert, et al. “Understanding Higher-Order Interactions in Information Space.” Entropy, vol. 26, no. 8, 637, MDPI, 2024, doi:10.3390/e26080637.","chicago":"Edelsbrunner, Herbert, Katharina Ölsböck, and Hubert Wagner. “Understanding Higher-Order Interactions in Information Space.” Entropy. MDPI, 2024. https://doi.org/10.3390/e26080637.","ista":"Edelsbrunner H, Ölsböck K, Wagner H. 2024. Understanding higher-order interactions in information space. Entropy. 26(8), 637.","apa":"Edelsbrunner, H., Ölsböck, K., & Wagner, H. (2024). Understanding higher-order interactions in information space. Entropy. MDPI. https://doi.org/10.3390/e26080637","ieee":"H. Edelsbrunner, K. Ölsböck, and H. Wagner, “Understanding higher-order interactions in information space,” Entropy, vol. 26, no. 8. MDPI, 2024.","ama":"Edelsbrunner H, Ölsböck K, Wagner H. Understanding higher-order interactions in information space. Entropy. 2024;26(8). doi:10.3390/e26080637"},"researchdata_availability":"yes","article_number":"637","abstract":[{"lang":"eng","text":"Abstract\r\nMethods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback–Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics."}],"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","last_name":"Ölsböck","first_name":"Katharina","full_name":"Ölsböck, Katharina","orcid":"0000-0002-4672-8297"},{"full_name":"Wagner, Hubert","orcid":"0009-0009-9111-8429","last_name":"Wagner","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert"}],"title":"Understanding higher-order interactions in information space","date_updated":"2026-10-02T11:50:23Z"},{"quality_controlled":"1","doi":"10.1007/978-3-031-49272-3_23","publication_identifier":{"eisbn":["9783031492723"],"issn":["0302-9743"],"eissn":["1611-3349"],"isbn":["9783031492716"]},"fulldoi":"https://doi.org/10.1007/978-3-031-49272-3_23","department":[{"_id":"HeEd"}],"ec_funded":1,"das_tickbox":"0","date_published":"2024-01-01T00:00:00Z","status":"public","month":"01","isi":1,"_id":"15012","oa_version":"Preprint","scopus_import":"1","page":"339-346","language":[{"iso":"eng"}],"publication":"31st International Symposium on Graph Drawing and Network Visualization","publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","related_material":{"record":[{"status":"public","relation":"later_version","id":"21253"}]},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.13201","open_access":"1"}],"conference":{"end_date":"2023-09-22","start_date":"2023-09-20","name":"GD: Graph Drawing and Network Visualization","location":"Isola delle Femmine, Palermo, Italy"},"publisher":"Springer Nature","type":"conference","article_processing_charge":"No","year":"2024","date_created":"2024-02-18T23:01:03Z","acknowledgement":"János Pach’s Research partially supported by European Research Council (ERC), grant “GeoScape” No. 882971 and by the Hungarian Science Foundation (NKFIH), grant K-131529. Work by Morteza Saghafian is partially supported by the European Research Council (ERC), grant No. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31.","oa":1,"external_id":{"isi":["001207939600023"],"arxiv":["2306.13201"]},"supplementarymaterial":"no","researchdata_availability":"no","arxiv":1,"day":"01","alternative_title":["LNCS"],"citation":{"mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” 31st International Symposium on Graph Drawing and Network Visualization, vol. 14465, Springer Nature, 2024, pp. 339–46, doi:10.1007/978-3-031-49272-3_23.","short":"J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” In 31st International Symposium on Graph Drawing and Network Visualization, 14465:339–46. Springer Nature, 2024. https://doi.org/10.1007/978-3-031-49272-3_23.","ista":"Pach J, Saghafian M, Schnider P. 2024. Decomposition of geometric graphs into star-forests. 31st International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LNCS, vol. 14465, 339–346.","apa":"Pach, J., Saghafian, M., & Schnider, P. (2024). Decomposition of geometric graphs into star-forests. In 31st International Symposium on Graph Drawing and Network Visualization (Vol. 14465, pp. 339–346). Isola delle Femmine, Palermo, Italy: Springer Nature. https://doi.org/10.1007/978-3-031-49272-3_23","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” in 31st International Symposium on Graph Drawing and Network Visualization, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp. 339–346.","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. In: 31st International Symposium on Graph Drawing and Network Visualization. Vol 14465. Springer Nature; 2024:339-346. doi:10.1007/978-3-031-49272-3_23"},"project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF"}],"intvolume":" 14465","volume":14465,"title":"Decomposition of geometric graphs into star-forests","date_updated":"2026-10-02T11:50:59Z","author":[{"full_name":"Pach, János","first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza"},{"full_name":"Schnider, Patrick","first_name":"Patrick","last_name":"Schnider"}],"abstract":[{"lang":"eng","text":"We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n-1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs."}]},{"article_processing_charge":"No","date_created":"2024-06-30T22:01:05Z","year":"2024","article_type":"original","supplementarymaterial":"no","acknowledgement":"The second author is partially supported by the Alexander von Humboldt Foundation. The sixth author is supported by the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement 754411, and by Austrian Science Fund(FWF) grant M-3073. All other authors are supported by European Research Council (ERC) grant 788183, by the Wittgenstein Prize, by Austrian Science Fund (FWF) grant Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF) grant I 02979-N35.","external_id":{"arxiv":["2204.01077"],"isi":["001292728600001"]},"oa":1,"day":"07","volume":38,"project":[{"_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","call_identifier":"H2020"},{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses","grant_number":"M03073"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"intvolume":" 38","citation":{"apa":"Edelsbrunner, H., Garber, A., Ghafaris, M., Heiss, T., Saghafiant, M., & Wintraecken, M. (2024). Brillouin zones of integer lattices and their perturbations. SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/22M1489071","ieee":"H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, and M. Wintraecken, “Brillouin zones of integer lattices and their perturbations,” SIAM Journal on Discrete Mathematics, vol. 38, no. 2. Society for Industrial and Applied Mathematics, pp. 1784–1807, 2024.","ista":"Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M. 2024. Brillouin zones of integer lattices and their perturbations. SIAM Journal on Discrete Mathematics. 38(2), 1784–1807.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafaris, Teresa Heiss, Morteza Saghafiant, and Mathijs Wintraecken. “Brillouin Zones of Integer Lattices and Their Perturbations.” SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics, 2024. https://doi.org/10.1137/22M1489071.","short":"H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, M. Wintraecken, SIAM Journal on Discrete Mathematics 38 (2024) 1784–1807.","mla":"Edelsbrunner, Herbert, et al. “Brillouin Zones of Integer Lattices and Their Perturbations.” SIAM Journal on Discrete Mathematics, vol. 38, no. 2, Society for Industrial and Applied Mathematics, 2024, pp. 1784–807, doi:10.1137/22M1489071.","ama":"Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M. Brillouin zones of integer lattices and their perturbations. SIAM Journal on Discrete Mathematics. 2024;38(2):1784-1807. doi:10.1137/22M1489071"},"researchdata_availability":"no","arxiv":1,"abstract":[{"text":"For a locally finite set, 𝐴⊆ℝ𝑑\r\n, the 𝑘\r\nth Brillouin zone of 𝑎∈𝐴\r\n is the region of points 𝑥∈ℝ𝑑\r\n for which ‖𝑥−𝑎‖\r\n is the 𝑘\r\nth smallest among the Euclidean distances between 𝑥\r\n and the points in 𝐴\r\n. If 𝐴\r\n is a lattice, the 𝑘\r\nth Brillouin zones of the points in 𝐴\r\n are translates of each other, and together they tile space. Depending on the value of 𝑘\r\n, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in ℝ2\r\n, and the convergence of the maximum volume of a chamber to zero for the integer lattice.","lang":"eng"}],"author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"last_name":"Ghafaris","first_name":"Mohadese","full_name":"Ghafaris, Mohadese"},{"full_name":"Heiss, Teresa","orcid":"0000-0002-1780-2689","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa"},{"first_name":"Morteza","last_name":"Saghafiant","full_name":"Saghafiant, Morteza"},{"full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","first_name":"Mathijs"}],"title":"Brillouin zones of integer lattices and their perturbations","date_updated":"2026-10-02T11:49:10Z","fulldoi":"https://doi.org/10.1137/22M1489071","department":[{"_id":"HeEd"}],"ec_funded":1,"quality_controlled":"1","corr_author":"1","doi":"10.1137/22M1489071","publication_identifier":{"issn":["0895-4801"]},"issue":"2","status":"public","date_published":"2024-06-07T00:00:00Z","das_tickbox":"0","page":"1784-1807","month":"06","oa_version":"Preprint","_id":"17190","scopus_import":"1","isi":1,"type":"journal_article","publisher":"Society for Industrial and Applied Mathematics","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","language":[{"iso":"eng"}],"publication":"SIAM Journal on Discrete Mathematics","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2204.01077","open_access":"1"}]},{"conference":{"end_date":"2024-09-20","start_date":"2024-09-18","name":"GD: Graph Drawing and Network Visualization","location":"Vienna, Austria"},"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","type":"conference","language":[{"iso":"eng"}],"publication":"32nd International Symposium on Graph Drawing and Network Visualization","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","ddc":["510"],"month":"10","isi":1,"_id":"18556","oa_version":"Published Version","scopus_import":"1","file_date_updated":"2024-11-18T07:49:25Z","status":"public","has_accepted_license":"1","das_tickbox":"0","date_published":"2024-10-28T00:00:00Z","OA_type":"gold","fulldoi":"https://doi.org/10.4230/LIPIcs.GD.2024.3","OA_place":"publisher","ec_funded":1,"department":[{"_id":"HeEd"}],"corr_author":"1","quality_controlled":"1","file":[{"checksum":"5f9b35e115c3d375e99be78da9054cb4","file_size":908541,"file_name":"2024_LIPIcs_CultreradiMontesano.pdf","date_updated":"2024-11-18T07:49:25Z","creator":"dernst","access_level":"open_access","file_id":"18560","date_created":"2024-11-18T07:49:25Z","content_type":"application/pdf","success":1,"relation":"main_file"}],"publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773430"]},"doi":"10.4230/LIPIcs.GD.2024.3","abstract":[{"lang":"eng","text":"Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25."}],"author":[{"full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","first_name":"Sebastiano"},{"orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej","first_name":"Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"title":"The Euclidean MST-ratio for bi-colored lattices","date_updated":"2026-10-02T11:52:23Z","day":"28","alternative_title":["LIPIcs"],"intvolume":" 320","citation":{"chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In 32nd International Symposium on Graph Drawing and Network Visualization, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.GD.2024.3.","mla":"Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” 32nd International Symposium on Graph Drawing and Network Visualization, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.GD.2024.3.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, in:, 32nd International Symposium on Graph Drawing and Network Visualization, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in 32nd International Symposium on Graph Drawing and Network Visualization, Vienna, Austria, 2024, vol. 320.","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In 32nd International Symposium on Graph Drawing and Network Visualization (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.GD.2024.3","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2024. The Euclidean MST-ratio for bi-colored lattices. 32nd International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LIPIcs, vol. 320, 3.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: 32nd International Symposium on Graph Drawing and Network Visualization. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.GD.2024.3"},"project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"volume":320,"researchdata_availability":"no","arxiv":1,"article_number":"3","supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"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. 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.","oa":1,"external_id":{"arxiv":["2403.10204"],"isi":["001540278400001"]},"article_processing_charge":"Yes","date_created":"2024-11-17T23:01:47Z","year":"2024"},{"article_number":"87","researchdata_availability":"no","volume":293,"citation":{"ama":"Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality construction for homotopy inference (media exposition). In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.87","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. The ultimate frontier: An optimality construction for homotopy inference (media exposition). 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 87.","ieee":"D. Attali et al., “The ultimate frontier: An optimality construction for homotopy inference (media exposition),” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293.","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., & Wintraecken, M. (2024). The ultimate frontier: An optimality construction for homotopy inference (media exposition). In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.87","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","mla":"Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” 40th International Symposium on Computational Geometry, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.87.","chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, Andre Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.87."},"project":[{"call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"},{"grant_number":"M03073","name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"intvolume":" 293","alternative_title":["LIPIcs"],"day":"06","date_updated":"2026-10-02T11:51:50Z","title":"The ultimate frontier: An optimality construction for homotopy inference (media exposition)","author":[{"full_name":"Attali, Dominique","last_name":"Attali","first_name":"Dominique"},{"first_name":"Hana","last_name":"Kourimska","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","full_name":"Kourimska, Hana","orcid":"0000-0001-7841-0091"},{"full_name":"Fillmore, Christopher D","first_name":"Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","last_name":"Fillmore"},{"last_name":"Ghosh","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","first_name":"Ishika","full_name":"Ghosh, Ishika"},{"first_name":"Andre","last_name":"Lieutier","full_name":"Lieutier, Andre"},{"first_name":"Elizabeth R","last_name":"Stephenson","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","full_name":"Stephenson, Elizabeth R","orcid":"0000-0002-6862-208X"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","first_name":"Mathijs","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs"}],"abstract":[{"lang":"eng","text":"In our companion paper \"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds\" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting."}],"year":"2024","date_created":"2024-09-19T10:29:48Z","article_processing_charge":"Yes","oa":1,"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I02979-N35. Mathijs Wintraecken: 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) grant No. M-3073, and the welcome package from IDEX of the Université Côte d’Azur.\r\nWe thank Jean-Daniel Boissonnat, Herbert Edelsbrunner, and Mariette Yvinec for discussion.","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"supplementarymaterial":"yes","oa_version":"Published Version","_id":"18097","month":"06","ddc":["000"],"publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"publication":"40th International Symposium on Computational Geometry","type":"conference","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","conference":{"location":"Athens, Greece","name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11","end_date":"2024-06-14"},"publication_identifier":{"isbn":["9783959773164"]},"file":[{"date_created":"2024-09-19T10:30:37Z","content_type":"application/pdf","success":1,"relation":"main_file","file_size":3507177,"checksum":"9355c2e60b8ec285e1b22719c5b73f1a","file_name":"2024_LIPICs_Attali.pdf","access_level":"open_access","date_updated":"2024-09-19T10:30:37Z","creator":"dernst","file_id":"18098"}],"doi":"10.4230/LIPIcs.SoCG.2024.87","quality_controlled":"1","corr_author":"1","department":[{"_id":"HeEd"}],"ec_funded":1,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.87","date_published":"2024-06-06T00:00:00Z","das_tickbox":"0","has_accepted_license":"1","file_date_updated":"2024-09-19T10:30:37Z","status":"public"},{"researchdata_availability":"no","day":"01","citation":{"apa":"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. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” Journal of Applied and Computational Topology, vol. 8. Springer Nature, pp. 1101–1119, 2024.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 8, 1101–1119.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-023-00126-9.","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology, vol. 8, Springer Nature, 2024, pp. 1101–19, doi:10.1007/s41468-023-00126-9.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 1101–1119.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 2024;8:1101-1119. doi:10.1007/s41468-023-00126-9"},"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Persistent Homology, Algorithms and Stochastic Geometry"},{"call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"}],"intvolume":" 8","volume":8,"title":"Geometric characterization of the persistence of 1D maps","date_updated":"2026-10-02T11:52:56Z","author":[{"last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita"},{"full_name":"Cultrera Di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","last_name":"Cultrera Di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"abstract":[{"lang":"eng","text":"We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram."}],"article_processing_charge":"Yes (via OA deal)","date_created":"2023-07-02T22:00:44Z","year":"2024","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.","oa":1,"external_id":{"pmid":["39678706"]},"supplementarymaterial":"no","article_type":"original","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"ddc":["000"],"month":"10","_id":"13182","oa_version":"Published Version","scopus_import":"1","page":"1101-1119","publication":"Journal of Applied and Computational Topology","language":[{"iso":"eng"}],"publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","related_material":{"record":[{"id":"11660","relation":"earlier_version","status":"public"},{"relation":"dissertation_contains","id":"15094","status":"public"}]},"publisher":"Springer Nature","type":"journal_article","corr_author":"1","quality_controlled":"1","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"doi":"10.1007/s41468-023-00126-9","file":[{"file_size":476896,"checksum":"d493df5088c222b88d9ca46b623ad0ee","file_name":"2024_JourApplCompTopo_Biswas.pdf","access_level":"open_access","date_updated":"2025-01-09T07:39:41Z","creator":"dernst","file_id":"18783","date_created":"2025-01-09T07:39:41Z","content_type":"application/pdf","success":1,"relation":"main_file"}],"OA_type":"hybrid","fulldoi":"https://doi.org/10.1007/s41468-023-00126-9","OA_place":"publisher","department":[{"_id":"HeEd"}],"ec_funded":1,"has_accepted_license":"1","das_tickbox":"0","date_published":"2024-10-01T00:00:00Z","pmid":1,"status":"public","file_date_updated":"2025-01-09T07:39:41Z"},{"title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","date_updated":"2026-10-02T11:55:12Z","author":[{"first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","last_name":"Biswas","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890"},{"orcid":"0000-0001-6249-0832","full_name":"Cultrera Di Montesano, Sebastiano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera Di Montesano"},{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"abstract":[{"lang":"eng","text":"The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements."}],"researchdata_availability":"no","day":"01","volume":8,"citation":{"ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 2024;8:557-578. doi:10.1007/s41468-024-00173-w","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.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” Journal of Applied and Computational Topology, vol. 8. Springer Nature, pp. 557–578, 2024.","apa":"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. Springer Nature. https://doi.org/10.1007/s41468-024-00173-w","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.” Journal of Applied and Computational Topology, vol. 8, Springer Nature, 2024, pp. 557–78, doi:10.1007/s41468-024-00173-w.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-024-00173-w."},"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF"}],"intvolume":" 8","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.","external_id":{"pmid":["39308789"]},"oa":1,"article_type":"original","supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"article_processing_charge":"Yes (via OA deal)","date_created":"2024-05-12T22:01:03Z","year":"2024","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","language":[{"iso":"eng"}],"publication":"Journal of Applied and Computational Topology","related_material":{"record":[{"relation":"earlier_version","id":"11658","status":"public"}]},"type":"journal_article","publisher":"Springer Nature","month":"09","ddc":["510"],"_id":"15380","oa_version":"Published Version","scopus_import":"1","page":"557-578","date_published":"2024-09-01T00:00:00Z","has_accepted_license":"1","das_tickbox":"0","pmid":1,"file_date_updated":"2025-04-23T08:01:36Z","status":"public","quality_controlled":"1","corr_author":"1","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"doi":"10.1007/s41468-024-00173-w","file":[{"content_type":"application/pdf","date_created":"2025-04-23T08:01:36Z","relation":"main_file","success":1,"file_name":"2024_JourApplCompTopo_BiswasRa.pdf","file_size":522831,"checksum":"0ee15c1493a6413cf356ab2f32c81a9e","file_id":"19612","access_level":"open_access","date_updated":"2025-04-23T08:01:36Z","creator":"dernst"}],"fulldoi":"https://doi.org/10.1007/s41468-024-00173-w","OA_type":"hybrid","OA_place":"publisher","ec_funded":1,"department":[{"_id":"HeEd"}]},{"author":[{"first_name":"Peter","last_name":"Frankl","full_name":"Frankl, Peter"},{"full_name":"Pach, János","first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach"},{"full_name":"Pálvölgyi, Dömötör","first_name":"Dömötör","last_name":"Pálvölgyi"}],"abstract":[{"lang":"eng","text":"Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant <2 such that every family of subsets of an n-element set that contains no odd-sunflower consists of at most n sets. We construct such families of size at least 1.5021n. We also characterize minimal odd-sunflowers of triples."}],"date_updated":"2026-10-02T11:54:47Z","title":"Odd-sunflowers","volume":206,"citation":{"mla":"Frankl, Peter, et al. “Odd-Sunflowers.” Journal of Combinatorial Theory, Series A, vol. 206, no. 8, 105889, Elsevier, 2024, doi:10.1016/j.jcta.2024.105889.","short":"P. Frankl, J. Pach, D. Pálvölgyi, Journal of Combinatorial Theory, Series A 206 (2024).","chicago":"Frankl, Peter, János Pach, and Dömötör Pálvölgyi. “Odd-Sunflowers.” Journal of Combinatorial Theory, Series A. Elsevier, 2024. https://doi.org/10.1016/j.jcta.2024.105889.","ista":"Frankl P, Pach J, Pálvölgyi D. 2024. Odd-sunflowers. Journal of Combinatorial Theory, Series A. 206(8), 105889.","apa":"Frankl, P., Pach, J., & Pálvölgyi, D. (2024). Odd-sunflowers. Journal of Combinatorial Theory, Series A. Elsevier. https://doi.org/10.1016/j.jcta.2024.105889","ieee":"P. Frankl, J. Pach, and D. Pálvölgyi, “Odd-sunflowers,” Journal of Combinatorial Theory, Series A, vol. 206, no. 8. Elsevier, 2024.","ama":"Frankl P, Pach J, Pálvölgyi D. Odd-sunflowers. Journal of Combinatorial Theory, Series A. 2024;206(8). doi:10.1016/j.jcta.2024.105889"},"intvolume":" 206","day":"01","article_number":"105889","arxiv":1,"researchdata_availability":"not applicable","tmp":{"name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)","image":"/images/cc_by_nc.png","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","short":"CC BY-NC (4.0)"},"article_type":"original","supplementarymaterial":"yes","external_id":{"arxiv":["2310.16701"],"isi":["001217739200001"]},"oa":1,"acknowledgement":"We are grateful to Balázs Keszegh, and to the members of the Miklós Schweitzer Competition committee of 2022 for valuable discussions, and Shira Zerbib for pointing out several important mathematical typos.","date_created":"2024-03-31T22:01:11Z","year":"2024","article_processing_charge":"No","type":"journal_article","publisher":"Elsevier","dataavailabilitystatement":"No data was used for the research described in the article.","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","publication":"Journal of Combinatorial Theory, Series A","language":[{"iso":"eng"}],"oa_version":"Published Version","_id":"15247","scopus_import":"1","isi":1,"month":"08","ddc":["510"],"status":"public","issue":"8","file_date_updated":"2025-01-09T08:37:20Z","date_published":"2024-08-01T00:00:00Z","das_tickbox":"1","has_accepted_license":"1","OA_place":"publisher","department":[{"_id":"HeEd"}],"fulldoi":"https://doi.org/10.1016/j.jcta.2024.105889","OA_type":"hybrid","file":[{"date_created":"2025-01-09T08:37:20Z","content_type":"application/pdf","success":1,"relation":"main_file","file_size":366029,"checksum":"ffc29d65e712849f0d31009271e06a63","file_name":"2024_JourCombiTheoryA_Frankl.pdf","creator":"dernst","date_updated":"2025-01-09T08:37:20Z","access_level":"open_access","file_id":"18791"}],"doi":"10.1016/j.jcta.2024.105889","publication_identifier":{"issn":["0097-3165"],"eissn":["1096-0899"]},"quality_controlled":"1","corr_author":"1"},{"article_processing_charge":"No","year":"2024","date_created":"2024-12-17T16:17:55Z","acknowledgement":"I was supported by the European Research Council (ERC) Horizon 2020 project\r\n“Alpha Shape Theory Extended” No. 788183 and by the Pöttinger Scholarship. In addition,\r\nI am very thankful for having been able to attend the second Workshop for Women in\r\nComputational Topology in July 2019, funded by the Mathematical Sciences Institute at\r\nANU, the US National Science Foundation through the award CCF-1841455, the Australian\r\nMathematical Sciences Institute and the Association for Women in Mathematics. Two of the\r\nprojects presented in this thesis started there. One of them reached completion thanks to\r\nfunding from the MSRI Summer Research in Mathematics program awarded to me and my\r\ncollaborators in 2020.","oa":1,"supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"researchdata_availability":"no","day":"17","alternative_title":["ISTA Thesis"],"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020"}],"citation":{"ista":"Heiss T. 2024. New methods for applying topological data analysis to materials science. Institute of Science and Technology Austria.","ieee":"T. Heiss, “New methods for applying topological data analysis to materials science,” Institute of Science and Technology Austria, 2024.","apa":"Heiss, T. (2024). New methods for applying topological data analysis to materials science. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18667","mla":"Heiss, Teresa. New Methods for Applying Topological Data Analysis to Materials Science. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18667.","short":"T. Heiss, New Methods for Applying Topological Data Analysis to Materials Science, Institute of Science and Technology Austria, 2024.","chicago":"Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials Science.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18667.","ama":"Heiss T. New methods for applying topological data analysis to materials science. 2024. doi:10.15479/at:ista:18667"},"title":"New methods for applying topological data analysis to materials science","date_updated":"2026-10-02T11:55:42Z","abstract":[{"lang":"eng","text":"Many chemical and physical properties of materials are determined by the material’s shape,\r\nfor example the size of its pores and the width of its tunnels. This makes materials science\r\na prime application area for geometrical and topological methods. Nevertheless many\r\nmethods in topological data analysis have not been satisfyingly extended to the needs of\r\nmaterials science. This thesis provides new methods and new mathematical theorems\r\ntargeted at those specific needs by answering four different research questions. While the\r\nmotivation for each of the research questions arises from materials science, the methods\r\nare versatile and can be applied in different areas as well. \r\n\r\nThe first research question is concerned with image data, for example a three-dimensional\r\ncomputed tomography (CT) scan of a material, like sand or stone. There are two commonly\r\nused topologies for digital images and depending on the application either of them might be\r\nrequired. However, software for computing the topological data analysis method persistence\r\nhomology, usually supports only one of the two topologies. We answer the question how to\r\ncompute persistent homology of an image with respect to one of the two topologies using\r\nsoftware that is intended for the other topology. \r\n\r\nThe second research question is concerned with image data as well, and asks how much\r\nof the topological information of an image is lost when the resolution is coarsened. As\r\ncomputer tomography scanners are more expensive the higher the resolution, it is an\r\nimportant question in materials science to know which resolution is enough to get satisfying\r\npersistent homology. We give theoretical bounds on the information loss based on different\r\ngeometrical properties of the object to be scanned. In addition, we conduct experiments on\r\nsand and stone CT image data. \r\n\r\nThe third research question is motivated by comparing crystalline materials efficiently. As\r\nthe atoms within a crystal repeat periodically, crystalline materials are either modeled by\r\nunmanageable infinite periodic point sets, or by one of their fundamental domains, which is\r\nunstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with\r\nappropriate properties such that comparing the crystals can be eased by comparing the\r\nfingerprints instead. We define the density fingerprint and prove the necessary properties. \r\n\r\nThe fourth research question is motivated by studying the hole-structure or connectedness,\r\ni.e. persistent homology or merge trees, of crystalline materials. A common way to deal\r\nwith periodicity is to take a fundamental domain and identify opposite boundaries to form a\r\ntorus. However, computing persistent homology or merge trees on that torus loses some\r\nof the information materials scientists are interested in and is additionally not stable under\r\ncertain noise. We therefore decorate the merge tree stemming from the torus with additional\r\ninformation describing the density and growth rate of the periodic copies of a component\r\nwithin a growing spherical window. We prove all desired properties, like stability and efficient\r\ncomputability."}],"author":[{"orcid":"0000-0002-1780-2689","full_name":"Heiss, Teresa","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss"}],"doi_confirm":"1","corr_author":"1","publication_identifier":{"issn":["2663-337X"],"isbn":["978-3-99078-052-7"]},"doi":"10.15479/at:ista:18667","file":[{"file_id":"18686","date_updated":"2024-12-19T10:24:46Z","creator":"theiss","access_level":"open_access","file_name":"Teresa_Heiss_PhD_Thesis_final.pdf","checksum":"247bb057aed2fba1cd4711917aaa2d77","file_size":7752253,"relation":"main_file","success":1,"content_type":"application/pdf","date_created":"2024-12-19T10:24:46Z"},{"creator":"theiss","date_updated":"2024-12-19T10:24:50Z","access_level":"closed","file_id":"18687","checksum":"9648b45c07a008ee11a07f99856a139d","file_size":17197731,"file_name":"PhD_Thesis.zip","relation":"source_file","date_created":"2024-12-19T10:24:50Z","content_type":"application/zip"}],"fulldoi":"https://doi.org/10.15479/at:ista:18667","OA_place":"publisher","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"ec_funded":1,"das_tickbox":"0","has_accepted_license":"1","date_published":"2024-12-17T00:00:00Z","degree_awarded":"PhD","status":"public","keyword":["persistent homology","topological data analysis","periodic","crystalline materials","images","fingerprint"],"file_date_updated":"2024-12-19T10:24:50Z","month":"12","ddc":["514","516","004"],"_id":"18667","oa_version":"Published Version","page":"111","language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","related_material":{"record":[{"status":"public","id":"10828","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"11440"},{"relation":"part_of_dissertation","id":"9345","status":"public"},{"id":"18673","relation":"part_of_dissertation","status":"public"}]},"supervisor":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"}],"publisher":"Institute of Science and Technology Austria","type":"dissertation"},{"abstract":[{"text":"Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in\r\ndiscrete geometry that have captivated mathematicians for centuries, if not millennia. This\r\nthesis seeks to cast new light on these structures by illustrating specific instances where a\r\ntopological perspective, specifically through discrete Morse theory and persistent homology,\r\nprovides valuable insights.\r\n\r\nAt first glance, the topology of these geometric objects might seem uneventful: point sets\r\nessentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which\r\nis a contractible space, and the topology of a network primarily involves the enumeration\r\nof connected components and cycles within the network. However, beneath this apparent\r\nsimplicity, there lies an array of intriguing structures, a small subset of which will be uncovered\r\nin this thesis.\r\n\r\nFocused on three case studies, each addressing one of the mentioned objects, this work\r\nwill showcase connections that intertwine topology with diverse fields such as combinatorial\r\ngeometry, algorithms and data structures, and emerging applications like spatial biology.\r\n\r\n","lang":"eng"}],"author":[{"full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","last_name":"Cultrera di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"}],"doi_confirm":"1","title":"Persistence and Morse theory for discrete geometric structures","date_updated":"2026-10-02T11:57:31Z","alternative_title":["ISTA Thesis"],"day":"08","citation":{"ama":"Cultrera di Montesano S. Persistence and Morse theory for discrete geometric structures. 2024. doi:10.15479/at:ista:15094","short":"S. Cultrera di Montesano, Persistence and Morse Theory for Discrete Geometric Structures, Institute of Science and Technology Austria, 2024.","mla":"Cultrera di Montesano, Sebastiano. Persistence and Morse Theory for Discrete Geometric Structures. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:15094.","chicago":"Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete Geometric Structures.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:15094.","ista":"Cultrera di Montesano S. 2024. Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria.","ieee":"S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric structures,” Institute of Science and Technology Austria, 2024.","apa":"Cultrera di Montesano, S. (2024). Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:15094"},"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF"},{"name":"Persistent Homology, Algorithms and Stochastic Geometry","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"researchdata_availability":"no","supplementarymaterial":"no","tmp":{"image":"/images/cc_by_nc_sa.png","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)"},"oa":1,"article_processing_charge":"No","date_created":"2024-03-08T15:28:10Z","year":"2024","supervisor":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert"}],"type":"dissertation","publisher":"Institute of Science and Technology Austria","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"part_of_dissertation","id":"15090","status":"public"},{"status":"public","id":"11658","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"11660"},{"id":"13182","relation":"part_of_dissertation","status":"public"},{"id":"15091","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"15093","relation":"part_of_dissertation"}]},"page":"108","ddc":["514","500","516"],"month":"03","_id":"15094","oa_version":"Published Version","degree_awarded":"PhD","file_date_updated":"2024-03-14T14:14:35Z","status":"public","date_published":"2024-03-08T00:00:00Z","das_tickbox":"0","has_accepted_license":"1","fulldoi":"https://doi.org/10.15479/at:ista:15094","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"OA_place":"publisher","ec_funded":1,"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","corr_author":"1","doi":"10.15479/at:ista:15094","file":[{"content_type":"application/pdf","date_created":"2024-03-14T08:55:07Z","relation":"main_file","success":1,"file_name":"Thesis Sebastiano.pdf","checksum":"1e468bfa42a7dcf04d89f4dadc621c87","file_size":4106872,"file_id":"15112","creator":"scultrer","date_updated":"2024-03-14T08:55:07Z","access_level":"open_access"},{"relation":"source_file","date_created":"2024-03-14T08:56:24Z","content_type":"application/zip","access_level":"closed","creator":"scultrer","date_updated":"2024-03-14T14:14:35Z","file_id":"15113","file_size":4746234,"checksum":"bcbd213490f5a7e68855a092bbce93f1","file_name":"Thesis (1).zip"}],"publication_identifier":{"issn":["2663-337X"]}},{"publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"publication":"Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)","main_file_link":[{"url":"https://arxiv.org/abs/2311.01115","open_access":"1"}],"related_material":{"record":[{"status":"public","id":"15094","relation":"dissertation_contains"}]},"editor":[{"first_name":"David P.","last_name":"Woodruff","full_name":"Woodruff, David P."}],"conference":{"start_date":"2024-01-07","end_date":"2024-01-10","location":"Alexandria, VA, USA","name":"SODA: Symposium on Discrete Algorithms"},"type":"conference","publisher":"Society for Industrial and Applied Mathematics","month":"01","scopus_import":"1","_id":"15093","oa_version":"Preprint","page":"243 - 295","date_published":"2024-01-04T00:00:00Z","das_tickbox":"0","status":"public","quality_controlled":"1","corr_author":"1","doi":"10.1137/1.9781611977912.11","publication_identifier":{"eisbn":["9781611977912"]},"fulldoi":"https://doi.org/10.1137/1.9781611977912.11","department":[{"_id":"HeEd"},{"_id":"MoHe"}],"ec_funded":1,"title":"Dynamically maintaining the persistent homology of time series","date_updated":"2026-10-02T11:57:32Z","abstract":[{"text":"We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.","lang":"eng"}],"author":[{"first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832","full_name":"Cultrera di Montesano, Sebastiano"},{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Henzinger, Monika H","orcid":"0000-0002-5008-6530","id":"540c9bbd-f2de-11ec-812d-d04a5be85630","last_name":"Henzinger","first_name":"Monika H"},{"first_name":"Lara","last_name":"Ost","full_name":"Ost, Lara"}],"arxiv":1,"researchdata_availability":"no","day":"04","citation":{"short":"S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, L. Ost, in:, D.P. Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Society for Industrial and Applied Mathematics, 2024, pp. 243–295.","mla":"Cultrera di Montesano, Sebastiano, et al. “Dynamically Maintaining the Persistent Homology of Time Series.” Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), edited by David P. Woodruff, Society for Industrial and Applied Mathematics, 2024, pp. 243–95, doi:10.1137/1.9781611977912.11.","chicago":"Cultrera di Montesano, Sebastiano, Herbert Edelsbrunner, Monika Henzinger, and Lara Ost. “Dynamically Maintaining the Persistent Homology of Time Series.” In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), edited by David P. Woodruff, 243–95. Society for Industrial and Applied Mathematics, 2024. https://doi.org/10.1137/1.9781611977912.11.","ista":"Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. 2024. Dynamically maintaining the persistent homology of time series. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SODA: Symposium on Discrete Algorithms, 243–295.","apa":"Cultrera di Montesano, S., Edelsbrunner, H., Henzinger, M., & Ost, L. (2024). Dynamically maintaining the persistent homology of time series. In D. P. Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (pp. 243–295). Alexandria, VA, USA: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611977912.11","ieee":"S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, and L. Ost, “Dynamically maintaining the persistent homology of time series,” in Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Alexandria, VA, USA, 2024, pp. 243–295.","ama":"Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. Dynamically maintaining the persistent homology of time series. In: Woodruff DP, ed. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society for Industrial and Applied Mathematics; 2024:243-295. doi:10.1137/1.9781611977912.11"},"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"name":"The design and evaluation of modern fully dynamic data structures","_id":"bd9ca328-d553-11ed-ba76-dc4f890cfe62","grant_number":"101019564","call_identifier":"H2020"},{"grant_number":"Z00422","name":"Efficient algorithms","_id":"34def286-11ca-11ed-8bc3-da5948e1613c"},{"_id":"bd9e3a2e-d553-11ed-ba76-8aa684ce17fe","name":"Fast Algorithms for a Reactive Network Layer","grant_number":"P33775"}],"acknowledgement":"The first and second authors are funded by the European Research Council under the European Union’s Horizon 2020 research and innovation programme, ERC grant no. 788183,“Alpha Shape Theory Extended (Alpha)”, by the Wittgenstein Prize, FWF grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.The third author received funding by the European Research Council under the European Union’s Horizon 2020research and innovation programme, ERC grant no. 101019564, “The Design of Modern Fully Dynamic DataStructures (MoDynStruct)”, and by the Austrian Science Fund through the Wittgenstein Prize with FWF grant no. Z 422-N, and also by FWF grant no. I 5982-N, and by FWF grant no. P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024. The fourth author is funded by the Vienna Graduate School on Computational Optimization, FWF project no. W1260-N35.","external_id":{"arxiv":["2311.01115"]},"oa":1,"supplementarymaterial":"no","article_processing_charge":"No","date_created":"2024-03-08T10:27:39Z","year":"2024"},{"title":"Discrete microlocal Morse theory","date_updated":"2026-10-02T11:57:00Z","author":[{"first_name":"Adam","last_name":"Brown","full_name":"Brown, Adam"},{"full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej"}],"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"}],"researchdata_availability":"no","arxiv":1,"day":"09","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"citation":{"chicago":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2209.14993.","mla":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” ArXiv, doi:10.48550/arXiv.2209.14993.","short":"A. Brown, O. Draganov, ArXiv (n.d.).","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” arXiv. .","apa":"Brown, A., & Draganov, O. (n.d.). Discrete microlocal Morse theory. arXiv. https://doi.org/10.48550/arXiv.2209.14993","ista":"Brown A, Draganov O. Discrete microlocal Morse theory. arXiv, 10.48550/arXiv.2209.14993.","ama":"Brown A, Draganov O. Discrete microlocal Morse theory. arXiv. doi:10.48550/arXiv.2209.14993"},"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European\r\nUnion’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize,\r\nAustrian Science Fund (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), grant no. I 02979-N35.","external_id":{"arxiv":["2209.14993"]},"oa":1,"supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"article_processing_charge":"No","date_created":"2025-01-31T17:03:04Z","year":"2024","publication_status":"draft","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication":"arXiv","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2209.14993"}],"related_material":{"record":[{"status":"public","id":"20323","relation":"later_version"},{"id":"18979","relation":"dissertation_contains","status":"public"}]},"type":"preprint","month":"06","_id":"18981","oa_version":"Preprint","date_published":"2024-06-09T00:00:00Z","das_tickbox":"0","status":"public","corr_author":"1","doi":"10.48550/arXiv.2209.14993","fulldoi":"https://doi.org/10.48550/arXiv.2209.14993","ec_funded":1,"department":[{"_id":"HeEd"}],"OA_place":"repository"},{"ec_funded":1,"department":[{"_id":"HeEd"}],"OA_place":"repository","fulldoi":"https://doi.org/10.48550/arXiv.2408.16575","doi":"10.48550/arXiv.2408.16575","corr_author":"1","status":"public","das_tickbox":"0","date_published":"2024-08-29T00:00:00Z","_id":"18673","oa_version":"Preprint","month":"08","type":"preprint","related_material":{"record":[{"status":"public","id":"18667","relation":"dissertation_contains"}]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2408.16575"}],"language":[{"iso":"eng"}],"publication":"arXiv","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"draft","year":"2024","date_created":"2024-12-18T14:06:57Z","article_processing_charge":"No","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"supplementarymaterial":"no","oa":1,"external_id":{"arxiv":["2408.16575"]},"acknowledgement":"Both authors are partially supported by the European Research Council (ERC) Horizon 2020 project\r\n‘Alpha Shape Theory Extended’, grant no. 788183. The first author is also partially supported by the DFG\r\nCollaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund\r\n(FWF), grant no. I 02979-N35.","citation":{"ieee":"H. Edelsbrunner and T. Heiss, “Merge trees of periodic filtrations,” arXiv. .","apa":"Edelsbrunner, H., & Heiss, T. (n.d.). Merge trees of periodic filtrations. arXiv. https://doi.org/10.48550/arXiv.2408.16575","ista":"Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv, 10.48550/arXiv.2408.16575.","chicago":"Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2408.16575.","short":"H. Edelsbrunner, T. Heiss, ArXiv (n.d.).","mla":"Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” ArXiv, doi:10.48550/arXiv.2408.16575.","ama":"Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv. doi:10.48550/arXiv.2408.16575"},"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"day":"29","researchdata_availability":"no","arxiv":1,"abstract":[{"text":"Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes O((n+m)logn) time, in which n and m are the numbers of vertices and edges in the quotient complex, respectively.\r\n","lang":"eng"}],"author":[{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Heiss, Teresa","orcid":"0000-0002-1780-2689","first_name":"Teresa","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"}],"date_updated":"2026-10-02T11:55:43Z","title":"Merge trees of periodic filtrations"},{"day":"07","citation":{"ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv. doi:10.48550/arXiv.2212.03128","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” arXiv. .","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). Chromatic alpha complexes. arXiv. https://doi.org/10.48550/arXiv.2212.03128","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv, 2212.03128.","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2212.03128.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” ArXiv, 2212.03128, doi:10.48550/arXiv.2212.03128.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.)."},"arxiv":1,"researchdata_availability":"no","article_number":"2212.03128","author":[{"first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832","full_name":"Cultrera di Montesano, Sebastiano"},{"first_name":"Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823"},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"abstract":[{"lang":"eng","text":"Motivated by applications in the medical sciences, we study finite chromatic\r\nsets in Euclidean space from a topological perspective. Based on the persistent\r\nhomology for images, kernels and cokernels, we design provably stable\r\nhomological quantifiers that describe the geometric micro- and macro-structure\r\nof how the color classes mingle. These can be efficiently computed using\r\nchromatic variants of Delaunay and alpha complexes, and code that does these\r\ncomputations is provided."}],"title":"Chromatic alpha complexes","date_updated":"2026-10-02T11:56:13Z","article_processing_charge":"No","date_created":"2024-03-08T10:13:59Z","year":"2024","supplementarymaterial":"no","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"external_id":{"arxiv":["2212.03128"]},"oa":1,"month":"02","_id":"15091","oa_version":"Preprint","type":"preprint","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"draft","publication":"arXiv","language":[{"iso":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/2212.03128","open_access":"1"}],"related_material":{"record":[{"status":"public","id":"18979","relation":"dissertation_contains"},{"id":"20585","relation":"later_version","status":"public"},{"id":"15094","relation":"dissertation_contains","status":"public"}]},"fulldoi":"https://doi.org/10.48550/arXiv.2212.03128","OA_place":"repository","department":[{"_id":"HeEd"}],"corr_author":"1","doi":"10.48550/arXiv.2212.03128","status":"public","date_published":"2024-02-07T00:00:00Z","das_tickbox":"0"}]