[{"title":"Decomposition of geometric graphs into star-forests","arxiv":1,"_id":"21253","language":[{"iso":"eng"}],"day":"01","OA_place":"repository","intvolume":"       129","publication_identifier":{"issn":["0925-7721"]},"date_published":"2025-12-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2026-10-02T11:50:58Z","external_id":{"arxiv":["2306.13201"]},"month":"12","oa":1,"author":[{"first_name":"János","last_name":"Pach","full_name":"Pach, János"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","first_name":"Morteza"},{"last_name":"Schnider","first_name":"Patrick","full_name":"Schnider, Patrick"}],"status":"public","year":"2025","OA_type":"green","citation":{"ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” <i>Computational Geometry</i>, vol. 129. Elsevier, 2025.","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>.","ista":"Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186.","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>, vol. 129, 102186, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>.","short":"J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025).","apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2025). Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. 2025;129. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>"},"date_created":"2026-02-16T15:48:42Z","department":[{"_id":"HeEd"}],"volume":129,"article_number":"102186","corr_author":"1","oa_version":"Preprint","publication_status":"published","publisher":"Elsevier","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2306.13201"}],"type":"journal_article","fulldoi":"https://doi.org/10.1016/j.comgeo.2025.102186","quality_controlled":"1","article_processing_charge":"No","publication":"Computational Geometry","article_type":"original","abstract":[{"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.","lang":"eng"}],"doi":"10.1016/j.comgeo.2025.102186","acknowledgement":"A preliminary version of this note has been published in the proceedings of the 31st International Symposium on Graph Drawing and Network Visualization, Palermo, 2023. The authors would like to thank the anonymous referees for their valuable comments.","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"15012"}]}},{"oa_version":"Published Version","publication_status":"published","corr_author":"1","alternative_title":["ISTA Thesis"],"page":"140","type":"dissertation","license":"https://creativecommons.org/licenses/by/4.0/","fulldoi":"https://doi.org/10.15479/at:ista:18979","publisher":"Institute of Science and Technology Austria","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"abstract":[{"lang":"eng","text":"Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe describe a construction of a chromatic alpha complex, which enables  efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot."}],"project":[{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science"}],"article_processing_charge":"No","acknowledgement":"The research presented in this thesis was funded with the Wittgenstein Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.\r\n","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"15091"},{"relation":"part_of_dissertation","status":"public","id":"18981"}]},"supervisor":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"}],"doi":"10.15479/at:ista:18979","file":[{"file_name":"Thesis.zip","creator":"odragano","file_size":11899491,"checksum":"af6567e5d35e5eb330b8925ae37f1998","file_id":"18983","relation":"source_file","date_created":"2025-01-31T16:58:30Z","date_updated":"2025-01-31T16:58:30Z","content_type":"application/zip","access_level":"closed"},{"file_size":8857514,"creator":"odragano","checksum":"c3fef68e35b9dc2020b2ca6006da6343","file_name":"Thesis.pdf","file_id":"19000","relation":"main_file","date_created":"2025-02-04T16:22:07Z","date_updated":"2025-02-04T16:22:07Z","content_type":"application/pdf","access_level":"open_access"}],"day":"03","language":[{"iso":"eng"}],"_id":"18979","keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"title":"Structures and computations in topological data analysis","date_updated":"2026-10-02T11:57:00Z","date_published":"2025-02-03T00:00:00Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","ddc":["514","004"],"publication_identifier":{"issn":["2663-337X"]},"file_date_updated":"2025-02-04T16:22:07Z","OA_place":"publisher","author":[{"last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823"}],"degree_awarded":"PhD","has_accepted_license":"1","oa":1,"month":"02","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"year":"2025","date_created":"2025-01-31T17:04:40Z","citation":{"ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025.","chicago":"Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>.","ista":"Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria.","short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","apa":"Draganov, O. (2025). <i>Structures and computations in topological data analysis</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>","ama":"Draganov O. Structures and computations in topological data analysis. 2025. doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>","mla":"Draganov, Ondrej. <i>Structures and Computations in Topological Data Analysis</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>."},"status":"public"},{"author":[{"first_name":"Adam","last_name":"Brown","id":"70B7FDF6-608D-11E9-9333-8535E6697425","full_name":"Brown, Adam"},{"orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej","first_name":"Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"}],"oa":1,"has_accepted_license":"1","month":"10","external_id":{"arxiv":["2209.14993"]},"department":[{"_id":"HeEd"}],"year":"2025","OA_type":"hybrid","citation":{"mla":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10, 108068, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>.","apa":"Brown, A., &#38; Draganov, O. (2025). Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>","short":"A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).","ama":"Brown A, Draganov O. Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. 2025;229(10). doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.","ista":"Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure and Applied Algebra. 229(10), 108068.","chicago":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>."},"date_created":"2025-09-10T05:40:09Z","status":"public","issue":"10","day":"01","PlanS_conform":"1","_id":"20323","language":[{"iso":"eng"}],"arxiv":1,"title":"Discrete microlocal Morse theory","ddc":["510"],"date_updated":"2026-10-02T11:57:00Z","date_published":"2025-10-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["0022-4049"]},"intvolume":"       229","file_date_updated":"2025-12-30T07:55:08Z","OA_place":"publisher","article_type":"original","abstract":[{"lang":"eng","text":"We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities."}],"project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"publication":"Journal of Pure and Applied Algebra","article_processing_charge":"Yes (via OA deal)","quality_controlled":"1","related_material":{"record":[{"id":"18981","status":"public","relation":"earlier_version"}]},"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35","doi":"10.1016/j.jpaa.2025.108068","file":[{"date_updated":"2025-12-30T07:55:08Z","date_created":"2025-12-30T07:55:08Z","content_type":"application/pdf","access_level":"open_access","file_name":"2025_JourPureAppliedAlgebra_Brown.pdf","file_size":3090836,"checksum":"39bcad462278c9322ef810af7db67f56","creator":"dernst","success":1,"file_id":"20886","relation":"main_file"}],"oa_version":"Published Version","publication_status":"published","corr_author":"1","volume":229,"article_number":"108068","type":"journal_article","ec_funded":1,"fulldoi":"https://doi.org/10.1016/j.jpaa.2025.108068","scopus_import":"1","publisher":"Elsevier","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"}},{"oa_version":"Published Version","publication_status":"published","corr_author":"1","page":"12080-12099","type":"conference","fulldoi":"https://doi.org/10.18653/v1/2024.findings-emnlp.705","scopus_import":"1","publisher":"Association for Computational Linguistics","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"abstract":[{"lang":"eng","text":"Word embeddings represent language vocabularies as clouds of d-dimensional points. We investigate how information is conveyed by the general shape of these clouds, instead of representing the semantic meaning of each token. Specifically, we use the notion of persistent homology from topological data analysis (TDA) to measure the distances between language pairs from the shape of their unlabeled embeddings. These distances quantify the degree of non-isometry of the embeddings. To distinguish whether these differences are random training errors or capture real information about the languages, we use the computed distance matrices to construct language phylogenetic trees over 81 Indo-European languages. Careful evaluation shows that our reconstructed trees exhibit strong and statistically-significant similarities to the reference."}],"publication":"Findings of the Association for Computational Linguistics: EMNLP 2024","article_processing_charge":"No","quality_controlled":"1","doi":"10.18653/v1/2024.findings-emnlp.705","file":[{"date_created":"2025-02-10T08:20:34Z","date_updated":"2025-02-10T08:20:34Z","content_type":"application/pdf","access_level":"open_access","checksum":"f4416a5962194f0181ab0dc7f9ef93c0","file_size":1312638,"creator":"dernst","file_name":"2024_EMNLP_Draganov.pdf","file_id":"19016","success":1,"relation":"main_file"}],"day":"01","language":[{"iso":"eng"}],"_id":"18998","arxiv":1,"title":"The shape of word embeddings: Quantifying non-isometry with topological data analysis","ddc":["500"],"date_published":"2024-11-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2025-02-10T08:21:37Z","file_date_updated":"2025-02-10T08:20:34Z","OA_place":"publisher","author":[{"first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov","full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823"},{"full_name":"Skiena, Steven","first_name":"Steven","last_name":"Skiena"}],"conference":{"start_date":"2024-11-12","end_date":"2024-11-16","name":"EMNLP: Conference on Empirical Methods in Natural Language Processing","location":"Miami, FL, United States"},"has_accepted_license":"1","oa":1,"month":"11","external_id":{"arxiv":["2404.00500"]},"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"year":"2024","OA_type":"gold","date_created":"2025-02-04T16:19:28Z","citation":{"mla":"Draganov, Ondrej, and Steven Skiena. “The Shape of Word Embeddings: Quantifying Non-Isometry with Topological Data Analysis.” <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>, Association for Computational Linguistics, 2024, pp. 12080–99, doi:<a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">10.18653/v1/2024.findings-emnlp.705</a>.","ama":"Draganov O, Skiena S. The shape of word embeddings: Quantifying non-isometry with topological data analysis. In: <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>. Association for Computational Linguistics; 2024:12080-12099. doi:<a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">10.18653/v1/2024.findings-emnlp.705</a>","apa":"Draganov, O., &#38; Skiena, S. (2024). The shape of word embeddings: Quantifying non-isometry with topological data analysis. In <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i> (pp. 12080–12099). Miami, FL, United States: Association for Computational Linguistics. <a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">https://doi.org/10.18653/v1/2024.findings-emnlp.705</a>","short":"O. Draganov, S. Skiena, in:, Findings of the Association for Computational Linguistics: EMNLP 2024, Association for Computational Linguistics, 2024, pp. 12080–12099.","chicago":"Draganov, Ondrej, and Steven Skiena. “The Shape of Word Embeddings: Quantifying Non-Isometry with Topological Data Analysis.” In <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>, 12080–99. Association for Computational Linguistics, 2024. <a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">https://doi.org/10.18653/v1/2024.findings-emnlp.705</a>.","ista":"Draganov O, Skiena S. 2024. The shape of word embeddings: Quantifying non-isometry with topological data analysis. Findings of the Association for Computational Linguistics: EMNLP 2024. EMNLP: Conference on Empirical Methods in Natural Language Processing, 12080–12099.","ieee":"O. Draganov and S. Skiena, “The shape of word embeddings: Quantifying non-isometry with topological data analysis,” in <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>, Miami, FL, United States, 2024, pp. 12080–12099."},"status":"public"},{"publication_status":"submitted","oa_version":"Preprint","corr_author":"1","article_number":"2406.04102","fulldoi":"https://doi.org/10.48550/ARXIV.2406.04102","type":"preprint","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2406.04102"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"abstract":[{"lang":"eng","text":"Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry."}],"publication":"arXiv","article_processing_charge":"No","doi":"10.48550/ARXIV.2406.04102","day":"06","language":[{"iso":"eng"}],"_id":"18999","arxiv":1,"title":"Chromatic topological data analysis","date_published":"2024-06-06T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"date_updated":"2025-02-10T08:14:27Z","OA_place":"repository","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"},{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov","first_name":"Ondrej","full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"has_accepted_license":"1","oa":1,"month":"06","external_id":{"arxiv":["2406.04102"]},"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"date_created":"2025-02-04T16:21:21Z","citation":{"ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic topological data analysis,” <i>arXiv</i>. .","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Topological Data Analysis.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">https://doi.org/10.48550/ARXIV.2406.04102</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv, 2406.04102.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Topological Data Analysis.” <i>ArXiv</i>, 2406.04102, doi:<a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">10.48550/ARXIV.2406.04102</a>.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Chromatic topological data analysis. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">https://doi.org/10.48550/ARXIV.2406.04102</a>","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">10.48550/ARXIV.2406.04102</a>"},"OA_type":"green","year":"2024","status":"public"},{"oa_version":"Published Version","publication_status":"published","volume":293,"alternative_title":["LIPIcs"],"page":"11:1-11:19","type":"conference","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.11","ec_funded":1,"scopus_import":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"abstract":[{"text":"In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. - In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. - The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction. We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.","lang":"eng"}],"project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses","grant_number":"M03073"}],"publication":"40th International Symposium on Computational Geometry","researchdata_availability":"no","article_processing_charge":"No","quality_controlled":"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. I 02979-N35.\r\nWintraecken, Mathijs: 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.","doi":"10.4230/LIPIcs.SoCG.2024.11","file":[{"relation":"main_file","creator":"cfillmor","file_name":"LIPIcs.SoCG.2024.11.pdf","file_size":20886142,"checksum":"6a2ddc8b51aa58f197a8b294750f1f8d","file_id":"17171","success":1,"content_type":"application/pdf","access_level":"open_access","date_updated":"2024-06-25T11:47:26Z","date_created":"2024-06-25T11:47:26Z"}],"day":"06","language":[{"iso":"eng"}],"_id":"17170","arxiv":1,"title":"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds","date_updated":"2026-10-02T11:23:43Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","ddc":["516"],"date_published":"2024-06-06T00:00:00Z","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773164"]},"supplementarymaterial":"no","intvolume":"       293","file_date_updated":"2024-06-25T11:47:26Z","das_tickbox":"0","author":[{"last_name":"Attali","first_name":"Dominique","full_name":"Attali, Dominique"},{"orcid":"0000-0001-7841-0091","full_name":"Kourimska, Hana","last_name":"Kourimska","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana"},{"last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","full_name":"Fillmore, Christopher D"},{"full_name":"Ghosh, Ishika","first_name":"Ishika","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","last_name":"Ghosh"},{"full_name":"Lieutier, André","first_name":"André","last_name":"Lieutier"},{"orcid":"0000-0002-6862-208X","full_name":"Stephenson, Elizabeth R","last_name":"Stephenson","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","first_name":"Elizabeth R"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220"}],"conference":{"end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","start_date":"2024-06-11"},"has_accepted_license":"1","oa":1,"month":"06","external_id":{"arxiv":["2206.10485"]},"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"year":"2024","date_created":"2024-06-25T11:45:58Z","citation":{"mla":"Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>.","ama":"Attali D, Kourimska H, Fillmore CD, et al. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>","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, p. 11:1-11:19.","chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” In <i>40th International Symposium on Computational Geometry</i>, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>.","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 11:1-11:19.","ieee":"D. Attali <i>et al.</i>, “Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293, p. 11:1-11:19."},"status":"public"},{"file":[{"date_updated":"2024-07-22T09:43:19Z","date_created":"2024-07-22T09:43:19Z","content_type":"application/pdf","access_level":"open_access","creator":"dernst","file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","file_size":892019,"checksum":"b207b4e00f904e8ea8a30e24f0251f79","success":1,"file_id":"17301","relation":"main_file"}],"doi":"10.1007/s00454-023-00566-1","acknowledgement":"Work by all authors but A. Garber is supported by the European Research Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation.","quality_controlled":"1","article_processing_charge":"Yes (via OA deal)","publication":"Discrete and Computational Geometry","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"researchdata_availability":"no","article_type":"original","abstract":[{"lang":"eng","text":"For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2  is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970))."}],"publisher":"Springer Nature","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"scopus_import":"1","type":"journal_article","fulldoi":"https://doi.org/10.1007/s00454-023-00566-1","ec_funded":1,"volume":72,"page":"29-48","corr_author":"1","oa_version":"Published Version","publication_status":"published","status":"public","year":"2024","date_created":"2023-09-17T22:01:10Z","citation":{"mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, vol. 72, Springer Nature, 2024, pp. 29–48, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2024;72:29-48. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete and Computational Geometry</i>, vol. 72. Springer Nature, pp. 29–48, 2024.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48."},"department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2204.01076"],"pmid":["39610762"],"isi":["001060727600004"]},"month":"07","oa":1,"has_accepted_license":"1","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Garber, Alexey","first_name":"Alexey","last_name":"Garber"},{"first_name":"Mohadese","last_name":"Ghafari","full_name":"Ghafari, Mohadese"},{"first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss","orcid":"0000-0002-1780-2689","full_name":"Heiss, Teresa"},{"first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"das_tickbox":"0","intvolume":"        72","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"supplementarymaterial":"yes","file_date_updated":"2024-07-22T09:43:19Z","date_published":"2024-07-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","ddc":["510"],"date_updated":"2026-10-02T11:45:50Z","title":"On angles in higher order Brillouin tessellations and related tilings in the plane","arxiv":1,"language":[{"iso":"eng"}],"_id":"14345","isi":1,"pmid":1,"day":"01"},{"oa":1,"has_accepted_license":"1","author":[{"orcid":"0000-0001-7841-0091","full_name":"Kourimska, Hana","last_name":"Kourimska","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana"},{"last_name":"Lieutier","first_name":"André","full_name":"Lieutier, André"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","first_name":"Mathijs"}],"conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry"},"external_id":{"arxiv":["2212.01118"]},"month":"06","year":"2024","date_created":"2024-06-16T22:01:06Z","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: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>","short":"H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","apa":"Kourimska, H., Lieutier, A., &#38; Wintraecken, M. (2024). The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>","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.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>.","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 <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>.","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.","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 <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293."},"department":[{"_id":"HeEd"}],"status":"public","language":[{"iso":"eng"}],"_id":"17144","day":"01","title":"The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms","arxiv":1,"date_updated":"2026-10-02T11:46:39Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2024-06-01T00:00:00Z","ddc":["510"],"das_tickbox":"0","intvolume":"       293","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"supplementarymaterial":"no","file_date_updated":"2024-06-17T08:33:40Z","researchdata_availability":"no","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses","grant_number":"M03073"}],"publication":"40th International Symposium on Computational Geometry","abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","article_processing_charge":"No","doi":"10.4230/LIPIcs.SoCG.2024.69","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.","file":[{"content_type":"application/pdf","access_level":"open_access","date_created":"2024-06-17T08:33:40Z","date_updated":"2024-06-17T08:33:40Z","relation":"main_file","checksum":"b40ff456c19294adb5d9613fcfd751c6","file_name":"2024_LIPICS_Kourimska.pdf","file_size":1612558,"creator":"dernst","file_id":"17150","success":1}],"oa_version":"Published Version","publication_status":"published","alternative_title":["LIPIcs"],"volume":293,"article_number":"69","type":"conference","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.69","ec_funded":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"scopus_import":"1"},{"arxiv":1,"title":"Maximum Betti numbers of Čech complexes","day":"01","_id":"17146","language":[{"iso":"eng"}],"file_date_updated":"2024-06-17T08:46:33Z","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"intvolume":"       293","supplementarymaterial":"no","das_tickbox":"0","date_updated":"2026-10-02T11:48:01Z","ddc":["510"],"date_published":"2024-06-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","month":"06","external_id":{"arxiv":["2310.14801"]},"conference":{"end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","start_date":"2024-06-11"},"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach","full_name":"Pach, János"}],"has_accepted_license":"1","oa":1,"status":"public","department":[{"_id":"HeEd"}],"date_created":"2024-06-16T22:01:06Z","citation":{"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.","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>.","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>","short":"H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","apa":"Edelsbrunner, H., &#38; Pach, J. (2024). Maximum Betti numbers of Čech complexes. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>"},"year":"2024","article_number":"53","alternative_title":["LIPIcs"],"volume":293,"publication_status":"published","oa_version":"Published Version","scopus_import":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","ec_funded":1,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.53","type":"conference","article_processing_charge":"No","quality_controlled":"1","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."}],"publication":"40th International Symposium on Computational Geometry","researchdata_availability":"no","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF"}],"file":[{"success":1,"file_id":"17152","creator":"dernst","checksum":"5442d44fb89d77477a87668d6e61aac9","file_size":766562,"file_name":"2024_LIPICS_Edelsbrunner.pdf","relation":"main_file","date_created":"2024-06-17T08:46:33Z","date_updated":"2024-06-17T08:46:33Z","access_level":"open_access","content_type":"application/pdf"}],"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.","related_material":{"record":[{"relation":"later_version","status":"public","id":"20657"}]},"doi":"10.4230/LIPIcs.SoCG.2024.53"},{"_id":"17145","language":[{"iso":"eng"}],"day":"01","title":"Grid peeling of parabolas","arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2024-06-01T00:00:00Z","ddc":["510"],"date_updated":"2026-10-02T11:47:18Z","das_tickbox":"0","file_date_updated":"2024-06-17T08:40:04Z","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773164"]},"supplementarymaterial":"no","intvolume":"       293","oa":1,"has_accepted_license":"1","conference":{"end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","start_date":"2024-06-11"},"author":[{"full_name":"Rote, Günter","last_name":"Rote","first_name":"Günter"},{"full_name":"Rüber, Moritz","last_name":"Rüber","first_name":"Moritz"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"external_id":{"arxiv":["2402.15787"]},"month":"06","citation":{"chicago":"Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>.","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.","ieee":"G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","mla":"Rote, Günter, et al. “Grid Peeling of Parabolas.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>.","ama":"Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>","short":"G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","apa":"Rote, G., Rüber, M., &#38; Saghafian, M. (2024). Grid peeling of parabolas. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>"},"date_created":"2024-06-16T22:01:06Z","year":"2024","department":[{"_id":"HeEd"}],"status":"public","publication_status":"published","oa_version":"Published Version","article_number":"76","alternative_title":["LIPIcs"],"volume":293,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.76","type":"conference","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","publication":"40th International Symposium on Computational Geometry","researchdata_availability":"no","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"}],"quality_controlled":"1","article_processing_charge":"No","doi":"10.4230/LIPIcs.SoCG.2024.76","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.","file":[{"date_updated":"2024-06-17T08:40:04Z","date_created":"2024-06-17T08:40:04Z","content_type":"application/pdf","access_level":"open_access","file_name":"2024_LIPICS_Rote.pdf","creator":"dernst","file_size":1430896,"checksum":"fbad1de06383a6b7e8a1cb3e8c7205ce","success":1,"file_id":"17151","relation":"main_file"}]},{"language":[{"iso":"eng"}],"_id":"17891","day":"01","pmid":1,"isi":1,"issue":"8","title":"Understanding higher-order interactions in information space","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2026-10-02T11:50:23Z","date_published":"2024-08-01T00:00:00Z","ddc":["510"],"das_tickbox":"1","file_date_updated":"2024-09-09T09:01:12Z","supplementarymaterial":"yes","intvolume":"        26","publication_identifier":{"eissn":["1099-4300"]},"has_accepted_license":"1","oa":1,"author":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Ölsböck","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina","full_name":"Ölsböck, Katharina","orcid":"0000-0002-4672-8297"},{"first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner","full_name":"Wagner, Hubert","orcid":"0009-0009-9111-8429"}],"external_id":{"isi":["001305543500001"],"pmid":["39202107"]},"month":"08","date_created":"2024-09-08T22:01:11Z","citation":{"ista":"Edelsbrunner H, Ölsböck K, Wagner H. 2024. Understanding higher-order interactions in information space. Entropy. 26(8), 637.","chicago":"Edelsbrunner, Herbert, Katharina Ölsböck, and Hubert Wagner. “Understanding Higher-Order Interactions in Information Space.” <i>Entropy</i>. MDPI, 2024. <a href=\"https://doi.org/10.3390/e26080637\">https://doi.org/10.3390/e26080637</a>.","ieee":"H. Edelsbrunner, K. Ölsböck, and H. Wagner, “Understanding higher-order interactions in information space,” <i>Entropy</i>, vol. 26, no. 8. MDPI, 2024.","mla":"Edelsbrunner, Herbert, et al. “Understanding Higher-Order Interactions in Information Space.” <i>Entropy</i>, vol. 26, no. 8, 637, MDPI, 2024, doi:<a href=\"https://doi.org/10.3390/e26080637\">10.3390/e26080637</a>.","ama":"Edelsbrunner H, Ölsböck K, Wagner H. Understanding higher-order interactions in information space. <i>Entropy</i>. 2024;26(8). doi:<a href=\"https://doi.org/10.3390/e26080637\">10.3390/e26080637</a>","apa":"Edelsbrunner, H., Ölsböck, K., &#38; Wagner, H. (2024). Understanding higher-order interactions in information space. <i>Entropy</i>. MDPI. <a href=\"https://doi.org/10.3390/e26080637\">https://doi.org/10.3390/e26080637</a>","short":"H. Edelsbrunner, K. Ölsböck, H. Wagner, Entropy 26 (2024)."},"year":"2024","department":[{"_id":"HeEd"}],"status":"public","publication_status":"published","oa_version":"Published Version","article_number":"637","volume":26,"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).","fulldoi":"https://doi.org/10.3390/e26080637","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publisher":"MDPI","scopus_import":"1","researchdata_availability":"yes","publication":"Entropy","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."}],"article_type":"original","quality_controlled":"1","article_processing_charge":"Yes","doi":"10.3390/e26080637","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.","related_material":{"link":[{"relation":"software","url":"https://git.ista.ac.at/katharina.oelsboeck/wrap_2_3-public/"}]},"file":[{"relation":"main_file","file_id":"17948","success":1,"file_name":"2024_Entropy_Edelsbrunner.pdf","creator":"dernst","file_size":8025139,"checksum":"624a9e2c5b49d6c38b88b0f675467ba3","access_level":"open_access","content_type":"application/pdf","date_updated":"2024-09-09T09:01:12Z","date_created":"2024-09-09T09:01:12Z"}]},{"publisher":"Springer Nature","scopus_import":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.13201","open_access":"1"}],"type":"conference","ec_funded":1,"fulldoi":"https://doi.org/10.1007/978-3-031-49272-3_23","volume":14465,"alternative_title":["LNCS"],"page":"339-346","oa_version":"Preprint","publication_status":"published","doi":"10.1007/978-3-031-49272-3_23","related_material":{"record":[{"relation":"later_version","status":"public","id":"21253"}]},"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.","quality_controlled":"1","article_processing_charge":"No","publication":"31st International Symposium on Graph Drawing and Network Visualization","researchdata_availability":"no","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"abstract":[{"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.","lang":"eng"}],"das_tickbox":"0","supplementarymaterial":"no","publication_identifier":{"eisbn":["9783031492723"],"eissn":["1611-3349"],"issn":["0302-9743"],"isbn":["9783031492716"]},"intvolume":"     14465","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2024-01-01T00:00:00Z","date_updated":"2026-10-02T11:50:59Z","title":"Decomposition of geometric graphs into star-forests","arxiv":1,"_id":"15012","language":[{"iso":"eng"}],"isi":1,"day":"01","status":"public","year":"2024","date_created":"2024-02-18T23:01:03Z","citation":{"ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” in <i>31st International Symposium on Graph Drawing and Network Visualization</i>, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp. 339–346.","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” In <i>31st International Symposium on Graph Drawing and Network Visualization</i>, 14465:339–46. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>.","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.","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>31st International Symposium on Graph Drawing and Network Visualization</i>, vol. 14465, Springer Nature, 2024, pp. 339–46, doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>.","apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2024). Decomposition of geometric graphs into star-forests. In <i>31st International Symposium on Graph Drawing and Network Visualization</i> (Vol. 14465, pp. 339–346). Isola delle Femmine, Palermo, Italy: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>","short":"J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. In: <i>31st International Symposium on Graph Drawing and Network Visualization</i>. Vol 14465. Springer Nature; 2024:339-346. doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>"},"department":[{"_id":"HeEd"}],"external_id":{"isi":["001207939600023"],"arxiv":["2306.13201"]},"month":"01","oa":1,"author":[{"full_name":"Pach, János","first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"},{"full_name":"Schnider, Patrick","last_name":"Schnider","first_name":"Patrick"}],"conference":{"start_date":"2023-09-20","location":"Isola delle Femmine, Palermo, Italy","name":"GD: Graph Drawing and Network Visualization","end_date":"2023-09-22"}},{"doi":"10.1137/22M1489071","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.","quality_controlled":"1","article_processing_charge":"No","researchdata_availability":"no","project":[{"_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","call_identifier":"H2020"},{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"M03073","name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"publication":"SIAM Journal on Discrete Mathematics","article_type":"original","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"}],"publisher":"Society for Industrial and Applied Mathematics","scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2204.01077"}],"type":"journal_article","fulldoi":"https://doi.org/10.1137/22M1489071","ec_funded":1,"volume":38,"page":"1784-1807","corr_author":"1","oa_version":"Preprint","publication_status":"published","status":"public","year":"2024","citation":{"apa":"Edelsbrunner, H., Garber, A., Ghafaris, M., Heiss, T., Saghafiant, M., &#38; Wintraecken, M. (2024). Brillouin zones of integer lattices and their perturbations. <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/22M1489071\">https://doi.org/10.1137/22M1489071</a>","short":"H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, M. Wintraecken, SIAM Journal on Discrete Mathematics 38 (2024) 1784–1807.","ama":"Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M. Brillouin zones of integer lattices and their perturbations. <i>SIAM Journal on Discrete Mathematics</i>. 2024;38(2):1784-1807. doi:<a href=\"https://doi.org/10.1137/22M1489071\">10.1137/22M1489071</a>","mla":"Edelsbrunner, Herbert, et al. “Brillouin Zones of Integer Lattices and Their Perturbations.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 38, no. 2, Society for Industrial and Applied Mathematics, 2024, pp. 1784–807, doi:<a href=\"https://doi.org/10.1137/22M1489071\">10.1137/22M1489071</a>.","ieee":"H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, and M. Wintraecken, “Brillouin zones of integer lattices and their perturbations,” <i>SIAM Journal on Discrete Mathematics</i>, 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.” <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics, 2024. <a href=\"https://doi.org/10.1137/22M1489071\">https://doi.org/10.1137/22M1489071</a>."},"date_created":"2024-06-30T22:01:05Z","department":[{"_id":"HeEd"}],"external_id":{"arxiv":["2204.01077"],"isi":["001292728600001"]},"month":"06","oa":1,"author":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"full_name":"Ghafaris, Mohadese","last_name":"Ghafaris","first_name":"Mohadese"},{"full_name":"Heiss, Teresa","orcid":"0000-0002-1780-2689","first_name":"Teresa","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saghafiant","first_name":"Morteza","full_name":"Saghafiant, Morteza"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken"}],"das_tickbox":"0","intvolume":"        38","publication_identifier":{"issn":["0895-4801"]},"supplementarymaterial":"no","date_published":"2024-06-07T00:00:00Z","date_updated":"2026-10-02T11:49:10Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"Brillouin zones of integer lattices and their perturbations","arxiv":1,"_id":"17190","language":[{"iso":"eng"}],"isi":1,"issue":"2","day":"07"},{"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.","doi":"10.4230/LIPIcs.GD.2024.3","file":[{"relation":"main_file","file_name":"2024_LIPIcs_CultreradiMontesano.pdf","checksum":"5f9b35e115c3d375e99be78da9054cb4","creator":"dernst","file_size":908541,"success":1,"file_id":"18560","content_type":"application/pdf","access_level":"open_access","date_updated":"2024-11-18T07:49:25Z","date_created":"2024-11-18T07:49:25Z"}],"abstract":[{"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.","lang":"eng"}],"publication":"32nd International Symposium on Graph Drawing and Network Visualization","researchdata_availability":"no","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"article_processing_charge":"Yes","quality_controlled":"1","type":"conference","fulldoi":"https://doi.org/10.4230/LIPIcs.GD.2024.3","ec_funded":1,"scopus_import":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"oa_version":"Published Version","publication_status":"published","corr_author":"1","volume":320,"alternative_title":["LIPIcs"],"article_number":"3","department":[{"_id":"HeEd"}],"OA_type":"gold","year":"2024","date_created":"2024-11-17T23:01:47Z","citation":{"chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">https://doi.org/10.4230/LIPIcs.GD.2024.3</a>.","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.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vienna, Austria, 2024, vol. 320.","mla":"Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">10.4230/LIPIcs.GD.2024.3</a>.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: <i>32nd International Symposium on Graph Drawing and Network Visualization</i>. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">10.4230/LIPIcs.GD.2024.3</a>","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.","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In <i>32nd International Symposium on Graph Drawing and Network Visualization</i> (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">https://doi.org/10.4230/LIPIcs.GD.2024.3</a>"},"status":"public","author":[{"first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"},{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov","first_name":"Ondrej","orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej"},{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"conference":{"name":"GD: Graph Drawing and Network Visualization","end_date":"2024-09-20","location":"Vienna, Austria","start_date":"2024-09-18"},"oa":1,"has_accepted_license":"1","month":"10","external_id":{"isi":["001540278400001"],"arxiv":["2403.10204"]},"date_updated":"2026-10-02T11:52:23Z","date_published":"2024-10-28T00:00:00Z","ddc":["510"],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","supplementarymaterial":"no","intvolume":"       320","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773430"]},"file_date_updated":"2024-11-18T07:49:25Z","das_tickbox":"0","OA_place":"publisher","isi":1,"day":"28","_id":"18556","language":[{"iso":"eng"}],"arxiv":1,"title":"The Euclidean MST-ratio for bi-colored lattices"},{"_id":"18097","language":[{"iso":"eng"}],"day":"06","title":"The ultimate frontier: An optimality construction for homotopy inference (media exposition)","ddc":["000"],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2026-10-02T11:51:50Z","date_published":"2024-06-06T00:00:00Z","das_tickbox":"0","file_date_updated":"2024-09-19T10:30:37Z","publication_identifier":{"isbn":["9783959773164"]},"supplementarymaterial":"yes","intvolume":"       293","oa":1,"has_accepted_license":"1","conference":{"end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","start_date":"2024-06-11"},"author":[{"last_name":"Attali","first_name":"Dominique","full_name":"Attali, Dominique"},{"first_name":"Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","last_name":"Kourimska","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"},{"id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","last_name":"Ghosh","first_name":"Ishika","full_name":"Ghosh, Ishika"},{"full_name":"Lieutier, Andre","last_name":"Lieutier","first_name":"Andre"},{"full_name":"Stephenson, Elizabeth R","orcid":"0000-0002-6862-208X","first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","last_name":"Stephenson"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs"}],"month":"06","date_created":"2024-09-19T10:29:48Z","citation":{"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 <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>.","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 <i>et al.</i>, “The ultimate frontier: An optimality construction for homotopy inference (media exposition),” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","mla":"Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>.","ama":"Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality construction for homotopy inference (media exposition). In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). The ultimate frontier: An optimality construction for homotopy inference (media exposition). In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>","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."},"year":"2024","department":[{"_id":"HeEd"}],"status":"public","corr_author":"1","publication_status":"published","oa_version":"Published Version","article_number":"87","volume":293,"alternative_title":["LIPIcs"],"ec_funded":1,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.87","type":"conference","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","researchdata_availability":"no","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073","name":"Learning and triangulating manifolds via collapses"}],"publication":"40th International Symposium on Computational Geometry","abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","article_processing_charge":"Yes","doi":"10.4230/LIPIcs.SoCG.2024.87","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.","file":[{"access_level":"open_access","content_type":"application/pdf","date_updated":"2024-09-19T10:30:37Z","date_created":"2024-09-19T10:30:37Z","relation":"main_file","success":1,"file_id":"18098","checksum":"9355c2e60b8ec285e1b22719c5b73f1a","creator":"dernst","file_size":3507177,"file_name":"2024_LIPICs_Attali.pdf"}]},{"file_date_updated":"2025-01-09T07:39:41Z","supplementarymaterial":"no","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"intvolume":"         8","das_tickbox":"0","OA_place":"publisher","date_updated":"2026-10-02T11:52:56Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2024-10-01T00:00:00Z","ddc":["000"],"title":"Geometric characterization of the persistence of 1D maps","day":"01","pmid":1,"language":[{"iso":"eng"}],"_id":"13182","status":"public","department":[{"_id":"HeEd"}],"citation":{"short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 1101–1119.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. 2024;8:1101-1119. doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 1101–19, doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 1101–1119, 2024.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>.","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."},"date_created":"2023-07-02T22:00:44Z","OA_type":"hybrid","year":"2024","month":"10","external_id":{"pmid":["39678706"]},"author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","last_name":"Biswas","first_name":"Ranita","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890"},{"full_name":"Cultrera Di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera Di Montesano"},{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"has_accepted_license":"1","oa":1,"scopus_import":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publisher":"Springer Nature","ec_funded":1,"fulldoi":"https://doi.org/10.1007/s41468-023-00126-9","type":"journal_article","page":"1101-1119","volume":8,"publication_status":"published","oa_version":"Published Version","corr_author":"1","file":[{"access_level":"open_access","content_type":"application/pdf","date_updated":"2025-01-09T07:39:41Z","date_created":"2025-01-09T07:39:41Z","relation":"main_file","success":1,"file_id":"18783","creator":"dernst","file_size":476896,"file_name":"2024_JourApplCompTopo_Biswas.pdf","checksum":"d493df5088c222b88d9ca46b623ad0ee"}],"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.","related_material":{"record":[{"id":"11660","status":"public","relation":"earlier_version"},{"id":"15094","status":"public","relation":"dissertation_contains"}]},"doi":"10.1007/s41468-023-00126-9","article_processing_charge":"Yes (via OA deal)","quality_controlled":"1","article_type":"original","abstract":[{"text":"We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram.","lang":"eng"}],"researchdata_availability":"no","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887","name":"Persistent Homology, Algorithms and Stochastic Geometry"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"}],"publication":"Journal of Applied and Computational Topology"},{"publisher":"Springer Nature","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"scopus_import":"1","type":"journal_article","ec_funded":1,"fulldoi":"https://doi.org/10.1007/s41468-024-00173-w","volume":8,"page":"557-578","corr_author":"1","oa_version":"Published Version","publication_status":"published","file":[{"date_created":"2025-04-23T08:01:36Z","date_updated":"2025-04-23T08:01:36Z","content_type":"application/pdf","access_level":"open_access","checksum":"0ee15c1493a6413cf356ab2f32c81a9e","creator":"dernst","file_size":522831,"file_name":"2024_JourApplCompTopo_BiswasRa.pdf","file_id":"19612","success":1,"relation":"main_file"}],"doi":"10.1007/s41468-024-00173-w","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"11658"}]},"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.","quality_controlled":"1","article_processing_charge":"Yes (via OA deal)","publication":"Journal of Applied and Computational Topology","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"researchdata_availability":"no","article_type":"original","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."}],"das_tickbox":"0","OA_place":"publisher","intvolume":"         8","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"supplementarymaterial":"no","file_date_updated":"2025-04-23T08:01:36Z","date_published":"2024-09-01T00:00:00Z","ddc":["510"],"date_updated":"2026-10-02T11:55:12Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","language":[{"iso":"eng"}],"_id":"15380","pmid":1,"day":"01","status":"public","OA_type":"hybrid","year":"2024","citation":{"mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 557–78, doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. 2024;8:557-578. doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>.","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,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 557–578, 2024."},"date_created":"2024-05-12T22:01:03Z","department":[{"_id":"HeEd"}],"external_id":{"pmid":["39308789"]},"month":"09","has_accepted_license":"1","oa":1,"author":[{"full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita"},{"last_name":"Cultrera Di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","full_name":"Cultrera Di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza"}]},{"arxiv":1,"title":"Odd-sunflowers","day":"01","issue":"8","isi":1,"language":[{"iso":"eng"}],"_id":"15247","file_date_updated":"2025-01-09T08:37:20Z","publication_identifier":{"eissn":["1096-0899"],"issn":["0097-3165"]},"supplementarymaterial":"yes","intvolume":"       206","das_tickbox":"1","OA_place":"publisher","date_published":"2024-08-01T00:00:00Z","date_updated":"2026-10-02T11:54:47Z","ddc":["510"],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","month":"08","external_id":{"arxiv":["2310.16701"],"isi":["001217739200001"]},"author":[{"first_name":"Peter","last_name":"Frankl","full_name":"Frankl, Peter"},{"last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János","full_name":"Pach, János"},{"last_name":"Pálvölgyi","first_name":"Dömötör","full_name":"Pálvölgyi, Dömötör"}],"oa":1,"has_accepted_license":"1","status":"public","department":[{"_id":"HeEd"}],"date_created":"2024-03-31T22:01:11Z","citation":{"ista":"Frankl P, Pach J, Pálvölgyi D. 2024. Odd-sunflowers. Journal of Combinatorial Theory, Series A. 206(8), 105889.","chicago":"Frankl, Peter, János Pach, and Dömötör Pálvölgyi. “Odd-Sunflowers.” <i>Journal of Combinatorial Theory, Series A</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">https://doi.org/10.1016/j.jcta.2024.105889</a>.","ieee":"P. Frankl, J. Pach, and D. Pálvölgyi, “Odd-sunflowers,” <i>Journal of Combinatorial Theory, Series A</i>, vol. 206, no. 8. Elsevier, 2024.","ama":"Frankl P, Pach J, Pálvölgyi D. Odd-sunflowers. <i>Journal of Combinatorial Theory, Series A</i>. 2024;206(8). doi:<a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">10.1016/j.jcta.2024.105889</a>","apa":"Frankl, P., Pach, J., &#38; Pálvölgyi, D. (2024). Odd-sunflowers. <i>Journal of Combinatorial Theory, Series A</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">https://doi.org/10.1016/j.jcta.2024.105889</a>","short":"P. Frankl, J. Pach, D. Pálvölgyi, Journal of Combinatorial Theory, Series A 206 (2024).","mla":"Frankl, Peter, et al. “Odd-Sunflowers.” <i>Journal of Combinatorial Theory, Series A</i>, vol. 206, no. 8, 105889, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.jcta.2024.105889\">10.1016/j.jcta.2024.105889</a>."},"year":"2024","OA_type":"hybrid","article_number":"105889","volume":206,"publication_status":"published","oa_version":"Published Version","corr_author":"1","scopus_import":"1","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","short":"CC BY-NC (4.0)","name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)","image":"/images/cc_by_nc.png"},"publisher":"Elsevier","fulldoi":"https://doi.org/10.1016/j.jcta.2024.105889","license":"https://creativecommons.org/licenses/by-nc/4.0/","type":"journal_article","dataavailabilitystatement":"No data was used for the research described in the article.","article_processing_charge":"No","quality_controlled":"1","article_type":"original","abstract":[{"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.","lang":"eng"}],"publication":"Journal of Combinatorial Theory, Series A","researchdata_availability":"not applicable","file":[{"relation":"main_file","creator":"dernst","file_size":366029,"checksum":"ffc29d65e712849f0d31009271e06a63","file_name":"2024_JourCombiTheoryA_Frankl.pdf","file_id":"18791","success":1,"content_type":"application/pdf","access_level":"open_access","date_updated":"2025-01-09T08:37:20Z","date_created":"2025-01-09T08:37:20Z"}],"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.","doi":"10.1016/j.jcta.2024.105889"},{"supervisor":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"}],"doi":"10.15479/at:ista:18667","related_material":{"record":[{"id":"10828","relation":"part_of_dissertation","status":"public"},{"relation":"part_of_dissertation","status":"public","id":"11440"},{"id":"9345","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"18673"}]},"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.","file":[{"relation":"main_file","success":1,"file_id":"18686","creator":"theiss","file_name":"Teresa_Heiss_PhD_Thesis_final.pdf","checksum":"247bb057aed2fba1cd4711917aaa2d77","file_size":7752253,"access_level":"open_access","content_type":"application/pdf","date_created":"2024-12-19T10:24:46Z","date_updated":"2024-12-19T10:24:46Z"},{"date_created":"2024-12-19T10:24:50Z","date_updated":"2024-12-19T10:24:50Z","access_level":"closed","content_type":"application/zip","file_id":"18687","file_name":"PhD_Thesis.zip","creator":"theiss","file_size":17197731,"checksum":"9648b45c07a008ee11a07f99856a139d","relation":"source_file"}],"researchdata_availability":"no","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"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."}],"article_processing_charge":"No","ec_funded":1,"fulldoi":"https://doi.org/10.15479/at:ista:18667","type":"dissertation","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publisher":"Institute of Science and Technology Austria","corr_author":"1","publication_status":"published","oa_version":"Published Version","page":"111","alternative_title":["ISTA Thesis"],"citation":{"mla":"Heiss, Teresa. <i>New Methods for Applying Topological Data Analysis to Materials Science</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:18667\">10.15479/at:ista:18667</a>.","ama":"Heiss T. New methods for applying topological data analysis to materials science. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:18667\">10.15479/at:ista:18667</a>","apa":"Heiss, T. (2024). <i>New methods for applying topological data analysis to materials science</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18667\">https://doi.org/10.15479/at:ista:18667</a>","short":"T. Heiss, New Methods for Applying Topological Data Analysis to Materials Science, Institute of Science and Technology Austria, 2024.","ista":"Heiss T. 2024. New methods for applying topological data analysis to materials science. Institute of Science and Technology Austria.","chicago":"Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials Science.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:18667\">https://doi.org/10.15479/at:ista:18667</a>.","ieee":"T. Heiss, “New methods for applying topological data analysis to materials science,” Institute of Science and Technology Austria, 2024."},"date_created":"2024-12-17T16:17:55Z","year":"2024","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"status":"public","has_accepted_license":"1","degree_awarded":"PhD","oa":1,"author":[{"first_name":"Teresa","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","full_name":"Heiss, Teresa","orcid":"0000-0002-1780-2689"}],"month":"12","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","ddc":["514","516","004"],"date_updated":"2026-10-02T11:55:42Z","date_published":"2024-12-17T00:00:00Z","OA_place":"publisher","das_tickbox":"0","file_date_updated":"2024-12-19T10:24:50Z","publication_identifier":{"issn":["2663-337X"],"isbn":["978-3-99078-052-7"]},"supplementarymaterial":"no","_id":"18667","language":[{"iso":"eng"}],"day":"17","doi_confirm":"1","title":"New methods for applying topological data analysis to materials science","keyword":["persistent homology","topological data analysis","periodic","crystalline materials","images","fingerprint"]},{"author":[{"last_name":"Cultrera di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"}],"degree_awarded":"PhD","has_accepted_license":"1","oa":1,"month":"03","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"year":"2024","date_created":"2024-03-08T15:28:10Z","citation":{"chicago":"Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete Geometric Structures.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:15094\">https://doi.org/10.15479/at:ista:15094</a>.","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.","ama":"Cultrera di Montesano S. Persistence and Morse theory for discrete geometric structures. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:15094\">10.15479/at:ista:15094</a>","short":"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). <i>Persistence and Morse theory for discrete geometric structures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:15094\">https://doi.org/10.15479/at:ista:15094</a>","mla":"Cultrera di Montesano, Sebastiano. <i>Persistence and Morse Theory for Discrete Geometric Structures</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:15094\">10.15479/at:ista:15094</a>."},"status":"public","day":"08","_id":"15094","language":[{"iso":"eng"}],"title":"Persistence and Morse theory for discrete geometric structures","doi_confirm":"1","ddc":["514","500","516"],"date_published":"2024-03-08T00:00:00Z","date_updated":"2026-10-02T11:57:31Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_identifier":{"issn":["2663-337X"]},"supplementarymaterial":"no","file_date_updated":"2024-03-14T14:14:35Z","das_tickbox":"0","OA_place":"publisher","abstract":[{"lang":"eng","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"}],"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"name":"Persistent Homology, Algorithms and Stochastic Geometry","grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"researchdata_availability":"no","article_processing_charge":"No","related_material":{"record":[{"id":"15090","status":"public","relation":"part_of_dissertation"},{"id":"11658","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"11660"},{"id":"13182","status":"public","relation":"part_of_dissertation"},{"id":"15091","status":"public","relation":"part_of_dissertation"},{"id":"15093","relation":"part_of_dissertation","status":"public"}]},"doi":"10.15479/at:ista:15094","supervisor":[{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"}],"file":[{"success":1,"file_id":"15112","file_size":4106872,"checksum":"1e468bfa42a7dcf04d89f4dadc621c87","file_name":"Thesis Sebastiano.pdf","creator":"scultrer","relation":"main_file","date_created":"2024-03-14T08:55:07Z","date_updated":"2024-03-14T08:55:07Z","access_level":"open_access","content_type":"application/pdf"},{"creator":"scultrer","file_name":"Thesis (1).zip","file_size":4746234,"checksum":"bcbd213490f5a7e68855a092bbce93f1","file_id":"15113","relation":"source_file","date_updated":"2024-03-14T14:14:35Z","date_created":"2024-03-14T08:56:24Z","content_type":"application/zip","access_level":"closed"}],"oa_version":"Published Version","publication_status":"published","corr_author":"1","alternative_title":["ISTA Thesis"],"page":"108","license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","type":"dissertation","fulldoi":"https://doi.org/10.15479/at:ista:15094","ec_funded":1,"publisher":"Institute of Science and Technology Austria","tmp":{"image":"/images/cc_by_nc_sa.png","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","short":"CC BY-NC-SA (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode"}}]