[{"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","_id":"15094","status":"public","ddc":["514","500","516"],"title":"Persistence and Morse theory for discrete geometric structures","oa_version":"Published Version","file":[{"file_size":4106872,"content_type":"application/pdf","creator":"scultrer","access_level":"open_access","file_name":"Thesis Sebastiano.pdf","checksum":"1e468bfa42a7dcf04d89f4dadc621c87","success":1,"date_updated":"2024-03-14T08:55:07Z","date_created":"2024-03-14T08:55:07Z","relation":"main_file","file_id":"15112"},{"date_created":"2024-03-14T08:56:24Z","date_updated":"2024-03-14T14:14:35Z","checksum":"bcbd213490f5a7e68855a092bbce93f1","relation":"source_file","file_id":"15113","content_type":"application/zip","file_size":4746234,"creator":"scultrer","file_name":"Thesis (1).zip","access_level":"closed"}],"type":"dissertation","alternative_title":["ISTA Thesis"],"abstract":[{"text":"Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in\r\ndiscrete geometry that have captivated mathematicians for centuries, if not millennia. This\r\nthesis seeks to cast new light on these structures by illustrating specific instances where a\r\ntopological perspective, specifically through discrete Morse theory and persistent homology,\r\nprovides valuable insights.\r\n\r\nAt first glance, the topology of these geometric objects might seem uneventful: point sets\r\nessentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which\r\nis a contractible space, and the topology of a network primarily involves the enumeration\r\nof connected components and cycles within the network. However, beneath this apparent\r\nsimplicity, there lies an array of intriguing structures, a small subset of which will be uncovered\r\nin this thesis.\r\n\r\nFocused on three case studies, each addressing one of the mentioned objects, this work\r\nwill showcase connections that intertwine topology with diverse fields such as combinatorial\r\ngeometry, algorithms and data structures, and emerging applications like spatial biology.\r\n\r\n","lang":"eng"}],"citation":{"ama":"Cultrera di Montesano S. Persistence and Morse theory for discrete geometric structures. 2024. doi:10.15479/at:ista:15094","apa":"Cultrera di Montesano, S. (2024). Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:15094","ieee":"S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric structures,” Institute of Science and Technology Austria, 2024.","ista":"Cultrera di Montesano S. 2024. Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria.","short":"S. Cultrera di Montesano, Persistence and Morse Theory for Discrete Geometric Structures, Institute of Science and Technology Austria, 2024.","mla":"Cultrera di Montesano, Sebastiano. Persistence and Morse Theory for Discrete Geometric Structures. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:15094.","chicago":"Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete Geometric Structures.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:15094."},"page":"108","date_published":"2024-03-08T00:00:00Z","day":"08","has_accepted_license":"1","article_processing_charge":"No","year":"2024","publication_status":"published","publisher":"Institute of Science and Technology Austria","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"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"}],"related_material":{"record":[{"id":"11660","relation":"part_of_dissertation","status":"public"},{"id":"11658","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"13182"},{"id":"15090","relation":"part_of_dissertation","status":"public"},{"relation":"part_of_dissertation","status":"public","id":"15091"},{"id":"15093","relation":"part_of_dissertation","status":"public"}]},"date_created":"2024-03-08T15:28:10Z","date_updated":"2024-03-20T09:36:57Z","file_date_updated":"2024-03-14T14:14:35Z","ec_funded":1,"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","oa":1,"tmp":{"name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","image":"/images/cc_by_nc_sa.png","short":"CC BY-NC-SA (4.0)"},"project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"The Wittgenstein Prize","call_identifier":"FWF"},{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"doi":"10.15479/at:ista:15094","supervisor":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert"}],"degree_awarded":"PhD","language":[{"iso":"eng"}],"month":"03","publication_identifier":{"issn":["2663 - 337X"]}},{"article_processing_charge":"No","day":"04","citation":{"chicago":"Cultrera di Montesano, Sebastiano, Herbert Edelsbrunner, Monika H Henzinger, and Lara Ost. “Dynamically Maintaining the Persistent Homology of Time Series.” In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), edited by David P. Woodruff, 243–95. Society for Industrial and Applied Mathematics, 2024. https://doi.org/10.1137/1.9781611977912.11.","mla":"Cultrera di Montesano, Sebastiano, et al. “Dynamically Maintaining the Persistent Homology of Time Series.” Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), edited by David P. Woodruff, Society for Industrial and Applied Mathematics, 2024, pp. 243–95, doi:10.1137/1.9781611977912.11.","short":"S. Cultrera di Montesano, H. Edelsbrunner, M.H. Henzinger, L. Ost, in:, D.P. Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Society for Industrial and Applied Mathematics, 2024, pp. 243–295.","ista":"Cultrera di Montesano S, Edelsbrunner H, Henzinger MH, Ost L. 2024. Dynamically maintaining the persistent homology of time series. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SODA: Symposium on Discrete Algorigthms, 243–295.","apa":"Cultrera di Montesano, S., Edelsbrunner, H., Henzinger, M. H., & Ost, L. (2024). Dynamically maintaining the persistent homology of time series. In D. P. Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (pp. 243–295). Alexandria, VA, USA: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611977912.11","ieee":"S. Cultrera di Montesano, H. Edelsbrunner, M. H. Henzinger, and L. Ost, “Dynamically maintaining the persistent homology of time series,” in Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Alexandria, VA, USA, 2024, pp. 243–295.","ama":"Cultrera di Montesano S, Edelsbrunner H, Henzinger MH, Ost L. Dynamically maintaining the persistent homology of time series. In: Woodruff DP, ed. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society for Industrial and Applied Mathematics; 2024:243-295. doi:10.1137/1.9781611977912.11"},"publication":"Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)","page":"243 - 295","date_published":"2024-01-04T00:00:00Z","type":"conference","abstract":[{"text":"We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.","lang":"eng"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","_id":"15093","status":"public","title":"Dynamically maintaining the persistent homology of time series","oa_version":"Preprint","publication_identifier":{"eisbn":["9781611977912"]},"month":"01","external_id":{"arxiv":["2311.01115"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2311.01115"}],"oa":1,"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"name":"The Wittgenstein Prize","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"The design and evaluation of modern fully dynamic data structures","call_identifier":"H2020","grant_number":"101019564","_id":"bd9ca328-d553-11ed-ba76-dc4f890cfe62"},{"grant_number":"Z00422","_id":"34def286-11ca-11ed-8bc3-da5948e1613c","name":"Wittgenstein Award - Monika Henzinger"},{"name":"Fast Algorithms for a Reactive Network Layer","grant_number":"P33775 ","_id":"bd9e3a2e-d553-11ed-ba76-8aa684ce17fe"}],"quality_controlled":"1","doi":"10.1137/1.9781611977912.11","conference":{"name":"SODA: Symposium on Discrete Algorigthms","start_date":"2024-01-07","location":"Alexandria, VA, USA","end_date":"2024-01-10"},"language":[{"iso":"eng"}],"ec_funded":1,"year":"2024","acknowledgement":"The first and second authors are funded by the European Research Council under the European Union’s Horizon 2020 research and innovation programme, ERC grant no. 788183,“Alpha Shape Theory Extended (Alpha)”, by the Wittgenstein Prize, FWF grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.The third author received funding by the European Research Council under the European Union’s Horizon 2020research and innovation programme, ERC grant no. 101019564, “The Design of Modern Fully Dynamic DataStructures (MoDynStruct)”, and by the Austrian Science Fund through the Wittgenstein Prize with FWF grant no. Z 422-N, and also by FWF grant no. I 5982-N, and by FWF grant no. P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024. The fourth author is funded by the Vienna Graduate School on Computational Optimization, FWF project no. W1260-N35.","editor":[{"full_name":"Woodruff, David P.","first_name":"David P.","last_name":"Woodruff"}],"publisher":"Society for Industrial and Applied Mathematics","department":[{"_id":"HeEd"},{"_id":"MoHe"}],"publication_status":"published","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"15094"}]},"author":[{"full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","last_name":"Cultrera di Montesano"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"id":"540c9bbd-f2de-11ec-812d-d04a5be85630","orcid":"0000-0002-5008-6530","first_name":"Monika H","last_name":"Henzinger","full_name":"Henzinger, Monika H"},{"first_name":"Lara","last_name":"Ost","full_name":"Ost, Lara"}],"date_updated":"2024-03-20T09:36:56Z","date_created":"2024-03-08T10:27:39Z"},{"month":"02","day":"07","article_processing_charge":"No","publication":"arXiv","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"arxiv":["2212.03128"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2212.03128"}],"oa":1,"citation":{"ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv, 2212.03128.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” arXiv. .","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). Chromatic alpha complexes. arXiv.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” ArXiv, 2212.03128.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” ArXiv, n.d."},"date_published":"2024-02-07T00:00:00Z","language":[{"iso":"eng"}],"article_number":"2212.03128","type":"preprint","abstract":[{"text":"Motivated by applications in the medical sciences, we study finite chromatic\r\nsets in Euclidean space from a topological perspective. Based on the persistent\r\nhomology for images, kernels and cokernels, we design provably stable\r\nhomological quantifiers that describe the geometric micro- and macro-structure\r\nof how the color classes mingle. These can be efficiently computed using\r\nchromatic variants of Delaunay and alpha complexes, and code that does these\r\ncomputations is provided.","lang":"eng"}],"license":"https://creativecommons.org/licenses/by/4.0/","year":"2024","_id":"15091","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","status":"public","title":"Chromatic alpha complexes","publication_status":"submitted","department":[{"_id":"HeEd"}],"author":[{"full_name":"Cultrera di Montesano, Sebastiano","first_name":"Sebastiano","last_name":"Cultrera di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"full_name":"Draganov, Ondrej","last_name":"Draganov","first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"related_material":{"record":[{"id":"15094","status":"public","relation":"dissertation_contains"}]},"date_created":"2024-03-08T10:13:59Z","date_updated":"2024-03-20T09:36:56Z","oa_version":"Preprint"},{"ddc":["000"],"title":"Geometric characterization of the persistence of 1D maps","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"13182","file":[{"relation":"main_file","file_id":"13185","checksum":"697249d5d1c61dea4410b9f021b70fce","success":1,"date_created":"2023-07-03T09:41:05Z","date_updated":"2023-07-03T09:41:05Z","access_level":"open_access","file_name":"2023_Journal of Applied and Computational Topology_Biswas.pdf","content_type":"application/pdf","file_size":487355,"creator":"alisjak"}],"oa_version":"Published Version","type":"journal_article","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"}],"article_type":"original","publication":"Journal of Applied and Computational Topology","citation":{"ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2023. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” Journal of Applied and Computational Topology. Springer Nature, 2023.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2023). Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 2023. doi:10.1007/s41468-023-00126-9","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology. Springer Nature, 2023. https://doi.org/10.1007/s41468-023-00126-9.","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology, Springer Nature, 2023, doi:10.1007/s41468-023-00126-9.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology (2023)."},"date_published":"2023-06-17T00:00:00Z","scopus_import":"1","day":"17","article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","publication_status":"epub_ahead","publisher":"Springer Nature","department":[{"_id":"HeEd"}],"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.","year":"2023","date_updated":"2024-03-20T09:36:56Z","date_created":"2023-07-02T22:00:44Z","author":[{"full_name":"Biswas, Ranita","first_name":"Ranita","last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890"},{"full_name":"Cultrera Di Montesano, Sebastiano","first_name":"Sebastiano","last_name":"Cultrera Di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"15094"}]},"file_date_updated":"2023-07-03T09:41:05Z","ec_funded":1,"quality_controlled":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"The Wittgenstein Prize"}],"oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"language":[{"iso":"eng"}],"doi":"10.1007/s41468-023-00126-9","month":"06","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]}},{"department":[{"_id":"HeEd"}],"publisher":"Springer Nature","publication_status":"published","acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria).","year":"2022","volume":67,"date_updated":"2023-08-02T14:31:25Z","date_created":"2022-02-20T23:01:34Z","author":[{"orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","last_name":"Biswas","first_name":"Ranita","full_name":"Biswas, Ranita"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","last_name":"Cultrera Di Montesano","full_name":"Cultrera Di Montesano, Sebastiano"},{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"file_date_updated":"2022-08-02T06:07:55Z","quality_controlled":"1","isi":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"external_id":{"isi":["000752175300002"]},"language":[{"iso":"eng"}],"doi":"10.1007/s00454-022-00371-2","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"month":"04","intvolume":" 67","ddc":["510"],"status":"public","title":"Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics","_id":"10773","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa_version":"Published Version","file":[{"creator":"dernst","file_size":2518111,"content_type":"application/pdf","file_name":"2022_DiscreteCompGeometry_Biswas.pdf","access_level":"open_access","date_updated":"2022-08-02T06:07:55Z","date_created":"2022-08-02T06:07:55Z","success":1,"checksum":"9383d3b70561bacee905e335dc922680","file_id":"11718","relation":"main_file"}],"type":"journal_article","abstract":[{"lang":"eng","text":"The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function."}],"page":"811-842","article_type":"original","citation":{"ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 2022;67:811-842. doi:10.1007/s00454-022-00371-2","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics,” Discrete and Computational Geometry, vol. 67. Springer Nature, pp. 811–842, 2022.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2022). Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00371-2","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 67, 811–842.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 67 (2022) 811–842.","mla":"Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry, vol. 67, Springer Nature, 2022, pp. 811–42, doi:10.1007/s00454-022-00371-2.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature, 2022. https://doi.org/10.1007/s00454-022-00371-2."},"publication":"Discrete and Computational Geometry","date_published":"2022-04-01T00:00:00Z","scopus_import":"1","has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","day":"01"},{"ec_funded":1,"file_date_updated":"2022-07-27T09:30:30Z","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"15094"}]},"author":[{"full_name":"Biswas, Ranita","first_name":"Ranita","last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890"},{"last_name":"Cultrera di Montesano","first_name":"Sebastiano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera di Montesano, Sebastiano"},{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"last_name":"Saghafian","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"date_created":"2022-07-27T09:31:15Z","date_updated":"2024-03-20T09:36:56Z","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. ","year":"2022","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","publication_status":"submitted","month":"07","language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"name":"The Wittgenstein Prize","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"}],"quality_controlled":"1","abstract":[{"lang":"eng","text":"We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. "}],"type":"journal_article","alternative_title":["LIPIcs"],"file":[{"checksum":"95903f9d1649e8e437a967b6f2f64730","date_created":"2022-07-27T09:30:30Z","date_updated":"2022-07-27T09:30:30Z","file_id":"11661","relation":"main_file","creator":"scultrer","content_type":"application/pdf","file_size":564836,"access_level":"open_access","file_name":"window 1.pdf"}],"oa_version":"Submitted Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"11660","title":"A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs","ddc":["510"],"status":"public","has_accepted_license":"1","article_processing_charge":"No","day":"25","date_published":"2022-07-25T00:00:00Z","citation":{"apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (n.d.). A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs,” LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, n.d.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, LIPIcs (n.d.).","mla":"Biswas, Ranita, et al. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik."},"publication":"LIPIcs"},{"type":"journal_article","abstract":[{"text":"The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.","lang":"eng"}],"title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","status":"public","ddc":["510"],"_id":"11658","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Submitted Version","file":[{"file_id":"11659","relation":"main_file","date_updated":"2022-07-27T09:25:53Z","date_created":"2022-07-27T09:25:53Z","checksum":"b2f511e8b1cae5f1892b0cdec341acac","file_name":"D-S-E.pdf","access_level":"open_access","creator":"scultrer","content_type":"application/pdf","file_size":639266}],"has_accepted_license":"1","article_processing_charge":"No","day":"27","citation":{"ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz Zentrum für Informatik.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz Zentrum für Informatik.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz Zentrum für Informatik, n.d.","mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Leibniz International Proceedings on Mathematics, Schloss Dagstuhl - Leibniz Zentrum für Informatik.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz International Proceedings on Mathematics (n.d.)."},"publication":"Leibniz International Proceedings on Mathematics","date_published":"2022-07-27T00:00:00Z","ec_funded":1,"file_date_updated":"2022-07-27T09:25:53Z","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publisher":"Schloss Dagstuhl - Leibniz Zentrum für Informatik","publication_status":"submitted","year":"2022","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.","date_created":"2022-07-27T09:27:34Z","date_updated":"2024-03-20T09:36:56Z","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"15094"}]},"author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890","first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita"},{"full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","last_name":"Cultrera di Montesano"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian"}],"month":"07","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"name":"The Wittgenstein Prize","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"language":[{"iso":"eng"}]},{"ec_funded":1,"abstract":[{"lang":"eng","text":"Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications."}],"type":"preprint","article_number":"2212.03121","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"15094"}]},"author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890","first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita"},{"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","first_name":"Ondrej","last_name":"Draganov","full_name":"Draganov, Ondrej"},{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert"},{"full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","first_name":"Morteza"}],"oa_version":"Preprint","date_created":"2024-03-08T09:54:20Z","date_updated":"2024-03-20T09:36:56Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","_id":"15090","year":"2022","department":[{"_id":"HeEd"}],"publication_status":"submitted","title":"On the size of chromatic Delaunay mosaics","status":"public","article_processing_charge":"No","day":"06","month":"12","date_published":"2022-12-06T00:00:00Z","language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"main_file_link":[{"url":"https://arxiv.org/abs/2212.03121","open_access":"1"}],"citation":{"ama":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv.","apa":"Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). On the size of chromatic Delaunay mosaics. arXiv.","ieee":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” arXiv. .","ista":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv, 2212.03121.","short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, 2212.03121.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, n.d."},"external_id":{"arxiv":["2212.03121"]},"oa":1,"publication":"arXiv","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887","name":"Discretization in Geometry and Dynamics"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF","name":"The Wittgenstein Prize"}]},{"abstract":[{"text":"Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.","lang":"eng"}],"alternative_title":["LIPIcs"],"type":"conference","oa_version":"Published Version","file":[{"file_name":"2021_LIPIcs_Biswas.pdf","access_level":"open_access","content_type":"application/pdf","file_size":727817,"creator":"asandaue","relation":"main_file","file_id":"9611","date_updated":"2021-06-28T13:11:39Z","date_created":"2021-06-28T13:11:39Z","checksum":"22b11a719018b22ecba2471b51f2eb40","success":1}],"intvolume":" 189","status":"public","title":"Counting cells of order-k voronoi tessellations in ℝ3 with morse theory","ddc":["516"],"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","_id":"9604","article_processing_charge":"No","has_accepted_license":"1","day":"02","scopus_import":"1","date_published":"2021-06-02T00:00:00Z","citation":{"ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. In: Leibniz International Proceedings in Informatics. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:10.4230/LIPIcs.SoCG.2021.16","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2021). Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. In Leibniz International Proceedings in Informatics (Vol. 189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2021.16","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Counting cells of order-k voronoi tessellations in ℝ3 with morse theory,” in Leibniz International Proceedings in Informatics, Online, 2021, vol. 189.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2021. Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 16.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.","mla":"Biswas, Ranita, et al. “Counting Cells of Order-k Voronoi Tessellations in ℝ3 with Morse Theory.” Leibniz International Proceedings in Informatics, vol. 189, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:10.4230/LIPIcs.SoCG.2021.16.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Counting Cells of Order-k Voronoi Tessellations in ℝ3 with Morse Theory.” In Leibniz International Proceedings in Informatics, Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.SoCG.2021.16."},"publication":"Leibniz International Proceedings in Informatics","ec_funded":1,"file_date_updated":"2021-06-28T13:11:39Z","article_number":"16","volume":189,"date_updated":"2023-02-23T14:02:28Z","date_created":"2021-06-27T22:01:48Z","author":[{"orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","last_name":"Biswas","first_name":"Ranita","full_name":"Biswas, Ranita"},{"full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","first_name":"Sebastiano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","department":[{"_id":"HeEd"}],"publication_status":"published","year":"2021","publication_identifier":{"isbn":["9783959771849"],"issn":["18688969"]},"month":"06","language":[{"iso":"eng"}],"doi":"10.4230/LIPIcs.SoCG.2021.16","conference":{"end_date":"2021-06-11","start_date":"2021-06-07","location":"Online","name":"SoCG: International Symposium on Computational Geometry"},"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887"}],"quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1}]