[{"isi":1,"language":[{"iso":"eng"}],"type":"journal_article","title":"On the size of chromatic Delaunay mosaics","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"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"article_type":"original","publication_status":"published","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"year":"2026","doi":"10.1007/s00454-025-00778-7","acknowledgement":"The fourth author thanks Boris Aronov for insightful discussions on the size of the overlay of Voronoi tessellations. Open 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.","date_created":"2025-10-12T22:01:26Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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."}],"file":[{"date_updated":"2026-01-05T13:21:20Z","creator":"dernst","file_size":570922,"date_created":"2026-01-05T13:21:20Z","content_type":"application/pdf","success":1,"checksum":"0addb5c1b78142f9fb453bfa04695400","relation":"main_file","access_level":"open_access","file_id":"20952","file_name":"2026_DiscreteCompGeom_Biswas.pdf"}],"external_id":{"arxiv":["2212.03121"],"isi":["001584166900001"]},"date_published":"2026-01-01T00:00:00Z","publication":"Discrete and Computational Geometry","day":"01","PlanS_conform":"1","author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890"},{"last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","orcid":"0000-0001-6249-0832"},{"last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","orcid":"0000-0003-0464-3823"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"oa_version":"Published Version","file_date_updated":"2026-01-05T13:21:20Z","department":[{"_id":"HeEd"}],"ddc":["510"],"arxiv":1,"scopus_import":"1","_id":"20456","corr_author":"1","status":"public","date_updated":"2026-01-05T13:21:56Z","oa":1,"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"15090"}]},"publisher":"Springer Nature","intvolume":"        75","month":"01","OA_place":"publisher","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"Yes (via OA deal)","ec_funded":1,"page":"24-47","citation":{"short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47.","ista":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 75, 24–47.","apa":"Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>","mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>, vol. 75, Springer Nature, 2026, pp. 24–47, doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>.","ieee":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” <i>Discrete and Computational Geometry</i>, vol. 75. Springer Nature, pp. 24–47, 2026.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>.","ama":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. 2026;75:24-47. doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>"},"has_accepted_license":"1","volume":75,"OA_type":"hybrid","quality_controlled":"1"},{"oa":1,"date_updated":"2026-03-09T11:31:29Z","_id":"21407","status":"public","file_date_updated":"2026-03-09T11:29:30Z","department":[{"_id":"HeEd"}],"ddc":["500"],"scopus_import":"1","arxiv":1,"citation":{"mla":"Edelsbrunner, Herbert, et al. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>, vol. 10, 5, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>.","apa":"Edelsbrunner, H., Kahle, M., &#38; Kanazawa, S. (2026). Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>","short":"H. Edelsbrunner, M. Kahle, S. Kanazawa, Journal of Applied and Computational Topology 10 (2026).","ista":"Edelsbrunner H, Kahle M, Kanazawa S. 2026. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 10, 5.","ama":"Edelsbrunner H, Kahle M, Kanazawa S. Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. 2026;10. doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>","chicago":"Edelsbrunner, Herbert, Matthew Kahle, and Shu Kanazawa. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>.","ieee":"H. Edelsbrunner, M. Kahle, and S. Kanazawa, “Maximum persistent Betti numbers of Čech complexes,” <i>Journal of Applied and Computational Topology</i>, vol. 10. Springer Nature, 2026."},"has_accepted_license":"1","OA_type":"hybrid","volume":10,"quality_controlled":"1","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"Yes (in subscription journal)","month":"03","OA_place":"publisher","publisher":"Springer Nature","intvolume":"        10","date_created":"2026-03-08T23:01:45Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_number":"5","year":"2026","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"acknowledgement":"The authors would like to thank Michael Lesnick and Primoz Skraba for their helpful comments regarding sparse approximations of filtrations. We are also grateful to the anonymous referees for their careful reading and constructive suggestions. The three authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35, the U.S. National Science Foundation (NSF-DMS), grant no. 2005630, and a JSPS Grant-in-Aid for Transformative Research Areas (A) (22H05107, Y.H.), EPSRC Research Grant EP/Y008642/1.","doi":"10.1007/s41468-026-00233-3","article_type":"original","publication_status":"published","language":[{"iso":"eng"}],"title":"Maximum persistent Betti numbers of Čech complexes","type":"journal_article","project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"oa_version":"Published Version","PlanS_conform":"1","day":"01","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"first_name":"Matthew","full_name":"Kahle, Matthew","last_name":"Kahle"},{"last_name":"Kanazawa","full_name":"Kanazawa, Shu","first_name":"Shu"}],"date_published":"2026-03-01T00:00:00Z","file":[{"creator":"dernst","date_updated":"2026-03-09T11:29:30Z","date_created":"2026-03-09T11:29:30Z","content_type":"application/pdf","success":1,"file_size":323111,"checksum":"0bf6dc430cafa40c08f260fe17d54595","file_id":"21416","file_name":"2026_JourAppliedCompTopology_Edelsbrunner.pdf","relation":"main_file","access_level":"open_access"}],"external_id":{"arxiv":["2409.05241"]},"publication":"Journal of Applied and Computational Topology","abstract":[{"lang":"eng","text":"This note proves that only a linear number of holes in a Cech complex of n points in R^d\r\ncan persist over an interval of constant length. Specifically, for any fixed dimension p <\r\nd and fixed ε > 0, the number of p-dimensional holes in the ˇ Cech complex at radius 1\r\nthat persist to radius 1+ε is bounded above by a constant times n,where n is the number\r\nof points. The proof uses a packing argument supported by relating theCˇ ech complexes\r\nwith corresponding snap complexes over the cells in a partition of space. The argument\r\nis self-contained and elementary, relying on geometric and combinatorial constructions\r\nrather than on the existing theory of sparse approximations or interleavings. The bound\r\nalso applies to Alpha complexes and Vietoris–Rips complexes. While our result can be\r\ninferred from prior work on sparse filtrations, to our knowledge, no explicit statement\r\nor direct proof of this bound appears in the literature."}]},{"corr_author":"1","_id":"19630","status":"public","ddc":["519","006"],"file_date_updated":"2025-04-30T15:49:16Z","department":[{"_id":"ChWo"},{"_id":"GradSch"}],"degree_awarded":"PhD","related_material":{"record":[{"id":"8135","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"17219"},{"id":"8384","status":"public","relation":"part_of_dissertation"}]},"oa":1,"date_updated":"2026-04-16T08:29:34Z","OA_place":"publisher","month":"04","supervisor":[{"orcid":"0000-0001-6646-5546","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","first_name":"Christopher J","last_name":"Wojtan","full_name":"Wojtan, Christopher J"}],"publisher":"Institute of Science and Technology Austria","has_accepted_license":"1","citation":{"mla":"Synak, Peter. <i>Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-19630\">10.15479/AT-ISTA-19630</a>.","apa":"Synak, P. (2025). <i>Methods for fluid simulation, surface tracking, and statistics of non-manifold structures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-19630\">https://doi.org/10.15479/AT-ISTA-19630</a>","short":"P. Synak, Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures, Institute of Science and Technology Austria, 2025.","ista":"Synak P. 2025. Methods for fluid simulation, surface tracking, and statistics of non-manifold structures. Institute of Science and Technology Austria.","chicago":"Synak, Peter. “Methods for Fluid Simulation, Surface Tracking, and Statistics of Non-Manifold Structures.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/AT-ISTA-19630\">https://doi.org/10.15479/AT-ISTA-19630</a>.","ama":"Synak P. Methods for fluid simulation, surface tracking, and statistics of non-manifold structures. 2025. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-19630\">10.15479/AT-ISTA-19630</a>","ieee":"P. Synak, “Methods for fluid simulation, surface tracking, and statistics of non-manifold structures,” Institute of Science and Technology Austria, 2025."},"ec_funded":1,"acknowledged_ssus":[{"_id":"ScienComp"}],"article_processing_charge":"No","page":"106","publication_status":"published","language":[{"iso":"eng"}],"project":[{"call_identifier":"H2020","_id":"2533E772-B435-11E9-9278-68D0E5697425","grant_number":"638176","name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales"},{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","grant_number":"101045083","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088"},{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"call_identifier":"H2020","_id":"2533E772-B435-11E9-9278-68D0E5697425","name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales","grant_number":"638176"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"title":"Methods for fluid simulation, surface tracking, and statistics of non-manifold structures","type":"dissertation","alternative_title":["ISTA Thesis"],"date_created":"2025-04-29T09:39:34Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","year":"2025","publication_identifier":{"issn":["2663-337X"]},"acknowledgement":"The project in Chapter 2 has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No. 638176. The project in Chapter 3 was funded in part by the European Union (ERC-2021-COG 101045083 CoDiNA). The project in Chapter 4 has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It was also partially supported by the DFG Collaborative Research Center TRR 109, 'Discretization in Geometry and Dynamics', through grant no. I02979-N35 of the Austrian Science Fund (FWF). Thank you for providing funds to support my work.","doi":"10.15479/AT-ISTA-19630","date_published":"2025-04-29T00:00:00Z","file":[{"file_size":60670543,"content_type":"application/x-zip-compressed","date_created":"2025-04-30T14:02:25Z","date_updated":"2025-04-30T14:02:25Z","creator":"cchlebak","access_level":"closed","relation":"source_file","file_id":"19633","file_name":"Thesis_source_Heiss_Synak.zip","checksum":"f00b519c27529daa0c3b2d4102b4fa7b"},{"checksum":"6e40a2fd3b1b881af1385670854a682e","file_name":"Thesis_PDFA_Heiss_Synak.pdf","file_id":"19634","access_level":"open_access","relation":"main_file","creator":"cchlebak","date_updated":"2025-04-30T15:49:16Z","content_type":"application/pdf","date_created":"2025-04-30T14:02:42Z","file_size":21319043}],"abstract":[{"text":"This thesis consists of three chapters, each corresponding to one publication. While each of these projects tackles a topic in a different area of research, they all share a common thread in the type of topological structure they handle - a partition of space into volumes separated by interfaces that meet in non-manifold junctions.\r\n\r\nIn Chapter 2, we study clusters of soap bubbles from a simulation perspective. In particular, we develop a surface-only algorithm that couples large scale motion and shape deformation of soap bubble clusters with the small scale evolution of the thin film's thickness, which is responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. We model film thickness as a reduced degree of freedom in the Navier-Stokes equations and from them derive three sets of equations governing normal and tangential motion of the soap film surface, as well as the evolution of the thin film thickness. We discretize these equations on a non-manifold triangle mesh, extending and adapting operators to handle complex topology. We also present an incompressible fluid solver for 2.5D films and an advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film.\r\n\r\nIn Chapter 3, we introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts mesh defects, such as overlaps, self-intersections, and inversions into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations, such as those presented in Chapter 2, but with an order of magnitude more interacting bubbles than what we could achieve before, and Boolean unions of non-manifold meshes consisting of millions of triangles.\r\n\r\nLastly, in Chapter 4, we utilize developments in the theory of random geometric complexes facilitated by observations from Discrete Morse theory. We survey the methods and results obtained with this new approach, and discuss some of its shortcomings. We use simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.","lang":"eng"}],"oa_version":"Published Version","day":"29","author":[{"last_name":"Synak","full_name":"Synak, Peter","id":"331776E2-F248-11E8-B48F-1D18A9856A87","first_name":"Peter"}]},{"date_updated":"2025-12-30T09:05:32Z","scopus_import":"1","department":[{"_id":"HeEd"}],"status":"public","corr_author":"1","_id":"19937","ec_funded":1,"article_processing_charge":"No","OA_type":"closed access","quality_controlled":"1","volume":719,"citation":{"chicago":"Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>.","ama":"Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. 2025;719(11). doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>","ieee":"M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality,” <i>Information Sciences</i>, vol. 719, no. 11. Elsevier, 2025.","mla":"Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>, vol. 719, no. 11, 122425, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>.","apa":"Mahini, M., Beigy, H., Qadami, S., &#38; Saghafian, M. (2025). Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>","ista":"Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. 719(11), 122425.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025)."},"intvolume":"       719","publisher":"Elsevier","month":"11","acknowledgement":"The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which improved this paper.\r\nWork by the first and fourth authors is partially 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.","doi":"10.1016/j.ins.2025.122425","year":"2025","article_number":"122425","publication_identifier":{"issn":["0020-0255"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-06-30T08:48:48Z","title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","type":"journal_article","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"language":[{"iso":"eng"}],"isi":1,"publication_status":"published","article_type":"original","author":[{"first_name":"Mohammad","last_name":"Mahini","full_name":"Mahini, Mohammad"},{"last_name":"Beigy","full_name":"Beigy, Hamid","first_name":"Hamid"},{"last_name":"Qadami","full_name":"Qadami, Salman","first_name":"Salman"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"day":"01","oa_version":"None","issue":"11","abstract":[{"text":"Simplets are elementary units within simplicial complexes and are fundamental for analyzing the structure of simplicial complexes. Previous efforts have mainly focused on accurately counting or approximating the number of simplets rather than studying their frequencies. However, analyzing simplet frequencies is more practical for large-scale simplicial complexes. This paper introduces the Simplet Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies in simplicial complexes. Additionally, we provide a bound on the sample complexity required to approximate the SFD vector using any uniform sampling-based algorithm accurately. We extend the definition of simplet frequency distribution to encompass simplices, allowing for the analysis of simplet frequencies within simplices of simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex. Furthermore, we present a bound on the sample complexity required for accurately approximating the SDV and SDC for a set of simplices using any uniform sampling-based algorithm. We also introduce algorithms for approximating SFD, geometric SFD, SDV, and SDC. We also validate the theoretical bounds with experiments on random simplicial complexes and demonstrate the practical application through a case study.","lang":"eng"}],"publication":"Information Sciences","external_id":{"isi":["001516170500002"]},"date_published":"2025-11-01T00:00:00Z"},{"date_created":"2025-07-13T22:01:22Z","alternative_title":["LIPIcs"],"conference":{"location":"Kanazawa, Japan","start_date":"2025-06-23","name":"SoCG: Symposium on Computational Geometry","end_date":"2025-06-27"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","article_number":"43","publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"acknowledgement":"Herbert Edelsbrunner: partially supported by the Wittgenstein Prize, Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35","doi":"10.4230/LIPIcs.SoCG.2025.43","publication_status":"published","language":[{"iso":"eng"}],"title":"On spheres with k points inside","type":"conference","project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"oa_version":"Published Version","day":"20","author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"file":[{"content_type":"application/pdf","date_created":"2025-07-14T07:24:22Z","success":1,"file_size":661893,"creator":"dernst","date_updated":"2025-07-14T07:24:22Z","file_id":"20016","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","relation":"main_file","access_level":"open_access","checksum":"b5313ed8575ea87913c71a6e3c7513c8"}],"external_id":{"arxiv":["2410.21204"]},"date_published":"2025-06-20T00:00:00Z","publication":"41st International Symposium on Computational Geometry","abstract":[{"text":"We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.","lang":"eng"}],"oa":1,"date_updated":"2025-07-14T07:26:14Z","corr_author":"1","_id":"20005","status":"public","department":[{"_id":"HeEd"}],"file_date_updated":"2025-07-14T07:24:22Z","ddc":["510"],"scopus_import":"1","arxiv":1,"citation":{"mla":"Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43.","ama":"Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. 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Saghafian, “On spheres with k points inside,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332."},"has_accepted_license":"1","volume":332,"quality_controlled":"1","OA_type":"gold","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"Yes","month":"06","OA_place":"publisher","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","intvolume":"       332"},{"month":"06","OA_place":"publisher","intvolume":"       332","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","OA_type":"gold","volume":332,"citation":{"ieee":"L. Ost, S. Cultrera di Montesano, and H. Edelsbrunner, “Banana trees for the persistence in time series experimentally,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","ama":"Ost L, Cultrera di Montesano S, Edelsbrunner H. Banana trees for the persistence in time series experimentally. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>","chicago":"Ost, Lara, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner. “Banana Trees for the Persistence in Time Series Experimentally.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>.","short":"L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","ista":"Ost L, Cultrera di Montesano S, Edelsbrunner H. 2025. Banana trees for the persistence in time series experimentally. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 71.","mla":"Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 71, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>.","apa":"Ost, L., Cultrera di Montesano, S., &#38; Edelsbrunner, H. (2025). Banana trees for the persistence in time series experimentally. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>"},"has_accepted_license":"1","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"Yes","status":"public","_id":"20006","corr_author":"1","arxiv":1,"scopus_import":"1","file_date_updated":"2025-07-14T08:23:38Z","department":[{"_id":"HeEd"}],"ddc":["000"],"related_material":{"link":[{"relation":"software","url":"https://github.com/laraost/BananaPersist"}]},"oa":1,"date_updated":"2025-12-30T11:04:33Z","publication":"41st International Symposium on Computational Geometry","external_id":{"arxiv":["2405.17920"]},"date_published":"2025-06-20T00:00:00Z","file":[{"checksum":"3a4a7a707a56e0cfdf51428782dee55a","access_level":"open_access","relation":"main_file","file_id":"20017","file_name":"2025_LIPIcs.SoCG_Ost.pdf","date_updated":"2025-07-14T08:23:38Z","creator":"dernst","file_size":834623,"date_created":"2025-07-14T08:23:38Z","content_type":"application/pdf","success":1}],"abstract":[{"text":"In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.","lang":"eng"}],"oa_version":"Published Version","author":[{"last_name":"Ost","full_name":"Ost, Lara","first_name":"Lara"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"day":"20","publication_status":"published","type":"conference","title":"Banana trees for the persistence in time series experimentally","project":[{"_id":"9B9290DE-BA93-11EA-9121-9846C619BF3A","grant_number":"W1260-N35","name":"Vienna Graduate School on Computational Optimization"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"language":[{"iso":"eng"}],"conference":{"start_date":"2025-06-23","name":"SoCG: Symposium on Computational Geometry","location":"Kanazawa, Japan","end_date":"2025-06-27"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","alternative_title":["LIPIcs"],"date_created":"2025-07-13T22:01:22Z","acknowledgement":"Lara Ost: Supported by the Vienna Graduate School on Computational Optimization\r\n(VGSCO), FWF project no. W1260-N35.\r\nSebastiano Cultrera di Montesano: Supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard.\r\nHerbert Edelsbrunner: Partially supported by the Wittgenstein Prize, FWF grant no. Z 342-N31,\r\nand by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.","doi":"10.4230/LIPIcs.SoCG.2025.71","year":"2025","article_number":"71","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773706"]}},{"oa_version":"Published Version","pmid":1,"issue":"8","author":[{"full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Ziga","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","full_name":"Virk, Ziga","last_name":"Virk"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","last_name":"Wagner","full_name":"Wagner, Hubert"}],"day":"01","PlanS_conform":"1","publication":"Entropy","file":[{"date_updated":"2025-09-08T07:55:48Z","creator":"dernst","file_size":379340,"success":1,"content_type":"application/pdf","date_created":"2025-09-08T07:55:48Z","checksum":"65c5399c4015d9c8abb8c7a96f3d7836","relation":"main_file","access_level":"open_access","file_name":"2025_Entropy_Akopyan.pdf","file_id":"20309"}],"external_id":{"pmid":["40870326"],"isi":["001557476000001"]},"date_published":"2025-08-01T00:00:00Z","abstract":[{"lang":"eng","text":"Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence."}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2025-09-07T22:01:33Z","doi":"10.3390/e27080854","acknowledgement":"This research received partial funding from the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis of Neural Networks’. The APC was waived.","article_number":"854","publication_identifier":{"eissn":["1099-4300"]},"year":"2025","publication_status":"published","DOAJ_listed":"1","article_type":"original","type":"journal_article","title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"isi":1,"language":[{"iso":"eng"}],"quality_controlled":"1","OA_type":"gold","volume":27,"citation":{"apa":"Akopyan, A., Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2025). Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. MDPI. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>","mla":"Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>, vol. 27, no. 8, 854, MDPI, 2025, doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>.","ista":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).","chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>. MDPI, 2025. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>.","ama":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. 2025;27(8). doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>","ieee":"A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between the Jensen–Shannon divergence and the minmax divergence,” <i>Entropy</i>, vol. 27, no. 8. MDPI, 2025."},"has_accepted_license":"1","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"ec_funded":1,"article_processing_charge":"Yes","month":"08","OA_place":"publisher","intvolume":"        27","publisher":"MDPI","oa":1,"date_updated":"2025-09-30T14:32:31Z","status":"public","_id":"20293","corr_author":"1","scopus_import":"1","department":[{"_id":"HeEd"}],"file_date_updated":"2025-09-08T07:55:48Z","ddc":["500"]},{"date_updated":"2025-12-30T07:55:21Z","oa":1,"related_material":{"record":[{"id":"18981","status":"public","relation":"earlier_version"}]},"department":[{"_id":"HeEd"}],"file_date_updated":"2025-12-30T07:55:08Z","ddc":["510"],"scopus_import":"1","arxiv":1,"corr_author":"1","_id":"20323","status":"public","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"Yes (via OA deal)","ec_funded":1,"citation":{"ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.","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>.","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>","short":"A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).","ista":"Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure and Applied Algebra. 229(10), 108068.","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>"},"has_accepted_license":"1","quality_controlled":"1","volume":229,"OA_type":"hybrid","publisher":"Elsevier","intvolume":"       229","month":"10","OA_place":"publisher","year":"2025","article_number":"108068","publication_identifier":{"issn":["0022-4049"]},"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","date_created":"2025-09-10T05:40:09Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"type":"journal_article","title":"Discrete microlocal Morse theory","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF"}],"article_type":"original","publication_status":"published","day":"01","PlanS_conform":"1","author":[{"first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","full_name":"Brown, Adam","last_name":"Brown"},{"orcid":"0000-0003-0464-3823","last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej"}],"issue":"10","oa_version":"Published Version","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."}],"date_published":"2025-10-01T00:00:00Z","external_id":{"arxiv":["2209.14993"]},"file":[{"checksum":"39bcad462278c9322ef810af7db67f56","file_name":"2025_JourPureAppliedAlgebra_Brown.pdf","file_id":"20886","relation":"main_file","access_level":"open_access","creator":"dernst","date_updated":"2025-12-30T07:55:08Z","success":1,"date_created":"2025-12-30T07:55:08Z","content_type":"application/pdf","file_size":3090836}],"publication":"Journal of Pure and Applied Algebra"},{"article_processing_charge":"No","ec_funded":1,"citation":{"ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025.","ama":"Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. 2025;461. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>","mla":"Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>, vol. 461, 110055, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>."},"OA_type":"green","volume":461,"quality_controlled":"1","publisher":"Elsevier","intvolume":"       461","month":"02","OA_place":"repository","date_updated":"2025-04-15T07:16:53Z","oa":1,"department":[{"_id":"HeEd"}],"scopus_import":"1","arxiv":1,"corr_author":"1","_id":"18626","status":"public","day":"01","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2310.18238"}],"oa_version":"Preprint","abstract":[{"text":"The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.","lang":"eng"}],"date_published":"2025-02-01T00:00:00Z","external_id":{"arxiv":["2310.18238"],"isi":["001370682500001"]},"publication":"Advances in Mathematics","year":"2025","publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"article_number":"110055","doi":"10.1016/j.aim.2024.110055","acknowledgement":"Work by the first and third authors is partially 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 the second author is partially supported by the Alexander von Humboldt Foundation.","date_created":"2024-12-08T23:01:54Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","isi":1,"language":[{"iso":"eng"}],"title":"Order-2 Delaunay triangulations optimize angles","type":"journal_article","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_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"}],"article_type":"original","publication_status":"published"},{"degree_awarded":"PhD","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"file_date_updated":"2025-02-04T16:22:07Z","ddc":["514","004"],"status":"public","_id":"18979","corr_author":"1","date_updated":"2026-04-07T11:47:30Z","oa":1,"related_material":{"record":[{"id":"15091","status":"public","relation":"part_of_dissertation"},{"id":"18981","relation":"part_of_dissertation","status":"public"}]},"supervisor":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"}],"publisher":"Institute of Science and Technology Austria","month":"02","OA_place":"publisher","page":"140","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"No","citation":{"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>.","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>","ieee":"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>","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>.","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."},"has_accepted_license":"1","type":"dissertation","title":"Structures and computations in topological data analysis","project":[{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"language":[{"iso":"eng"}],"publication_status":"published","doi":"10.15479/at:ista:18979","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","keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"publication_identifier":{"issn":["2663-337X"]},"year":"2025","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_created":"2025-01-31T17:04:40Z","alternative_title":["ISTA Thesis"],"abstract":[{"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.","lang":"eng"}],"date_published":"2025-02-03T00:00:00Z","file":[{"access_level":"closed","relation":"source_file","file_name":"Thesis.zip","file_id":"18983","checksum":"af6567e5d35e5eb330b8925ae37f1998","file_size":11899491,"date_created":"2025-01-31T16:58:30Z","content_type":"application/zip","date_updated":"2025-01-31T16:58:30Z","creator":"odragano"},{"creator":"odragano","date_updated":"2025-02-04T16:22:07Z","content_type":"application/pdf","date_created":"2025-02-04T16:22:07Z","file_size":8857514,"checksum":"c3fef68e35b9dc2020b2ca6006da6343","file_id":"19000","file_name":"Thesis.pdf","access_level":"open_access","relation":"main_file"}],"author":[{"last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","orcid":"0000-0003-0464-3823"}],"day":"03","oa_version":"Published Version"},{"OA_place":"repository","month":"10","intvolume":"       132","publisher":"Elsevier","OA_type":"green","quality_controlled":"1","volume":132,"citation":{"ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” <i>European Journal of Combinatorics</i>, vol. 132. Elsevier, 2025.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. 2025;132. doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2025).","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2025). Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>","mla":"Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>, vol. 132, 104248, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>."},"article_processing_charge":"No","ec_funded":1,"status":"public","_id":"20490","corr_author":"1","scopus_import":"1","arxiv":1,"department":[{"_id":"HeEd"}],"oa":1,"date_updated":"2025-12-01T12:57:29Z","publication":"European Journal of Combinatorics","date_published":"2025-10-10T00:00:00Z","external_id":{"arxiv":["2212.11380"],"isi":["001599061500002"]},"abstract":[{"lang":"eng","text":"We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n\r\n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.\r\n"}],"oa_version":"Preprint","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2212.11380","open_access":"1"}],"author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"first_name":"Alexey","last_name":"Garber","full_name":"Garber, Alexey"},{"full_name":"Ghafari, Mohadese","last_name":"Ghafari","first_name":"Mohadese"},{"orcid":"0000-0002-1780-2689","last_name":"Heiss","full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"day":"10","publication_status":"epub_ahead","article_type":"original","project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"type":"journal_article","title":"Flips in two-dimensional hypertriangulations","isi":1,"language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-10-19T22:01:31Z","doi":"10.1016/j.ejc.2025.104248","acknowledgement":"Work by all authors but the second 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 the second author is partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation . The second author thanks Jesús A. De Loera for useful discussions on flips and non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic graphs.","publication_identifier":{"issn":["0195-6698"]},"year":"2025","article_number":"104248"},{"corr_author":"1","_id":"20585","status":"public","department":[{"_id":"HeEd"}],"arxiv":1,"scopus_import":"1","related_material":{"record":[{"id":"15091","status":"public","relation":"earlier_version"}]},"date_updated":"2025-11-04T12:25:47Z","month":"03","OA_place":"repository","publisher":"American Institute of Mathematical Sciences","intvolume":"         8","citation":{"apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2025). Chromatic alpha complexes. <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>, vol. 8, American Institute of Mathematical Sciences, 2025, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2025) 30–62.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2025;8:30-62. doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences, 2025. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>Foundations of Data Science</i>, vol. 8. American Institute of Mathematical Sciences, pp. 30–62, 2025."},"volume":8,"quality_controlled":"1","OA_type":"green","article_processing_charge":"No","ec_funded":1,"page":"30-62","article_type":"original","publication_status":"epub_ahead","language":[{"iso":"eng"}],"type":"journal_article","title":"Chromatic alpha complexes","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"date_created":"2025-11-02T23:01:33Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","publication_identifier":{"eissn":["2639-8001"]},"acknowledgement":"This project has received funding from the European Research\r\nCouncil (ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.","doi":"10.3934/fods.2025003","external_id":{"arxiv":["2212.03128"]},"date_published":"2025-03-01T00:00:00Z","publication":"Foundations of Data Science","abstract":[{"text":"Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.","lang":"eng"}],"oa_version":"Preprint","day":"01","author":[{"full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej","last_name":"Draganov","first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","last_name":"Saghafian"}]},{"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"ec_funded":1,"article_processing_charge":"Yes (via OA deal)","OA_type":"hybrid","quality_controlled":"1","citation":{"ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. 2025. doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>.","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025.","apa":"Edelsbrunner, H., &#38; Pach, J. (2025). Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>.","ista":"Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry.","short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry (2025)."},"has_accepted_license":"1","publisher":"Springer Nature","month":"11","OA_place":"publisher","date_updated":"2025-12-01T15:19:21Z","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"17146"}]},"oa":1,"arxiv":1,"scopus_import":"1","department":[{"_id":"HeEd"}],"ddc":["510"],"status":"public","corr_author":"1","_id":"20657","author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"last_name":"Pach","full_name":"Pach, János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János"}],"day":"10","PlanS_conform":"1","oa_version":"Published Version","main_file_link":[{"url":"https://doi.org/10.1007/s00454-025-00796-5","open_access":"1"}],"abstract":[{"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. ","lang":"eng"}],"publication":"Discrete & Computational Geometry","date_published":"2025-11-10T00:00:00Z","external_id":{"isi":["001610592600001"],"arxiv":["2310.14801"]},"doi":"10.1007/s00454-025-00796-5","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.","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-11-19T09:44:58Z","title":"Maximum Betti numbers of Čech complexes","type":"journal_article","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF"}],"isi":1,"language":[{"iso":"eng"}],"publication_status":"epub_ahead","article_type":"original"},{"abstract":[{"lang":"eng","text":"The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes."}],"publication":"Discrete & Computational Geometry","external_id":{"pmid":["39974750"],"isi":["001238566200004"],"arxiv":["2012.03350"]},"date_published":"2025-03-01T00:00:00Z","file":[{"success":1,"content_type":"application/pdf","date_created":"2025-04-23T07:31:32Z","file_size":283443,"creator":"dernst","date_updated":"2025-04-23T07:31:32Z","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","file_id":"19610","relation":"main_file","access_level":"open_access","checksum":"ffb0c818222138f9f113f4bbea41e834"}],"author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko","orcid":"0000-0002-0659-3201"}],"day":"01","oa_version":"Published Version","pmid":1,"type":"journal_article","title":"Average and expected distortion of Voronoi paths and scapes","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF"}],"isi":1,"language":[{"iso":"eng"}],"publication_status":"published","article_type":"original","acknowledgement":"The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on computational experiments that motivated the work reported in this paper. The authors also thank Daniel Bonnema for proofreading and noticing an issue with the original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).\r\nThis 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":"10.1007/s00454-024-00660-y","year":"2025","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2024-06-16T22:01:07Z","intvolume":"        73","publisher":"Springer Nature","month":"03","OA_place":"publisher","page":"490-499","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"Yes (via OA deal)","ec_funded":1,"quality_controlled":"1","OA_type":"hybrid","volume":73,"citation":{"apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>.","ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","ama":"Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. 2025;73:490-499. doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>.","ieee":"H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” <i>Discrete &#38; Computational Geometry</i>, vol. 73. Springer Nature, pp. 490–499, 2025."},"has_accepted_license":"1","scopus_import":"1","arxiv":1,"department":[{"_id":"HeEd"}],"file_date_updated":"2025-04-23T07:31:32Z","ddc":["510"],"status":"public","corr_author":"1","_id":"17149","date_updated":"2026-02-16T12:18:50Z","oa":1},{"oa_version":"Published Version","author":[{"full_name":"Attali, Dominique","last_name":"Attali","first_name":"Dominique"},{"last_name":"Kourimska","full_name":"Kourimska, Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","orcid":"0000-0001-7841-0091"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","last_name":"Fillmore","full_name":"Fillmore, Christopher D"},{"full_name":"Ghosh, Ishika","last_name":"Ghosh","first_name":"Ishika","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff"},{"first_name":"Andre","last_name":"Lieutier","full_name":"Lieutier, Andre"},{"orcid":"0000-0002-6862-208X","first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","full_name":"Stephenson, Elizabeth R","last_name":"Stephenson"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","orcid":"0000-0002-7472-2220"}],"day":"06","publication":"40th International Symposium on Computational Geometry","file":[{"date_updated":"2024-09-19T10:30:37Z","creator":"dernst","file_size":3507177,"date_created":"2024-09-19T10:30:37Z","content_type":"application/pdf","success":1,"checksum":"9355c2e60b8ec285e1b22719c5b73f1a","relation":"main_file","access_level":"open_access","file_id":"18098","file_name":"2024_LIPICs_Attali.pdf"}],"date_published":"2024-06-06T00:00:00Z","abstract":[{"lang":"eng","text":"In our companion paper \"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds\" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","conference":{"start_date":"2024-06-11","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","end_date":"2024-06-14"},"date_created":"2024-09-19T10:29:48Z","alternative_title":["LIPIcs"],"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.","publication_identifier":{"isbn":["9783959773164"]},"year":"2024","article_number":"87","publication_status":"published","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF"},{"call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"name":"Learning and triangulating manifolds via collapses","grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"type":"conference","title":"The ultimate frontier: An optimality construction for homotopy inference (media exposition)","language":[{"iso":"eng"}],"volume":293,"quality_controlled":"1","has_accepted_license":"1","citation":{"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.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","mla":"Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” <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>.","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>","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.","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>","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>."},"ec_funded":1,"article_processing_charge":"Yes","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"month":"06","intvolume":"       293","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","oa":1,"date_updated":"2025-04-15T07:16:58Z","status":"public","corr_author":"1","_id":"18097","ddc":["000"],"department":[{"_id":"HeEd"}],"file_date_updated":"2024-09-19T10:30:37Z"},{"publication_status":"published","type":"conference","title":"The Euclidean MST-ratio for bi-colored lattices","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_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"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"language":[{"iso":"eng"}],"isi":1,"conference":{"name":"GD: Graph Drawing and Network Visualization","start_date":"2024-09-18","location":"Vienna, Austria","end_date":"2024-09-20"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2024-11-17T23:01:47Z","alternative_title":["LIPIcs"],"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","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773430"]},"year":"2024","article_number":"3","publication":"32nd International Symposium on Graph Drawing and Network Visualization","external_id":{"arxiv":["2403.10204"],"isi":["001540278400001"]},"date_published":"2024-10-28T00:00:00Z","file":[{"file_name":"2024_LIPIcs_CultreradiMontesano.pdf","file_id":"18560","access_level":"open_access","relation":"main_file","checksum":"5f9b35e115c3d375e99be78da9054cb4","success":1,"content_type":"application/pdf","date_created":"2024-11-18T07:49:25Z","file_size":908541,"creator":"dernst","date_updated":"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"}],"oa_version":"Published Version","author":[{"orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano"},{"id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","last_name":"Draganov","full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823"},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"day":"28","status":"public","_id":"18556","corr_author":"1","scopus_import":"1","arxiv":1,"department":[{"_id":"HeEd"}],"file_date_updated":"2024-11-18T07:49:25Z","ddc":["510"],"oa":1,"date_updated":"2025-12-02T13:50:50Z","month":"10","OA_place":"publisher","intvolume":"       320","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","OA_type":"gold","volume":320,"quality_controlled":"1","citation":{"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>","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>.","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.","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.","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>","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>.","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."},"has_accepted_license":"1","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"Yes","ec_funded":1},{"department":[{"_id":"HeEd"}],"arxiv":1,"_id":"18673","corr_author":"1","status":"public","date_updated":"2026-04-07T12:54:09Z","related_material":{"record":[{"id":"18667","relation":"dissertation_contains","status":"public"}]},"oa":1,"month":"08","OA_place":"repository","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"article_processing_charge":"No","ec_funded":1,"citation":{"ama":"Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2408.16575\">10.48550/arXiv.2408.16575</a>","chicago":"Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2408.16575\">https://doi.org/10.48550/arXiv.2408.16575</a>.","ieee":"H. Edelsbrunner and T. Heiss, “Merge trees of periodic filtrations,” <i>arXiv</i>. .","mla":"Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/arXiv.2408.16575\">10.48550/arXiv.2408.16575</a>.","apa":"Edelsbrunner, H., &#38; Heiss, T. (n.d.). Merge trees of periodic filtrations. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2408.16575\">https://doi.org/10.48550/arXiv.2408.16575</a>","short":"H. Edelsbrunner, T. Heiss, ArXiv (n.d.).","ista":"Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv, <a href=\"https://doi.org/10.48550/arXiv.2408.16575\">10.48550/arXiv.2408.16575</a>."},"language":[{"iso":"eng"}],"title":"Merge trees of periodic filtrations","type":"preprint","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"publication_status":"draft","year":"2024","acknowledgement":"Both authors are partially supported by the European Research Council (ERC) Horizon 2020 project\r\n‘Alpha Shape Theory Extended’, grant no. 788183. The first author is also partially supported by the DFG\r\nCollaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund\r\n(FWF), grant no. I 02979-N35.","doi":"10.48550/arXiv.2408.16575","date_created":"2024-12-18T14:06:57Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes O((n+m)logn) time, in which n and m are the numbers of vertices and edges in the quotient complex, respectively.\r\n","lang":"eng"}],"external_id":{"arxiv":["2408.16575"]},"date_published":"2024-08-29T00:00:00Z","publication":"arXiv","day":"29","author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-1780-2689","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","full_name":"Heiss, Teresa","last_name":"Heiss"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2408.16575"}],"oa_version":"Preprint"},{"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European\r\nUnion’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.","doi":"10.48550/arXiv.2209.14993","year":"2024","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-01-31T17:03:04Z","title":"Discrete microlocal Morse theory","type":"preprint","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"language":[{"iso":"eng"}],"publication_status":"draft","author":[{"first_name":"Adam","full_name":"Brown, Adam","last_name":"Brown"},{"first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","last_name":"Draganov","orcid":"0000-0003-0464-3823"}],"day":"09","oa_version":"Preprint","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2209.14993","open_access":"1"}],"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."}],"publication":"arXiv","external_id":{"arxiv":["2209.14993"]},"date_published":"2024-06-09T00:00:00Z","date_updated":"2026-04-07T11:47:29Z","oa":1,"related_material":{"record":[{"id":"20323","status":"public","relation":"later_version"},{"id":"18979","relation":"dissertation_contains","status":"public"}]},"arxiv":1,"department":[{"_id":"HeEd"}],"status":"public","_id":"18981","corr_author":"1","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"ec_funded":1,"article_processing_charge":"No","citation":{"ista":"Brown A, Draganov O. Discrete microlocal Morse theory. arXiv, <a href=\"https://doi.org/10.48550/arXiv.2209.14993\">10.48550/arXiv.2209.14993</a>.","short":"A. Brown, O. Draganov, ArXiv (n.d.).","apa":"Brown, A., &#38; Draganov, O. (n.d.). Discrete microlocal Morse theory. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2209.14993\">https://doi.org/10.48550/arXiv.2209.14993</a>","mla":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/arXiv.2209.14993\">10.48550/arXiv.2209.14993</a>.","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>arXiv</i>. .","ama":"Brown A, Draganov O. Discrete microlocal Morse theory. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2209.14993\">10.48550/arXiv.2209.14993</a>","chicago":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2209.14993\">https://doi.org/10.48550/arXiv.2209.14993</a>."},"month":"06","OA_place":"repository"},{"date_updated":"2026-04-07T12:58:48Z","oa":1,"related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"15091"},{"id":"11660","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"15090"},{"id":"15093","status":"public","relation":"part_of_dissertation"},{"id":"13182","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"11658"}]},"ddc":["514","500","516"],"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"file_date_updated":"2024-03-14T14:14:35Z","degree_awarded":"PhD","corr_author":"1","_id":"15094","status":"public","ec_funded":1,"article_processing_charge":"No","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","image":"/images/cc_by_nc_sa.png","short":"CC BY-NC-SA (4.0)"},"page":"108","has_accepted_license":"1","citation":{"short":"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.","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>.","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>","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>."},"publisher":"Institute of Science and Technology Austria","supervisor":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833"}],"OA_place":"publisher","month":"03","publication_identifier":{"issn":["2663-337X"]},"year":"2024","doi":"10.15479/at:ista:15094","date_created":"2024-03-08T15:28:10Z","alternative_title":["ISTA Thesis"],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","language":[{"iso":"eng"}],"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"},{"grant_number":"I4887","name":"Persistent Homology, Algorithms and Stochastic Geometry","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"type":"dissertation","title":"Persistence and Morse theory for discrete geometric structures","publication_status":"published","day":"08","author":[{"orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano"}],"oa_version":"Published Version","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"}],"date_published":"2024-03-08T00:00:00Z","file":[{"date_updated":"2024-03-14T08:55:07Z","creator":"scultrer","file_size":4106872,"content_type":"application/pdf","date_created":"2024-03-14T08:55:07Z","success":1,"checksum":"1e468bfa42a7dcf04d89f4dadc621c87","access_level":"open_access","relation":"main_file","file_id":"15112","file_name":"Thesis Sebastiano.pdf"},{"creator":"scultrer","date_updated":"2024-03-14T14:14:35Z","content_type":"application/zip","date_created":"2024-03-14T08:56:24Z","file_size":4746234,"checksum":"bcbd213490f5a7e68855a092bbce93f1","file_id":"15113","file_name":"Thesis (1).zip","relation":"source_file","access_level":"closed"}]},{"publication_status":"published","article_type":"original","title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","type":"journal_article","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2024-05-12T22:01:03Z","doi":"10.1007/s41468-024-00173-w","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.","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"year":"2024","publication":"Journal of Applied and Computational Topology","file":[{"file_id":"19612","file_name":"2024_JourApplCompTopo_BiswasRa.pdf","access_level":"open_access","relation":"main_file","checksum":"0ee15c1493a6413cf356ab2f32c81a9e","date_created":"2025-04-23T08:01:36Z","content_type":"application/pdf","success":1,"file_size":522831,"creator":"dernst","date_updated":"2025-04-23T08:01:36Z"}],"external_id":{"pmid":["39308789"]},"date_published":"2024-09-01T00:00:00Z","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."}],"oa_version":"Published Version","pmid":1,"author":[{"orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita"},{"orcid":"0000-0001-6249-0832","last_name":"Cultrera Di Montesano","full_name":"Cultrera Di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"day":"01","status":"public","corr_author":"1","_id":"15380","scopus_import":"1","file_date_updated":"2025-04-23T08:01:36Z","department":[{"_id":"HeEd"}],"ddc":["510"],"related_material":{"record":[{"id":"11658","status":"public","relation":"earlier_version"}]},"oa":1,"date_updated":"2025-05-14T09:27:57Z","month":"09","OA_place":"publisher","intvolume":"         8","publisher":"Springer Nature","OA_type":"hybrid","volume":8,"quality_controlled":"1","citation":{"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>","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>.","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.","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>.","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>","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578.","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."},"has_accepted_license":"1","page":"557-578","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"ec_funded":1,"article_processing_charge":"Yes (via OA deal)"}]