--- OA_place: repository OA_type: green _id: '18626' abstract: - lang: eng 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." 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. article_number: '110055' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 2025;461. doi:10.1016/j.aim.2024.110055 apa: Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2024.110055 chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” Advances in Mathematics. Elsevier, 2025. https://doi.org/10.1016/j.aim.2024.110055. ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” Advances in Mathematics, vol. 461. Elsevier, 2025. ista: Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055. mla: Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” Advances in Mathematics, vol. 461, 110055, Elsevier, 2025, doi:10.1016/j.aim.2024.110055. short: H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025). corr_author: '1' date_created: 2024-12-08T23:01:54Z date_published: 2025-02-01T00:00:00Z date_updated: 2025-04-15T07:16:53Z day: '01' department: - _id: HeEd doi: 10.1016/j.aim.2024.110055 ec_funded: 1 external_id: arxiv: - '2310.18238' isi: - '001370682500001' intvolume: ' 461' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2310.18238 month: '02' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Advances in Mathematics publication_identifier: eissn: - 1090-2082 issn: - 0001-8708 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Order-2 Delaunay triangulations optimize angles type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 461 year: '2025' ... --- OA_place: publisher _id: '18979' abstract: - lang: eng text: "Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe describe a construction of a chromatic alpha complex, which enables efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot." 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" alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 citation: ama: Draganov O. Structures and computations in topological data analysis. 2025. doi:10.15479/at:ista:18979 apa: Draganov, O. (2025). Structures and computations in topological data analysis. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18979 chicago: Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.” Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/at:ista:18979. ieee: O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025. ista: Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria. mla: Draganov, Ondrej. Structures and Computations in Topological Data Analysis. Institute of Science and Technology Austria, 2025, doi:10.15479/at:ista:18979. short: O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025. corr_author: '1' date_created: 2025-01-31T17:04:40Z date_published: 2025-02-03T00:00:00Z date_updated: 2026-04-07T11:47:30Z day: '03' ddc: - '514' - '004' degree_awarded: PhD department: - _id: GradSch - _id: HeEd doi: 10.15479/at:ista:18979 file: - access_level: closed checksum: af6567e5d35e5eb330b8925ae37f1998 content_type: application/zip creator: odragano date_created: 2025-01-31T16:58:30Z date_updated: 2025-01-31T16:58:30Z file_id: '18983' file_name: Thesis.zip file_size: 11899491 relation: source_file - access_level: open_access checksum: c3fef68e35b9dc2020b2ca6006da6343 content_type: application/pdf creator: odragano date_created: 2025-02-04T16:22:07Z date_updated: 2025-02-04T16:22:07Z file_id: '19000' file_name: Thesis.pdf file_size: 8857514 relation: main_file file_date_updated: 2025-02-04T16:22:07Z has_accepted_license: '1' keyword: - topological data analysis - chromatic point set - alpha complex - persistent homology - six pack - sheaf - microlocal discrete Morse - injective resolution - collapse - knot - discrete Morse theory language: - iso: eng month: '02' oa: 1 oa_version: Published Version page: '140' project: - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '15091' relation: part_of_dissertation status: public - id: '18981' relation: part_of_dissertation status: public status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: Structures and computations in topological data analysis tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: dissertation user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd year: '2025' ... --- OA_place: repository OA_type: green _id: '20490' 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" 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. article_number: '104248' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Mohadese full_name: Ghafari, Mohadese last_name: Ghafari - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 2025;132. doi:10.1016/j.ejc.2025.104248 apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2025). Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. Elsevier. https://doi.org/10.1016/j.ejc.2025.104248 chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” European Journal of Combinatorics. Elsevier, 2025. https://doi.org/10.1016/j.ejc.2025.104248. ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” European Journal of Combinatorics, vol. 132. Elsevier, 2025. ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248. mla: Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” European Journal of Combinatorics, vol. 132, 104248, Elsevier, 2025, doi:10.1016/j.ejc.2025.104248. short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2025). corr_author: '1' date_created: 2025-10-19T22:01:31Z date_published: 2025-10-10T00:00:00Z date_updated: 2025-12-01T12:57:29Z day: '10' department: - _id: HeEd doi: 10.1016/j.ejc.2025.104248 ec_funded: 1 external_id: arxiv: - '2212.11380' isi: - '001599061500002' intvolume: ' 132' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2212.11380 month: '10' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: European Journal of Combinatorics publication_identifier: issn: - 0195-6698 publication_status: epub_ahead publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Flips in two-dimensional hypertriangulations type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 132 year: '2025' ... --- OA_place: repository OA_type: green _id: '20585' abstract: - lang: eng 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. 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." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. Foundations of Data Science. 2025;8:30-62. doi:10.3934/fods.2025003 apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2025). Chromatic alpha complexes. Foundations of Data Science. American Institute of Mathematical Sciences. https://doi.org/10.3934/fods.2025003 chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” Foundations of Data Science. American Institute of Mathematical Sciences, 2025. https://doi.org/10.3934/fods.2025003. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” Foundations of Data Science, vol. 8. American Institute of Mathematical Sciences, pp. 30–62, 2025. ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62. mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” Foundations of Data Science, vol. 8, American Institute of Mathematical Sciences, 2025, pp. 30–62, doi:10.3934/fods.2025003. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2025) 30–62. corr_author: '1' date_created: 2025-11-02T23:01:33Z date_published: 2025-03-01T00:00:00Z date_updated: 2025-11-04T12:25:47Z day: '01' department: - _id: HeEd doi: 10.3934/fods.2025003 ec_funded: 1 external_id: arxiv: - '2212.03128' intvolume: ' 8' language: - iso: eng month: '03' oa_version: Preprint page: 30-62 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Foundations of Data Science publication_identifier: eissn: - 2639-8001 publication_status: epub_ahead publisher: American Institute of Mathematical Sciences quality_controlled: '1' related_material: record: - id: '15091' relation: earlier_version status: public scopus_import: '1' status: public title: Chromatic alpha complexes type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 8 year: '2025' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '20657' abstract: - lang: eng text: 'The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². ' 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. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: János full_name: Pach, János id: E62E3130-B088-11EA-B919-BF823C25FEA4 last_name: Pach citation: ama: Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. Discrete & Computational Geometry. 2025. doi:10.1007/s00454-025-00796-5 apa: Edelsbrunner, H., & Pach, J. (2025). Maximum Betti numbers of Čech complexes. Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00796-5 chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-025-00796-5. ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” Discrete & Computational Geometry. Springer Nature, 2025. ista: Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete & Computational Geometry. mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” Discrete & Computational Geometry, Springer Nature, 2025, doi:10.1007/s00454-025-00796-5. short: H. Edelsbrunner, J. Pach, Discrete & Computational Geometry (2025). corr_author: '1' date_created: 2025-11-19T09:44:58Z date_published: 2025-11-10T00:00:00Z date_updated: 2025-12-01T15:19:21Z day: '10' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-025-00796-5 ec_funded: 1 external_id: arxiv: - '2310.14801' isi: - '001610592600001' has_accepted_license: '1' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.1007/s00454-025-00796-5 month: '11' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: Discrete & Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: epub_ahead publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '17146' relation: earlier_version status: public scopus_import: '1' status: public title: Maximum Betti numbers of Čech complexes tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2025' ... --- OA_place: repository OA_type: green _id: '20658' abstract: - lang: eng text: The medial axis of a smoothly embedded surface in R^3 consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided. alternative_title: - LNCS article_processing_charge: No arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Elizabeth R full_name: Stephenson, Elizabeth R id: 2D04F932-F248-11E8-B48F-1D18A9856A87 last_name: Stephenson orcid: 0000-0002-6862-208X - first_name: Martin H full_name: Thoresen, Martin H id: 47CB1472-F248-11E8-B48F-1D18A9856A87 last_name: Thoresen citation: ama: 'Edelsbrunner H, Stephenson ER, Thoresen MH. The mid-sphere cousin of the medial axis transform. In: 4th International Joint Conference on Discrete Geometry and Mathematical Morphology. Vol 16296. Springer Nature; 2025:133-147. doi:10.1007/978-3-032-09544-2_10' apa: 'Edelsbrunner, H., Stephenson, E. R., & Thoresen, M. H. (2025). The mid-sphere cousin of the medial axis transform. In 4th International Joint Conference on Discrete Geometry and Mathematical Morphology (Vol. 16296, pp. 133–147). Groningen, The Netherlands: Springer Nature. https://doi.org/10.1007/978-3-032-09544-2_10' chicago: Edelsbrunner, Herbert, Elizabeth R Stephenson, and Martin H Thoresen. “The Mid-Sphere Cousin of the Medial Axis Transform.” In 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, 16296:133–47. Springer Nature, 2025. https://doi.org/10.1007/978-3-032-09544-2_10. ieee: H. Edelsbrunner, E. R. Stephenson, and M. H. Thoresen, “The mid-sphere cousin of the medial axis transform,” in 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, Groningen, The Netherlands, 2025, vol. 16296, pp. 133–147. ista: 'Edelsbrunner H, Stephenson ER, Thoresen MH. 2025. The mid-sphere cousin of the medial axis transform. 4th International Joint Conference on Discrete Geometry and Mathematical Morphology. DGMM: Discrete Geometry and Mathematical Morphology, LNCS, vol. 16296, 133–147.' mla: Edelsbrunner, Herbert, et al. “The Mid-Sphere Cousin of the Medial Axis Transform.” 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, vol. 16296, Springer Nature, 2025, pp. 133–47, doi:10.1007/978-3-032-09544-2_10. short: H. Edelsbrunner, E.R. Stephenson, M.H. Thoresen, in:, 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, Springer Nature, 2025, pp. 133–147. conference: end_date: 2025-11-06 location: Groningen, The Netherlands name: 'DGMM: Discrete Geometry and Mathematical Morphology' start_date: 2025-11-03 date_created: 2025-11-23T23:01:37Z date_published: 2025-11-01T00:00:00Z date_updated: 2025-11-24T10:05:11Z day: '01' department: - _id: HeEd doi: 10.1007/978-3-032-09544-2_10 external_id: arxiv: - '2504.14743' intvolume: ' 16296' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2504.14743 month: '11' oa: 1 oa_version: Preprint page: 133-147 publication: 4th International Joint Conference on Discrete Geometry and Mathematical Morphology publication_identifier: eissn: - 1611-3349 isbn: - '9783032095435' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: The mid-sphere cousin of the medial axis transform type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 16296 year: '2025' ... --- OA_place: publisher OA_type: hybrid _id: '20729' abstract: - lang: eng text: 'Persistence modules (defined as a sequence of vector spaces and linear maps between them) are a key tool in topological data analysis. They are easy to interpret and fast to compute. However, when considering persistence maps (i.e. maps between persistence modules), these properties are lost. We propose a new invariant for persistence maps consisting of a partial matching such that: it is easy to interpret, it is more discriminative than the image of the persistence map, and can be calculated with cubical complexity.' acknowledgement: Álvaro Torras-Casas contract is funded by the French Agence Nationale de la Recherche through the project reference ANR-22-CPJ1-0047-01. Rocio Gonzalez-Diaz is partially funded by the European Union under grant agreement no. 101070028-2 (REXASI-PRO). article_processing_charge: Yes (in subscription journal) author: - first_name: Rocio full_name: Gonzalez-Diaz, Rocio last_name: Gonzalez-Diaz - first_name: Manuel full_name: Soriano Trigueros, Manuel id: 15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8 last_name: Soriano Trigueros orcid: 0000-0003-2449-1433 - first_name: Alvaro full_name: Torras-Casas, Alvaro last_name: Torras-Casas citation: ama: 'Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. Additive partial matchings for persistent homology. In: Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery; 2025:188-196. doi:10.1145/3747199.3747561' apa: 'Gonzalez-Diaz, R., Soriano Trigueros, M., & Torras-Casas, A. (2025). Additive partial matchings for persistent homology. In Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation (pp. 188–196). Guanajuato, Mexico: Association for Computing Machinery. https://doi.org/10.1145/3747199.3747561' chicago: Gonzalez-Diaz, Rocio, Manuel Soriano Trigueros, and Alvaro Torras-Casas. “Additive Partial Matchings for Persistent Homology.” In Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation, 188–96. Association for Computing Machinery, 2025. https://doi.org/10.1145/3747199.3747561. ieee: R. Gonzalez-Diaz, M. Soriano Trigueros, and A. Torras-Casas, “Additive partial matchings for persistent homology,” in Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation, Guanajuato, Mexico, 2025, pp. 188–196. ista: 'Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. 2025. Additive partial matchings for persistent homology. Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation. ISSAC: International Symposium on Symbolic and Algebraic Computation, 188–196.' mla: Gonzalez-Diaz, Rocio, et al. “Additive Partial Matchings for Persistent Homology.” Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation, Association for Computing Machinery, 2025, pp. 188–96, doi:10.1145/3747199.3747561. short: R. Gonzalez-Diaz, M. Soriano Trigueros, A. Torras-Casas, in:, Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation, Association for Computing Machinery, 2025, pp. 188–196. conference: end_date: 2025-08-01 location: Guanajuato, Mexico name: 'ISSAC: International Symposium on Symbolic and Algebraic Computation' start_date: 2025-07-28 corr_author: '1' date_created: 2025-12-07T23:02:01Z date_published: 2025-11-10T00:00:00Z date_updated: 2025-12-09T13:46:42Z day: '10' ddc: - '510' department: - _id: HeEd doi: 10.1145/3747199.3747561 file: - access_level: open_access checksum: 1c299cca165a20e2518afe4fda63dbf1 content_type: application/pdf creator: dernst date_created: 2025-12-09T13:43:17Z date_updated: 2025-12-09T13:43:17Z file_id: '20751' file_name: 2025_ISSAC_GonzalezDiaz.pdf file_size: 761617 relation: main_file success: 1 file_date_updated: 2025-12-09T13:43:17Z has_accepted_license: '1' language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: 188-196 publication: Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation publication_identifier: isbn: - '9798400720758' publication_status: published publisher: Association for Computing Machinery quality_controlled: '1' scopus_import: '1' status: public title: Additive partial matchings for persistent homology tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2025' ... --- OA_place: publisher OA_type: diamond PlanS_conform: '1' _id: '20867' abstract: - lang: eng text: We discuss the embeddability of subspaces of the Gromov–Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension n(n−1)∕2. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both nonembeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean–Hausdorff distance. acknowledgement: "The author was supported by the FWF Grant, Project number I4245-N35. The author would like to thank Thomas Weighill for the helpful discussions around Theorem 3.10, and Takamitsu Yamauchi for bringing to my attention the fundamental reference [35]. Furthermore, the author\r\nis thankful for the detailed and helpful comments of the reviewer of this manuscript." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Nicolò full_name: Zava, Nicolò id: c8b3499c-7a77-11eb-b046-aa368cbbf2ad last_name: Zava orcid: 0000-0001-8686-1888 citation: ama: Zava N. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. Algebraic & Geometric Topology. 2025;25(8):5153-5174. doi:10.2140/agt.2025.25.5153 apa: Zava, N. (2025). Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. Algebraic & Geometric Topology. Mathematical Sciences Publishers. https://doi.org/10.2140/agt.2025.25.5153 chicago: Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” Algebraic & Geometric Topology. Mathematical Sciences Publishers, 2025. https://doi.org/10.2140/agt.2025.25.5153. ieee: N. Zava, “Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces,” Algebraic & Geometric Topology, vol. 25, no. 8. Mathematical Sciences Publishers, pp. 5153–5174, 2025. ista: Zava N. 2025. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces. Algebraic & Geometric Topology. 25(8), 5153–5174. mla: Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff Space into Hilbert Spaces.” Algebraic & Geometric Topology, vol. 25, no. 8, Mathematical Sciences Publishers, 2025, pp. 5153–74, doi:10.2140/agt.2025.25.5153. short: N. Zava, Algebraic & Geometric Topology 25 (2025) 5153–5174. corr_author: '1' date_created: 2025-12-29T12:09:09Z date_published: 2025-11-20T00:00:00Z date_updated: 2026-01-05T12:19:09Z day: '20' ddc: - '500' department: - _id: HeEd doi: 10.2140/agt.2025.25.5153 external_id: arxiv: - '2303.04730' file: - access_level: open_access checksum: 1e05b4f17a44500ae1ae1e21bc636f6a content_type: application/pdf creator: dernst date_created: 2026-01-05T12:16:38Z date_updated: 2026-01-05T12:16:38Z file_id: '20943' file_name: 2025_AlgebraicGeomTopology_Zava.pdf file_size: 574389 relation: main_file success: 1 file_date_updated: 2026-01-05T12:16:38Z has_accepted_license: '1' intvolume: ' 25' issue: '8' language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: 5153-5174 project: - _id: 26AD5D90-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I04245 name: Algebraic Footprints of Geometric Features in Homology publication: Algebraic & Geometric Topology publication_identifier: eissn: - 1472-2739 issn: - 1472-2747 publication_status: published publisher: Mathematical Sciences Publishers quality_controlled: '1' scopus_import: '1' status: public title: Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 25 year: '2025' ... --- OA_place: repository _id: '21016' abstract: - lang: eng text: Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times. acknowledgement: "Y. B. B. and L. F. were funded by the Independent Research Fund Denmark, grant\r\nnumber 1026-00037. T. H. was partially supported by the European Research Council\r\n(ERC) Horizon 2020, grant number 788183." article_number: '2505.17858' article_processing_charge: No arxiv: 1 author: - first_name: Yossi full_name: Bleile, Yossi id: 920a7385-7995-11ef-9bfd-8c434cd8f3c2 last_name: Bleile orcid: 0000-0002-4861-9174 - first_name: Lisbeth full_name: Fajstrup, Lisbeth last_name: Fajstrup - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 - first_name: Anne Marie full_name: Svane, Anne Marie last_name: Svane - first_name: Søren Strandskov full_name: Sørensen, Søren Strandskov last_name: Sørensen citation: ama: Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. arXiv. doi:10.48550/arXiv.2505.17858 apa: Bokor Bleile, Y., Fajstrup, L., Heiss, T., Svane, A. M., & Sørensen, S. S. (n.d.). Identifying cobordisms using kernel persistence. arXiv. https://doi.org/10.48550/arXiv.2505.17858 chicago: Bokor Bleile, Yossi, Lisbeth Fajstrup, Teresa Heiss, Anne Marie Svane, and Søren Strandskov Sørensen. “Identifying Cobordisms Using Kernel Persistence.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2505.17858. ieee: Y. Bokor Bleile, L. Fajstrup, T. Heiss, A. M. Svane, and S. S. Sørensen, “Identifying cobordisms using kernel persistence,” arXiv. . ista: Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. arXiv, 2505.17858. mla: Bokor Bleile, Yossi, et al. “Identifying Cobordisms Using Kernel Persistence.” ArXiv, 2505.17858, doi:10.48550/arXiv.2505.17858. short: Y. Bokor Bleile, L. Fajstrup, T. Heiss, A.M. Svane, S.S. Sørensen, ArXiv (n.d.). date_created: 2026-01-20T10:12:21Z date_published: 2025-05-23T00:00:00Z date_updated: 2026-06-11T11:51:13Z day: '23' department: - _id: HeEd doi: 10.48550/arXiv.2505.17858 ec_funded: 1 external_id: arxiv: - '2505.17858' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2505.17858 month: '05' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended publication: arXiv publication_status: submitted status: public title: Identifying cobordisms using kernel persistence type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2025' ... --- OA_place: repository _id: '21050' abstract: - lang: eng text: "In 1873, James C. Maxwell conjectured that the electric field generated by $n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov and Shapiro, who also posed two additional interesting conjectures.\r\n In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to a conjecture of Gabrielov, Novikov and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges.\r\n Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find smaller bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day." article_processing_charge: No arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Christopher D full_name: Fillmore, Christopher D id: 35638A5C-AAC7-11E9-B0BF-5503E6697425 last_name: Fillmore - first_name: Gonçalo full_name: Olivera, Gonçalo last_name: Olivera citation: ama: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv. doi:10.48550/ARXIV.2501.05315 apa: Edelsbrunner, H., Fillmore, C. D., & Olivera, G. (n.d.). Counting equilibria of the electrostatic potential. arXiv. https://doi.org/10.48550/ARXIV.2501.05315 chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting Equilibria of the Electrostatic Potential.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2501.05315. ieee: H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the electrostatic potential,” arXiv. . ista: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv, 10.48550/ARXIV.2501.05315. mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” ArXiv, doi:10.48550/ARXIV.2501.05315. short: H. Edelsbrunner, C.D. Fillmore, G. Olivera, ArXiv (n.d.). corr_author: '1' date_created: 2026-01-27T14:29:27Z date_published: 2025-03-20T00:00:00Z date_updated: 2026-06-02T09:24:17Z day: '20' department: - _id: HeEd doi: 10.48550/ARXIV.2501.05315 external_id: arxiv: - '2501.05315' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2501.05315 month: '03' oa: 1 oa_version: Preprint publication: arXiv publication_status: draft related_material: record: - id: '21021' relation: dissertation_contains status: public - id: '21931' relation: later_version status: public status: public title: Counting equilibria of the electrostatic potential tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2025' ... --- OA_place: repository OA_type: green _id: '21253' abstract: - lang: eng text: We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n - 1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs. acknowledgement: A preliminary version of this note has been published in the proceedings of the 31st International Symposium on Graph Drawing and Network Visualization, Palermo, 2023. The authors would like to thank the anonymous referees for their valuable comments. article_number: '102186' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: János full_name: Pach, János last_name: Pach - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian - first_name: Patrick full_name: Schnider, Patrick last_name: Schnider citation: ama: Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. Computational Geometry. 2025;129. doi:10.1016/j.comgeo.2025.102186 apa: Pach, J., Saghafian, M., & Schnider, P. (2025). Decomposition of geometric graphs into star-forests. Computational Geometry. Elsevier. https://doi.org/10.1016/j.comgeo.2025.102186 chicago: Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” Computational Geometry. Elsevier, 2025. https://doi.org/10.1016/j.comgeo.2025.102186. ieee: J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” Computational Geometry, vol. 129. Elsevier, 2025. ista: Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186. mla: Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” Computational Geometry, vol. 129, 102186, Elsevier, 2025, doi:10.1016/j.comgeo.2025.102186. short: J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025). corr_author: '1' date_created: 2026-02-16T15:48:42Z date_published: 2025-12-01T00:00:00Z date_updated: 2026-04-16T09:12:36Z day: '01' department: - _id: HeEd doi: 10.1016/j.comgeo.2025.102186 external_id: arxiv: - '2306.13201' intvolume: ' 129' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2306.13201 month: '12' oa: 1 oa_version: Preprint publication: Computational Geometry publication_identifier: issn: - 0925-7721 publication_status: published publisher: Elsevier quality_controlled: '1' related_material: record: - id: '15012' relation: earlier_version status: public status: public title: Decomposition of geometric graphs into star-forests type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 129 year: '2025' ... --- OA_place: publisher OA_type: hybrid _id: '17149' 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. 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." article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Anton full_name: Nikitenko, Anton id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87 last_name: Nikitenko orcid: 0000-0002-0659-3201 citation: ama: Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. Discrete & Computational Geometry. 2025;73:490-499. doi:10.1007/s00454-024-00660-y apa: Edelsbrunner, H., & Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-024-00660-y chicago: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-024-00660-y. ieee: H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” Discrete & Computational Geometry, vol. 73. Springer Nature, pp. 490–499, 2025. ista: Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete & Computational Geometry. 73, 490–499. mla: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” Discrete & Computational Geometry, vol. 73, Springer Nature, 2025, pp. 490–99, doi:10.1007/s00454-024-00660-y. short: H. Edelsbrunner, A. Nikitenko, Discrete & Computational Geometry 73 (2025) 490–499. corr_author: '1' date_created: 2024-06-16T22:01:07Z date_published: 2025-03-01T00:00:00Z date_updated: 2026-02-16T12:18:50Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-024-00660-y ec_funded: 1 external_id: arxiv: - '2012.03350' isi: - '001238566200004' pmid: - '39974750' file: - access_level: open_access checksum: ffb0c818222138f9f113f4bbea41e834 content_type: application/pdf creator: dernst date_created: 2025-04-23T07:31:32Z date_updated: 2025-04-23T07:31:32Z file_id: '19610' file_name: 2025_DiscreteComputGeom_EdelsbrunnerHe.pdf file_size: 283443 relation: main_file success: 1 file_date_updated: 2025-04-23T07:31:32Z has_accepted_license: '1' intvolume: ' 73' isi: 1 language: - iso: eng month: '03' oa: 1 oa_version: Published Version page: 490-499 pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Discrete & Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Average and expected distortion of Voronoi paths and scapes tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 73 year: '2025' ... --- _id: '18097' 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 \U0001D4AE (either manifold or general set with positive reach) such that one can infer the homotopy of \U0001D4AE 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." 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." alternative_title: - LIPIcs article_number: '87' article_processing_charge: Yes author: - first_name: Dominique full_name: Attali, Dominique last_name: Attali - first_name: Hana full_name: Kourimska, Hana id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E last_name: Kourimska orcid: 0000-0001-7841-0091 - first_name: Christopher D full_name: Fillmore, Christopher D id: 35638A5C-AAC7-11E9-B0BF-5503E6697425 last_name: Fillmore - first_name: Ishika full_name: Ghosh, Ishika id: ee449b28-344d-11ef-a6d5-9ca430e9e9ff last_name: Ghosh - first_name: Andre full_name: Lieutier, Andre last_name: Lieutier - first_name: Elizabeth R full_name: Stephenson, Elizabeth R id: 2D04F932-F248-11E8-B48F-1D18A9856A87 last_name: Stephenson orcid: 0000-0002-6862-208X - first_name: Mathijs full_name: Wintraecken, Mathijs id: 307CFBC8-F248-11E8-B48F-1D18A9856A87 last_name: Wintraecken orcid: 0000-0002-7472-2220 citation: ama: 'Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality construction for homotopy inference (media exposition). In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.87' apa: 'Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., & Wintraecken, M. (2024). The ultimate frontier: An optimality construction for homotopy inference (media exposition). In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.87' chicago: 'Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, Andre Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.87.' ieee: 'D. Attali et al., “The ultimate frontier: An optimality construction for homotopy inference (media exposition),” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293.' 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.' mla: 'Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” 40th International Symposium on Computational Geometry, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.87.' 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. conference: end_date: 2024-06-14 location: Athens, Greece name: 'SoCG: Symposium on Computational Geometry' start_date: 2024-06-11 corr_author: '1' date_created: 2024-09-19T10:29:48Z date_published: 2024-06-06T00:00:00Z date_updated: 2025-04-15T07:16:58Z day: '06' ddc: - '000' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2024.87 ec_funded: 1 file: - access_level: open_access checksum: 9355c2e60b8ec285e1b22719c5b73f1a content_type: application/pdf creator: dernst date_created: 2024-09-19T10:30:37Z date_updated: 2024-09-19T10:30:37Z file_id: '18098' file_name: 2024_LIPICs_Attali.pdf file_size: 3507177 relation: main_file success: 1 file_date_updated: 2024-09-19T10:30:37Z has_accepted_license: '1' intvolume: ' 293' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 260C2330-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '754411' name: ISTplus - Postdoctoral Fellowships - _id: fc390959-9c52-11eb-aca3-afa58bd282b2 grant_number: M03073 name: Learning and triangulating manifolds via collapses publication: 40th International Symposium on Computational Geometry publication_identifier: isbn: - '9783959773164' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' status: public title: 'The ultimate frontier: An optimality construction for homotopy inference (media exposition)' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 293 year: '2024' ... --- OA_place: publisher OA_type: gold _id: '18556' abstract: - lang: eng text: Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25. 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. alternative_title: - LIPIcs article_number: '3' article_processing_charge: Yes arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: 32nd International Symposium on Graph Drawing and Network Visualization. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.GD.2024.3' apa: 'Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In 32nd International Symposium on Graph Drawing and Network Visualization (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.GD.2024.3' chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In 32nd International Symposium on Graph Drawing and Network Visualization, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.GD.2024.3. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in 32nd International Symposium on Graph Drawing and Network Visualization, Vienna, Austria, 2024, vol. 320. 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.' mla: Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” 32nd International Symposium on Graph Drawing and Network Visualization, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.GD.2024.3. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, in:, 32nd International Symposium on Graph Drawing and Network Visualization, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. conference: end_date: 2024-09-20 location: Vienna, Austria name: 'GD: Graph Drawing and Network Visualization' start_date: 2024-09-18 corr_author: '1' date_created: 2024-11-17T23:01:47Z date_published: 2024-10-28T00:00:00Z date_updated: 2025-12-02T13:50:50Z day: '28' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.GD.2024.3 ec_funded: 1 external_id: arxiv: - '2403.10204' isi: - '001540278400001' file: - access_level: open_access checksum: 5f9b35e115c3d375e99be78da9054cb4 content_type: application/pdf creator: dernst date_created: 2024-11-18T07:49:25Z date_updated: 2024-11-18T07:49:25Z file_id: '18560' file_name: 2024_LIPIcs_CultreradiMontesano.pdf file_size: 908541 relation: main_file success: 1 file_date_updated: 2024-11-18T07:49:25Z has_accepted_license: '1' intvolume: ' 320' isi: 1 language: - iso: eng month: '10' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: 32nd International Symposium on Graph Drawing and Network Visualization publication_identifier: isbn: - '9783959773430' issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: The Euclidean MST-ratio for bi-colored lattices tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 320 year: '2024' ... --- DOAJ_listed: '1' OA_place: publisher OA_type: gold _id: '18604' abstract: - lang: eng text: 'A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a randomized FPT-time algorithm where the parameter is the number of popular faces.' acknowledgement: "This work was initiated at the 16th European Research Week on Geometric Graphs in Strobl in 2019. A.W. has been supported by the Austrian Science Fund (FWF): W1230. S.T. has been funded by the Vienna Science and Technology Fund (WWTF) [10.47379/ICT19035] and by the NWO Gravitation project NETWORKS under grant no. 024.002.003. Part of the work was done while A.W. was emplyed at Graz University of Technology. Preliminary versions of this work have been presented at the 38th European Workshop on Computational Geometry (EuroCG\r\n2022) in Perugia [10] and at the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023) in Isola delle Femmine [11]." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Phoebe full_name: De Nooijer, Phoebe last_name: De Nooijer - first_name: Soeren full_name: Terziadis, Soeren last_name: Terziadis - first_name: Alexandra full_name: Weinberger, Alexandra last_name: Weinberger - first_name: Zuzana full_name: Masárová, Zuzana id: 45CFE238-F248-11E8-B48F-1D18A9856A87 last_name: Masárová orcid: 0000-0002-6660-1322 - first_name: Tamara full_name: Mchedlidze, Tamara last_name: Mchedlidze - first_name: Maarten full_name: Löffler, Maarten last_name: Löffler - first_name: Günter full_name: Rote, Günter last_name: Rote citation: ama: De Nooijer P, Terziadis S, Weinberger A, et al. Removing popular faces in curve arrangements. Journal of Graph Algorithms and Applications. 2024;28(2):47-82. doi:10.7155/jgaa.v28i2.2988 apa: De Nooijer, P., Terziadis, S., Weinberger, A., Masárová, Z., Mchedlidze, T., Löffler, M., & Rote, G. (2024). Removing popular faces in curve arrangements. Journal of Graph Algorithms and Applications. Brown University. https://doi.org/10.7155/jgaa.v28i2.2988 chicago: De Nooijer, Phoebe, Soeren Terziadis, Alexandra Weinberger, Zuzana Masárová, Tamara Mchedlidze, Maarten Löffler, and Günter Rote. “Removing Popular Faces in Curve Arrangements.” Journal of Graph Algorithms and Applications. Brown University, 2024. https://doi.org/10.7155/jgaa.v28i2.2988. ieee: P. De Nooijer et al., “Removing popular faces in curve arrangements,” Journal of Graph Algorithms and Applications, vol. 28, no. 2. Brown University, pp. 47–82, 2024. ista: De Nooijer P, Terziadis S, Weinberger A, Masárová Z, Mchedlidze T, Löffler M, Rote G. 2024. Removing popular faces in curve arrangements. Journal of Graph Algorithms and Applications. 28(2), 47–82. mla: De Nooijer, Phoebe, et al. “Removing Popular Faces in Curve Arrangements.” Journal of Graph Algorithms and Applications, vol. 28, no. 2, Brown University, 2024, pp. 47–82, doi:10.7155/jgaa.v28i2.2988. short: P. De Nooijer, S. Terziadis, A. Weinberger, Z. Masárová, T. Mchedlidze, M. Löffler, G. Rote, Journal of Graph Algorithms and Applications 28 (2024) 47–82. corr_author: '1' date_created: 2024-12-01T23:01:54Z date_published: 2024-11-03T00:00:00Z date_updated: 2024-12-03T09:49:18Z day: '03' ddc: - '510' department: - _id: UlWa - _id: HeEd doi: 10.7155/jgaa.v28i2.2988 external_id: arxiv: - '2202.12175' file: - access_level: open_access checksum: be611da6f9d790dc980d6fb7283fe889 content_type: application/pdf creator: dernst date_created: 2024-12-03T09:45:00Z date_updated: 2024-12-03T09:45:00Z file_id: '18609' file_name: 2024_JourGraphAlgorithms_deNooijer.pdf file_size: 1582493 relation: main_file success: 1 file_date_updated: 2024-12-03T09:45:00Z has_accepted_license: '1' intvolume: ' 28' issue: '2' language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: 47-82 publication: Journal of Graph Algorithms and Applications publication_identifier: issn: - 1526-1719 publication_status: published publisher: Brown University quality_controlled: '1' scopus_import: '1' status: public title: Removing popular faces in curve arrangements tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 28 year: '2024' ... --- OA_place: publisher _id: '18667' abstract: - lang: eng text: "Many chemical and physical properties of materials are determined by the material’s shape,\r\nfor example the size of its pores and the width of its tunnels. This makes materials science\r\na prime application area for geometrical and topological methods. Nevertheless many\r\nmethods in topological data analysis have not been satisfyingly extended to the needs of\r\nmaterials science. This thesis provides new methods and new mathematical theorems\r\ntargeted at those specific needs by answering four different research questions. While the\r\nmotivation for each of the research questions arises from materials science, the methods\r\nare versatile and can be applied in different areas as well. \r\n\r\nThe first research question is concerned with image data, for example a three-dimensional\r\ncomputed tomography (CT) scan of a material, like sand or stone. There are two commonly\r\nused topologies for digital images and depending on the application either of them might be\r\nrequired. However, software for computing the topological data analysis method persistence\r\nhomology, usually supports only one of the two topologies. We answer the question how to\r\ncompute persistent homology of an image with respect to one of the two topologies using\r\nsoftware that is intended for the other topology. \r\n\r\nThe second research question is concerned with image data as well, and asks how much\r\nof the topological information of an image is lost when the resolution is coarsened. As\r\ncomputer tomography scanners are more expensive the higher the resolution, it is an\r\nimportant question in materials science to know which resolution is enough to get satisfying\r\npersistent homology. We give theoretical bounds on the information loss based on different\r\ngeometrical properties of the object to be scanned. In addition, we conduct experiments on\r\nsand and stone CT image data. \r\n\r\nThe third research question is motivated by comparing crystalline materials efficiently. As\r\nthe atoms within a crystal repeat periodically, crystalline materials are either modeled by\r\nunmanageable infinite periodic point sets, or by one of their fundamental domains, which is\r\nunstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with\r\nappropriate properties such that comparing the crystals can be eased by comparing the\r\nfingerprints instead. We define the density fingerprint and prove the necessary properties. \r\n\r\nThe fourth research question is motivated by studying the hole-structure or connectedness,\r\ni.e. persistent homology or merge trees, of crystalline materials. A common way to deal\r\nwith periodicity is to take a fundamental domain and identify opposite boundaries to form a\r\ntorus. However, computing persistent homology or merge trees on that torus loses some\r\nof the information materials scientists are interested in and is additionally not stable under\r\ncertain noise. We therefore decorate the merge tree stemming from the torus with additional\r\ninformation describing the density and growth rate of the periodic copies of a component\r\nwithin a growing spherical window. We prove all desired properties, like stability and efficient\r\ncomputability." acknowledgement: "I was supported by the European Research Council (ERC) Horizon 2020 project\r\n“Alpha Shape Theory Extended” No. 788183 and by the Pöttinger Scholarship. In addition,\r\nI am very thankful for having been able to attend the second Workshop for Women in\r\nComputational Topology in July 2019, funded by the Mathematical Sciences Institute at\r\nANU, the US National Science Foundation through the award CCF-1841455, the Australian\r\nMathematical Sciences Institute and the Association for Women in Mathematics. Two of the\r\nprojects presented in this thesis started there. One of them reached completion thanks to\r\nfunding from the MSRI Summer Research in Mathematics program awarded to me and my\r\ncollaborators in 2020." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 citation: ama: Heiss T. New methods for applying topological data analysis to materials science. 2024. doi:10.15479/at:ista:18667 apa: Heiss, T. (2024). New methods for applying topological data analysis to materials science. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18667 chicago: Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials Science.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18667. ieee: T. Heiss, “New methods for applying topological data analysis to materials science,” Institute of Science and Technology Austria, 2024. ista: Heiss T. 2024. New methods for applying topological data analysis to materials science. Institute of Science and Technology Austria. mla: Heiss, Teresa. New Methods for Applying Topological Data Analysis to Materials Science. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18667. short: T. Heiss, New Methods for Applying Topological Data Analysis to Materials Science, Institute of Science and Technology Austria, 2024. corr_author: '1' date_created: 2024-12-17T16:17:55Z date_published: 2024-12-17T00:00:00Z date_updated: 2026-04-07T12:54:10Z day: '17' ddc: - '514' - '516' - '004' degree_awarded: PhD department: - _id: GradSch - _id: HeEd doi: 10.15479/at:ista:18667 ec_funded: 1 file: - access_level: open_access checksum: 247bb057aed2fba1cd4711917aaa2d77 content_type: application/pdf creator: theiss date_created: 2024-12-19T10:24:46Z date_updated: 2024-12-19T10:24:46Z file_id: '18686' file_name: Teresa_Heiss_PhD_Thesis_final.pdf file_size: 7752253 relation: main_file success: 1 - access_level: closed checksum: 9648b45c07a008ee11a07f99856a139d content_type: application/zip creator: theiss date_created: 2024-12-19T10:24:50Z date_updated: 2024-12-19T10:24:50Z file_id: '18687' file_name: PhD_Thesis.zip file_size: 17197731 relation: source_file file_date_updated: 2024-12-19T10:24:50Z has_accepted_license: '1' keyword: - persistent homology - topological data analysis - periodic - crystalline materials - images - fingerprint language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: '111' project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended publication_identifier: isbn: - 978-3-99078-052-7 issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '10828' relation: part_of_dissertation status: public - id: '11440' relation: part_of_dissertation status: public - id: '18673' relation: part_of_dissertation status: public - id: '9345' relation: part_of_dissertation status: public status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: New methods for applying topological data analysis to materials science tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: dissertation user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd year: '2024' ... --- OA_place: repository _id: '18673' abstract: - lang: eng 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" 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." article_processing_charge: No arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 citation: ama: Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv. doi:10.48550/arXiv.2408.16575 apa: Edelsbrunner, H., & Heiss, T. (n.d.). Merge trees of periodic filtrations. arXiv. https://doi.org/10.48550/arXiv.2408.16575 chicago: Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2408.16575. ieee: H. Edelsbrunner and T. Heiss, “Merge trees of periodic filtrations,” arXiv. . ista: Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv, 10.48550/arXiv.2408.16575. mla: Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” ArXiv, doi:10.48550/arXiv.2408.16575. short: H. Edelsbrunner, T. Heiss, ArXiv (n.d.). corr_author: '1' date_created: 2024-12-18T14:06:57Z date_published: 2024-08-29T00:00:00Z date_updated: 2026-04-07T12:54:09Z day: '29' department: - _id: HeEd doi: 10.48550/arXiv.2408.16575 ec_funded: 1 external_id: arxiv: - '2408.16575' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2408.16575 month: '08' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: arXiv publication_status: draft related_material: record: - id: '18667' relation: dissertation_contains status: public status: public title: Merge trees of periodic filtrations tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- OA_place: repository _id: '18981' 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. 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." article_processing_charge: No arxiv: 1 author: - first_name: Adam full_name: Brown, Adam last_name: Brown - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 citation: ama: Brown A, Draganov O. Discrete microlocal Morse theory. arXiv. doi:10.48550/arXiv.2209.14993 apa: Brown, A., & Draganov, O. (n.d.). Discrete microlocal Morse theory. arXiv. https://doi.org/10.48550/arXiv.2209.14993 chicago: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2209.14993. ieee: A. Brown and O. Draganov, “Discrete microlocal Morse theory,” arXiv. . ista: Brown A, Draganov O. Discrete microlocal Morse theory. arXiv, 10.48550/arXiv.2209.14993. mla: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” ArXiv, doi:10.48550/arXiv.2209.14993. short: A. Brown, O. Draganov, ArXiv (n.d.). corr_author: '1' date_created: 2025-01-31T17:03:04Z date_published: 2024-06-09T00:00:00Z date_updated: 2026-04-07T11:47:29Z day: '09' department: - _id: HeEd doi: 10.48550/arXiv.2209.14993 ec_funded: 1 external_id: arxiv: - '2209.14993' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2209.14993 month: '06' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: arXiv publication_status: draft related_material: record: - id: '20323' relation: later_version status: public - id: '18979' relation: dissertation_contains status: public status: public title: Discrete microlocal Morse theory tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- OA_place: publisher OA_type: gold _id: '18998' abstract: - lang: eng text: Word embeddings represent language vocabularies as clouds of d-dimensional points. We investigate how information is conveyed by the general shape of these clouds, instead of representing the semantic meaning of each token. Specifically, we use the notion of persistent homology from topological data analysis (TDA) to measure the distances between language pairs from the shape of their unlabeled embeddings. These distances quantify the degree of non-isometry of the embeddings. To distinguish whether these differences are random training errors or capture real information about the languages, we use the computed distance matrices to construct language phylogenetic trees over 81 Indo-European languages. Careful evaluation shows that our reconstructed trees exhibit strong and statistically-significant similarities to the reference. article_processing_charge: No arxiv: 1 author: - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Steven full_name: Skiena, Steven last_name: Skiena citation: ama: 'Draganov O, Skiena S. The shape of word embeddings: Quantifying non-isometry with topological data analysis. In: Findings of the Association for Computational Linguistics: EMNLP 2024. Association for Computational Linguistics; 2024:12080-12099. doi:10.18653/v1/2024.findings-emnlp.705' apa: 'Draganov, O., & Skiena, S. (2024). The shape of word embeddings: Quantifying non-isometry with topological data analysis. In Findings of the Association for Computational Linguistics: EMNLP 2024 (pp. 12080–12099). Miami, FL, United States: Association for Computational Linguistics. https://doi.org/10.18653/v1/2024.findings-emnlp.705' chicago: 'Draganov, Ondrej, and Steven Skiena. “The Shape of Word Embeddings: Quantifying Non-Isometry with Topological Data Analysis.” In Findings of the Association for Computational Linguistics: EMNLP 2024, 12080–99. Association for Computational Linguistics, 2024. https://doi.org/10.18653/v1/2024.findings-emnlp.705.' ieee: 'O. Draganov and S. Skiena, “The shape of word embeddings: Quantifying non-isometry with topological data analysis,” in Findings of the Association for Computational Linguistics: EMNLP 2024, Miami, FL, United States, 2024, pp. 12080–12099.' ista: 'Draganov O, Skiena S. 2024. The shape of word embeddings: Quantifying non-isometry with topological data analysis. Findings of the Association for Computational Linguistics: EMNLP 2024. EMNLP: Conference on Empirical Methods in Natural Language Processing, 12080–12099.' mla: 'Draganov, Ondrej, and Steven Skiena. “The Shape of Word Embeddings: Quantifying Non-Isometry with Topological Data Analysis.” Findings of the Association for Computational Linguistics: EMNLP 2024, Association for Computational Linguistics, 2024, pp. 12080–99, doi:10.18653/v1/2024.findings-emnlp.705.' short: 'O. Draganov, S. Skiena, in:, Findings of the Association for Computational Linguistics: EMNLP 2024, Association for Computational Linguistics, 2024, pp. 12080–12099.' conference: end_date: 2024-11-16 location: Miami, FL, United States name: 'EMNLP: Conference on Empirical Methods in Natural Language Processing' start_date: 2024-11-12 corr_author: '1' date_created: 2025-02-04T16:19:28Z date_published: 2024-11-01T00:00:00Z date_updated: 2025-02-10T08:21:37Z day: '01' ddc: - '500' department: - _id: GradSch - _id: HeEd doi: 10.18653/v1/2024.findings-emnlp.705 external_id: arxiv: - '2404.00500' file: - access_level: open_access checksum: f4416a5962194f0181ab0dc7f9ef93c0 content_type: application/pdf creator: dernst date_created: 2025-02-10T08:20:34Z date_updated: 2025-02-10T08:20:34Z file_id: '19016' file_name: 2024_EMNLP_Draganov.pdf file_size: 1312638 relation: main_file success: 1 file_date_updated: 2025-02-10T08:20:34Z has_accepted_license: '1' language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: 12080-12099 publication: 'Findings of the Association for Computational Linguistics: EMNLP 2024' publication_status: published publisher: Association for Computational Linguistics quality_controlled: '1' scopus_import: '1' status: public title: 'The shape of word embeddings: Quantifying non-isometry with topological data analysis' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- OA_place: repository OA_type: green _id: '18999' abstract: - lang: eng text: Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry. article_number: '2406.04102' article_processing_charge: No arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv. doi:10.48550/ARXIV.2406.04102 apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). Chromatic topological data analysis. arXiv. https://doi.org/10.48550/ARXIV.2406.04102 chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Topological Data Analysis.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2406.04102. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic topological data analysis,” arXiv. . ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv, 2406.04102. mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Topological Data Analysis.” ArXiv, 2406.04102, doi:10.48550/ARXIV.2406.04102. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.). corr_author: '1' date_created: 2025-02-04T16:21:21Z date_published: 2024-06-06T00:00:00Z date_updated: 2025-02-10T08:14:27Z day: '06' ddc: - '510' department: - _id: GradSch - _id: HeEd doi: 10.48550/ARXIV.2406.04102 external_id: arxiv: - '2406.04102' has_accepted_license: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2406.04102 month: '06' oa: 1 oa_version: Preprint publication: arXiv publication_status: submitted status: public title: Chromatic topological data analysis tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ...