--- OA_place: publisher OA_type: gold _id: '22002' abstract: - lang: eng text: Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c⋅n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex. acknowledgement: "Jakub Leśkiewicz wants to thank his supervisor, Prof. Marian Mrozek, forscientific guidance, patience, and opportunity to delay the rest of his duties while writing this work.\r\nThe author also extends thanks to his entire family, to Zuzanna Świątek, and to Mikołaj Kardyś,\r\nBEng, MSc, for providing meals during the most intensive periods of work. Jakub Leśkiewicz: The research was partially funded by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874 and Opus Grant 2025/57/B/ST1/00550. Bartosz Furmanek: The research was partially funded by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874 and Opus Grant 2025/57/B/ST1/00550. Michał Lipiński: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. \r\nDmitriy Morozov: This work was supported in part by the U.S. Department of Energy, Office\r\nof Science, Office of Advanced Scientific Computing Research, under Contract No. DE-AC02-\r\n05CH11231." alternative_title: - LIPIcs article_number: 72:1-72:17 article_processing_charge: No arxiv: 1 author: - first_name: Jakub full_name: Leśkiewicz, Jakub last_name: Leśkiewicz - first_name: Bartosz full_name: Furmanek, Bartosz last_name: Furmanek - first_name: Michał full_name: Lipiński, Michał id: dfffb474-4317-11ee-8f5c-fe3fc95a425e last_name: Lipiński orcid: 0000-0001-9789-9750 - first_name: Dmitriy full_name: Morozov, Dmitriy last_name: Morozov citation: ama: 'Leśkiewicz J, Furmanek B, Lipiński M, Morozov D. Topological simplification guided by forbidden regions. In: 42nd International Symposium on Computational Geometry. Vol 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2026. doi:10.4230/LIPIcs.SoCG.2026.72' apa: 'Leśkiewicz, J., Furmanek, B., Lipiński, M., & Morozov, D. (2026). Topological simplification guided by forbidden regions. In 42nd International Symposium on Computational Geometry (Vol. 367). New Brunswick, NJ, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2026.72' chicago: Leśkiewicz, Jakub, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov. “Topological Simplification Guided by Forbidden Regions.” In 42nd International Symposium on Computational Geometry, Vol. 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. https://doi.org/10.4230/LIPIcs.SoCG.2026.72. ieee: J. Leśkiewicz, B. Furmanek, M. Lipiński, and D. Morozov, “Topological simplification guided by forbidden regions,” in 42nd International Symposium on Computational Geometry, New Brunswick, NJ, United States, 2026, vol. 367. ista: 'Leśkiewicz J, Furmanek B, Lipiński M, Morozov D. 2026. Topological simplification guided by forbidden regions. 42nd International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 367, 72:1-72:17.' mla: Leśkiewicz, Jakub, et al. “Topological Simplification Guided by Forbidden Regions.” 42nd International Symposium on Computational Geometry, vol. 367, 72:1-72:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026, doi:10.4230/LIPIcs.SoCG.2026.72. short: J. Leśkiewicz, B. Furmanek, M. Lipiński, D. Morozov, in:, 42nd International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. conference: end_date: 2026-06-05 location: New Brunswick, NJ, United States name: 'SoCG: Symposium on Computational Geometry' start_date: 2026-06-02 corr_author: '1' das_tickbox: '0' date_created: 2026-06-14T22:01:43Z date_published: 2026-05-27T00:00:00Z date_updated: 2026-06-22T07:45:36Z day: '27' ddc: - '500' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2026.72 ec_funded: 1 external_id: arxiv: - '2603.16416' file: - access_level: open_access checksum: 3be91c06fdf716c8735b6af64a09a921 content_type: application/pdf creator: dernst date_created: 2026-06-22T07:39:21Z date_updated: 2026-06-22T07:39:21Z file_id: '22110' file_name: 2026_LIPIcSSoCG_Leskiewicz.pdf file_size: 2052749 relation: main_file success: 1 file_date_updated: 2026-06-22T07:39:21Z has_accepted_license: '1' intvolume: ' 367' keyword: - persistent homology - topological simplification - depth posets language: - iso: eng month: '05' oa: 1 oa_version: Published Version project: - _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c call_identifier: H2020 grant_number: '101034413' name: 'IST-BRIDGE: International postdoctoral program' publication: 42nd International Symposium on Computational Geometry publication_identifier: eissn: - 1868-8969 isbn: - '9783959774185' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: Topological simplification guided by forbidden regions 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: 367 year: '2026' ... --- OA_place: publisher OA_type: gold _id: '22003' abstract: - lang: eng text: 'Let G be a finite, connected metric graph and let X be a subset of G. If X is sufficiently dense in G, we show that the Gromov-Hausdorff distance matches the Hausdorff distance, namely d_GH(G,X) = d_H(G,X). When the metric graph is the circle G = S¹ with circumference 2π, a recent study established the equality d_GH(S¹,X) = d_H(S¹,X) whenever d_GH(S¹,X) < π/6. Our results relax this hypothesis to d_GH(S¹,X) < π/3, and furthermore, we show that the constant π/3 is the best possible. We lower bound the Gromov-Hausdorff distance d_GH(G,X) by the Hausdorff distance d_H(G,X) via a simple topological obstruction: the existence of a possibly discontinuous function f: G → X with too small distortion contradicts the connectedness of G.' acknowledgement: "Funding Henry Adams: Simons Foundation Travel Support for Mathematicians.\r\nŽiga Virk: Slovene research agency grant P1-0292.\r\nNicolò Zava: FWF Grant, Project number I4245-N35.\r\n" alternative_title: - LIPIcs article_number: 3:1-3:16 article_processing_charge: Yes arxiv: 1 author: - first_name: Henry full_name: Adams, Henry last_name: Adams - first_name: Sushovan full_name: Majhi, Sushovan last_name: Majhi - first_name: Fedor full_name: Manin, Fedor last_name: Manin - first_name: Ziga full_name: Virk, Ziga id: 2E36B656-F248-11E8-B48F-1D18A9856A87 last_name: Virk - first_name: Nicolò full_name: Zava, Nicolò id: c8b3499c-7a77-11eb-b046-aa368cbbf2ad last_name: Zava orcid: 0000-0001-8686-1888 citation: ama: 'Adams H, Majhi S, Manin F, Virk Z, Zava N. Lower bounding the Gromov–Hausdorff distance in metric graphs. In: 42nd International Symposium on Computational Geometry. Vol 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2026. doi:10.4230/LIPIcs.SoCG.2026.3' apa: 'Adams, H., Majhi, S., Manin, F., Virk, Z., & Zava, N. (2026). Lower bounding the Gromov–Hausdorff distance in metric graphs. In 42nd International Symposium on Computational Geometry (Vol. 367). New Brunswick, NJ, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2026.3' chicago: Adams, Henry, Sushovan Majhi, Fedor Manin, Ziga Virk, and Nicolò Zava. “Lower Bounding the Gromov–Hausdorff Distance in Metric Graphs.” In 42nd International Symposium on Computational Geometry, Vol. 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. https://doi.org/10.4230/LIPIcs.SoCG.2026.3. ieee: H. Adams, S. Majhi, F. Manin, Z. Virk, and N. Zava, “Lower bounding the Gromov–Hausdorff distance in metric graphs,” in 42nd International Symposium on Computational Geometry, New Brunswick, NJ, United States, 2026, vol. 367. ista: 'Adams H, Majhi S, Manin F, Virk Z, Zava N. 2026. Lower bounding the Gromov–Hausdorff distance in metric graphs. 42nd International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 367, 3:1-3:16.' mla: Adams, Henry, et al. “Lower Bounding the Gromov–Hausdorff Distance in Metric Graphs.” 42nd International Symposium on Computational Geometry, vol. 367, 3:1-3:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026, doi:10.4230/LIPIcs.SoCG.2026.3. short: H. Adams, S. Majhi, F. Manin, Z. Virk, N. Zava, in:, 42nd International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. conference: end_date: 2026-06-05 location: New Brunswick, NJ, United States name: 'SoCG: Symposium on Computational Geometry' start_date: 2026-06-02 corr_author: '1' das_tickbox: '0' date_created: 2026-06-14T22:01:44Z date_published: 2026-05-27T00:00:00Z date_updated: 2026-06-22T08:49:17Z day: '27' ddc: - '500' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2026.3 external_id: arxiv: - '2411.09182' file: - access_level: open_access checksum: 25d27c016409563196b8aecfe5bfdf41 content_type: application/pdf creator: dernst date_created: 2026-06-22T08:43:47Z date_updated: 2026-06-22T08:43:47Z file_id: '22115' file_name: 2026_LIPIcSSoCG_Adams.pdf file_size: 1091310 relation: main_file success: 1 file_date_updated: 2026-06-22T08:43:47Z has_accepted_license: '1' intvolume: ' 367' keyword: - Gromov–Hausdorff distance - distortion - connectedness - Borsuk–Ulam theorem language: - iso: eng month: '05' oa: 1 oa_version: Published Version project: - _id: 26AD5D90-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I04245 name: Algebraic Footprints of Geometric Features in Homology publication: 42nd International Symposium on Computational Geometry publication_identifier: eissn: - 1868-8969 isbn: - '9783959774185' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: Lower bounding the Gromov–Hausdorff distance in metric graphs 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: 367 year: '2026' ... --- OA_place: repository OA_type: green _id: '20980' abstract: - lang: eng text: 'Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix. These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures—such as attractors, repellers, and orbits—in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of Dey et al. and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.' acknowledgement: "This research was supported by NSF grants DMS-2301360 and CCF-2437030 as well as from the European Union's Horizon 2020 research and innovation programme under Marie Sk\\lodowska-Curie grant 101034413.\r\n" article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Tamal K. full_name: Dey, Tamal K. last_name: Dey - first_name: Andrew full_name: Haas, Andrew last_name: Haas - first_name: Michał full_name: Lipiński, Michał id: dfffb474-4317-11ee-8f5c-fe3fc95a425e last_name: Lipiński orcid: 0000-0001-9789-9750 citation: ama: Dey TK, Haas A, Lipiński M. Computing a connection matrix and persistence efficiently from a morse decomposition. SIAM Journal on Applied Dynamical Systems. 2026;25(1):108-130. doi:10.1137/25m1739406 apa: Dey, T. K., Haas, A., & Lipiński, M. (2026). Computing a connection matrix and persistence efficiently from a morse decomposition. SIAM Journal on Applied Dynamical Systems. Society for Industrial & Applied Mathematics. https://doi.org/10.1137/25m1739406 chicago: Dey, Tamal K., Andrew Haas, and Michał Lipiński. “Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition.” SIAM Journal on Applied Dynamical Systems. Society for Industrial & Applied Mathematics, 2026. https://doi.org/10.1137/25m1739406. ieee: T. K. Dey, A. Haas, and M. Lipiński, “Computing a connection matrix and persistence efficiently from a morse decomposition,” SIAM Journal on Applied Dynamical Systems, vol. 25, no. 1. Society for Industrial & Applied Mathematics, pp. 108–130, 2026. ista: Dey TK, Haas A, Lipiński M. 2026. Computing a connection matrix and persistence efficiently from a morse decomposition. SIAM Journal on Applied Dynamical Systems. 25(1), 108–130. mla: Dey, Tamal K., et al. “Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition.” SIAM Journal on Applied Dynamical Systems, vol. 25, no. 1, Society for Industrial & Applied Mathematics, 2026, pp. 108–30, doi:10.1137/25m1739406. short: T.K. Dey, A. Haas, M. Lipiński, SIAM Journal on Applied Dynamical Systems 25 (2026) 108–130. date_created: 2026-01-12T11:17:06Z date_published: 2026-01-01T00:00:00Z date_updated: 2026-01-20T07:40:39Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1137/25m1739406 ec_funded: 1 external_id: arxiv: - '2502.19369' intvolume: ' 25' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2502.19369 month: '01' oa: 1 oa_version: Preprint page: 108-130 project: - _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c call_identifier: H2020 grant_number: '101034413' name: 'IST-BRIDGE: International postdoctoral program' publication: SIAM Journal on Applied Dynamical Systems publication_identifier: issn: - 1536-0040 publication_status: published publisher: Society for Industrial & Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: Computing a connection matrix and persistence efficiently from a morse decomposition type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 25 year: '2026' ... --- OA_place: publisher _id: '21021' abstract: - lang: eng text: This thesis examines how geometry and topology intersect in the representation, transformation, and analysis of complex shapes. It considers how continuous manifolds relate to their discrete analogues, how topological structures evolve in persistence vineyards, and how tools from topological data analysis can illuminate problems in mathematical physics. Central to this exploration is the question of how structure, both geometric and topological, persists or changes under approximation, sampling, or deformation. The work develops new approaches to skeletal and grid-based representations of surfaces, reveals the full expressive capacity of persistence vineyards, and applies topological methods to the longstanding problem of equilibria in electrostatic fields. These threads braid together into a broader understanding of how topology and geometry inform one another across theory, computation, and application. acknowledged_ssus: - _id: M-Shop - _id: ScienComp acknowledgement: "The research presented in this thesis was funded by the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’.\r\n" alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Christopher D full_name: Fillmore, Christopher D id: 35638A5C-AAC7-11E9-B0BF-5503E6697425 last_name: Fillmore citation: ama: Fillmore CD. Braiding geometry and topology to study shapes and data. 2026. doi:10.15479/AT-ISTA-21021 apa: Fillmore, C. D. (2026). Braiding geometry and topology to study shapes and data. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-21021 chicago: Fillmore, Christopher D. “Braiding Geometry and Topology to Study Shapes and Data.” Institute of Science and Technology Austria, 2026. https://doi.org/10.15479/AT-ISTA-21021. ieee: C. D. Fillmore, “Braiding geometry and topology to study shapes and data,” Institute of Science and Technology Austria, 2026. ista: Fillmore CD. 2026. Braiding geometry and topology to study shapes and data. Institute of Science and Technology Austria. mla: Fillmore, Christopher D. Braiding Geometry and Topology to Study Shapes and Data. Institute of Science and Technology Austria, 2026, doi:10.15479/AT-ISTA-21021. short: C.D. Fillmore, Braiding Geometry and Topology to Study Shapes and Data, Institute of Science and Technology Austria, 2026. corr_author: '1' date_created: 2026-01-20T21:38:40Z date_published: 2026-01-21T00:00:00Z date_updated: 2026-04-07T11:42:49Z day: '21' ddc: - '514' - '516' degree_awarded: PhD department: - _id: GradSch - _id: HeEd - _id: UlWa doi: 10.15479/AT-ISTA-21021 file: - access_level: open_access checksum: 4c0889130095c31d4e5088c5b8dfd607 content_type: application/pdf creator: cfillmor date_created: 2026-01-26T19:44:46Z date_updated: 2026-01-30T11:40:09Z file_id: '21046' file_name: 2025_Fillmore_Christopher_Thesis.pdf file_size: 55954297 relation: main_file - access_level: closed checksum: d69afb71d82ab98f856886126ee7303a content_type: application/x-zip-compressed creator: cfillmor date_created: 2026-01-26T19:46:20Z date_updated: 2026-01-26T19:46:20Z file_id: '21047' file_name: Thesis.zip file_size: 166080788 relation: source_file file_date_updated: 2026-01-30T11:40:09Z has_accepted_license: '1' language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: '122' publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '20260' relation: part_of_dissertation status: public - id: '21050' relation: part_of_dissertation status: public - id: '21051' 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 - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 title: Braiding geometry and topology to study shapes and data 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: '2026' ... --- OA_place: repository _id: '21051' abstract: - lang: eng text: 'In this work, we introduce and study what we believe is an intriguing and, to the best of our knowledge, previously unknown connection between two areas in computational topology, topological data analysis (TDA) and knot theory. Given a function from a topological space to $\mathbb{R}$, TDA provides tools to simplify and study the importance of topological features: in particular, the $l^{th}$-dimensional persistence diagram encodes the $l$-homology in the sublevel set as the function value increases as a set of points in the plane. Given a continuous one-parameter family of such functions, we can combine the persistence diagrams into an object known as a vineyard, which track the evolution of points in the persistence diagram. If we further restrict that family of functions to be periodic, we identify the two ends of the vineyard, yielding a closed vineyard. This allows the study of monodromy, which in this context means that following the family of functions for a period permutes the set of points in a non-trivial way. In this work, given a link and value $l$, we construct a topological space and periodic family of functions such that the closed $l$-vineyard contains this link. This shows that vineyards are topologically as rich as one could possibly hope. Importantly, it has at least two immediate consequences: First, monodromy of any periodicity can occur in a $l$-vineyard, answering a variant of a question by [Arya et al 2024]. To exhibit this, we also reformulate monodromy in a more geometric way, which may be of interest in itself. Second, distinguishing vineyards is likely to be difficult given the known difficulty of knot and link recognition, which have strong connections to many NP-hard problems.' article_processing_charge: No arxiv: 1 author: - first_name: Erin full_name: ' Chambers, Erin' last_name: ' Chambers' - first_name: Christopher D full_name: Fillmore, Christopher D id: 35638A5C-AAC7-11E9-B0BF-5503E6697425 last_name: Fillmore - 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: Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards. arXiv. doi:10.48550/ARXIV.2504.11203 apa: Chambers, E., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M. (n.d.). Braiding vineyards. arXiv. https://doi.org/10.48550/ARXIV.2504.11203 chicago: Chambers, Erin, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Braiding Vineyards.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2504.11203. ieee: E. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding vineyards,” arXiv. . ista: Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards. arXiv, 10.48550/ARXIV.2504.11203. mla: Chambers, Erin, et al. “Braiding Vineyards.” ArXiv, doi:10.48550/ARXIV.2504.11203. short: E. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, ArXiv (n.d.). corr_author: '1' date_created: 2026-01-27T14:41:44Z date_published: 2026-01-02T00:00:00Z date_updated: 2026-04-07T11:42:48Z day: '02' department: - _id: HeEd doi: 10.48550/ARXIV.2504.11203 external_id: arxiv: - '2504.11203' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2504.11203 month: '01' oa: 1 oa_version: Preprint publication: arXiv publication_status: draft related_material: record: - id: '21056' relation: later_version status: public - id: '21021' relation: dissertation_contains status: public status: public title: Braiding vineyards 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: '2026' ... --- OA_place: repository OA_type: green _id: '21056' abstract: - lang: eng text: "In this work, we introduce and study what we believe is an intriguing, and, to the best of our knowledge, previously unknown connection between two fundamental areas in computational topology, namely topological data analysis (TDA) and knot theory. Given a function from a topological space to ℝ, TDA provides tools to simplify and study the importance of topological features: in particular, the \U0001D459^\U0001D461⁢ℎ-dimensional persistence diagram encodes the topological changes (or \U0001D459-homology) in the sublevel set as the function value increases into a set of points in the plane. Given a continuous one parameter family of such functions, we can combine the persistence diagrams into an object known as a vineyard, which tracks the evolution of points in the persistence diagram as the function changes. If we further restrict that family of functions to be periodic, we identify the two ends of the vineyard, yielding a closed vineyard. This allows the study of monodromy, which in this context means that following the family of functions for a period permutes the set of points in a non-trivial way. Recent work has studied monodromy in the directional persistent homology transform, demonstrating some interesting connections between an input shape and monodromy in the persistent homology transform for 0-dimensional homology embedded in ℝ^2.\r\nIn this work, given a link and a value \U0001D459, we construct a topological space (based on the given link) and periodic family of functions on this space (based on the Euclidean distance function), such that the closed \U0001D459-vineyard contains this link. This shows that vineyards are topologically as rich as one could possibly hope, suggesting many future directions of work. Importantly, it has at least two immediate consequences we explicitly point out:\r\n1.\tMonodromy of any periodicity can occur in a \U0001D459-vineyard for any \U0001D459. This answers a variant of a question by Arya and collaborators. To exhibit this as a consequence of our first main result we also reformulate monodromy in a more geometric way, which may be of interest in itself.\r\n2.\tTopologically distinguishing closed vineyards is likely to be difficult (from a complexity theory as well as from a practical perspective) because of the difficulty of knot and link recognition, which have strong connections to many NP-hard problems." acknowledgement: We thank the reviewers for both SODA and ATMCS for their comments, whichimproved the exposition. We thank Kate Turner for discussion and Clément Maria for pointing out thatAlexander’s theorem was already (well) known. Mathijs Wintraecken would like to express his gratitude tothe administrative support he received from University of Notre Dame during his visit and from Sophie Honnoratand Stephanie Verdonck at Inria in general.This work has been supported by the ANR grant StratMesh, ANR-24-CE48-1899, by NSF award 2444309, andthe welcome package from IDEX of the Université Côte d’Azur, ANR-15-IDEX-01. article_processing_charge: No arxiv: 1 author: - first_name: Erin W. full_name: Chambers, Erin W. last_name: Chambers - first_name: Christopher D full_name: Fillmore, Christopher D id: 35638A5C-AAC7-11E9-B0BF-5503E6697425 last_name: Fillmore - 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: 'Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Braiding Vineyards. In: Green Larsen K, Saha B, eds. Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA, United States: Society for Industrial and Applied Mathematics; 2026:6240-6263. doi:10.1137/1.9781611978971.225' apa: 'Chambers, E. W., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M. (2026). Braiding Vineyards. In K. Green Larsen & B. Saha (Eds.), Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 6240–6263). Philadelphia, PA, United States: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611978971.225' chicago: 'Chambers, Erin W., Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Braiding Vineyards.” In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, edited by Kasper Green Larsen and Barna Saha, 6240–63. Philadelphia, PA, United States: Society for Industrial and Applied Mathematics, 2026. https://doi.org/10.1137/1.9781611978971.225.' ieee: 'E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding Vineyards,” in Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, K. Green Larsen and B. Saha, Eds. Philadelphia, PA, United States: Society for Industrial and Applied Mathematics, 2026, pp. 6240–6263.' ista: 'Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2026.Braiding Vineyards. In: Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms. , 6240–6263.' mla: Chambers, Erin W., et al. “Braiding Vineyards.” Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, edited by Kasper Green Larsen and Barna Saha, Society for Industrial and Applied Mathematics, 2026, pp. 6240–63, doi:10.1137/1.9781611978971.225. short: E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, in:, K. Green Larsen, B. Saha (Eds.), Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, United States, 2026, pp. 6240–6263. date_created: 2026-01-28T12:58:16Z date_published: 2026-01-07T00:00:00Z date_updated: 2026-02-16T08:06:23Z day: '07' department: - _id: HeEd doi: 10.1137/1.9781611978971.225 editor: - first_name: Kasper full_name: Green Larsen, Kasper last_name: Green Larsen - first_name: Barna full_name: Saha, Barna last_name: Saha external_id: arxiv: - '2504.11203' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2504.11203 month: '01' oa: 1 oa_version: Preprint page: 6240-6263 place: Philadelphia, PA, United States publication: Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms publication_identifier: eisbn: - '9781611978971' publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' related_material: record: - id: '21051' relation: earlier_version status: public status: public title: Braiding Vineyards type: book_chapter user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2026' ... --- DOAJ_listed: '1' OA_place: publisher OA_type: gold PlanS_conform: '1' _id: '21115' abstract: - lang: eng text: Quantifying cell morphology is central to understanding cellular regulation, fate, and heterogeneity, yet conventional image-based analyses often struggle with diverse or irregular shapes. We present a computational framework that uses topological data analysis to characterise and compare single-cell morphologies from fluorescence microscopy. Each cell is represented by its contour together with the position of its nucleus, from which we construct a filtration based on a radial distance function and derive a persistence diagram encoding the shape’s topological evolution. The similarity between two cells is quantified using the 2-Wasserstein distance between their diagrams, yielding a shape distance we call the PH distance. We apply this method to two representative experimental systems—primary human mesenchymal stem cells (hMSCs) and HeLa cells—and show that PH distances enable the detection of outliers in those systems, the identification of sub-populations, and the quantification of shape heterogeneity. We benchmark PH against three established contour-based distances (aspect ratio, Fourier descriptors, and elastic shape analysis) and show that PH offers better separation between cell types and greater robustness when clustering heterogeneous populations. Together, these results demonstrate that persistent-homology-based signatures provide a principled and sensitive approach for analysing cell morphology in settings where traditional geometric or image-based descriptors are insufficient. acknowledgement: We thank Stephan Huckemann, Katharine Turner, Benjamin Eltzner, Stephan Tillmann, Fariza Rashid, Vanessa Robins, and Lamiae Azizi for many useful discussions at various stages of this project. FR and PY gratefully acknowledge Matthias Weiss (Experimental Physics I, University of Bayreuth, Germany) for granting access to cell culture and laboratories, as well as funding consumables and the fruitful discussion that contributed to this work. For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission. article_number: e1013890 article_processing_charge: Yes article_type: original author: - first_name: Yossi full_name: Bleile, Yossi id: 920a7385-7995-11ef-9bfd-8c434cd8f3c2 last_name: Bleile orcid: 0000-0002-4861-9174 - first_name: Pooja full_name: Yadav, Pooja last_name: Yadav - first_name: Patrice full_name: Koehl, Patrice last_name: Koehl - first_name: Florian full_name: Rehfeldt, Florian last_name: Rehfeldt citation: ama: 'Bokor Bleile Y, Yadav P, Koehl P, Rehfeldt F. Persistence diagrams as morphological signatures of cells: A method to measure and compare cells within a population. PLoS Computational Biology. 2026;22. doi:10.1371/journal.pcbi.1013890' apa: 'Bokor Bleile, Y., Yadav, P., Koehl, P., & Rehfeldt, F. (2026). Persistence diagrams as morphological signatures of cells: A method to measure and compare cells within a population. PLoS Computational Biology. Public Library of Science. https://doi.org/10.1371/journal.pcbi.1013890' chicago: 'Bokor Bleile, Yossi, Pooja Yadav, Patrice Koehl, and Florian Rehfeldt. “Persistence Diagrams as Morphological Signatures of Cells: A Method to Measure and Compare Cells within a Population.” PLoS Computational Biology. Public Library of Science, 2026. https://doi.org/10.1371/journal.pcbi.1013890.' ieee: 'Y. Bokor Bleile, P. Yadav, P. Koehl, and F. Rehfeldt, “Persistence diagrams as morphological signatures of cells: A method to measure and compare cells within a population,” PLoS Computational Biology, vol. 22. Public Library of Science, 2026.' ista: 'Bokor Bleile Y, Yadav P, Koehl P, Rehfeldt F. 2026. Persistence diagrams as morphological signatures of cells: A method to measure and compare cells within a population. PLoS Computational Biology. 22, e1013890.' mla: 'Bokor Bleile, Yossi, et al. “Persistence Diagrams as Morphological Signatures of Cells: A Method to Measure and Compare Cells within a Population.” PLoS Computational Biology, vol. 22, e1013890, Public Library of Science, 2026, doi:10.1371/journal.pcbi.1013890.' short: Y. Bokor Bleile, P. Yadav, P. Koehl, F. Rehfeldt, PLoS Computational Biology 22 (2026). corr_author: '1' date_created: 2026-01-30T10:36:32Z date_published: 2026-01-28T00:00:00Z date_updated: 2026-06-11T11:51:13Z day: '28' ddc: - '000' department: - _id: HeEd doi: 10.1371/journal.pcbi.1013890 external_id: pmid: - '41604421' file: - access_level: open_access checksum: 3899d929ee9be0453c95524e49992d72 content_type: application/pdf creator: dernst date_created: 2026-02-10T07:13:06Z date_updated: 2026-02-10T07:13:06Z file_id: '21204' file_name: 2026_PloSCompBio_Bleile.pdf file_size: 8908746 relation: main_file success: 1 file_date_updated: 2026-02-10T07:13:06Z has_accepted_license: '1' intvolume: ' 22' language: - iso: eng month: '01' oa: 1 oa_version: Published Version pmid: 1 publication: PLoS Computational Biology publication_identifier: issn: - 1553-7358 publication_status: published publisher: Public Library of Science quality_controlled: '1' related_material: link: - relation: software url: https://github.com/yossibokorbleile/correa scopus_import: '1' status: public title: 'Persistence diagrams as morphological signatures of cells: A method to measure and compare cells within a population' 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: 22 year: '2026' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '21232' abstract: - lang: eng text: "Abstract\r\n In this paper, we consider a simple class of stratified spaces – 2-complexes. We present an algorithm that learns the abstract structure of an embedded 2-complex from a point cloud sampled from it. We use tools and inspiration from computational geometry, algebraic topology, and topological data analysis and prove the correctness of the identified abstract structure under assumptions on the embedding." acknowledgement: The author would like to thank Kate Turner, Chris Williams, Jonathan Spreer, Stephan Tillmann, Vanessa Robins, Vigleik Angeltveit, Martin Helmer, and James Morgan for very helpful discussions; and thanks Sara Kališnik Hintz and Paul Bendich for comments on an earlier version. Additonally, the author would like to thank both reviewers for their very insightful and helpful comments, without which the paper would be infinitely less coherent than it currently is. Open access funding provided by Institute of Science and Technology (IST Austria). The work in this paper was supported by an Australian Federal Government Grant, 2019-2022, Stratified Space Learning. article_number: '17' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Yossi full_name: Bleile, Yossi id: 920a7385-7995-11ef-9bfd-8c434cd8f3c2 last_name: Bleile orcid: 0000-0002-4861-9174 citation: ama: 'Bokor Bleile Y. Towards stratified space learning: 2-complexes. La Matematica. 2026;5. doi:10.1007/s44007-025-00183-9' apa: 'Bokor Bleile, Y. (2026). Towards stratified space learning: 2-complexes. La Matematica. Springer Nature. https://doi.org/10.1007/s44007-025-00183-9' chicago: 'Bokor Bleile, Yossi. “Towards Stratified Space Learning: 2-Complexes.” La Matematica. Springer Nature, 2026. https://doi.org/10.1007/s44007-025-00183-9.' ieee: 'Y. Bokor Bleile, “Towards stratified space learning: 2-complexes,” La Matematica, vol. 5. Springer Nature, 2026.' ista: 'Bokor Bleile Y. 2026. Towards stratified space learning: 2-complexes. La Matematica. 5, 17.' mla: 'Bokor Bleile, Yossi. “Towards Stratified Space Learning: 2-Complexes.” La Matematica, vol. 5, 17, Springer Nature, 2026, doi:10.1007/s44007-025-00183-9.' short: Y. Bokor Bleile, La Matematica 5 (2026). corr_author: '1' date_created: 2026-02-16T10:44:44Z date_published: 2026-02-08T00:00:00Z date_updated: 2026-06-11T11:51:14Z day: '08' ddc: - '510' department: - _id: HeEd doi: 10.1007/s44007-025-00183-9 external_id: arxiv: - '2305.02724' file: - access_level: open_access checksum: 6cae2efb47b025af22a8539c606a4e09 content_type: application/pdf creator: dernst date_created: 2026-02-23T10:18:52Z date_updated: 2026-02-23T10:18:52Z file_id: '21347' file_name: 2026_LaMatematica_Bleile.pdf file_size: 15051582 relation: main_file success: 1 file_date_updated: 2026-02-23T10:18:52Z has_accepted_license: '1' intvolume: ' 5' language: - iso: eng month: '02' oa: 1 oa_version: Published Version publication: La Matematica publication_identifier: issn: - 2730-9657 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: 'Towards stratified space learning: 2-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 volume: 5 year: '2026' ... --- OA_place: repository OA_type: green _id: '21374' abstract: - lang: eng text: "Let . S be a set of distinct points in general position in the\r\nEuclidean plane. A plane Hamiltonian path on . S is a crossing-free geometric path such that every point of .S is a vertex of the path. It is\r\nknown that, if. S is sufficiently large, there exist three edge-disjoint plane\r\nHamiltonian paths on . S. In this paper we study an edge-constrained\r\nversion of the problem of finding Hamiltonian paths on a point set. We\r\nfirst consider the problem of finding a single plane Hamiltonian path . π\r\nwith endpoints .s, t ∈ S and constraints given by a segment . ab, where\r\n.a, b ∈ S. We consider the following scenarios: (i) .ab ∈ π; (ii) .ab π. We\r\ncharacterize those quintuples . S, a, b, s, t for which . π exists. Secondly,\r\nwe consider the problem of finding two plane Hamiltonian paths . π1, π2\r\non a set . S with constraints given by a segment . ab, where .a, b ∈ S. We\r\nconsider the following scenarios: (i) .π1 and .π2 share no edges and .ab is\r\nan edge of . π1; (ii) .π1 and .π2 share no edges and none of them includes\r\n.ab as an edge; (iii) both .π1 and .π2 include .ab as an edge and share no\r\nother edges. In all cases, we characterize those triples . S, a, b for which\r\n.π1 and .π2 exist." acknowledgement: "We thank the organizers of the HOMONOLO 2024 workshop in Nová Louka, Czech Republic, for the fruitful atmosphere where the research on this project was initiated.\r\n\r\nT. Antić, A. Džuklevski, J. Kratochvíl and M. Saumell received funding from GAČR grant 23–04949X, T.A and A.Dž were additionally supported by GAUK grant SVV–2025–260822. G. Liotta was supported in part by MUR of Italy, PRIN Project no. 2022TS4Y3N – EXPAND and PON Project ARS01_00540. J. Fiala was in part supported by GAČR grant 25-16847S." alternative_title: - LNCS article_processing_charge: No arxiv: 1 author: - first_name: Todor full_name: Antić, Todor last_name: Antić - first_name: Aleksa full_name: Džuklevski, Aleksa last_name: Džuklevski - first_name: Jiří full_name: Fiala, Jiří last_name: Fiala - first_name: Jan full_name: Kratochvíl, Jan last_name: Kratochvíl - first_name: Giuseppe full_name: Liotta, Giuseppe last_name: Liotta - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian - first_name: Maria full_name: Saumell, Maria last_name: Saumell - first_name: Johannes full_name: Zink, Johannes last_name: Zink citation: ama: 'Antić T, Džuklevski A, Fiala J, et al. Edge-constrained Hamiltonian paths on a point set. In: 51st International Conference on Current Trends in Theory and Practice of Computer Science. Vol 16448. Springer Nature; 2026:532-546. doi:10.1007/978-3-032-17801-5_39' apa: 'Antić, T., Džuklevski, A., Fiala, J., Kratochvíl, J., Liotta, G., Saghafian, M., … Zink, J. (2026). Edge-constrained Hamiltonian paths on a point set. In 51st International Conference on Current Trends in Theory and Practice of Computer Science (Vol. 16448, pp. 532–546). Krakow, Poland: Springer Nature. https://doi.org/10.1007/978-3-032-17801-5_39' chicago: Antić, Todor, Aleksa Džuklevski, Jiří Fiala, Jan Kratochvíl, Giuseppe Liotta, Morteza Saghafian, Maria Saumell, and Johannes Zink. “Edge-Constrained Hamiltonian Paths on a Point Set.” In 51st International Conference on Current Trends in Theory and Practice of Computer Science, 16448:532–46. Springer Nature, 2026. https://doi.org/10.1007/978-3-032-17801-5_39. ieee: T. Antić et al., “Edge-constrained Hamiltonian paths on a point set,” in 51st International Conference on Current Trends in Theory and Practice of Computer Science, Krakow, Poland, 2026, vol. 16448, pp. 532–546. ista: 'Antić T, Džuklevski A, Fiala J, Kratochvíl J, Liotta G, Saghafian M, Saumell M, Zink J. 2026. Edge-constrained Hamiltonian paths on a point set. 51st International Conference on Current Trends in Theory and Practice of Computer Science. SOFSEM: Conference on Current Trends in Theory and Practice of Computer Science, LNCS, vol. 16448, 532–546.' mla: Antić, Todor, et al. “Edge-Constrained Hamiltonian Paths on a Point Set.” 51st International Conference on Current Trends in Theory and Practice of Computer Science, vol. 16448, Springer Nature, 2026, pp. 532–46, doi:10.1007/978-3-032-17801-5_39. short: T. Antić, A. Džuklevski, J. Fiala, J. Kratochvíl, G. Liotta, M. Saghafian, M. Saumell, J. Zink, in:, 51st International Conference on Current Trends in Theory and Practice of Computer Science, Springer Nature, 2026, pp. 532–546. conference: end_date: 2026-02-13 location: Krakow, Poland name: 'SOFSEM: Conference on Current Trends in Theory and Practice of Computer Science' start_date: 2026-02-09 date_created: 2026-03-01T23:01:40Z date_published: 2026-02-13T00:00:00Z date_updated: 2026-03-02T08:49:20Z day: '13' department: - _id: HeEd doi: 10.1007/978-3-032-17801-5_39 external_id: arxiv: - '2511.22526' intvolume: ' 16448' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2511.22526 month: '02' oa: 1 oa_version: Preprint page: 532-546 publication: 51st International Conference on Current Trends in Theory and Practice of Computer Science publication_identifier: eissn: - 1611-3349 isbn: - '9783032178008' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Edge-constrained Hamiltonian paths on a point set type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 16448 year: '2026' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '21407' 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." 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. article_number: '5' article_processing_charge: Yes (in subscription journal) 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: Matthew full_name: Kahle, Matthew last_name: Kahle - first_name: Shu full_name: Kanazawa, Shu last_name: Kanazawa citation: ama: Edelsbrunner H, Kahle M, Kanazawa S. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 2026;10. doi:10.1007/s41468-026-00233-3 apa: Edelsbrunner, H., Kahle, M., & Kanazawa, S. (2026). Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-026-00233-3 chicago: Edelsbrunner, Herbert, Matthew Kahle, and Shu Kanazawa. “Maximum Persistent Betti Numbers of Čech Complexes.” Journal of Applied and Computational Topology. Springer Nature, 2026. https://doi.org/10.1007/s41468-026-00233-3. ieee: H. Edelsbrunner, M. Kahle, and S. Kanazawa, “Maximum persistent Betti numbers of Čech complexes,” Journal of Applied and Computational Topology, vol. 10. Springer Nature, 2026. ista: Edelsbrunner H, Kahle M, Kanazawa S. 2026. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 10, 5. mla: Edelsbrunner, Herbert, et al. “Maximum Persistent Betti Numbers of Čech Complexes.” Journal of Applied and Computational Topology, vol. 10, 5, Springer Nature, 2026, doi:10.1007/s41468-026-00233-3. short: H. Edelsbrunner, M. Kahle, S. Kanazawa, Journal of Applied and Computational Topology 10 (2026). date_created: 2026-03-08T23:01:45Z date_published: 2026-03-01T00:00:00Z date_updated: 2026-03-09T11:31:29Z day: '01' ddc: - '500' department: - _id: HeEd doi: 10.1007/s41468-026-00233-3 external_id: arxiv: - '2409.05241' file: - access_level: open_access checksum: 0bf6dc430cafa40c08f260fe17d54595 content_type: application/pdf creator: dernst date_created: 2026-03-09T11:29:30Z date_updated: 2026-03-09T11:29:30Z file_id: '21416' file_name: 2026_JourAppliedCompTopology_Edelsbrunner.pdf file_size: 323111 relation: main_file success: 1 file_date_updated: 2026-03-09T11:29:30Z has_accepted_license: '1' intvolume: ' 10' language: - iso: eng month: '03' oa: 1 oa_version: Published Version project: - _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: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Maximum persistent 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 volume: 10 year: '2026' ... --- OA_place: repository OA_type: green _id: '21410' abstract: - lang: eng text: Given a finite set of red and blue points in R^d, the MST-ratio is defined as the total length of the Euclidean minimum spanning trees of the red points and the blue points, divided by the length of the Euclidean minimum spanning tree of their union. The MST-ratio has recently gained attention due to its direct interpretation in topological models for studying point sets with applications in spatial biology. The maximum MST-ratio of a point set is the maximum MST-ratio over all proper colorings of its points by red and blue. We prove that finding the maximum MST-ratio of a given point set is NP-hard when the dimension is part of the input. Moreover, we present a quadratic-time 3-approximation algorithm for this problem. As part of the proof, we show that in any metric space, the maximum MST-ratio is smaller than 3. Furthermore, we study the average MST-ratio over all colorings of a set of n points. We show that this average is always at least n-2/n-1, and for n random points uniformly distributed in a d-dimensional unit cube, the average tends to (math formular) in expectation as n approaches infinity. acknowledgement: "A. J. Ameli—Supported by the project COALESCE (ERC grant no. 853234).\r\nM. Saghafian—Partially supported by the European Research Council (ERC), grant no. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31." alternative_title: - LNCS article_processing_charge: No arxiv: 1 author: - first_name: Afrouz full_name: Jabal Ameli, Afrouz last_name: Jabal Ameli - first_name: Faezeh full_name: Motiei, Faezeh last_name: Motiei - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Jabal Ameli A, Motiei F, Saghafian M. On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In: 20th International Conference and Workshops on Algorithms and Computation. Vol 16444. Springer Nature; 2026:386-401. doi:10.1007/978-981-95-7127-7_26' apa: 'Jabal Ameli, A., Motiei, F., & Saghafian, M. (2026). On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In 20th International Conference and Workshops on Algorithms and Computation (Vol. 16444, pp. 386–401). Perugia, Italy: Springer Nature. https://doi.org/10.1007/978-981-95-7127-7_26' chicago: 'Jabal Ameli, Afrouz, Faezeh Motiei, and Morteza Saghafian. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” In 20th International Conference and Workshops on Algorithms and Computation, 16444:386–401. Springer Nature, 2026. https://doi.org/10.1007/978-981-95-7127-7_26.' ieee: 'A. Jabal Ameli, F. Motiei, and M. Saghafian, “On the MST-ratio: Theoretical bounds and complexity of finding the maximum,” in 20th International Conference and Workshops on Algorithms and Computation, Perugia, Italy, 2026, vol. 16444, pp. 386–401.' ista: 'Jabal Ameli A, Motiei F, Saghafian M. 2026. On the MST-ratio: Theoretical bounds and complexity of finding the maximum. 20th International Conference and Workshops on Algorithms and Computation. WALCOM: International Conference and Workshops on Algorithms and Computation, LNCS, vol. 16444, 386–401.' mla: 'Jabal Ameli, Afrouz, et al. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” 20th International Conference and Workshops on Algorithms and Computation, vol. 16444, Springer Nature, 2026, pp. 386–401, doi:10.1007/978-981-95-7127-7_26.' short: A. Jabal Ameli, F. Motiei, M. Saghafian, in:, 20th International Conference and Workshops on Algorithms and Computation, Springer Nature, 2026, pp. 386–401. conference: end_date: 2026-03-06 location: Perugia, Italy name: 'WALCOM: International Conference and Workshops on Algorithms and Computation' start_date: 2026-03-04 date_created: 2026-03-08T23:01:45Z date_published: 2026-02-14T00:00:00Z date_updated: 2026-03-09T10:25:41Z day: '14' department: - _id: HeEd doi: 10.1007/978-981-95-7127-7_26 ec_funded: 1 external_id: arxiv: - '2409.11079' intvolume: ' 16444' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2409.11079 month: '02' oa: 1 oa_version: Preprint page: 386-401 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 publication: 20th International Conference and Workshops on Algorithms and Computation publication_identifier: eissn: - 1611-3349 isbn: - '9789819571260' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: 'On the MST-ratio: Theoretical bounds and complexity of finding the maximum' type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 16444 year: '2026' ... --- OA_place: repository OA_type: green _id: '21781' abstract: - lang: eng text: "Given a set A of n points (vertices) in general position in the plane, the complete geometric graph \r\nKn[A] consists of all (n2) segments (edges) between the elements of A. It is known that the edge set of every complete geometric graph on n vertices can be partitioned into O(n3∕2) crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set A of n randomly selected points, uniformly distributed in [0,1]2, with probability tending to 1 as n→∞, the edge set of Kn[A] can be covered by O(nlogn) crossing-free paths and by O(n√logn) crossing-free matchings. On the other hand, we construct n-element point sets such that covering the edge set of Kn[A] requires a quadratic number of monotone paths." acknowledgement: "Research partially supported by ERC Advanced Grant \"GeoScape\", no. 882971 and\r\nHungarian NKFIH grant no. K-131529. Work by the third author is supported by EPSRC grant\r\nEP/X013642/1. Work by the third author is partially supported by the European Research Council (ERC), grant no. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Adrian full_name: Dumitrescu, Adrian last_name: Dumitrescu - 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: Alex full_name: Scott, Alex last_name: Scott citation: ama: Dumitrescu A, Pach J, Saghafian M, Scott A. Covering complete geometric graphs by monotone paths. Combinatorics and Number Theory. 2026;15(1):73-82. doi:10.2140/cnt.2026.15.73 apa: Dumitrescu, A., Pach, J., Saghafian, M., & Scott, A. (2026). Covering complete geometric graphs by monotone paths. Combinatorics and Number Theory. Mathematical Sciences Publishers. https://doi.org/10.2140/cnt.2026.15.73 chicago: Dumitrescu, Adrian, János Pach, Morteza Saghafian, and Alex Scott. “Covering Complete Geometric Graphs by Monotone Paths.” Combinatorics and Number Theory. Mathematical Sciences Publishers, 2026. https://doi.org/10.2140/cnt.2026.15.73. ieee: A. Dumitrescu, J. Pach, M. Saghafian, and A. Scott, “Covering complete geometric graphs by monotone paths,” Combinatorics and Number Theory, vol. 15, no. 1. Mathematical Sciences Publishers, pp. 73–82, 2026. ista: Dumitrescu A, Pach J, Saghafian M, Scott A. 2026. Covering complete geometric graphs by monotone paths. Combinatorics and Number Theory. 15(1), 73–82. mla: Dumitrescu, Adrian, et al. “Covering Complete Geometric Graphs by Monotone Paths.” Combinatorics and Number Theory, vol. 15, no. 1, Mathematical Sciences Publishers, 2026, pp. 73–82, doi:10.2140/cnt.2026.15.73. short: A. Dumitrescu, J. Pach, M. Saghafian, A. Scott, Combinatorics and Number Theory 15 (2026) 73–82. date_created: 2026-05-03T22:01:37Z date_published: 2026-04-17T00:00:00Z date_updated: 2026-05-07T07:45:24Z day: '17' department: - _id: HeEd doi: 10.2140/cnt.2026.15.73 ec_funded: 1 external_id: arxiv: - '2507.10840' intvolume: ' 15' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2507.10840 month: '04' oa: 1 oa_version: Preprint page: 73-82 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 publication: Combinatorics and Number Theory publication_identifier: eissn: - 2996-220X issn: - 2996-2196 publication_status: published publisher: Mathematical Sciences Publishers quality_controlled: '1' scopus_import: '1' status: public title: Covering complete geometric graphs by monotone paths type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 15 year: '2026' ... --- OA_place: repository OA_type: green _id: '21931' 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 (nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. 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 Conjecture 1.8 by Gabrielov, Novikov, and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower 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_number: e70163 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: Christopher D full_name: Fillmore, Christopher D id: 35638A5C-AAC7-11E9-B0BF-5503E6697425 last_name: Fillmore - first_name: Goncalo full_name: Oliveira, Goncalo id: 58abbde8-f455-11eb-a497-98c8fd71b905 last_name: Oliveira citation: ama: Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. 2026;132(5). doi:10.1112/plms.70163 apa: Edelsbrunner, H., Fillmore, C. D., & Oliveira, G. (2026). Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. Wiley. https://doi.org/10.1112/plms.70163 chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting Equilibria of the Electrostatic Potential.” Proceedings of the London Mathematical Society. Wiley, 2026. https://doi.org/10.1112/plms.70163. ieee: H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of the electrostatic potential,” Proceedings of the London Mathematical Society, vol. 132, no. 5. Wiley, 2026. ista: Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. 132(5), e70163. mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” Proceedings of the London Mathematical Society, vol. 132, no. 5, e70163, Wiley, 2026, doi:10.1112/plms.70163. short: H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical Society 132 (2026). corr_author: '1' date_created: 2026-05-31T22:02:13Z date_published: 2026-05-01T00:00:00Z date_updated: 2026-06-02T09:24:18Z day: '01' department: - _id: HeEd - _id: TaHa doi: 10.1112/plms.70163 external_id: arxiv: - '2501.05315' intvolume: ' 132' issue: '5' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2501.05315 month: '05' oa: 1 oa_version: Preprint publication: Proceedings of the London Mathematical Society publication_identifier: eissn: - 1460-244X issn: - 0024-6115 publication_status: published publisher: Wiley quality_controlled: '1' related_material: record: - id: '21050' relation: earlier_version status: public scopus_import: '1' status: public title: Counting equilibria of the electrostatic potential type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 132 year: '2026' ... --- _id: '21971' abstract: - lang: eng text: "A Rust library for analyzing dendritic structures using quadric matrices. This project provides efficient tools for representing dendritic trees, computing quadric error metrics, and visualizing eigenvalue distributions on hexagonal plots.\r\n\r\nThis library implements quadric-based geometric analysis of dendritic structures, commonly found in neuroscience applications. Key features include:\r\n\r\nTree data structures: Hierarchical vertex and edge representations for dendritic trees\r\nQuadric matrices: Computation of quadric error metrics for edges and vertices\r\nVisualisation: Hexagonal plot generation using NormPolar transformations\r\nInteractive tools: Desktop application with plotting capabilities" author: - first_name: Yossi full_name: Bleile, Yossi id: 920a7385-7995-11ef-9bfd-8c434cd8f3c2 last_name: Bleile orcid: 0000-0002-4861-9174 - first_name: Emanuele full_name: Cortinovis, Emanuele last_name: Cortinovis citation: ama: Bokor Bleile Y, Cortinovis E. Quadrix. 2026. doi:10.15479/AT-ISTA-21971 apa: Bokor Bleile, Y., & Cortinovis, E. (2026). Quadrix. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-21971 chicago: Bokor Bleile, Yossi, and Emanuele Cortinovis. “Quadrix.” Institute of Science and Technology Austria, 2026. https://doi.org/10.15479/AT-ISTA-21971. ieee: Y. Bokor Bleile and E. Cortinovis, “Quadrix.” Institute of Science and Technology Austria, 2026. ista: Bokor Bleile Y, Cortinovis E. 2026. Quadrix, Institute of Science and Technology Austria, 10.15479/AT-ISTA-21971. mla: Bokor Bleile, Yossi, and Emanuele Cortinovis. Quadrix. Institute of Science and Technology Austria, 2026, doi:10.15479/AT-ISTA-21971. short: Y. Bokor Bleile, E. Cortinovis, (2026). corr_author: '1' date_created: 2026-06-09T19:19:13Z date_published: 2026-06-15T00:00:00Z date_updated: 2026-06-15T23:00:03Z day: '15' department: - _id: HeEd doi: 10.15479/AT-ISTA-21971 file: - access_level: open_access checksum: 48f633b6767c4b15dd6220ca2b4dc175 content_type: application/octet-stream creator: ybleile date_created: 2026-06-09T19:16:02Z date_updated: 2026-06-09T19:16:02Z file_id: '21974' file_name: LICENSE file_size: 1081 relation: main_file success: 1 - access_level: open_access checksum: de25d0b224acbde3d38f837fdd8f97d5 content_type: application/octet-stream creator: ybleile date_created: 2026-06-09T19:16:27Z date_updated: 2026-06-09T19:16:27Z file_id: '21975' file_name: quadrix-x64.exe file_size: 11308032 relation: main_file success: 1 - access_level: open_access checksum: a7b94a7380dc178e76ebdba9f1fa45c2 content_type: application/octet-stream creator: ybleile date_created: 2026-06-09T19:16:28Z date_updated: 2026-06-09T19:16:28Z file_id: '21976' file_name: quadrix-arm64.exe file_size: 10655744 relation: main_file success: 1 - access_level: open_access checksum: 2404aa8619a56668bd95032791ee1250 content_type: application/zip creator: ybleile date_created: 2026-06-09T19:16:27Z date_updated: 2026-06-09T19:16:27Z file_id: '21977' file_name: Quadrix Desktop.app.zip file_size: 2032 relation: main_file success: 1 - access_level: open_access checksum: 106930f81563c5c719a5f4030b5ca5ed content_type: application/octet-stream creator: ybleile date_created: 2026-06-09T19:16:40Z date_updated: 2026-06-09T19:16:40Z file_id: '21978' file_name: quadrix-arm64 file_size: 12187896 relation: main_file success: 1 - access_level: open_access checksum: 0e6ba129318446676f220087e7e6ff41 content_type: application/octet-stream creator: ybleile date_created: 2026-06-09T19:16:52Z date_updated: 2026-06-09T19:16:52Z file_id: '21979' file_name: quadrix-x64 file_size: 20587592 relation: main_file success: 1 - access_level: open_access checksum: f0b03385d17df049219465ab7403fe09 content_type: application/gzip creator: pub-gitlab-bot date_created: 2026-06-09T19:19:12Z date_updated: 2026-06-09T19:19:12Z file_id: '21972' file_name: Quadrix.zip file_size: 1914198 relation: main_file - access_level: open_access checksum: ede0bbb24bf41ab4009cf1b6a9009671 content_type: application/zip creator: ybleile date_created: 2026-06-10T19:09:38Z date_updated: 2026-06-10T19:09:38Z file_id: '21993' file_name: THIRD_PARTY_LICENSES.zip file_size: 37557 relation: supplementary_material - access_level: open_access checksum: f3c5fcc62c88e449ab5c660244df5aef content_type: text/markdown creator: ybleile date_created: 2026-06-15T08:13:32Z date_updated: 2026-06-15T08:13:32Z file_id: '22009' file_name: README.md file_size: 3839 relation: main_file success: 1 - access_level: open_access checksum: aa74828c3165aafcdee4ddcc9ecd37ac content_type: application/gzip creator: pub-gitlab-bot date_created: 2026-06-15T08:14:24Z date_updated: 2026-06-15T08:14:24Z file_id: '22008' file_name: Quadrix.zip file_size: 1912923 relation: main_file file_date_updated: 2026-06-15T08:14:24Z has_accepted_license: '1' keyword: - quadratics - mathematics - dendrites - geometry - topology license: https://opensource.org/licenses/MIT month: '06' oa: 1 project: - _id: 9106a876-16d5-11f0-9cad-bbf11c9952f9 grant_number: ESP 9584724 name: Quantitative Unbiased Shape Analysis with Geometry & Topology publisher: Institute of Science and Technology Austria status: public title: Quadrix tmp: legal_code_url: https://opensource.org/licenses/MIT name: The MIT License short: MIT type: software user_id: 68b8ca59-c5b3-11ee-8790-cd641c68093d year: '2026' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '20456' 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. 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. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - 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: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 2026;75:24-47. doi:10.1007/s00454-025-00778-7 apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00778-7 chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature, 2026. https://doi.org/10.1007/s00454-025-00778-7. ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” Discrete and Computational Geometry, vol. 75. Springer Nature, pp. 24–47, 2026. 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. mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” Discrete and Computational Geometry, vol. 75, Springer Nature, 2026, pp. 24–47, doi:10.1007/s00454-025-00778-7. short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47. corr_author: '1' date_created: 2025-10-12T22:01:26Z date_published: 2026-01-01T00:00:00Z date_updated: 2026-01-05T13:21:56Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-025-00778-7 ec_funded: 1 external_id: arxiv: - '2212.03121' isi: - '001584166900001' file: - access_level: open_access checksum: 0addb5c1b78142f9fb453bfa04695400 content_type: application/pdf creator: dernst date_created: 2026-01-05T13:21:20Z date_updated: 2026-01-05T13:21:20Z file_id: '20952' file_name: 2026_DiscreteCompGeom_Biswas.pdf file_size: 570922 relation: main_file success: 1 file_date_updated: 2026-01-05T13:21:20Z has_accepted_license: '1' intvolume: ' 75' isi: 1 language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 24-47 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 and Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '15090' relation: earlier_version status: public scopus_import: '1' status: public title: On the size of chromatic Delaunay mosaics 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: 75 year: '2026' ... --- 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' ...