[{"has_accepted_license":"1","publication_status":"published","author":[{"first_name":"Christopher D","full_name":"Fillmore, Christopher D","last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"}],"file":[{"file_name":"2025_Fillmore_Christopher_Thesis.pdf","content_type":"application/pdf","access_level":"open_access","date_updated":"2026-01-30T11:40:09Z","creator":"cfillmor","checksum":"4c0889130095c31d4e5088c5b8dfd607","date_created":"2026-01-26T19:44:46Z","relation":"main_file","file_size":55954297,"file_id":"21046"},{"checksum":"d69afb71d82ab98f856886126ee7303a","creator":"cfillmor","date_updated":"2026-01-26T19:46:20Z","access_level":"closed","content_type":"application/x-zip-compressed","file_name":"Thesis.zip","file_id":"21047","file_size":166080788,"relation":"source_file","date_created":"2026-01-26T19:46:20Z"}],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa":1,"OA_place":"publisher","publication_identifier":{"issn":["2663-337X"]},"language":[{"iso":"eng"}],"day":"21","year":"2026","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","date_published":"2026-01-21T00:00:00Z","article_processing_charge":"No","acknowledged_ssus":[{"_id":"M-Shop"},{"_id":"ScienComp"}],"doi":"10.15479/AT-ISTA-21021","abstract":[{"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.","lang":"eng"}],"corr_author":"1","date_created":"2026-01-20T21:38:40Z","oa_version":"Published Version","supervisor":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"last_name":"Wagner","full_name":"Wagner, Uli","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568"}],"ddc":["514","516"],"file_date_updated":"2026-01-30T11:40:09Z","citation":{"ieee":"C. D. Fillmore, “Braiding geometry and topology to study shapes and data,” Institute of Science and Technology Austria, 2026.","apa":"Fillmore, C. D. (2026). <i>Braiding geometry and topology to study shapes and data</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-21021\">https://doi.org/10.15479/AT-ISTA-21021</a>","ista":"Fillmore CD. 2026. Braiding geometry and topology to study shapes and data. Institute of Science and Technology Austria.","short":"C.D. Fillmore, Braiding Geometry and Topology to Study Shapes and Data, Institute of Science and Technology Austria, 2026.","chicago":"Fillmore, Christopher D. “Braiding Geometry and Topology to Study Shapes and Data.” Institute of Science and Technology Austria, 2026. <a href=\"https://doi.org/10.15479/AT-ISTA-21021\">https://doi.org/10.15479/AT-ISTA-21021</a>.","mla":"Fillmore, Christopher D. <i>Braiding Geometry and Topology to Study Shapes and Data</i>. Institute of Science and Technology Austria, 2026, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-21021\">10.15479/AT-ISTA-21021</a>.","ama":"Fillmore CD. Braiding geometry and topology to study shapes and data. 2026. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-21021\">10.15479/AT-ISTA-21021</a>"},"date_updated":"2026-04-07T11:42:49Z","degree_awarded":"PhD","page":"122","related_material":{"record":[{"id":"20260","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"21050","relation":"part_of_dissertation"},{"status":"public","id":"21051","relation":"part_of_dissertation"}]},"department":[{"_id":"GradSch"},{"_id":"HeEd"},{"_id":"UlWa"}],"publisher":"Institute of Science and Technology Austria","title":"Braiding geometry and topology to study shapes and data","status":"public","type":"dissertation","month":"01","alternative_title":["ISTA Thesis"],"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)"},"_id":"21021"},{"arxiv":1,"year":"2026","doi":"10.48550/ARXIV.2504.11203","article_processing_charge":"No","publication":"arXiv","date_published":"2026-01-02T00:00:00Z","related_material":{"record":[{"relation":"later_version","id":"21056","status":"public"},{"relation":"dissertation_contains","id":"21021","status":"public"}]},"department":[{"_id":"HeEd"}],"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."}],"corr_author":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2504.11203"}],"type":"preprint","month":"01","status":"public","title":"Braiding vineyards","date_created":"2026-01-27T14:41:44Z","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)"},"oa_version":"Preprint","_id":"21051","citation":{"short":"E.  Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, ArXiv (n.d.).","ista":"Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards. arXiv, <a href=\"https://doi.org/10.48550/ARXIV.2504.11203\">10.48550/ARXIV.2504.11203</a>.","apa":"Chambers, E., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (n.d.). Braiding vineyards. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2504.11203\">https://doi.org/10.48550/ARXIV.2504.11203</a>","ieee":"E.  Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding vineyards,” <i>arXiv</i>. .","mla":"Chambers, Erin, et al. “Braiding Vineyards.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/ARXIV.2504.11203\">10.48550/ARXIV.2504.11203</a>.","ama":"Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2504.11203\">10.48550/ARXIV.2504.11203</a>","chicago":"Chambers, Erin, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Braiding Vineyards.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2504.11203\">https://doi.org/10.48550/ARXIV.2504.11203</a>."},"publication_status":"draft","author":[{"last_name":" Chambers","first_name":"Erin","full_name":" Chambers, Erin"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","last_name":"Fillmore","first_name":"Christopher D","full_name":"Fillmore, Christopher D"},{"last_name":"Stephenson","full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6862-208X"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","first_name":"Mathijs"}],"external_id":{"arxiv":["2504.11203"]},"date_updated":"2026-04-07T11:42:48Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","language":[{"iso":"eng"}],"day":"02","OA_place":"repository","oa":1},{"editor":[{"last_name":"Green Larsen","full_name":"Green Larsen, Kasper","first_name":"Kasper"},{"last_name":"Saha","full_name":"Saha, Barna","first_name":"Barna"}],"page":"6240-6263","place":"Philadelphia, PA, United States","date_updated":"2026-02-16T08:06:23Z","citation":{"apa":"Chambers, E. W., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2026). Braiding Vineyards. In K. Green Larsen &#38; B. Saha (Eds.), <i>Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms</i> (pp. 6240–6263). Philadelphia, PA, United States: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611978971.225\">https://doi.org/10.1137/1.9781611978971.225</a>","ieee":"E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding Vineyards,” in <i>Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, K. Green Larsen and B. Saha, Eds. Philadelphia, PA, United States: Society for Industrial and Applied Mathematics, 2026, pp. 6240–6263.","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.","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.","chicago":"Chambers, Erin W., Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Braiding Vineyards.” In <i>Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, edited by Kasper Green Larsen and Barna Saha, 6240–63. Philadelphia, PA, United States: Society for Industrial and Applied Mathematics, 2026. <a href=\"https://doi.org/10.1137/1.9781611978971.225\">https://doi.org/10.1137/1.9781611978971.225</a>.","mla":"Chambers, Erin W., et al. “Braiding Vineyards.” <i>Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, edited by Kasper Green Larsen and Barna Saha, Society for Industrial and Applied Mathematics, 2026, pp. 6240–63, doi:<a href=\"https://doi.org/10.1137/1.9781611978971.225\">10.1137/1.9781611978971.225</a>.","ama":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Braiding Vineyards. In: Green Larsen K, Saha B, eds. <i>Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Philadelphia, PA, United States: Society for Industrial and Applied Mathematics; 2026:6240-6263. doi:<a href=\"https://doi.org/10.1137/1.9781611978971.225\">10.1137/1.9781611978971.225</a>"},"_id":"21056","status":"public","title":"Braiding Vineyards","type":"book_chapter","month":"01","publisher":"Society for Industrial and Applied Mathematics","quality_controlled":"1","department":[{"_id":"HeEd"}],"related_material":{"record":[{"status":"public","id":"21051","relation":"earlier_version"}]},"OA_place":"repository","oa":1,"publication_identifier":{"eisbn":["9781611978971"]},"language":[{"iso":"eng"}],"day":"07","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2504.11203"]},"author":[{"last_name":"Chambers","full_name":"Chambers, Erin W.","first_name":"Erin W."},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","full_name":"Fillmore, Christopher D","first_name":"Christopher D","last_name":"Fillmore"},{"full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","last_name":"Stephenson","orcid":"0000-0002-6862-208X","id":"2D04F932-F248-11E8-B48F-1D18A9856A87"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","first_name":"Mathijs"}],"OA_type":"green","publication_status":"published","oa_version":"Preprint","date_created":"2026-01-28T12:58:16Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2504.11203"}],"abstract":[{"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 𝑙^𝑡⁢ℎ-dimensional persistence diagram encodes the topological changes (or 𝑙-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 𝑙, 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 𝑙-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 𝑙-vineyard for any 𝑙. 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.","lang":"eng"}],"publication":"Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms","date_published":"2026-01-07T00:00:00Z","article_processing_charge":"No","doi":"10.1137/1.9781611978971.225","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.","arxiv":1,"year":"2026"},{"quality_controlled":"1","department":[{"_id":"HeEd"},{"_id":"TaHa"}],"related_material":{"record":[{"id":"21050","relation":"earlier_version","status":"public"}]},"_id":"21931","status":"public","title":"Counting equilibria of the electrostatic potential","type":"journal_article","month":"05","publisher":"Wiley","date_updated":"2026-06-02T09:24:18Z","citation":{"chicago":"Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting Equilibria of the Electrostatic Potential.” <i>Proceedings of the London Mathematical Society</i>. Wiley, 2026. <a href=\"https://doi.org/10.1112/plms.70163\">https://doi.org/10.1112/plms.70163</a>.","mla":"Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” <i>Proceedings of the London Mathematical Society</i>, vol. 132, no. 5, e70163, Wiley, 2026, doi:<a href=\"https://doi.org/10.1112/plms.70163\">10.1112/plms.70163</a>.","ama":"Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic potential. <i>Proceedings of the London Mathematical Society</i>. 2026;132(5). doi:<a href=\"https://doi.org/10.1112/plms.70163\">10.1112/plms.70163</a>","apa":"Edelsbrunner, H., Fillmore, C. D., &#38; Oliveira, G. (2026). Counting equilibria of the electrostatic potential. <i>Proceedings of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/plms.70163\">https://doi.org/10.1112/plms.70163</a>","ieee":"H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of the electrostatic potential,” <i>Proceedings of the London Mathematical Society</i>, vol. 132, no. 5. Wiley, 2026.","short":"H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical Society 132 (2026).","ista":"Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. 132(5), e70163."},"issue":"5","article_type":"original","abstract":[{"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.","lang":"eng"}],"date_published":"2026-05-01T00:00:00Z","publication":"Proceedings of the London Mathematical Society","article_processing_charge":"No","doi":"10.1112/plms.70163","intvolume":"       132","year":"2026","arxiv":1,"oa_version":"Preprint","volume":132,"article_number":"e70163","date_created":"2026-05-31T22:02:13Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2501.05315","open_access":"1"}],"corr_author":"1","scopus_import":"1","external_id":{"arxiv":["2501.05315"]},"author":[{"last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","full_name":"Fillmore, Christopher D","last_name":"Fillmore"},{"id":"58abbde8-f455-11eb-a497-98c8fd71b905","last_name":"Oliveira","full_name":"Oliveira, Goncalo","first_name":"Goncalo"}],"publication_status":"published","OA_type":"green","publication_identifier":{"issn":["0024-6115"],"eissn":["1460-244X"]},"oa":1,"OA_place":"repository","language":[{"iso":"eng"}],"day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"author":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner"},{"last_name":"Fillmore","full_name":"Fillmore, Christopher D","first_name":"Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"full_name":"Olivera, Gonçalo","first_name":"Gonçalo","last_name":"Olivera"}],"external_id":{"arxiv":["2501.05315"]},"date_updated":"2026-06-02T09:24:17Z","publication_status":"draft","citation":{"chicago":"Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting Equilibria of the Electrostatic Potential.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">https://doi.org/10.48550/ARXIV.2501.05315</a>.","ama":"Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">10.48550/ARXIV.2501.05315</a>","mla":"Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">10.48550/ARXIV.2501.05315</a>.","ieee":"H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the electrostatic potential,” <i>arXiv</i>. .","apa":"Edelsbrunner, H., Fillmore, C. D., &#38; Olivera, G. (n.d.). Counting equilibria of the electrostatic potential. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">https://doi.org/10.48550/ARXIV.2501.05315</a>","ista":"Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv, <a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">10.48550/ARXIV.2501.05315</a>.","short":"H. Edelsbrunner, C.D. Fillmore, G. Olivera, ArXiv (n.d.)."},"language":[{"iso":"eng"}],"day":"20","oa":1,"OA_place":"repository","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","related_material":{"record":[{"id":"21021","relation":"dissertation_contains","status":"public"},{"status":"public","relation":"later_version","id":"21931"}]},"department":[{"_id":"HeEd"}],"abstract":[{"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.","lang":"eng"}],"doi":"10.48550/ARXIV.2501.05315","article_processing_charge":"No","publication":"arXiv","date_published":"2025-03-20T00:00:00Z","arxiv":1,"year":"2025","_id":"21050","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)"},"oa_version":"Preprint","month":"03","type":"preprint","status":"public","title":"Counting equilibria of the electrostatic potential","date_created":"2026-01-27T14:29:27Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2501.05315","open_access":"1"}],"corr_author":"1"},{"volume":4,"oa_version":"Published Version","project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"754411","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses","grant_number":"M03073"}],"ddc":["510"],"corr_author":"1","PlanS_conform":"1","date_created":"2025-08-31T22:01:33Z","doi":"10.1007/s44007-025-00170-0","article_processing_charge":"Yes (via OA deal)","date_published":"2025-12-01T00:00:00Z","publication":"La Matematica","abstract":[{"text":"The medial axis of a set consists of the points in the ambient space without a unique closest point in the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a skeleton topologically equivalent to the original set. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities, various prunings of the medial axis have been proposed in the computational geometry community. Here, we examine one type of pruning, called burning. Because of the good experimental results it was hoped that the burning method of simplifying the medial axis would be stable. In this work, we show a simple example that dashes such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of a shape where the medial axis goes from collapsible to non-collapsible. More precisely, we consider the standard deformation retract from the closed ball to Bing’s house with two rooms, but stop just short of the point where Bing’s house becomes two dimensional. This way we obtain an isotopy from the 3-ball to a thickened version of Bing’s house. Under this isotopy, the medial axis goes from collapsible to non-collapsible. We stress that this isotopy can be made generic, in the sense of singularity theory, as developed by Arnol’d and Thom.","lang":"eng"}],"intvolume":"         4","year":"2025","acknowledgement":"We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie Yan, and Tao Ju for sharing code to generate the examples. We further thank Abigail Thompson for discussion on the conjecture and James Damon for sharing his insight in singularity theory. We thank the reviewers for their detailed reviews, which helped to improve the exposition.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). Partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ and the European Research Council (ERC), grant no. 788183, ‘Alpha Shape Theory Extended’. The first author was supported in part by the National Science Foundation through grants DBI-1759807, CCF-1907612, and CCF-2444309. The fourth author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) M-3073, ANR grant StratMesh, ANR-24-CE48-1899, and the welcome package from IDEX of the Université Côte d’Azur, ANR-15-IDEX-01.","language":[{"iso":"eng"}],"day":"01","OA_place":"publisher","publication_identifier":{"eissn":["2730-9657"]},"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"file_size":2678640,"relation":"main_file","date_created":"2025-12-30T07:52:58Z","success":1,"file_id":"20885","creator":"dernst","date_updated":"2025-12-30T07:52:58Z","access_level":"open_access","content_type":"application/pdf","file_name":"2025_LaMatematica_Chambers.pdf","checksum":"e2043259194bfcdf3d74c4da8a5a853f"}],"publication_status":"published","OA_type":"hybrid","has_accepted_license":"1","author":[{"first_name":"Erin Wolf","full_name":"Chambers, Erin Wolf","last_name":"Chambers"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","full_name":"Fillmore, Christopher D","last_name":"Fillmore"},{"last_name":"Stephenson","full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6862-208X"},{"last_name":"Wintraecken","first_name":"Mathijs","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220"}],"scopus_import":"1","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)"},"_id":"20260","publisher":"Springer Nature","type":"journal_article","month":"12","status":"public","title":"Burning or collapsing the medial axis is unstable","department":[{"_id":"HeEd"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"21021"}]},"quality_controlled":"1","article_type":"original","page":"811-828","ec_funded":1,"citation":{"mla":"Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>, vol. 4, Springer Nature, 2025, pp. 811–28, doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>.","ama":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. 2025;4:811-828. doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>","chicago":"Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>.","short":"E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica 4 (2025) 811–828.","ista":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica. 4, 811–828.","apa":"Chambers, E. W., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2025). Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>","ieee":"E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning or collapsing the medial axis is unstable,” <i>La Matematica</i>, vol. 4. Springer Nature, pp. 811–828, 2025."},"file_date_updated":"2025-12-30T07:52:58Z","date_updated":"2026-04-07T11:42:48Z"},{"ddc":["516"],"project":[{"call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342"},{"call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"grant_number":"M03073","name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"volume":293,"oa_version":"Published Version","date_created":"2024-06-25T11:45:58Z","abstract":[{"text":"In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. - In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. - The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction. We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.","lang":"eng"}],"article_processing_charge":"No","publication":"40th International Symposium on Computational Geometry","date_published":"2024-06-06T00:00:00Z","doi":"10.4230/LIPIcs.SoCG.2024.11","acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nWintraecken, Mathijs: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d'Azur.","arxiv":1,"intvolume":"       293","year":"2024","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773164"]},"oa":1,"language":[{"iso":"eng"}],"day":"06","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"creator":"cfillmor","date_updated":"2024-06-25T11:47:26Z","content_type":"application/pdf","access_level":"open_access","file_name":"LIPIcs.SoCG.2024.11.pdf","checksum":"6a2ddc8b51aa58f197a8b294750f1f8d","file_size":20886142,"relation":"main_file","date_created":"2024-06-25T11:47:26Z","success":1,"file_id":"17171"}],"external_id":{"arxiv":["2206.10485"]},"scopus_import":"1","author":[{"last_name":"Attali","full_name":"Attali, Dominique","first_name":"Dominique"},{"orcid":"0000-0001-7841-0091","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","full_name":"Kourimska, Hana","last_name":"Kourimska"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","last_name":"Fillmore","first_name":"Christopher D","full_name":"Fillmore, Christopher D"},{"id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","first_name":"Ishika","full_name":"Ghosh, Ishika","last_name":"Ghosh"},{"last_name":"Lieutier","full_name":"Lieutier, André","first_name":"André"},{"full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","last_name":"Stephenson","orcid":"0000-0002-6862-208X","id":"2D04F932-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Wintraecken","first_name":"Mathijs","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220"}],"has_accepted_license":"1","publication_status":"published","_id":"17170","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)"},"status":"public","title":"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds","month":"06","type":"conference","alternative_title":["LIPIcs"],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"ec_funded":1,"page":"11:1-11:19","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11","location":"Athens, Greece","end_date":"2024-06-14"},"date_updated":"2025-04-15T07:16:57Z","file_date_updated":"2024-06-25T11:47:26Z","citation":{"chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” In <i>40th International Symposium on Computational Geometry</i>, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>.","mla":"Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>.","ama":"Attali D, Kourimska H, Fillmore CD, et al. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>","ieee":"D. Attali <i>et al.</i>, “Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293, p. 11:1-11:19.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19.","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 11:1-11:19."}},{"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)"},"_id":"18097","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","type":"conference","alternative_title":["LIPIcs"],"month":"06","status":"public","title":"The ultimate frontier: An optimality construction for homotopy inference (media exposition)","department":[{"_id":"HeEd"}],"quality_controlled":"1","conference":{"location":"Athens, Greece","end_date":"2024-06-14","start_date":"2024-06-11","name":"SoCG: Symposium on Computational Geometry"},"ec_funded":1,"citation":{"mla":"Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>.","ama":"Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality construction for homotopy inference (media exposition). In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>","chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, Andre Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>.","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. The ultimate frontier: An optimality construction for homotopy inference (media exposition). 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 87.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","ieee":"D. Attali <i>et al.</i>, “The ultimate frontier: An optimality construction for homotopy inference (media exposition),” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). The ultimate frontier: An optimality construction for homotopy inference (media exposition). In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>"},"file_date_updated":"2024-09-19T10:30:37Z","date_updated":"2025-04-15T07:16:58Z","article_number":"87","oa_version":"Published Version","volume":293,"project":[{"grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","grant_number":"754411"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses","grant_number":"M03073"}],"ddc":["000"],"corr_author":"1","date_created":"2024-09-19T10:29:48Z","doi":"10.4230/LIPIcs.SoCG.2024.87","date_published":"2024-06-06T00:00:00Z","publication":"40th International Symposium on Computational Geometry","article_processing_charge":"Yes","abstract":[{"lang":"eng","text":"In our companion paper \"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds\" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting."}],"year":"2024","intvolume":"       293","acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I02979-N35. Mathijs Wintraecken: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d’Azur.\r\nWe thank Jean-Daniel Boissonnat, Herbert Edelsbrunner, and Mariette Yvinec for discussion.","day":"06","language":[{"iso":"eng"}],"publication_identifier":{"isbn":["9783959773164"]},"oa":1,"file":[{"date_updated":"2024-09-19T10:30:37Z","file_name":"2024_LIPICs_Attali.pdf","content_type":"application/pdf","access_level":"open_access","creator":"dernst","checksum":"9355c2e60b8ec285e1b22719c5b73f1a","date_created":"2024-09-19T10:30:37Z","file_size":3507177,"relation":"main_file","success":1,"file_id":"18098"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","has_accepted_license":"1","author":[{"last_name":"Attali","first_name":"Dominique","full_name":"Attali, Dominique"},{"last_name":"Kourimska","full_name":"Kourimska, Hana","first_name":"Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","orcid":"0000-0001-7841-0091"},{"last_name":"Fillmore","full_name":"Fillmore, Christopher D","first_name":"Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"first_name":"Ishika","full_name":"Ghosh, Ishika","last_name":"Ghosh","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff"},{"last_name":"Lieutier","first_name":"Andre","full_name":"Lieutier, Andre"},{"last_name":"Stephenson","full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6862-208X"},{"orcid":"0000-0002-7472-2220","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","first_name":"Mathijs","last_name":"Wintraecken"}]},{"has_accepted_license":"1","publication_status":"published","scopus_import":"1","author":[{"first_name":"Erin","full_name":"Chambers, Erin","last_name":"Chambers"},{"id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","last_name":"Fillmore","full_name":"Fillmore, Christopher D","first_name":"Christopher D"},{"orcid":"0000-0002-6862-208X","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","last_name":"Stephenson"},{"orcid":"0000-0002-7472-2220","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","full_name":"Wintraecken, Mathijs","first_name":"Mathijs","last_name":"Wintraecken"}],"file":[{"success":1,"file_id":"11437","date_created":"2022-06-07T07:58:30Z","file_size":17580705,"relation":"main_file","checksum":"b25ce40fade4ebc0bcaae176db4f5f1f","date_updated":"2022-06-07T07:58:30Z","file_name":"2022_LIPICs_Chambers.pdf","content_type":"application/pdf","access_level":"open_access","creator":"dernst"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["1868-8969"],"isbn":["978-3-95977-227-3"]},"oa":1,"day":"01","language":[{"iso":"eng"}],"year":"2022","intvolume":"       224","acknowledgement":"Partially supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics” and the European Research Council (ERC), grant no. 788183, “Alpha Shape Theory Extended”. Erin Chambers: Supported in part by the National Science Foundation through grants DBI-1759807, CCF-1907612, and CCF-2106672. Mathijs Wintraecken: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411. The Austrian science fund (FWF) M-3073 Acknowledgements We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie Yan and Tao Ju for sharing code to generate the examples.","date_published":"2022-06-01T00:00:00Z","article_processing_charge":"No","publication":"38th International Symposium on Computational Geometry","doi":"10.4230/LIPIcs.SoCG.2022.66","abstract":[{"lang":"eng","text":"The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible."}],"corr_author":"1","date_created":"2022-06-01T14:18:04Z","volume":224,"oa_version":"Published Version","ddc":["510"],"project":[{"grant_number":"M03073","name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"},{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","grant_number":"754411"}],"file_date_updated":"2022-06-07T07:58:30Z","citation":{"apa":"Chambers, E., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2022). A cautionary tale: Burning the medial axis is unstable. In X. Goaoc &#38; M. Kerber (Eds.), <i>38th International Symposium on Computational Geometry</i> (Vol. 224, p. 66:1-66:9). Berlin, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">https://doi.org/10.4230/LIPIcs.SoCG.2022.66</a>","ieee":"E. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “A cautionary tale: Burning the medial axis is unstable,” in <i>38th International Symposium on Computational Geometry</i>, Berlin, Germany, 2022, vol. 224, p. 66:1-66:9.","short":"E. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, in:, X. Goaoc, M. Kerber (Eds.), 38th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022, p. 66:1-66:9.","ista":"Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. 2022. A cautionary tale: Burning the medial axis is unstable. 38th International Symposium on Computational Geometry. SoCG: Symposium on Computational GeometryLIPIcs vol. 224, 66:1-66:9.","chicago":"Chambers, Erin, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “A Cautionary Tale: Burning the Medial Axis Is Unstable.” In <i>38th International Symposium on Computational Geometry</i>, edited by Xavier Goaoc and Michael Kerber, 224:66:1-66:9. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">https://doi.org/10.4230/LIPIcs.SoCG.2022.66</a>.","ama":"Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. A cautionary tale: Burning the medial axis is unstable. In: Goaoc X, Kerber M, eds. <i>38th International Symposium on Computational Geometry</i>. Vol 224. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2022:66:1-66:9. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">10.4230/LIPIcs.SoCG.2022.66</a>","mla":"Chambers, Erin, et al. “A Cautionary Tale: Burning the Medial Axis Is Unstable.” <i>38th International Symposium on Computational Geometry</i>, edited by Xavier Goaoc and Michael Kerber, vol. 224, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022, p. 66:1-66:9, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2022.66\">10.4230/LIPIcs.SoCG.2022.66</a>."},"date_updated":"2025-04-14T07:43:57Z","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2022-06-07","end_date":"2022-06-10","location":"Berlin, Germany"},"ec_funded":1,"page":"66:1-66:9","editor":[{"last_name":"Goaoc","full_name":"Goaoc, Xavier","first_name":"Xavier"},{"last_name":"Kerber","full_name":"Kerber, Michael","first_name":"Michael"}],"quality_controlled":"1","department":[{"_id":"HeEd"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","title":"A cautionary tale: Burning the medial axis is unstable","status":"public","month":"06","type":"conference","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)"},"series_title":"LIPIcs","_id":"11428"}]