[{"author":[{"last_name":"Koval","id":"2eed1f3b-896a-11ed-bdf8-93c7c4bf159e","full_name":"Koval, Illya","first_name":"Illya"}],"day":"01","OA_place":"publisher","ec_funded":1,"page":"221-298","das_tickbox":"0","supplementarymaterial":"yes","volume":244,"article_type":"original","OA_type":"hybrid","file":[{"file_size":2256345,"creator":"dernst","file_id":"22394","content_type":"application/pdf","success":1,"relation":"main_file","date_updated":"2026-07-23T10:55:24Z","checksum":"487fa9113e1bbf32a6c70e6d1e8f63bc","date_created":"2026-07-23T10:55:24Z","access_level":"open_access","file_name":"2026_InventionesMath_Koval.pdf"}],"quality_controlled":"1","fulldoi":"https://doi.org/10.1007/s00222-025-01397-y","publication_status":"published","arxiv":1,"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publication":"Inventiones Mathematicae","date_created":"2023-09-06T08:35:43Z","date_published":"2026-04-01T00:00:00Z","oa":1,"researchdata_availability":"no","year":"2026","PlanS_conform":"1","status":"public","intvolume":"       244","mathsc":["37C83","35J05","37J70","74J25"],"type":"journal_article","language":[{"iso":"eng"}],"month":"04","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","oa_version":"Published Version","project":[{"grant_number":"885707","name":"Spectral rigidity and integrability for billiards and geodesic flows","_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","call_identifier":"H2020"}],"corr_author":"1","department":[{"_id":"GradSch"},{"_id":"VaKa"}],"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","publication_identifier":{"issn":["0020-9910"],"eissn":["1432-1297"]},"abstract":[{"text":"The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral uniqueness of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish local uniqueness for almost all of them.","lang":"eng"}],"file_date_updated":"2026-07-23T10:55:24Z","citation":{"apa":"Koval, I. (2026). Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse. <i>Inventiones Mathematicae</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00222-025-01397-y\">https://doi.org/10.1007/s00222-025-01397-y</a>","mla":"Koval, Illya. “Local Strong Birkhoff Conjecture and Local Spectral Rigidity of Almost Every Ellipse.” <i>Inventiones Mathematicae</i>, vol. 244, Springer Nature, 2026, pp. 221–98, doi:<a href=\"https://doi.org/10.1007/s00222-025-01397-y\">10.1007/s00222-025-01397-y</a>.","ama":"Koval I. Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse. <i>Inventiones Mathematicae</i>. 2026;244:221-298. doi:<a href=\"https://doi.org/10.1007/s00222-025-01397-y\">10.1007/s00222-025-01397-y</a>","short":"I. Koval, Inventiones Mathematicae 244 (2026) 221–298.","chicago":"Koval, Illya. “Local Strong Birkhoff Conjecture and Local Spectral Rigidity of Almost Every Ellipse.” <i>Inventiones Mathematicae</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00222-025-01397-y\">https://doi.org/10.1007/s00222-025-01397-y</a>.","ista":"Koval I. 2026. Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse. Inventiones Mathematicae. 244, 221–298.","ieee":"I. Koval, “Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse,” <i>Inventiones Mathematicae</i>, vol. 244. Springer Nature, pp. 221–298, 2026."},"doi":"10.1007/s00222-025-01397-y","title":"Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse","date_updated":"2026-07-23T10:58:59Z","ddc":["510"],"_id":"14278","publisher":"Springer Nature","external_id":{"arxiv":["2111.12171"]},"acknowledgement":"The author acknowledges the partial support of the European Research Council Grant #885707. He also thanks Vadim Kaloshin for proposing the idea of the project and greatly aiding the implementation. The author is also grateful to Hamid Hezari, Amir Vig, Steve Zelditch, Comlan E. Koudjinan, Corentin Fierobe, Ngo Nhok Tkhai Shon and Roman Sarapin for useful discussions. The author also acknowledges partial support of ISTern summer program. The project started in the summer of 2021, when the author was an intern at ISTA. Open access funding provided by Institute of Science and Technology (IST Austria)."},{"project":[{"name":"Randomness and structure in combinatorics","_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777"}],"scopus_import":"1","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"month":"06","oa_version":"Published Version","title":"Exponentially many graphs are determined by their spectrum","external_id":{"isi":["001249741500001"],"arxiv":["2309.09788"]},"acknowledgement":"Matthew Kwan was supported by ERC Starting Grant ‘RANDSTRUCT’ No. 101076777.","date_updated":"2026-10-16T10:14:59Z","ddc":["500"],"_id":"17475","publisher":"Oxford University Press","department":[{"_id":"MaKw"},{"_id":"VaKa"}],"has_accepted_license":"1","corr_author":"1","publication_identifier":{"issn":["0033-5606"],"eissn":["1464-3847"]},"abstract":[{"text":"As a discrete analogue of Kac’s celebrated question on ‘hearing the shape of a drum’ and towards a practical\r\ngraph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by\r\ntheir spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers,\r\nis that ‘almost all graphs are determined by their spectrum’, meaning that the fraction of unlabelled n-vertex\r\ngraphs which are determined by their spectrum converges to 1 as n → ∞.\r\nIn this paper, we make a step towards this conjecture, showing that there are exponentially many n-vertex\r\ngraphs which are determined by their spectrum. This improves on previous bounds (of shape e\r\nc\r\n√\r\nn\r\n). We also\r\npropose a number of further directions of research.\r\n","lang":"eng"}],"citation":{"apa":"Koval, I., &#38; Kwan, M. A. (2024). Exponentially many graphs are determined by their spectrum. <i>Quarterly Journal of Mathematics</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/qmath/haae030\">https://doi.org/10.1093/qmath/haae030</a>","mla":"Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined by Their Spectrum.” <i>Quarterly Journal of Mathematics</i>, vol. 75, no. 3, Oxford University Press, 2024, pp. 869–99, doi:<a href=\"https://doi.org/10.1093/qmath/haae030\">10.1093/qmath/haae030</a>.","ama":"Koval I, Kwan MA. Exponentially many graphs are determined by their spectrum. <i>Quarterly Journal of Mathematics</i>. 2024;75(3):869-899. doi:<a href=\"https://doi.org/10.1093/qmath/haae030\">10.1093/qmath/haae030</a>","short":"I. Koval, M.A. Kwan, Quarterly Journal of Mathematics 75 (2024) 869–899.","chicago":"Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined by Their Spectrum.” <i>Quarterly Journal of Mathematics</i>. Oxford University Press, 2024. <a href=\"https://doi.org/10.1093/qmath/haae030\">https://doi.org/10.1093/qmath/haae030</a>.","ista":"Koval I, Kwan MA. 2024. Exponentially many graphs are determined by their spectrum. Quarterly Journal of Mathematics. 75(3), 869–899.","ieee":"I. Koval and M. A. Kwan, “Exponentially many graphs are determined by their spectrum,” <i>Quarterly Journal of Mathematics</i>, vol. 75, no. 3. Oxford University Press, pp. 869–899, 2024."},"file_date_updated":"2024-09-06T12:23:57Z","doi":"10.1093/qmath/haae030","article_processing_charge":"Yes (via OA deal)","supplementarymaterial":"no","volume":75,"file":[{"checksum":"abf200d37ad69e6f2c0750a30296ad97","date_updated":"2024-09-06T12:23:57Z","relation":"main_file","access_level":"open_access","date_created":"2024-09-06T12:23:57Z","file_name":"2024_QuJofMath_Koval.pdf","file_size":946411,"creator":"cchlebak","file_id":"17851","content_type":"application/pdf","success":1}],"quality_controlled":"1","article_type":"original","day":"19","isi":1,"author":[{"first_name":"Illya","full_name":"Koval, Illya","last_name":"Koval","id":"2eed1f3b-896a-11ed-bdf8-93c7c4bf159e"},{"first_name":"Matthew Alan","full_name":"Kwan, Matthew Alan","last_name":"Kwan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","orcid":"0000-0002-4003-7567"}],"page":"869-899","das_tickbox":"0","date_published":"2024-06-19T00:00:00Z","oa":1,"date_created":"2024-09-01T22:01:07Z","type":"journal_article","issue":"3","year":"2024","status":"public","researchdata_availability":"no","intvolume":"        75","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publication":"Quarterly Journal of Mathematics","fulldoi":"https://doi.org/10.1093/qmath/haae030","publication_status":"published","arxiv":1}]