[{"language":[{"iso":"eng"}],"abstract":[{"text":"The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.","lang":"eng"}],"_id":"18626","date_updated":"2025-04-15T07:16:53Z","department":[{"_id":"HeEd"}],"date_published":"2025-02-01T00:00:00Z","citation":{"mla":"Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>, vol. 461, 110055, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025.","ama":"Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. 2025;461. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>."},"isi":1,"oa":1,"article_number":"110055","type":"journal_article","oa_version":"Preprint","date_created":"2024-12-08T23:01:54Z","scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Order-2 Delaunay triangulations optimize angles","article_type":"original","intvolume":"       461","author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"last_name":"Saghafian","first_name":"Morteza","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"article_processing_charge":"No","year":"2025","OA_type":"green","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"doi":"10.1016/j.aim.2024.110055","month":"02","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2310.18238","open_access":"1"}],"ec_funded":1,"OA_place":"repository","corr_author":"1","publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"arxiv":1,"status":"public","day":"01","publication_status":"published","publisher":"Elsevier","volume":461,"publication":"Advances in Mathematics","acknowledgement":"Work by the first and third authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation.","external_id":{"isi":["001370682500001"],"arxiv":["2310.18238"]},"quality_controlled":"1"},{"citation":{"mla":"Wang, Victor. “Asymptotic Growth of Translation-Dilation Orbits.” <i>Advances in Mathematics</i>, vol. 475, 110341, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2025.110341\">10.1016/j.aim.2025.110341</a>.","short":"V. Wang, Advances in Mathematics 475 (2025).","apa":"Wang, V. (2025). Asymptotic growth of translation-dilation orbits. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2025.110341\">https://doi.org/10.1016/j.aim.2025.110341</a>","ieee":"V. Wang, “Asymptotic growth of translation-dilation orbits,” <i>Advances in Mathematics</i>, vol. 475. Elsevier, 2025.","ama":"Wang V. Asymptotic growth of translation-dilation orbits. <i>Advances in Mathematics</i>. 2025;475. doi:<a href=\"https://doi.org/10.1016/j.aim.2025.110341\">10.1016/j.aim.2025.110341</a>","ista":"Wang V. 2025. Asymptotic growth of translation-dilation orbits. Advances in Mathematics. 475, 110341.","chicago":"Wang, Victor. “Asymptotic Growth of Translation-Dilation Orbits.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2025.110341\">https://doi.org/10.1016/j.aim.2025.110341</a>."},"date_published":"2025-07-01T00:00:00Z","department":[{"_id":"TiBr"}],"_id":"19727","abstract":[{"text":"By studying some Clausen-like multiple Dirichlet series, we complete the proof of Manin's conjecture for sufficiently split smooth equivariant compactifications of the translation-dilation group over the rationals. Secondary terms remain elusive in general.","lang":"eng"}],"date_updated":"2025-12-30T08:30:30Z","has_accepted_license":"1","language":[{"iso":"eng"}],"isi":1,"oa":1,"date_created":"2025-05-25T22:16:41Z","oa_version":"Published Version","type":"journal_article","article_number":"110341","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Asymptotic growth of translation-dilation orbits","scopus_import":"1","PlanS_conform":"1","intvolume":"       475","author":[{"id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","full_name":"Wang, Victor","first_name":"Victor","orcid":"0000-0002-0704-7026","last_name":"Wang"}],"article_type":"original","year":"2025","article_processing_charge":"Yes (via OA deal)","doi":"10.1016/j.aim.2025.110341","file_date_updated":"2025-12-30T08:30:17Z","OA_type":"hybrid","project":[{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","call_identifier":"H2020"}],"month":"07","ec_funded":1,"OA_place":"publisher","corr_author":"1","day":"01","status":"public","arxiv":1,"file":[{"success":1,"file_name":"2025_AdvMathematics_Wang.pdf","access_level":"open_access","file_size":1592341,"checksum":"01f2589b678ba840d6a4066c1d8d7642","content_type":"application/pdf","date_updated":"2025-12-30T08:30:17Z","creator":"dernst","file_id":"20895","date_created":"2025-12-30T08:30:17Z","relation":"main_file"}],"publication_identifier":{"issn":["0001-8708"],"eissn":["1090-2082"]},"publication":"Advances in Mathematics","volume":475,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"publisher":"Elsevier","publication_status":"published","quality_controlled":"1","ddc":["510"],"external_id":{"isi":["001495142300002"],"arxiv":["2309.07626"]},"acknowledgement":"I thank Yuri Tschinkel for introducing me to the beautiful paper [53] and associated open questions, and thank him as well as Ramin Takloo-Bighash and Sho Tanimoto for their encouragement and comments. Also, I thank Tim Browning and Dan Loughran for comments and suggestions concerning Manin–Peyre, homogeneous spaces, and splitness. Thanks also to Anshul Adve, Peter Sarnak, Philip Tosteson, Katy Woo, and Nina Zubrilina for some interesting discussions. I thank the Browning Group and Andy O'Desky for many conversations. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 101034413. Finally, I thank the editors and referees for their detailed input, which substantially improved the paper."},{"scopus_import":"1","title":"The average number of integral points on the congruent number curves","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"Yes (via OA deal)","year":"2024","article_type":"original","intvolume":"       457","author":[{"first_name":"Yik Tung","last_name":"Chan","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1"}],"language":[{"iso":"eng"}],"has_accepted_license":"1","abstract":[{"text":" We show that the total number of non-torsion integral points on the elliptic curves ED : y\r\n2 = x3 − D2x, where D ranges over positive squarefree integers less than N, is O(N(log N)\r\n−1/4+ǫ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown’s method on estimating the average size of the 2-Selmer group of the curves in this family.","lang":"eng"}],"_id":"18064","date_updated":"2025-01-13T08:54:36Z","citation":{"ista":"Chan S. 2024. The average number of integral points on the congruent number curves. Advances in Mathematics. 457(11), 109946.","chicago":"Chan, Stephanie. “The Average Number of Integral Points on the Congruent Number Curves.” <i>Advances in Mathematics</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.aim.2024.109946\">https://doi.org/10.1016/j.aim.2024.109946</a>.","ieee":"S. Chan, “The average number of integral points on the congruent number curves,” <i>Advances in Mathematics</i>, vol. 457, no. 11. Elsevier, 2024.","ama":"Chan S. The average number of integral points on the congruent number curves. <i>Advances in Mathematics</i>. 2024;457(11). doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109946\">10.1016/j.aim.2024.109946</a>","mla":"Chan, Stephanie. “The Average Number of Integral Points on the Congruent Number Curves.” <i>Advances in Mathematics</i>, vol. 457, no. 11, 109946, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109946\">10.1016/j.aim.2024.109946</a>.","short":"S. Chan, Advances in Mathematics 457 (2024).","apa":"Chan, S. (2024). The average number of integral points on the congruent number curves. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.109946\">https://doi.org/10.1016/j.aim.2024.109946</a>"},"department":[{"_id":"TiBr"}],"date_published":"2024-11-01T00:00:00Z","article_number":"109946","type":"journal_article","oa_version":"Published Version","date_created":"2024-09-15T22:01:39Z","oa":1,"issue":"11","publication_status":"published","publisher":"Elsevier","volume":457,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"publication":"Advances in Mathematics","publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"file":[{"file_id":"18829","relation":"main_file","date_created":"2025-01-13T08:54:09Z","access_level":"open_access","file_name":"2024_AdvancesMath_Chan.pdf","success":1,"file_size":564386,"content_type":"application/pdf","checksum":"f555742540ad91a3040aeafd68b1fcde","date_updated":"2025-01-13T08:54:09Z","creator":"dernst"}],"arxiv":1,"status":"public","day":"01","external_id":{"arxiv":["2112.01615"]},"ddc":["510"],"quality_controlled":"1","month":"11","OA_type":"hybrid","file_date_updated":"2025-01-13T08:54:09Z","doi":"10.1016/j.aim.2024.109946","corr_author":"1","OA_place":"publisher"},{"oa_version":"Published Version","date_created":"2024-09-15T22:01:39Z","article_number":"109943","type":"journal_article","oa":1,"isi":1,"_id":"18065","abstract":[{"text":"We establish a close connection between acceleration and dynamical degree for one-frequency quasi-periodic compact cocycles, by showing that two vectors derived separately from each coincide. Based on this, we provide a dynamical classification of one-frequency quasi-periodic  SO(3, R)-cocycles.","lang":"eng"}],"date_updated":"2025-09-08T09:44:19Z","date_published":"2024-11-01T00:00:00Z","citation":{"apa":"Hou, X., Pan, Y., &#38; Zhou, Q. (2024). Dynamical classification of analytic one-frequency quasi-periodic SO(3,R)-cocycles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.109943\">https://doi.org/10.1016/j.aim.2024.109943</a>","mla":"Hou, Xuanji, et al. “Dynamical Classification of Analytic One-Frequency Quasi-Periodic SO(3,R)-Cocycles.” <i>Advances in Mathematics</i>, vol. 457, 109943, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109943\">10.1016/j.aim.2024.109943</a>.","short":"X. Hou, Y. Pan, Q. Zhou, Advances in Mathematics 457 (2024).","ieee":"X. Hou, Y. Pan, and Q. Zhou, “Dynamical classification of analytic one-frequency quasi-periodic SO(3,R)-cocycles,” <i>Advances in Mathematics</i>, vol. 457. Elsevier, 2024.","ama":"Hou X, Pan Y, Zhou Q. Dynamical classification of analytic one-frequency quasi-periodic SO(3,R)-cocycles. <i>Advances in Mathematics</i>. 2024;457. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109943\">10.1016/j.aim.2024.109943</a>","chicago":"Hou, Xuanji, Yi Pan, and Qi Zhou. “Dynamical Classification of Analytic One-Frequency Quasi-Periodic SO(3,R)-Cocycles.” <i>Advances in Mathematics</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.aim.2024.109943\">https://doi.org/10.1016/j.aim.2024.109943</a>.","ista":"Hou X, Pan Y, Zhou Q. 2024. Dynamical classification of analytic one-frequency quasi-periodic SO(3,R)-cocycles. Advances in Mathematics. 457, 109943."},"department":[{"_id":"VaKa"}],"language":[{"iso":"eng"}],"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","year":"2024","intvolume":"       457","author":[{"first_name":"Xuanji","last_name":"Hou","full_name":"Hou, Xuanji"},{"last_name":"Pan","first_name":"Yi","id":"1e21c7f7-9070-11eb-847d-8b04c7169523","full_name":"Pan, Yi"},{"last_name":"Zhou","first_name":"Qi","full_name":"Zhou, Qi"}],"article_type":"original","title":"Dynamical classification of analytic one-frequency quasi-periodic SO(3,R)-cocycles","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","scopus_import":"1","corr_author":"1","OA_place":"publisher","ec_funded":1,"month":"11","file_date_updated":"2025-01-13T08:29:27Z","doi":"10.1016/j.aim.2024.109943","project":[{"_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","grant_number":"885707","name":"Spectral rigidity and integrability for billiards and geodesic flows","call_identifier":"H2020"}],"OA_type":"hybrid","quality_controlled":"1","acknowledgement":"X. Hou is partially supported by National Natural Science Foundation of China (Grant \r\n12071083) and Funds for Distinguished Youths of Hubei Province of China (\r\n2019CFA680). Y. Pan is supported by ERC Advanced Grant (#885707). Q. Zhou is partially supported by National Key R&D Program of China (2020YFA0713300), NSFC grant (\r\n12071232) and Nankai Zhide Foundation.","ddc":["510"],"external_id":{"arxiv":["2311.17537"],"isi":["001315306500001"]},"volume":457,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"publication":"Advances in Mathematics","publication_status":"published","publisher":"Elsevier","status":"public","day":"01","publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"file":[{"file_name":"2024_AdvancesMath_Hou.pdf","access_level":"open_access","success":1,"file_size":713659,"content_type":"application/pdf","checksum":"1c80b844a91d93cf4799f4a65873b18d","creator":"dernst","date_updated":"2025-01-13T08:29:27Z","file_id":"18826","relation":"main_file","date_created":"2025-01-13T08:29:27Z"}],"arxiv":1},{"ec_funded":1,"corr_author":"1","OA_place":"publisher","project":[{"call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"OA_type":"hybrid","doi":"10.1016/j.aim.2024.109616","file_date_updated":"2024-07-22T12:10:03Z","month":"05","ddc":["510"],"external_id":{"isi":["001216128200001"],"arxiv":["2302.02817"]},"acknowledgement":"Shiyu Shen has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 101034413.","quality_controlled":"1","arxiv":1,"file":[{"file_id":"17315","date_created":"2024-07-22T12:10:03Z","relation":"main_file","access_level":"open_access","file_name":"2024_AdvancesMath_Shen.pdf","success":1,"file_size":702889,"content_type":"application/pdf","checksum":"68f2f08136ccf547891a16a2c0621e97","date_updated":"2024-07-22T12:10:03Z","creator":"dernst"}],"publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"day":"01","status":"public","publisher":"Elsevier","publication_status":"published","publication":"Advances in Mathematics","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"volume":443,"isi":1,"oa":1,"issue":"5","type":"journal_article","article_number":"109616","oa_version":"Published Version","date_created":"2024-03-31T22:01:11Z","has_accepted_license":"1","language":[{"iso":"eng"}],"citation":{"ista":"Shen S. 2024. Mirror symmetry for parabolic Higgs bundles via p-adic integration. Advances in Mathematics. 443(5), 109616.","chicago":"Shen, Shiyu. “Mirror Symmetry for Parabolic Higgs Bundles via P-Adic Integration.” <i>Advances in Mathematics</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.aim.2024.109616\">https://doi.org/10.1016/j.aim.2024.109616</a>.","ama":"Shen S. Mirror symmetry for parabolic Higgs bundles via p-adic integration. <i>Advances in Mathematics</i>. 2024;443(5). doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109616\">10.1016/j.aim.2024.109616</a>","ieee":"S. Shen, “Mirror symmetry for parabolic Higgs bundles via p-adic integration,” <i>Advances in Mathematics</i>, vol. 443, no. 5. Elsevier, 2024.","mla":"Shen, Shiyu. “Mirror Symmetry for Parabolic Higgs Bundles via P-Adic Integration.” <i>Advances in Mathematics</i>, vol. 443, no. 5, 109616, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109616\">10.1016/j.aim.2024.109616</a>.","short":"S. Shen, Advances in Mathematics 443 (2024).","apa":"Shen, S. (2024). Mirror symmetry for parabolic Higgs bundles via p-adic integration. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.109616\">https://doi.org/10.1016/j.aim.2024.109616</a>"},"department":[{"_id":"TaHa"}],"date_published":"2024-05-01T00:00:00Z","abstract":[{"text":"Applying the technique of p-adic integration, we prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli spaces of (strongly) parabolic Higgs bundles for the structure groups SLn and PGLn, building on previous work of Groechenig-Wyss-Ziegler on the non-parabolic case. We also prove the E-polynomial of the smooth moduli space of parabolic GLn-Higgs bundles is independent of the degree of the underlying vector bundles.","lang":"eng"}],"_id":"15248","date_updated":"2025-09-04T13:21:18Z","article_type":"original","author":[{"full_name":"Shen, Shiyu","id":"544cccd3-9005-11ec-87bc-94aef1c5b814","first_name":"Shiyu","last_name":"Shen","orcid":"0000-0002-4444-8718"}],"intvolume":"       443","year":"2024","article_processing_charge":"Yes (via OA deal)","scopus_import":"1","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"Mirror symmetry for parabolic Higgs bundles via p-adic integration"},{"month":"10","file_date_updated":"2023-02-02T07:39:09Z","doi":"10.1016/j.aim.2022.108591","project":[{"grant_number":"885707","name":"Spectral rigidity and integrability for billiards and geodesic flows","call_identifier":"H2020","_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A"}],"corr_author":"1","ec_funded":1,"volume":408,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"publication":"Advances in Mathematics","publication_status":"published","publisher":"Elsevier","status":"public","day":"29","publication_identifier":{"issn":["0001-8708"]},"file":[{"file_size":2164036,"success":1,"file_name":"2022_AdvancesMathematics_Drach.pdf","access_level":"open_access","creator":"dernst","date_updated":"2023-02-02T07:39:09Z","checksum":"2710e6f5820f8c20a676ddcbb30f0e8d","content_type":"application/pdf","file_id":"12474","relation":"main_file","date_created":"2023-02-02T07:39:09Z"}],"quality_controlled":"1","acknowledgement":"We are grateful to a number of colleagues for helpful and inspiring discussions during the time when we worked on this project, in particular Dima Dudko, Misha Hlushchanka, John Hubbard, Misha Lyubich, Oleg Kozlovski, and Sebastian van Strien. Finally, we would like to thank our dynamics research group for numerous helpful and enjoyable discussions: Konstantin Bogdanov, Roman Chernov, Russell Lodge, Steffen Maaß, David Pfrang, Bernhard Reinke, Sergey Shemyakov, and Maik Sowinski. We gratefully acknowledge support by the Advanced Grant “HOLOGRAM” (#695 621) of the European Research Council (ERC), as well as hospitality of Cornell University in the spring of 2018 while much of this work was prepared. The first-named author also acknowledges the support of the ERC Advanced Grant “SPERIG” (#885 707).","ddc":["510"],"external_id":{"isi":["000860924200005"]},"date_updated":"2025-04-14T07:53:45Z","_id":"11717","abstract":[{"lang":"eng","text":"We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set.\r\nIn the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable “in the same way”.\r\nOur main tool is a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations."}],"citation":{"ista":"Drach K, Schleicher D. 2022. Rigidity of Newton dynamics. Advances in Mathematics. 408(Part A), 108591.","chicago":"Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” <i>Advances in Mathematics</i>. Elsevier, 2022. <a href=\"https://doi.org/10.1016/j.aim.2022.108591\">https://doi.org/10.1016/j.aim.2022.108591</a>.","ama":"Drach K, Schleicher D. Rigidity of Newton dynamics. <i>Advances in Mathematics</i>. 2022;408(Part A). doi:<a href=\"https://doi.org/10.1016/j.aim.2022.108591\">10.1016/j.aim.2022.108591</a>","ieee":"K. Drach and D. Schleicher, “Rigidity of Newton dynamics,” <i>Advances in Mathematics</i>, vol. 408, no. Part A. Elsevier, 2022.","short":"K. Drach, D. Schleicher, Advances in Mathematics 408 (2022).","mla":"Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” <i>Advances in Mathematics</i>, vol. 408, no. Part A, 108591, Elsevier, 2022, doi:<a href=\"https://doi.org/10.1016/j.aim.2022.108591\">10.1016/j.aim.2022.108591</a>.","apa":"Drach, K., &#38; Schleicher, D. (2022). Rigidity of Newton dynamics. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2022.108591\">https://doi.org/10.1016/j.aim.2022.108591</a>"},"department":[{"_id":"VaKa"}],"date_published":"2022-10-29T00:00:00Z","language":[{"iso":"eng"}],"has_accepted_license":"1","date_created":"2022-08-01T17:08:16Z","oa_version":"Published Version","article_number":"108591","type":"journal_article","issue":"Part A","oa":1,"isi":1,"title":"Rigidity of Newton dynamics","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","scopus_import":"1","article_processing_charge":"Yes (via OA deal)","year":"2022","keyword":["General Mathematics"],"author":[{"orcid":"0000-0002-9156-8616","last_name":"Drach","first_name":"Kostiantyn","full_name":"Drach, Kostiantyn","id":"fe8209e2-906f-11eb-847d-950f8fc09115"},{"first_name":"Dierk","last_name":"Schleicher","full_name":"Schleicher, Dierk"}],"intvolume":"       408","article_type":"original"},{"author":[{"full_name":"Cao, Yang","first_name":"Yang","last_name":"Cao"},{"first_name":"Zhizhong","last_name":"Huang","id":"21f1b52f-2fd1-11eb-a347-a4cdb9b18a51","full_name":"Huang, Zhizhong"}],"intvolume":"       398","article_type":"original","year":"2022","article_processing_charge":"No","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics","scopus_import":"1","issue":"3","isi":1,"oa":1,"date_created":"2022-02-20T23:01:30Z","oa_version":"Preprint","type":"journal_article","article_number":"108236","citation":{"apa":"Cao, Y., &#38; Huang, Z. (2022). Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2022.108236\">https://doi.org/10.1016/j.aim.2022.108236</a>","short":"Y. Cao, Z. Huang, Advances in Mathematics 398 (2022).","mla":"Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood Property and Geometric Sieve for Affine Quadrics.” <i>Advances in Mathematics</i>, vol. 398, no. 3, 108236, Elsevier, 2022, doi:<a href=\"https://doi.org/10.1016/j.aim.2022.108236\">10.1016/j.aim.2022.108236</a>.","chicago":"Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood Property and Geometric Sieve for Affine Quadrics.” <i>Advances in Mathematics</i>. Elsevier, 2022. <a href=\"https://doi.org/10.1016/j.aim.2022.108236\">https://doi.org/10.1016/j.aim.2022.108236</a>.","ista":"Cao Y, Huang Z. 2022. Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics. Advances in Mathematics. 398(3), 108236.","ieee":"Y. Cao and Z. Huang, “Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics,” <i>Advances in Mathematics</i>, vol. 398, no. 3. Elsevier, 2022.","ama":"Cao Y, Huang Z. Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics. <i>Advances in Mathematics</i>. 2022;398(3). doi:<a href=\"https://doi.org/10.1016/j.aim.2022.108236\">10.1016/j.aim.2022.108236</a>"},"department":[{"_id":"TiBr"}],"date_published":"2022-03-26T00:00:00Z","date_updated":"2024-10-09T21:01:35Z","_id":"10765","abstract":[{"lang":"eng","text":"We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x1,...,xn)=m, where q is a non-degenerate integral quadratic form in  n>3 variables and m is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics."}],"language":[{"iso":"eng"}],"quality_controlled":"1","external_id":{"arxiv":["2003.07287"],"isi":["000792517300014"]},"acknowledgement":"We are grateful to Mikhail Borovoi, Zeev Rudnick and Olivier Wienberg for their interest in our\r\nwork. We would like to address our gratitude to Ulrich Derenthal for his generous support at Leibniz Universitat Hannover. We are in debt to Tim Browning for an enlightening discussion and to the anonymous referees for critical comments, which lead to overall improvements of various preliminary versions of this paper. Part of this work was carried out and reported during a visit to the University of Science and Technology of China. We thank Yongqi Liang for offering warm hospitality. The first author was supported by a Humboldt-Forschungsstipendium. The second author was supported by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft.","day":"26","status":"public","arxiv":1,"publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"publication":"Advances in Mathematics","volume":398,"publisher":"Elsevier","publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2003.07287"}],"corr_author":"1","doi":"10.1016/j.aim.2022.108236","month":"03"},{"isi":1,"oa":1,"oa_version":"Published Version","date_created":"2021-09-21T15:58:59Z","article_number":"107992","type":"journal_article","_id":"10033","date_updated":"2025-04-14T09:09:35Z","abstract":[{"lang":"eng","text":"The ⊗*-monoidal structure on the category of sheaves on the Ran space is not pro-nilpotent in the sense of [3]. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the Ran space and integrating along the Ran space, i.e. taking factorization homology. Based on ideas sketched in [4], we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah-Bott formula given in [7] and [5]."}],"citation":{"short":"Q.P. Ho, Advances in Mathematics 392 (2021).","mla":"Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul Duality.” <i>Advances in Mathematics</i>, vol. 392, 107992, Elsevier, 2021, doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107992\">10.1016/j.aim.2021.107992</a>.","apa":"Ho, Q. P. (2021). The Atiyah-Bott formula and connectivity in chiral Koszul duality. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2021.107992\">https://doi.org/10.1016/j.aim.2021.107992</a>","ieee":"Q. P. Ho, “The Atiyah-Bott formula and connectivity in chiral Koszul duality,” <i>Advances in Mathematics</i>, vol. 392. Elsevier, 2021.","ama":"Ho QP. The Atiyah-Bott formula and connectivity in chiral Koszul duality. <i>Advances in Mathematics</i>. 2021;392. doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107992\">10.1016/j.aim.2021.107992</a>","ista":"Ho QP. 2021. The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. 392, 107992.","chicago":"Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul Duality.” <i>Advances in Mathematics</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.aim.2021.107992\">https://doi.org/10.1016/j.aim.2021.107992</a>."},"date_published":"2021-09-21T00:00:00Z","department":[{"_id":"TaHa"}],"language":[{"iso":"eng"}],"has_accepted_license":"1","author":[{"orcid":"0000-0001-6889-1418","last_name":"Ho","first_name":"Quoc P","full_name":"Ho, Quoc P","id":"3DD82E3C-F248-11E8-B48F-1D18A9856A87"}],"intvolume":"       392","article_type":"original","article_processing_charge":"Yes (via OA deal)","keyword":["Chiral algebras","Chiral homology","Factorization algebras","Koszul duality","Ran space"],"year":"2021","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"The Atiyah-Bott formula and connectivity in chiral Koszul duality","scopus_import":"1","corr_author":"1","file_date_updated":"2021-09-21T15:58:52Z","doi":"10.1016/j.aim.2021.107992","project":[{"call_identifier":"FWF","grant_number":"M02751","name":"Algebro-Geometric Applications of Factorization Homology","_id":"26B96266-B435-11E9-9278-68D0E5697425"}],"month":"09","quality_controlled":"1","acknowledgement":"The author would like to express his gratitude to D. Gaitsgory, without whose tireless guidance and encouragement in pursuing this problem, this work would not have been possible. The author is grateful to his advisor B.C. Ngô for many years of patient guidance and support. This paper is revised while the author is a postdoc in Hausel group at IST Austria. We thank him and the group for providing a wonderful research environment. The author also gratefully acknowledges the support of the Lise Meitner fellowship “Algebro-Geometric Applications of Factorization Homology,” Austrian Science Fund (FWF): M 2751.","ddc":["514"],"external_id":{"arxiv":["1610.00212"],"isi":["000707040300031"]},"status":"public","day":"21","publication_identifier":{"eissn":["1090-2082"],"issn":["0001-8708"]},"arxiv":1,"file":[{"date_created":"2021-09-21T15:58:52Z","relation":"main_file","file_id":"10034","content_type":"application/pdf","checksum":"f3c0086d41af11db31c00014efb38072","date_updated":"2021-09-21T15:58:52Z","creator":"qho","file_name":"1-s2.0-S000187082100431X-main.pdf","access_level":"open_access","file_size":840635}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"volume":392,"publication":"Advances in Mathematics","publication_status":"published","publisher":"Elsevier"},{"month":"03","doi":"10.1016/j.aim.2021.107595","project":[{"_id":"26A455A6-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Geometric study of Wasserstein spaces and free probability","grant_number":"846294"}],"ec_funded":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.10447"}],"volume":380,"publication":"Advances in Mathematics","publication_status":"published","publisher":"Elsevier","status":"public","day":"26","publication_identifier":{"issn":["0001-8708"]},"arxiv":1,"quality_controlled":"1","acknowledgement":"D. Virosztek was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","external_id":{"isi":["000619676100035"],"arxiv":["1910.10447"]},"date_updated":"2025-04-14T07:50:40Z","_id":"9036","abstract":[{"lang":"eng","text":"In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space."}],"department":[{"_id":"LaEr"}],"citation":{"mla":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” <i>Advances in Mathematics</i>, vol. 380, no. 3, 107595, Elsevier, 2021, doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107595\">10.1016/j.aim.2021.107595</a>.","short":"D. Virosztek, Advances in Mathematics 380 (2021).","apa":"Virosztek, D. (2021). The metric property of the quantum Jensen-Shannon divergence. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2021.107595\">https://doi.org/10.1016/j.aim.2021.107595</a>","ista":"Virosztek D. 2021. The metric property of the quantum Jensen-Shannon divergence. Advances in Mathematics. 380(3), 107595.","chicago":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” <i>Advances in Mathematics</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.aim.2021.107595\">https://doi.org/10.1016/j.aim.2021.107595</a>.","ieee":"D. Virosztek, “The metric property of the quantum Jensen-Shannon divergence,” <i>Advances in Mathematics</i>, vol. 380, no. 3. Elsevier, 2021.","ama":"Virosztek D. The metric property of the quantum Jensen-Shannon divergence. <i>Advances in Mathematics</i>. 2021;380(3). doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107595\">10.1016/j.aim.2021.107595</a>"},"date_published":"2021-03-26T00:00:00Z","language":[{"iso":"eng"}],"oa_version":"Preprint","date_created":"2021-01-22T17:55:17Z","article_number":"107595","type":"journal_article","issue":"3","isi":1,"oa":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"The metric property of the quantum Jensen-Shannon divergence","scopus_import":"1","article_processing_charge":"No","keyword":["General Mathematics"],"year":"2021","intvolume":"       380","author":[{"id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","full_name":"Virosztek, Daniel","first_name":"Daniel","orcid":"0000-0003-1109-5511","last_name":"Virosztek"}],"article_type":"original"},{"month":"06","file_date_updated":"2020-07-14T12:47:27Z","doi":"10.1016/j.aim.2019.04.031","ddc":["512"],"external_id":{"isi":["000468857300025"],"arxiv":["1810.08426"]},"quality_controlled":"1","publication_status":"published","publisher":"Elsevier","volume":349,"publication":"Advances in Mathematics","publication_identifier":{"issn":["0001-8708"],"eissn":["1090-2082"]},"file":[{"file_size":379158,"file_name":"wliqun.pdf","access_level":"open_access","creator":"tbrownin","date_updated":"2020-07-14T12:47:27Z","content_type":"application/pdf","checksum":"a63594a3a91b4ba6e2a1b78b0720b3d0","file_id":"6311","relation":"main_file","date_created":"2019-04-16T09:12:20Z"}],"arxiv":1,"status":"public","day":"20","type":"journal_article","date_created":"2019-04-16T09:13:25Z","oa_version":"Submitted Version","oa":1,"isi":1,"language":[{"iso":"eng"}],"has_accepted_license":"1","_id":"6310","abstract":[{"lang":"eng","text":"An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariskiopen subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy–Littlewood circle method."}],"date_updated":"2025-07-10T11:53:19Z","citation":{"ama":"Browning TD, Hu LQ. Counting rational points on biquadratic hypersurfaces. <i>Advances in Mathematics</i>. 2019;349:920-940. doi:<a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">10.1016/j.aim.2019.04.031</a>","ieee":"T. D. Browning and L. Q. Hu, “Counting rational points on biquadratic hypersurfaces,” <i>Advances in Mathematics</i>, vol. 349. Elsevier, pp. 920–940, 2019.","ista":"Browning TD, Hu LQ. 2019. Counting rational points on biquadratic hypersurfaces. Advances in Mathematics. 349, 920–940.","chicago":"Browning, Timothy D, and L.Q. Hu. “Counting Rational Points on Biquadratic Hypersurfaces.” <i>Advances in Mathematics</i>. Elsevier, 2019. <a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">https://doi.org/10.1016/j.aim.2019.04.031</a>.","short":"T.D. Browning, L.Q. Hu, Advances in Mathematics 349 (2019) 920–940.","mla":"Browning, Timothy D., and L. Q. Hu. “Counting Rational Points on Biquadratic Hypersurfaces.” <i>Advances in Mathematics</i>, vol. 349, Elsevier, 2019, pp. 920–40, doi:<a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">10.1016/j.aim.2019.04.031</a>.","apa":"Browning, T. D., &#38; Hu, L. Q. (2019). Counting rational points on biquadratic hypersurfaces. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">https://doi.org/10.1016/j.aim.2019.04.031</a>"},"date_published":"2019-06-20T00:00:00Z","page":"920-940","department":[{"_id":"TiBr"}],"article_processing_charge":"No","year":"2019","author":[{"first_name":"Timothy D","orcid":"0000-0002-8314-0177","last_name":"Browning","full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Hu, L.Q.","last_name":"Hu","first_name":"L.Q."}],"intvolume":"       349","scopus_import":"1","title":"Counting rational points on biquadratic hypersurfaces","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"language":[{"iso":"eng"}],"date_published":"2017-02-21T00:00:00Z","department":[{"_id":"HeEd"}],"citation":{"ieee":"A. Akopyan, I. Bárány, and S. Robins, “Algebraic vertices of non-convex polyhedra,” <i>Advances in Mathematics</i>, vol. 308. Academic Press, pp. 627–644, 2017.","ama":"Akopyan A, Bárány I, Robins S. Algebraic vertices of non-convex polyhedra. <i>Advances in Mathematics</i>. 2017;308:627-644. doi:<a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">10.1016/j.aim.2016.12.026</a>","ista":"Akopyan A, Bárány I, Robins S. 2017. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 308, 627–644.","chicago":"Akopyan, Arseniy, Imre Bárány, and Sinai Robins. “Algebraic Vertices of Non-Convex Polyhedra.” <i>Advances in Mathematics</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">https://doi.org/10.1016/j.aim.2016.12.026</a>.","short":"A. Akopyan, I. Bárány, S. Robins, Advances in Mathematics 308 (2017) 627–644.","mla":"Akopyan, Arseniy, et al. “Algebraic Vertices of Non-Convex Polyhedra.” <i>Advances in Mathematics</i>, vol. 308, Academic Press, 2017, pp. 627–44, doi:<a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">10.1016/j.aim.2016.12.026</a>.","apa":"Akopyan, A., Bárány, I., &#38; Robins, S. (2017). Algebraic vertices of non-convex polyhedra. <i>Advances in Mathematics</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">https://doi.org/10.1016/j.aim.2016.12.026</a>"},"page":"627 - 644","_id":"1180","abstract":[{"text":"In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform.","lang":"eng"}],"date_updated":"2025-06-04T08:45:48Z","oa":1,"isi":1,"type":"journal_article","oa_version":"Submitted Version","date_created":"2018-12-11T11:50:34Z","scopus_import":"1","title":"Algebraic vertices of non-convex polyhedra","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","orcid":"0000-0002-2548-617X","first_name":"Arseniy"},{"full_name":"Bárány, Imre","first_name":"Imre","last_name":"Bárány"},{"first_name":"Sinai","last_name":"Robins","full_name":"Robins, Sinai"}],"intvolume":"       308","year":"2017","article_processing_charge":"No","project":[{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"doi":"10.1016/j.aim.2016.12.026","month":"02","main_file_link":[{"url":"https://arxiv.org/abs/1508.07594","open_access":"1"}],"ec_funded":1,"publist_id":"6173","arxiv":1,"publication_identifier":{"issn":["0001-8708"]},"day":"21","status":"public","publisher":"Academic Press","publication_status":"published","publication":"Advances in Mathematics","volume":308,"external_id":{"arxiv":["1508.07594"],"isi":["000409292900015"]},"quality_controlled":"1"},{"year":"2017","article_processing_charge":"No","extern":"1","author":[{"first_name":"Afonso S.","last_name":"Bandeira","full_name":"Bandeira, Afonso S."},{"last_name":"Ferber","first_name":"Asaf","full_name":"Ferber, Asaf"},{"first_name":"Matthew Alan","last_name":"Kwan","orcid":"0000-0002-4003-7567","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","full_name":"Kwan, Matthew Alan"}],"intvolume":"       319","article_type":"original","user_id":"6785fbc1-c503-11eb-8a32-93094b40e1cf","title":"Resilience for the Littlewood–Offord problem","scopus_import":"1","oa_version":"Preprint","date_created":"2021-06-22T11:51:27Z","type":"journal_article","oa":1,"citation":{"chicago":"Bandeira, Afonso S., Asaf Ferber, and Matthew Alan Kwan. “Resilience for the Littlewood–Offord Problem.” <i>Advances in Mathematics</i>. Elsevier, 2017. <a href=\"https://doi.org/10.1016/j.aim.2017.08.031\">https://doi.org/10.1016/j.aim.2017.08.031</a>.","ista":"Bandeira AS, Ferber A, Kwan MA. 2017. Resilience for the Littlewood–Offord problem. Advances in Mathematics. 319, 292–312.","ama":"Bandeira AS, Ferber A, Kwan MA. Resilience for the Littlewood–Offord problem. <i>Advances in Mathematics</i>. 2017;319:292-312. doi:<a href=\"https://doi.org/10.1016/j.aim.2017.08.031\">10.1016/j.aim.2017.08.031</a>","ieee":"A. S. Bandeira, A. Ferber, and M. A. Kwan, “Resilience for the Littlewood–Offord problem,” <i>Advances in Mathematics</i>, vol. 319. Elsevier, pp. 292–312, 2017.","apa":"Bandeira, A. S., Ferber, A., &#38; Kwan, M. A. (2017). Resilience for the Littlewood–Offord problem. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2017.08.031\">https://doi.org/10.1016/j.aim.2017.08.031</a>","short":"A.S. Bandeira, A. Ferber, M.A. Kwan, Advances in Mathematics 319 (2017) 292–312.","mla":"Bandeira, Afonso S., et al. “Resilience for the Littlewood–Offord Problem.” <i>Advances in Mathematics</i>, vol. 319, Elsevier, 2017, pp. 292–312, doi:<a href=\"https://doi.org/10.1016/j.aim.2017.08.031\">10.1016/j.aim.2017.08.031</a>."},"page":"292-312","date_published":"2017-10-15T00:00:00Z","date_updated":"2023-02-23T14:01:57Z","_id":"9588","abstract":[{"lang":"eng","text":"Consider the sum  X(ξ)=∑ni=1aiξi , where  a=(ai)ni=1  is a sequence of non-zero reals and  ξ=(ξi)ni=1  is a sequence of i.i.d. Rademacher random variables (that is,  Pr[ξi=1]=Pr[ξi=−1]=1/2 ). The classical Littlewood-Offord problem asks for the best possible upper bound on the concentration probabilities  Pr[X=x] . In this paper we study a resilience version of the Littlewood-Offord problem: how many of the  ξi  is an adversary typically allowed to change without being able to force concentration on a particular value? We solve this problem asymptotically, and present a few interesting open problems."}],"language":[{"iso":"eng"}],"quality_controlled":"1","external_id":{"arxiv":["1609.08136"]},"publication":"Advances in Mathematics","volume":319,"publisher":"Elsevier","publication_status":"published","day":"15","status":"public","arxiv":1,"publication_identifier":{"issn":["0001-8708"]},"main_file_link":[{"url":"https://arxiv.org/abs/1609.08136","open_access":"1"}],"month":"10","doi":"10.1016/j.aim.2017.08.031"},{"doi":"10.1016/j.aim.2006.03.012","language":[{"iso":"eng"}],"month":"01","abstract":[{"lang":"eng","text":"Here we study an amazing phenomenon discovered by Newhouse [S. Newhouse, Non-density of Axiom A(a) on S2, in: Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 191–202; S. Newhouse,\r\nDiffeomorphisms with infinitely many sinks, Topology 13 (1974) 9–18; S. Newhouse, The abundance of\r\nwild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, Publ. Math. Inst. Hautes Études Sci.\r\n50 (1979) 101–151]. It turns out that in the space of Cr smooth diffeomorphisms Diffr(M) of a compact\r\nsurface M there is an open set U such that a Baire generic diffeomorphism f ∈ U has infinitely many coexisting sinks. In this paper we make a step towards understanding “how often does a surface diffeomorphism\r\nhave infinitely many sinks.” Our main result roughly says that with probability one for any positive D a\r\nsurface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those\r\nwhose period is relatively large compared to its cyclicity. It verifies a particular case of Palis’ Conjecture\r\nsaying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have\r\nprobability zero.\r\nOne of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number\r\nof periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched\r\nexponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II,\r\npreprint, 85 pp.]."}],"_id":"8511","date_updated":"2021-01-12T08:19:47Z","page":"710-797","date_published":"2007-01-30T00:00:00Z","citation":{"mla":"Gorodetski, A., and Vadim Kaloshin. “How Often Surface Diffeomorphisms Have Infinitely Many Sinks and Hyperbolicity of Periodic Points near a Homoclinic Tangency.” <i>Advances in Mathematics</i>, vol. 208, no. 2, Elsevier, 2007, pp. 710–97, doi:<a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">10.1016/j.aim.2006.03.012</a>.","short":"A. Gorodetski, V. Kaloshin, Advances in Mathematics 208 (2007) 710–797.","apa":"Gorodetski, A., &#38; Kaloshin, V. (2007). How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">https://doi.org/10.1016/j.aim.2006.03.012</a>","ieee":"A. Gorodetski and V. Kaloshin, “How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency,” <i>Advances in Mathematics</i>, vol. 208, no. 2. Elsevier, pp. 710–797, 2007.","ama":"Gorodetski A, Kaloshin V. How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. <i>Advances in Mathematics</i>. 2007;208(2):710-797. doi:<a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">10.1016/j.aim.2006.03.012</a>","ista":"Gorodetski A, Kaloshin V. 2007. How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. Advances in Mathematics. 208(2), 710–797.","chicago":"Gorodetski, A., and Vadim Kaloshin. “How Often Surface Diffeomorphisms Have Infinitely Many Sinks and Hyperbolicity of Periodic Points near a Homoclinic Tangency.” <i>Advances in Mathematics</i>. Elsevier, 2007. <a href=\"https://doi.org/10.1016/j.aim.2006.03.012\">https://doi.org/10.1016/j.aim.2006.03.012</a>."},"issue":"2","type":"journal_article","oa_version":"None","date_created":"2020-09-18T10:48:27Z","publication_identifier":{"issn":["0001-8708"]},"title":"How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","day":"30","publication_status":"published","publisher":"Elsevier","volume":208,"publication":"Advances in Mathematics","article_type":"original","quality_controlled":"1","author":[{"full_name":"Gorodetski, A.","first_name":"A.","last_name":"Gorodetski"},{"orcid":"0000-0002-6051-2628","last_name":"Kaloshin","first_name":"Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","full_name":"Kaloshin, Vadim"}],"intvolume":"       208","extern":"1","article_processing_charge":"No","year":"2007","keyword":["General Mathematics"]}]