Advances in Mathematics最新文献

{"title":"p-typical curves on p-adic Tate twists and de Rham–Witt forms","authors":"Sanath K. Devalapurkar,&nbsp;Shubhodip Mondal","doi":"10.1016/j.aim.2025.110448","DOIUrl":"10.1016/j.aim.2025.110448","url":null,"abstract":"<div><div>We show that de Rham–Witt forms are naturally isomorphic to <em>p</em>-typical curves on <em>p</em>-adic Tate twists, which revisits a question of Artin–Mazur from 1977 pursued in the earlier work of Bloch and Kato. We show this more generally by refining a result of Hesselholt on topological cyclic homology with the motivic filtrations introduced by Bhatt–Morrow–Scholze.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110448"},"PeriodicalIF":1.5,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Logarithmic prismatic cohomology II","authors":"Teruhisa Koshikawa ,&nbsp;Zijian Yao","doi":"10.1016/j.aim.2025.110446","DOIUrl":"10.1016/j.aim.2025.110446","url":null,"abstract":"<div><div>We continue to study the logarithmic prismatic cohomology defined by the first author, and complete the proof of the de Rham comparison and étale comparison generalizing those of Bhatt and Scholze. We prove these comparisons for a derived version of logarithmic prismatic cohomology, and, along the way, we construct a suitable Nygaard filtration and explain a relation between <em>F</em>-crystals and <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-local systems in the logarithmic setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110446"},"PeriodicalIF":1.5,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gromov's ellipticity of principal Gm-bundles","authors":"Sh. Kaliman","doi":"10.1016/j.aim.2025.110444","DOIUrl":"10.1016/j.aim.2025.110444","url":null,"abstract":"<div><div>We prove that every nontrivial principal <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>-bundle over a complete stably uniformly rational variety is algebraically elliptic in the sense of Gromov.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110444"},"PeriodicalIF":1.5,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal asymptotic lower bound for stability of fractional Sobolev inequality and the global stability of log-Sobolev inequality on the sphere","authors":"Lu Chen ,&nbsp;Guozhen Lu ,&nbsp;Hanli Tang","doi":"10.1016/j.aim.2025.110438","DOIUrl":"10.1016/j.aim.2025.110438","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we establish the optimal asymptotic lower bound for the stability of fractional Sobolev inequality:&lt;span&gt;&lt;span&gt;&lt;span&gt;(0.1)&lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the set of maximizers of the fractional Sobolev inequality of order &lt;em&gt;s&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; denotes the optimal lower bound of stability. We prove that the optimal lower bound &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; behaves asymptotically at the order of &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; when &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for any fixed &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. This extends the work by Dolbeault-Esteban-Figalli-Frank-Loss &lt;span&gt;&lt;span&gt;[22]&lt;/span&gt;&lt;/span&gt; on the stability of the first order Sobolev inequality (i.e., &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;) and quantify the asymptotic behavior for lower bound of stability of fractional Sobolev inequality established by the current author's previous work in &lt;span&gt;&lt;span&gt;[15]&lt;/span&gt;&lt;/span&gt; in the case of &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; behaves asymptotically at the order of &lt;em&gt;s&lt;/em&gt; when &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; for any given dimension &lt;em&gt;n&lt;/em&gt;. (See &lt;span&gt;&lt;span&gt;Theorem 1.1&lt;/span&gt;&lt;/span&gt; for these asymptotic estimates.) As an important application of this asymptotic estimate as &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we derive the global stability for the log-Sobolev inequality on the sphere established by Beckner in &lt;span&gt;&lt;span&gt;[3]&lt;/span&gt;&lt;/span&gt;, &lt;span&gt;&lt;span&gt;[5]&lt;/span&gt;&lt;/span&gt; with the optimal asymptotic low","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110438"},"PeriodicalIF":1.5,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lifting of locally initial objects and universal (co)acting Hopf algebras","authors":"A.L. Agore ,&nbsp;A.S. Gordienko ,&nbsp;J. Vercruysse","doi":"10.1016/j.aim.2025.110442","DOIUrl":"10.1016/j.aim.2025.110442","url":null,"abstract":"<div><div>The universal (co)acting bi/Hopf algebras introduced by Yu.I. Manin, M. Sweedler and D. Tambara, the universal Hopf algebra of a given (co)module structure, as well as the universal group of a grading, introduced by J. Patera and H. Zassenhaus, find their applications in the classification of quantum symmetries. Typically, universal (co)acting objects are defined as initial or terminal in the corresponding categories and, as such, they do not always exist. In order to ensure their existence, we introduce the support of a given object, which generalizes the support of a grading and is used to restrict the class of objects under consideration. The existence problems for universal objects are formulated and studied in a purely categorical manner by seeing them as particular cases of the lifting problem for a locally initial object. We prove the existence of a lifting and, consequently, of the universal (co)acting objects under some assumptions on the base (braided or symmetric monoidal) category. In contrast to existing constructions, our approach is self-dual in the sense that we can use the same proof to obtain the existence of universal actions and coactions. In particular, when the base category is the category of vector spaces over a field, the category of sets or their duals, we recover known existence results for the aforementioned universal objects. The proposed approach allows us to apply our results not only to the classical categories of sets and vectors spaces and their duals but also to (co)modules over bi/Hopf algebras, differential graded vector spaces, <em>G</em>-sets and graded sets.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110442"},"PeriodicalIF":1.5,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Injectivity radius lower bound of convex sum of tame Riemannian metrics and applications to symplectic topology","authors":"Jaeyoung Choi ,&nbsp;Yong-Geun Oh","doi":"10.1016/j.aim.2025.110443","DOIUrl":"10.1016/j.aim.2025.110443","url":null,"abstract":"<div><div>Motivated by the aspect of large-scale symplectic topology, we prove that for any pair <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of smooth complete Riemannian metrics of bounded curvature and <em>of injectivity radius bounded away from zero</em>, the convex sum <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>s</mi><mo>)</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>s</mi><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> also has bounded curvature depending only on the curvature bounds <span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msub></math></span> of <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, and that the injectivity radii of <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> have uniform lower bound depending only on the derivative bounds <span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></msub><mo>=</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msub></math></span>. A main technical ingredient to establish the injectivity radius lower bound is an application of the <em>quantitative inverse function theorem</em>. Using these estimates, we prove that each <em>quasi-isometry</em> class of tame metrics is convex <em>for all finite regularity class of</em> <span><math><mn>3</mn><mo>≤</mo><mi>r</mi><mo>&lt;</mo><mo>∞</mo></math></span><em>.</em> Using this Riemannian geometry result, we prove that the set of smooth <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span><em>-tame</em> almost complex structures inside the same quasi-isometry class associated to the symplectic form <em>ω</em> is contractible in strong <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> topology for all <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>r</mi></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110443"},"PeriodicalIF":1.5,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds","authors":"Kazuhiro Ishige,&nbsp;Asuka Takatsu,&nbsp;Haruto Tokunaga","doi":"10.1016/j.aim.2025.110439","DOIUrl":"10.1016/j.aim.2025.110439","url":null,"abstract":"<div><div>We prove that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110439"},"PeriodicalIF":1.5,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A spectral radius for matrices over an operator space","authors":"Orr Moshe Shalit ,&nbsp;Eli Shamovich","doi":"10.1016/j.aim.2025.110449","DOIUrl":"10.1016/j.aim.2025.110449","url":null,"abstract":"&lt;div&gt;&lt;div&gt;With every operator space structure &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, we associate a spectral radius function &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; on &lt;em&gt;d&lt;/em&gt;-tuples of operators. For a &lt;em&gt;d&lt;/em&gt;-tuple &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of matrices we show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; if and only if &lt;em&gt;X&lt;/em&gt; is jointly similar to a tuple in the open unit ball of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, that is, there is an invertible matrix &lt;em&gt;S&lt;/em&gt; such that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. More generally, for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; we show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; if and only if there exists an invertible &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. When &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the row operator space, for example, our spectral radius coincides with the joint spectral radius considered by Bunce, Popescu, and others, and we recover the condition for a tuple of matrices to be simultaneously similar to a strict row contraction. When &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the minimal operator space &lt;span&gt;&lt;math&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/m","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110449"},"PeriodicalIF":1.5,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Countable Borel treeable equivalence relations are classifiable by ℓ1","authors":"Shaun Allison","doi":"10.1016/j.aim.2025.110452","DOIUrl":"10.1016/j.aim.2025.110452","url":null,"abstract":"<div><div>In <span><span>[8]</span></span>, it was shown that any countable Borel equivalence relation (CBER) induced by a countable abelian Polish group is hyperfinite. This prompted Hjorth to ask if this is in fact true for all CBERs classifiable by (uncountable) abelian Polish groups.</div><div>We describe reductions involving <em>free Banach spaces</em> to show that every treeable CBER is classifiable by an abelian Polish group. As there exist treeable CBERs that are not hyperfinite, this answers Hjorth's question in the negative.</div><div>On the other hand, we show that any CBER classifiable by a countable product of locally compact abelian Polish groups (such as <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>) is indeed hyperfinite. We use a small fragment of the <em>Hjorth analysis</em> of Polish group actions, which is Hjorth's generalization of the Scott analysis of countable structures to Polish group actions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110452"},"PeriodicalIF":1.5,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature","authors":"Jingyi Chen ,&nbsp;Yuxiang Li","doi":"10.1016/j.aim.2025.110436","DOIUrl":"10.1016/j.aim.2025.110436","url":null,"abstract":"<div><div>We prove uniform convergence of metrics <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> on a closed surface with bounded integral curvature (measure) in the sense of A.D. Alexandrov, under the assumption that the curvature measures <span><math><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>=</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>, where <span><math><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> are nonnegative Radon measures converging weakly to measures <span><math><msup><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>,</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> respectively, and <span><math><msup><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> is less than 2<em>π</em> at each point (no cusps). This is the global version of Yu. G. Reshetnyak's well-known result on uniform convergence of metrics on a domain in <span><math><mi>C</mi></math></span>, and answers affirmatively the open question on the metric convergence on a closed surface. We also give an analytic proof of the fact that a (singular) metric <span><math><mi>g</mi><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>u</mi></mrow></msup><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> with bounded integral curvature on a closed Riemannian surface <span><math><mo>(</mo><mi>Σ</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> can be approximated by smooth metrics in the fixed conformal class <span><math><mo>[</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>]</mo></math></span>. Results on a closed surface with varying conformal classes and on complete noncompact surfaces are obtained as well.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110436"},"PeriodicalIF":1.5,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144695035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}