Advances in Mathematics最新文献

{"title":"Borel graphable equivalence relations","authors":"Tyler Arant ,&nbsp;Alexander S. Kechris ,&nbsp;Patrick Lutz","doi":"10.1016/j.aim.2025.110765","DOIUrl":"10.1016/j.aim.2025.110765","url":null,"abstract":"<div><div>This paper is devoted to the study of analytic equivalence relations which are <em>Borel graphable</em>, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in <span><math><mi>ZFC</mi></math></span>) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as <em>graphic groups.</em> We show that besides <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypergraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110765"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978919","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":"Extremal norms of potentials from fixed eigenvalues or eigenvalues ratio for Camassa-Holm equations","authors":"Jifeng Chu ,&nbsp;Gang Meng ,&nbsp;Zhi Zhang","doi":"10.1016/j.aim.2025.110766","DOIUrl":"10.1016/j.aim.2025.110766","url":null,"abstract":"<div><div>The aim of this paper is to study a kind of reverse spectral problems for Camassa-Holm equations. We prove the explicit solutions for the extremal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norms of the potentials, given the first fixed periodic eigenvalue or any fixed Dirichlet eigenvalue, or fixed ratios of Dirichlet eigenvalues. We use the setting of measure differential equations to understand such problems because the solution will lead to Dirac measure distributions of potentials. The explicit expressions for the sharp bounds are given as some elementary functions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110766"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928075","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":"Powers of ghost ideals","authors":"S. Estrada ,&nbsp;X.H. Fu ,&nbsp;I. Herzog ,&nbsp;S. Odabaşı","doi":"10.1016/j.aim.2025.110777","DOIUrl":"10.1016/j.aim.2025.110777","url":null,"abstract":"<div><div>A cocomplete additive category <span><math><mi>A</mi></math></span> may be equipped with the transfinite filtration of inductive ordinal powers of an ideal <span><math><mi>I</mi><mspace></mspace><mo>◃</mo><mspace></mspace><mi>A</mi></math></span>. If <span><math><mi>S</mi><mo>⊆</mo><mi>A</mi></math></span> is a subcategory of an exact category, the <span><math><mi>S</mi></math></span>-ghost morphisms, i.e., the morphisms right Ext-orthogonal to <span><math><mi>S</mi></math></span>, form such an ideal and the corresponding transfinite filtration is bounded below by the ideal of morphisms that factor through an object in the subcategory <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> of right Ext-perpendicular objects. The question of convergence for this filtration yields a transfinite formulation of the Generating Hypothesis. For an ordinal <em>λ</em>, the Generalized <em>λ</em>-Generating Hypothesis is the proposition that the <em>λ</em>-th power of the ideal of <span><math><mi>S</mi></math></span>-ghost morphisms is the (object) ideal of morphisms that factor through an object in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span>. It is shown to hold when the category <span><math><mi>A</mi></math></span> is a locally <em>λ</em>-presentable Grothendieck category and <span><math><mi>S</mi></math></span> is a set of <em>λ</em>-presentable objects.</div><div>Two cases of interest are treated: when the exact category is the category <span><math><mi>C</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of chain complexes of left <em>R</em>-modules, then the ideal of morphisms that are trivial on homology are the ghosts with respect to the subcategory of Cartan-Eilenberg projectives and the Generalized <em>ω</em>-Generating Hypothesis is shown to hold; when the exact category is the module category <em>R</em>-Mod for a ring whose left pure projective modules are closed under extension, then an analysis of the transfinite filtration induced by the FP-ghost ideal shows that every left FP-projective module is pure projective.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110777"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145950181","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":"The Lyubashenko modular functor for Drinfeld centers via non-semisimple string-nets","authors":"Lukas Müller ,&nbsp;Christoph Schweigert ,&nbsp;Lukas Woike ,&nbsp;Yang Yang","doi":"10.1016/j.aim.2025.110770","DOIUrl":"10.1016/j.aim.2025.110770","url":null,"abstract":"<div><div>The Levin-Wen string-nets of a spherical fusion category <span><math><mi>C</mi></math></span> describe, by results of Kirillov and Bartlett, the representations of mapping class groups of closed surfaces obtained from the Turaev-Viro construction applied to <span><math><mi>C</mi></math></span>. We provide a far-reaching generalization of this statement to arbitrary pivotal finite tensor categories, including non-semisimple or non-spherical ones: We show that the finitely cocompleted string-net modular functor built from the projective objects of a pivotal finite tensor category is equivalent to Lyubashenko's modular functor built from the Drinfeld center <span><math><mi>Z</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110770"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981069","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":"On Akemann-Weaver conjecture","authors":"Marcin Bownik","doi":"10.1016/j.aim.2025.110772","DOIUrl":"10.1016/j.aim.2025.110772","url":null,"abstract":"<div><div>Akemann and Weaver <span><span>[3]</span></span> showed Lyapunov-type theorem for rank one positive semidefinite matrices which is an extension of Weaver's KS₂ conjecture <span><span>[34]</span></span> that was proven by Marcus, Spielman, and Srivastava <span><span>[30]</span></span> in their breakthrough solution of the Kadison-Singer problem <span><span>[27]</span></span>. They conjectured that a similar result holds for higher rank matrices. We prove the conjecture of Akemann and Weaver by establishing Lyapunov-type theorem for trace class operators. In the process we prove a matrix discrepancy result for sums of hermitian matrices. This extends rank one result of Kyng, Luh, and Song <span><span>[28]</span></span> who established an improved bound in Lyapunov-type theorem of Akemann and Weaver.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110772"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927996","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":"Measures that violate the generalized continuum hypothesis","authors":"Tom Benhamou ,&nbsp;Gabriel Goldberg","doi":"10.1016/j.aim.2025.110778","DOIUrl":"10.1016/j.aim.2025.110778","url":null,"abstract":"<div><div>A simple <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span>-point on a regular cardinal <em>κ</em> is a uniform ultrafilter on <em>κ</em> with a mod-bounded decreasing generating sequence of length <em>λ</em>. We prove that if there is a simple <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span>-point ultrafilter over <span><math><mi>κ</mi><mo>&gt;</mo><mi>ω</mi></math></span>, then <span><math><mi>λ</mi><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span>. We show that such ultrafilters appear in the models of <span><span>[3]</span></span>, <span><span>[13]</span></span>. We improve the lower bound for the consistency strength of the existence of a <span><math><msub><mrow><mi>P</mi></mrow><mrow><msup><mrow><mi>κ</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup></mrow></msub></math></span>-point to a 2-strong cardinal. Finally, we apply our arguments to obtain non-trivial lower bounds for (1) the statement that the generalized tower number <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span> is greater than <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and <em>κ</em> is measurable, (2) the preservation of measurability after the generalized Mathias forcing, and (3) variations of filter games of <span><span>[28]</span></span>, <span><span>[22]</span></span>, <span><span>[18]</span></span> in the case <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>κ</mi></mrow></msup><mo>&gt;</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110778"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928078","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":"Ranks of matrices of logarithms of algebraic numbers II: The matrix coefficient conjecture","authors":"Samit Dasgupta,&nbsp;Mahesh Kakde","doi":"10.1016/j.aim.2025.110753","DOIUrl":"10.1016/j.aim.2025.110753","url":null,"abstract":"<div><div>Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or <em>p</em>-adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant, then after a rational change of basis on the left and right, it can be made to have a vanishing coefficient.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110753"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145847610","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":"Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature","authors":"Zhu Ye","doi":"10.1016/j.aim.2026.110786","DOIUrl":"10.1016/j.aim.2026.110786","url":null,"abstract":"<div><div>Let <em>M</em> be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of <em>M</em> is always greater than or equal to the dimension of some (or every) asymptotic cone of <em>M</em>.</div><div>Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of <em>M</em> equals <em>k</em>, then there exists an asymptotic cone of <em>M</em> whose upper box dimension is at most <em>k</em>. In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature.</div><div>In the subsequent part of the paper, we extend or partially extend Sormani's results concerning <em>M</em> with linear volume growth to more relaxed volume growth conditions. Our approach also allows us to present a new proof of Sormani's sublinear diameter growth theorem for open manifolds with <span><math><mrow><mi>Ric</mi></mrow><mo>≥</mo><mn>0</mn></math></span> and linear volume growth.</div><div>Finally, we construct an example of an open <em>n</em>-manifold <em>M</em> with <span><math><msub><mrow><mi>sec</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>≥</mo><mn>0</mn></math></span> whose volume growth order oscillates between 1 and <em>n</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110786"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981226","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":"Discrete fractals: Dimensions, quasi-isometric invariance and self-similarity","authors":"Kan Jiang ,&nbsp;Junjie Miao ,&nbsp;Lifeng Xi","doi":"10.1016/j.aim.2026.110791","DOIUrl":"10.1016/j.aim.2026.110791","url":null,"abstract":"<div><div>It is well-known that fractal dimensions are invariant under bi-Lipschitz mappings on Euclidean spaces, and therefore, bi-Lipschitz mappings are important in the classification of fractal sets. On locally finite discrete metric spaces, bi-Lipschitz mappings are a class of special quasi-isometries which constitute a fundamental concept in geometric group theory.</div><div>In this paper, we extend discrete fractal dimensions to locally finite discrete metric spaces, establishing their quasi-isometric invariance. For discrete self-similar sets with integer digits, we prove a complete classification of bi-Lipschitz and quasi-isometric equivalences, providing a discrete analogue to Falconer and Marsh's seminal results on Lipschitz equivalence of self-similar fractals. Our main theorem shows that two non-trivial such sets are quasi-isometric if and only if the logarithm of their scaling ratios and digit set cardinalities are rationally proportional. Furthermore, the bi-Lipschitz equivalence of these structures is strictly determined by the inclusion of zero in their digit sets, distinguishing them from standard self-similar fractals.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110791"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039199","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":"The equifibered approach to ∞-properads","authors":"Shaul Barkan,&nbsp;Jan Steinebrunner","doi":"10.1016/j.aim.2025.110755","DOIUrl":"10.1016/j.aim.2025.110755","url":null,"abstract":"<div><div>We define a notion of ∞-properads that generalizes ∞-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on ∞-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our ∞-properads as Segal presheaves on a category of graphs by comparing them to the Segal ∞-properads of Hackney–Robertson–Yau. Combining these two approaches yields a flexible tool for doing higher algebra with operations that have multiple inputs and outputs. Crucially, this allows for a definition of algebras over an ∞-properad such that, for example, topological field theories are algebras over the bordism ∞-properad.</div><div>The key ingredient to this paper is the notion of an <em>equifibered</em> map between <span><math><msub><mrow><mi>E</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-monoids, which is a well-behaved generalization of free maps. We also use this to prove facts about free <span><math><msub><mrow><mi>E</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-monoids, for example that free <span><math><msub><mrow><mi>E</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-monoids are closed under pullbacks along arbitrary maps.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110755"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886095","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}