Pub Date : 2026-02-01Epub Date: 2025-11-27DOI: 10.1016/j.aim.2025.110714
Lorenzo Notaro
In 1984, Ditor asked two questions:
(A)
For each and infinite cardinal κ, is there a join-semilattice of breadth and cardinality whose principal ideals have cardinality <κ?
(B)
For each , is there a lower finite lattice of cardinality whose elements have at most lower covers?
We show that both questions have positive answers under the axiom of constructibility, and hence consistently with . More specifically, we derive the positive answers from assuming that holds for enough κ's.
{"title":"Ladders and squares","authors":"Lorenzo Notaro","doi":"10.1016/j.aim.2025.110714","DOIUrl":"10.1016/j.aim.2025.110714","url":null,"abstract":"<div><div>In 1984, Ditor asked two questions:<ul><li><span>(A)</span><span><div>For each <span><math><mi>n</mi><mo>∈</mo><mi>ω</mi></math></span> and infinite cardinal <em>κ</em>, is there a join-semilattice of breadth <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> and cardinality <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mo>+</mo><mi>n</mi></mrow></msup></math></span> whose principal ideals have cardinality <<em>κ</em>?</div></span></li><li><span>(B)</span><span><div>For each <span><math><mi>n</mi><mo>∈</mo><mi>ω</mi></math></span>, is there a lower finite lattice of cardinality <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> whose elements have at most <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> lower covers?</div></span></li></ul> We show that both questions have positive answers under the axiom of constructibility, and hence consistently with <span><math><mi>ZFC</mi></math></span>. More specifically, we derive the positive answers from assuming that <span><math><msub><mrow><mo>□</mo></mrow><mrow><mi>κ</mi></mrow></msub></math></span> holds for enough <em>κ</em>'s.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110714"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600402","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}
Pub Date : 2026-02-01Epub Date: 2025-12-17DOI: 10.1016/j.aim.2025.110740
Hua-Lin Huang , Gongxiang Liu , Yuping Yang , Yu Ye
Using a variety of methods developed in the theory of finite-dimensional quasi-Hopf algebras, we classify all finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups. As a consequence, we partially confirm the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.
{"title":"On the classification of finite quasi-quantum groups over abelian groups","authors":"Hua-Lin Huang , Gongxiang Liu , Yuping Yang , Yu Ye","doi":"10.1016/j.aim.2025.110740","DOIUrl":"10.1016/j.aim.2025.110740","url":null,"abstract":"<div><div>Using a variety of methods developed in the theory of finite-dimensional quasi-Hopf algebras, we classify all finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups. As a consequence, we partially confirm the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110740"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760672","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}
Pub Date : 2026-02-01Epub Date: 2025-12-29DOI: 10.1016/j.aim.2025.110748
Myeonggi Kwon , Takahiro Oba
We prove the uniqueness, up to diffeomorphism, of symplectically aspherical fillings of the unit cotangent bundle of odd-dimensional spheres. As applications, we first show the non-existence of exact symplectic cobordisms between some 5-dimensional Brieskorn manifolds. We also determine the diffeomorphism types of closed symplectic 6-manifolds with certain codimension 2 symplectic submanifolds.
{"title":"Symplectic fillings of unit cotangent bundles of spheres and applications","authors":"Myeonggi Kwon , Takahiro Oba","doi":"10.1016/j.aim.2025.110748","DOIUrl":"10.1016/j.aim.2025.110748","url":null,"abstract":"<div><div>We prove the uniqueness, up to diffeomorphism, of symplectically aspherical fillings of the unit cotangent bundle of odd-dimensional spheres. As applications, we first show the non-existence of exact symplectic cobordisms between some 5-dimensional Brieskorn manifolds. We also determine the diffeomorphism types of closed symplectic 6-manifolds with certain codimension 2 symplectic submanifolds.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110748"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885208","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}
Pub Date : 2026-02-01Epub Date: 2025-12-22DOI: 10.1016/j.aim.2025.110743
Xiaxing Cai , Gangsong Leng , Yuchi Wu , Dongmeng Xi
In a previous work in affine convex geometry, an affine contravariant family of geometric measures (called affine dual curvature measures) was introduced. In that work, the authors solved a related affine dual Minkowski problem. The new affine family of Minkowski problems includes the logarithmic Minkowski problem as a special case.
In that spirit, this work introduces a series of geometric measures (called centro-section measures) that are derived from random sections. The centro-section measures serve to unify dual curvature measures and their affine analogs. Additionally, sufficient conditions are offered to solve the even Minkowski problem for the centro-section measures.
{"title":"Minkowski problems of centro-section measures","authors":"Xiaxing Cai , Gangsong Leng , Yuchi Wu , Dongmeng Xi","doi":"10.1016/j.aim.2025.110743","DOIUrl":"10.1016/j.aim.2025.110743","url":null,"abstract":"<div><div>In a previous work in affine convex geometry, an affine contravariant family of geometric measures (called <em>affine dual curvature measures</em>) was introduced. In that work, the authors solved a related affine dual Minkowski problem. The new affine family of Minkowski problems includes the logarithmic Minkowski problem as a special case.</div><div>In that spirit, this work introduces a series of geometric measures (called <em>centro-section measures</em>) that are derived from random sections. The centro-section measures serve to unify dual curvature measures and their affine analogs. Additionally, sufficient conditions are offered to solve the even Minkowski problem for the centro-section measures.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110743"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841828","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}
Pub Date : 2026-02-01Epub Date: 2025-12-19DOI: 10.1016/j.aim.2025.110745
Zikang Dong , Weijia Wang , Hao Zhang
This paper investigates the analytic properties of L-functions associated with almost meromorphic modular forms, extending classical results on L-functions of holomorphic modular forms. By generalizing the regularized Mellin transform, we define these L-functions and examine their properties, especially the distributions of zeros on the critical line. We prove that under certain singularity conditions, these L-functions have infinitely many zeros on the critical line. Additionally, we establish converse theorems for almost meromorphic modular forms, showing that their L-functions uniquely determine the forms. Numerical evidence is also included to support these results.
{"title":"Almost meromorphic modular forms and their associated L-functions","authors":"Zikang Dong , Weijia Wang , Hao Zhang","doi":"10.1016/j.aim.2025.110745","DOIUrl":"10.1016/j.aim.2025.110745","url":null,"abstract":"<div><div>This paper investigates the analytic properties of <em>L</em>-functions associated with almost meromorphic modular forms, extending classical results on <em>L</em>-functions of holomorphic modular forms. By generalizing the regularized Mellin transform, we define these <em>L</em>-functions and examine their properties, especially the distributions of zeros on the critical line. We prove that under certain singularity conditions, these <em>L</em>-functions have infinitely many zeros on the critical line. Additionally, we establish converse theorems for almost meromorphic modular forms, showing that their <em>L</em>-functions uniquely determine the forms. Numerical evidence is also included to support these results.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110745"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792175","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}
Pub Date : 2026-02-01Epub Date: 2025-12-11DOI: 10.1016/j.aim.2025.110733
Severin Barmeier , Zhengfang Wang
Extended Khovanov arc algebras are graded associative algebras which naturally appear in a variety of contexts, from knot and link homology, low-dimensional topology and topological quantum field theory to representation theory and symplectic geometry. C. Stroppel conjectured in her ICM 2010 address that the bigraded Hochschild cohomology groups of vanish in a certain range, implying that the algebras admit no nontrivial A∞ deformations, in particular that the algebras are intrinsically formal.
Whereas Stroppel's conjecture is known to hold for the algebras and by work of Seidel and Thomas, we show that does in fact admit nontrivial A∞ deformations with nonvanishing higher products for all .
We describe both the extended Khovanov arc algebras and their Koszul duals concretely as path algebras of quivers with relations. This allows us to give an explicit algebraic construction of A∞ deformations of by using the correspondence between A∞ deformations of a Koszul algebra and filtered associative deformations of its Koszul dual.
{"title":"A∞ deformations of extended Khovanov arc algebras and Stroppel's conjecture","authors":"Severin Barmeier , Zhengfang Wang","doi":"10.1016/j.aim.2025.110733","DOIUrl":"10.1016/j.aim.2025.110733","url":null,"abstract":"<div><div>Extended Khovanov arc algebras <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> are graded associative algebras which naturally appear in a variety of contexts, from knot and link homology, low-dimensional topology and topological quantum field theory to representation theory and symplectic geometry. C. Stroppel conjectured in her ICM 2010 address that the bigraded Hochschild cohomology groups of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> vanish in a certain range, implying that the algebras <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> admit no nontrivial A<sub>∞</sub> deformations, in particular that the algebras are intrinsically formal.</div><div>Whereas Stroppel's conjecture is known to hold for the algebras <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> by work of Seidel and Thomas, we show that <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> does in fact admit nontrivial A<sub>∞</sub> deformations with nonvanishing higher products for all <span><math><mi>m</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>.</div><div>We describe both the extended Khovanov arc algebras <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and their Koszul duals concretely as path algebras of quivers with relations. This allows us to give an explicit algebraic construction of A<sub>∞</sub> deformations of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> by using the correspondence between A<sub>∞</sub> deformations of a Koszul algebra and filtered associative deformations of its Koszul dual.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110733"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749188","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}
This paper is concerned with the stability and large-time behavior of 3D incompressible MHD equations with only vertical dissipation near a background magnetic field. By making full use of the dissipation generated by the background magnetic field, we first establish the global stability of the solutions in -norm. Then, the optimal decay rates of the solutions are obtained, which are consistent with the 2D classical heat equation. Moreover, some enhanced decay rates of are also achieved. In other words, the decay estimates of the second or third component of velocity/magnetic field coincide with those of 2D heat kernel, while the first component behaves like the 3D heat kernel. This is mainly due to the divergence-free condition and the anisotropic structure.
{"title":"Global stability and sharp decay estimates for 3D MHD equations with only vertical dissipation near a background magnetic field","authors":"Suhua Lai , Jiahong Wu , Jianwen Zhang , Xiaokui Zhao","doi":"10.1016/j.aim.2025.110747","DOIUrl":"10.1016/j.aim.2025.110747","url":null,"abstract":"<div><div>This paper is concerned with the stability and large-time behavior of 3D incompressible MHD equations with only vertical dissipation near a background magnetic field. By making full use of the dissipation generated by the background magnetic field, we first establish the global stability of the solutions in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>-norm. Then, the optimal decay rates of the solutions are obtained, which are consistent with the 2D classical heat equation. Moreover, some enhanced decay rates of <span><math><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> are also achieved. In other words, the decay estimates of the second or third component of velocity/magnetic field coincide with those of 2D heat kernel, while the first component behaves like the 3D heat kernel. This is mainly due to the divergence-free condition and the anisotropic structure.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110747"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841831","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}
Pub Date : 2026-02-01Epub Date: 2025-12-01DOI: 10.1016/j.aim.2025.110699
Mario Garcia-Fernandez , Raul Gonzalez Molina , Jeffrey Streets
We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding analytical results, we prove a priori estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's estimate for the complex Monge-Ampère equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.
{"title":"Pluriclosed flow and the Hull-Strominger system","authors":"Mario Garcia-Fernandez , Raul Gonzalez Molina , Jeffrey Streets","doi":"10.1016/j.aim.2025.110699","DOIUrl":"10.1016/j.aim.2025.110699","url":null,"abstract":"<div><div>We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding analytical results, we prove a priori <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> estimate for the complex Monge-Ampère equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110699"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694482","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}
Pub Date : 2026-02-01Epub Date: 2025-12-16DOI: 10.1016/j.aim.2025.110737
Bo Wang , Zhizhang Wang
In this paper, we consider the Hessian equations in some exterior domain with prescribed asymptotic behavior at infinity and Dirichlet-Neumann conditions on its interior boundary. We obtain that there exists a unique bounded domain such that the over-determined problem admits a unique strictly convex solution.
{"title":"A Serrin-type over-determined problem for Hessian equations in the exterior domain","authors":"Bo Wang , Zhizhang Wang","doi":"10.1016/j.aim.2025.110737","DOIUrl":"10.1016/j.aim.2025.110737","url":null,"abstract":"<div><div>In this paper, we consider the Hessian equations in some exterior domain with prescribed asymptotic behavior at infinity and Dirichlet-Neumann conditions on its interior boundary. We obtain that there exists a unique bounded domain such that the over-determined problem admits a unique strictly convex solution.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110737"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797145","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}
Pub Date : 2026-02-01Epub Date: 2025-12-08DOI: 10.1016/j.aim.2025.110712
Daniel Panazzolo , Maja Resman , Loïc Teyssier
A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classified by their Poincaré first-return map. We also prove that they are formally rigid when the Poincaré map is multivalued. Finally, we provide a list of all analytic classes of Liouville-integrable saddle loops.
{"title":"Rigidity of saddle loops","authors":"Daniel Panazzolo , Maja Resman , Loïc Teyssier","doi":"10.1016/j.aim.2025.110712","DOIUrl":"10.1016/j.aim.2025.110712","url":null,"abstract":"<div><div>A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classified by their Poincaré first-return map. We also prove that they are formally rigid when the Poincaré map is multivalued. Finally, we provide a list of all analytic classes of Liouville-integrable saddle loops.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110712"},"PeriodicalIF":1.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749191","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}
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