Advances in Mathematics最新文献

{"title":"ω-weak equivalences between weak ω-categories","authors":"Soichiro Fujii ,&nbsp;Keisuke Hoshino ,&nbsp;Yuki Maehara","doi":"10.1016/j.aim.2025.110490","DOIUrl":"10.1016/j.aim.2025.110490","url":null,"abstract":"<div><div>We study <em>ω</em>-weak equivalences between weak <em>ω</em>-categories in the sense of Batanin–Leinster. Our <em>ω</em>-weak equivalences are strict <em>ω</em>-functors satisfying essential surjectivity in every dimension, and when restricted to those between strict <em>ω</em>-categories, they coincide with the weak equivalences in the model category of strict <em>ω</em>-categories defined by Lafont, Métayer, and Worytkiewicz. We show that the class of <em>ω</em>-weak equivalences has the 2-out-of-3 property. We also consider a generalisation of <em>ω</em>-weak equivalences, defined as weak <em>ω</em>-functors (in the sense of Garner) satisfying essential surjectivity, and show that this class also has the 2-out-of-3 property.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110490"},"PeriodicalIF":1.5,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863887","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":"Polygonal surfaces in pseudo-hyperbolic spaces","authors":"Alex Moriani","doi":"10.1016/j.aim.2025.110484","DOIUrl":"10.1016/j.aim.2025.110484","url":null,"abstract":"<div><div>A polygonal surface in the pseudo-hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>n</mi></mrow></msup></math></span> is a complete maximal surface bounded by a lightlike polygon in the Einstein universe <span><math><msup><mrow><mi>Ein</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msup></math></span> with finitely many vertices. In this article, we give several characterizations of them. Polygonal surfaces are characterized by finiteness of their total curvature and by asymptotic flatness. They have parabolic type and polynomial quartic differential. Our result relies on a comparison between three ideal boundaries associated with a maximal surface, corresponding to three distinct distances naturally defined on the maximal surface.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110484"},"PeriodicalIF":1.5,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865611","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":"Cluster algebras and monotone Lagrangian tori","authors":"Yunhyung Cho ,&nbsp;Myungho Kim ,&nbsp;Yoosik Kim ,&nbsp;Euiyong Park","doi":"10.1016/j.aim.2025.110481","DOIUrl":"10.1016/j.aim.2025.110481","url":null,"abstract":"<div><div>Motivated by the construction of Newton–Okounkov bodies and toric degenerations via cluster algebras in <span><span>[37]</span></span>, <span><span>[27]</span></span>, we consider a family of Newton–Okounkov polytopes of a complex smooth Fano variety <em>X</em> related by a composition of tropicalized cluster mutations. According to the work of <span><span>[44]</span></span>, the toric degeneration associated with each Newton–Okounkov polytope Δ in the family produces a completely integrable system of <em>X</em> over Δ. We investigate circumstances in which each completely integrable system possesses a monotone Lagrangian torus fiber. We provide a sufficient condition, based on the data of tropical integer points and exchange matrices, for the family of constructed monotone Lagrangian tori to contain infinitely many monotone Lagrangian tori, no two of which are related by any symplectomorphism. By employing this criterion and exploiting the correspondence between the tropical integer points and the dual canonical basis elements, we generate infinitely many distinct monotone Lagrangian tori on flag manifolds of arbitrary type except in a few cases.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110481"},"PeriodicalIF":1.5,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865723","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":"Planar networks and simple Lie groups beyond type A","authors":"Anton Izosimov","doi":"10.1016/j.aim.2025.110482","DOIUrl":"10.1016/j.aim.2025.110482","url":null,"abstract":"<div><div>The general linear group <span><math><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, along with its adjoint simple group <span><math><mi>P</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, can be described by means of weighted planar networks. In this paper, we give a network description for simple Lie groups of types <em>B</em> and <em>C</em>. The corresponding networks are axially symmetric modulo a sequence of cluster mutations along the axis of symmetry. We extend to this setting the result of Gekhtman, Shapiro, and Vainshtein on the Poisson property of Postnikov's boundary measurement map. We also show that <em>B</em> and <em>C</em> type networks with positive weights parametrize the totally nonnegative part of the respective group. Finally, we construct network parametrizations of double Bruhat cells in symplectic and odd-dimensional orthogonal groups, and identify the corresponding face weights with Fock-Goncharov cluster coordinates.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110482"},"PeriodicalIF":1.5,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865722","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-semisimple sl2 quantum invariants of fibred links","authors":"Daniel López Neumann,&nbsp;Roland van der Veen","doi":"10.1016/j.aim.2025.110494","DOIUrl":"10.1016/j.aim.2025.110494","url":null,"abstract":"<div><div>The Akutsu-Deguchi-Ohtsuki (ADO) invariants are the most studied quantum link invariants coming from a non-semisimple tensor category. We show that, for fibred links in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, the degree of the ADO invariant is determined by the genus and the top coefficient is a root of unity. More precisely, we prove that the top coefficient is determined by the Hopf invariant of the plane field of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> associated to the fiber surface. Our proof is based on the genus bounds established in our previous work, together with a theorem of Giroux-Goodman stating that fiber surfaces in the three-sphere can be obtained from a disk by plumbing/deplumbing Hopf bands.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110494"},"PeriodicalIF":1.5,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865724","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":"Ergodicity of continuous and discrete quaternionic semigroups and Tauberian theorems","authors":"Chao Wang ,&nbsp;Tianyang Xu ,&nbsp;Jibin Li","doi":"10.1016/j.aim.2025.110485","DOIUrl":"10.1016/j.aim.2025.110485","url":null,"abstract":"<div><div>In this paper, the ergodic theory of continuous and discrete quaternionic semigroups and Tauberian theory are developed under the quaternionic setting. The notions of Cesàro and Abel convergence for the quaternionic Banach-valued locally integrable functions are introduced and the corresponding Tauberian conditions are established. Moreover, the Cesàro convergence is studied based on the slice hyperholomorphic extension and the relations between Cesàro and Abel convergence are formulated. We weaken the boundary condition of the slice hypercomplex domain of Cauchy integral via the geometric techniques of cutting rectifiable Jordan curves by finite circles and prove the quaternionic Cesàro mean ergodic theorem. Furthermore, the Tauberian theorems for the quaternionic power series are deduced. On the other hand, the notions of Cesàro and Abel ergodicity of the continuous and discrete quaternionic semigroups are proposed and the <em>S</em>-resolvent and spectral conditions for Cesàro ergodicity of these semigroups are obtained. In addition, some basic properties of Cesàro and Abel-ergodic projections are derived and Abel ergodic and Cesàro mean ergodic theorems are established. Besides, some equivalent characterizations of the quaternionic operator matrix as the semigroup generator are formulated and proved and the ergodic theorems of the quaternionic semigroup generated by quaternionic operator matrix are demonstrated. Finally, the ergodicity of solutions for the inhomogeneous Cauchy problem with periodic inhomogeneity in quaternionic setting is achieved.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110485"},"PeriodicalIF":1.5,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863885","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":"Topological classification of insulators: I. Non-interacting spectrally-gapped one-dimensional systems","authors":"Jui-Hui Chung ,&nbsp;Jacob Shapiro","doi":"10.1016/j.aim.2025.110486","DOIUrl":"10.1016/j.aim.2025.110486","url":null,"abstract":"<div><div>We study non-interacting electrons in disordered one-dimensional materials that exhibit a spectral gap, in each of the ten Altland-Zirnbauer symmetry classes. We define an appropriate topology on the space of Hamiltonians, such that the so-called strong topological invariants become complete invariants, yielding the one-dimensional column of the Kitaev periodic table, but now derived <em>without</em> recourse to K-theory. We thus confirm the conjecture regarding a one-to-one correspondence between topological phases of gapped non-interacting 1D systems and the respective Abelian groups <span><math><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo><mi>Z</mi><mo>,</mo><mn>2</mn><mi>Z</mi><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in the spectral-gap regime. The main tool we develop is an equivariant theory of homotopies of <em>local</em> unitaries and orthogonal projections. Moreover, we discuss an extension of the unitary theory to partial isometries, to provide a perspective toward the understanding of strongly-disordered, mobility-gapped materials.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110486"},"PeriodicalIF":1.5,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830187","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":"Global Ricci curvature behaviour for the Kähler-Ricci flow with finite time singularities","authors":"Alexander Bednarek","doi":"10.1016/j.aim.2025.110465","DOIUrl":"10.1016/j.aim.2025.110465","url":null,"abstract":"<div><div>We consider the Kähler-Ricci flow <span><math><msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></msub></math></span> on a compact manifold where the time of singularity, <em>T</em>, is finite. We assume the existence of a holomorphic map from the Kähler manifold <em>X</em> to some analytic variety <em>Y</em> which admits a Kähler metric on a neighbourhood of the image of <em>X</em> and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., <span><math><mo>[</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>]</mo><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>X</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span>. Under these assumptions we prove an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>-spacetime estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type <em>I</em> with respect to the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110465"},"PeriodicalIF":1.5,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830186","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":"Degree of the Grassmannian as an affine variety","authors":"Lek-Heng Lim ,&nbsp;Ke Ye","doi":"10.1016/j.aim.2025.110459","DOIUrl":"10.1016/j.aim.2025.110459","url":null,"abstract":"<div><div>The degree of the Grassmannian with respect to the Plücker embedding is well-known. However, the Plücker embedding, while ubiquitous in pure mathematics, is almost never used in applied mathematics. In applied mathematics, the Grassmannian is usually embedded as projection matrices <span><math><mi>Gr</mi><mo>(</mo><mi>k</mi><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>≅</mo><mo>{</mo><mi>P</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup><mo>:</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>T</mi></mrow></msup><mo>=</mo><mi>P</mi><mo>=</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><mi>tr</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span> or as involution matrices <span><math><mi>Gr</mi><mo>(</mo><mi>k</mi><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>≅</mo><mo>{</mo><mi>X</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>T</mi></mrow></msup><mo>=</mo><mi>X</mi><mo>,</mo><mspace></mspace><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>I</mi><mo>,</mo><mspace></mspace><mi>tr</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>n</mi><mo>}</mo></math></span>. We will determine an explicit expression for the degree of the Grassmannian with respect to these embeddings. In so doing, we resolved a conjecture of Devriendt, Friedman, Reinke, and Sturmfels about the degree of <span><math><mi>Gr</mi><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> and in fact generalized it to <span><math><mi>Gr</mi><mo>(</mo><mi>k</mi><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. We also proved a set-theoretic variant of another conjecture of theirs about the limit of <span><math><mi>Gr</mi><mo>(</mo><mi>k</mi><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> in the sense of Gröbner degeneration.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110459"},"PeriodicalIF":1.5,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826308","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":"Nowhere vanishing holomorphic one-forms and fibrations over abelian varieties","authors":"Nathan Chen ,&nbsp;Benjamin Church ,&nbsp;Feng Hao","doi":"10.1016/j.aim.2025.110463","DOIUrl":"10.1016/j.aim.2025.110463","url":null,"abstract":"<div><div>A result of Popa and Schnell shows that any holomorphic one-form on a smooth complex projective variety of general type admits zeros. More generally, given a variety <em>X</em> which admits <em>g</em> pointwise linearly independent holomorphic one-forms, they prove that <em>X</em> has Kodaira dimension <span><math><mi>κ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>dim</mi><mo>⁡</mo><mi>X</mi><mo>−</mo><mi>g</mi></math></span>. In the extremal case where <span><math><mi>κ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>dim</mi><mo>⁡</mo><mi>X</mi><mo>−</mo><mi>g</mi></math></span> and <em>X</em> is minimal, we prove that <em>X</em> admits a smooth morphism to an abelian variety, and classify all such <em>X</em> by showing they arise as diagonal quotients of the product of an abelian variety with a variety of general type. The case <span><math><mi>g</mi><mo>=</mo><mn>1</mn></math></span> was first proved by the third author, and classification results about surfaces and threefolds carrying nowhere vanishing forms have appeared in work of Schreieder and subsequent joint work with the third author. We also prove a birational version of this classification which holds without the minimal assumption, and establish additional cases of a conjecture of the third author.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110463"},"PeriodicalIF":1.5,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826305","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}