Advances in Mathematics最新文献

{"title":"New bounds for Stein's square functions in higher dimensions","authors":"Shengwen Gan ,&nbsp;Changkeun Oh ,&nbsp;Shukun Wu","doi":"10.1016/j.aim.2025.110342","DOIUrl":"10.1016/j.aim.2025.110342","url":null,"abstract":"<div><div>We improve the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> bounds on Stein's square function to the best-known range of the Fourier restriction problem when <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. Applications, including certain local smoothing estimates, are also discussed.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110342"},"PeriodicalIF":1.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942831","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":"Curvature homogeneous hypersurfaces in space forms","authors":"Robert Bryant ,&nbsp;Luis Florit ,&nbsp;Wolfgang Ziller","doi":"10.1016/j.aim.2025.110338","DOIUrl":"10.1016/j.aim.2025.110338","url":null,"abstract":"<div><div>We provide a classification of curvature homogeneous hypersurfaces in space forms by classifying the ones in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>. In higher dimensions, besides the isoparametric and the constant curvature ones, there is a single one in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span>. Besides the obvious examples, we show that there exists an isolated hypersurface with a circle of symmetries and a one parameter family admitting no continuous symmetries. Outside the set of minimal points, which only exists in the case of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>, every example is, locally and up to covers, of this form.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110338"},"PeriodicalIF":1.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942832","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":"Convex elements and Steinberg's cross-sections","authors":"Sian Nie ,&nbsp;Panjun Tan ,&nbsp;Qingchao Yu","doi":"10.1016/j.aim.2025.110346","DOIUrl":"10.1016/j.aim.2025.110346","url":null,"abstract":"<div><div>In this paper, we study convex elements in a (twisted) Weyl group introduced by Ivanov and the first named author. We show that each conjugacy class of the twisted Weyl group contains a convex element, and moreover, the Steinberg cross-sections exist for all convex elements. This result strictly enlarges the cases of Steinberg cross-sections from a new perspective, and will play an essential role in the study of higher Deligne-Lusztig representations.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110346"},"PeriodicalIF":1.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942830","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":"Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors","authors":"Jürgen Fuchs ,&nbsp;Gregor Schaumann ,&nbsp;Christoph Schweigert ,&nbsp;Simon Wood","doi":"10.1016/j.aim.2025.110325","DOIUrl":"10.1016/j.aim.2025.110325","url":null,"abstract":"<div><div>We develop the theory of module categories over a Grothendieck-Verdier category <span><math><mi>C</mi></math></span>, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category comes with two monoidal structures ⊗ and <figure><img></figure> which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory. In any Grothendieck-Verdier module category <span><math><mi>M</mi></math></span> we find two distinguished subcategories <figure><img></figure> and <span><math><msup><mrow><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mo>⊗</mo></mrow></msup></math></span>, which can be characterized by certain structure morphisms being actually invertible. The internal Hom <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mspace></mspace><mo>:</mo><mo>=</mo><munder><mrow><mrow><mi>Hom</mi></mrow></mrow><mo>_</mo></munder><mo>(</mo><mi>m</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> of an object <em>m</em> in <span><math><msup><mrow><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mo>⊗</mo></mrow></msup></math></span> that is a <span><math><mi>C</mi></math></span>-generator is an algebra such that <span><math><mrow><mi>mod</mi></mrow><mtext>-</mtext><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> is equivalent to <span><math><mi>M</mi></math></span> as a module category. Crucially, the subcategories <figure><img></figure> and <span><math><msup><mrow><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mo>⊗</mo></mrow></msup></math></span> are precisely those on which a relative Serre functor can be defined. This relative Serre functor furnishes an equivalence <figure><img></figure>, and any isomorphism <span><math><mi>m</mi><mspace></mspace><mover><mrow><mo>→</mo></mrow><mrow><mspace></mspace><mo>≅</mo><mspace></mspace></mrow></mover><mspace></mspace><mi>S</mi><mo>(</mo><mi>m</mi><mo>)</mo></math></span> endows the algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> with the structure of a Grothendieck-Verdier Frobenius algebra.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110325"},"PeriodicalIF":1.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942829","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 Ricci iteration towards cscK metrics","authors":"Kewei Zhang","doi":"10.1016/j.aim.2025.110340","DOIUrl":"10.1016/j.aim.2025.110340","url":null,"abstract":"<div><div>Motivated by the problem of finding constant scalar curvature Kähler metrics, we investigate a Ricci iteration sequence of Rubinstein that discretizes the pseudo-Calabi flow. While the long time existence of the flow is still an open question, we show that the iteration sequence does exist for all steps, along which the K-energy decreases. We further show that the iteration sequence, modulo automorphisms, converges smoothly to a constant scalar curvature Kähler metric if there is one, thus confirming a conjecture of Rubinstein from 2007 and extending results of Darvas–Rubinstein to arbitrary Kähler classes.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110340"},"PeriodicalIF":1.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942833","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":"Cox rings of nef anticanonical rational surfaces","authors":"Michela Artebani,&nbsp;Sofía Pérez Garbayo","doi":"10.1016/j.aim.2025.110328","DOIUrl":"10.1016/j.aim.2025.110328","url":null,"abstract":"<div><div>This paper deals with the problem of computing a generating set for the Cox ring <span><math><mi>R</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of a smooth projective rational surface <em>X</em> with nef anticanonical class. In case <span><math><mi>R</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is finitely generated, we show that the degrees of its generators are either classes of negative curves, elements of the Hilbert basis of the nef cone or certain ample classes of anticanonical degree one, which only appear when <em>X</em> is a rational elliptic surface of Halphen index <span><math><mi>m</mi><mo>&gt;</mo><mn>2</mn></math></span>. Moreover, we partially characterize which elements of the Hilbert basis of the nef cone are irredundant for generating <span><math><mi>R</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. We apply this result to compute explicit minimal generating sets for Cox rings of some rational elliptic surfaces of Halphen index &gt;1.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110328"},"PeriodicalIF":1.5,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936092","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":"Corrigendum to “Cauchy convergence in V-normed categories” [Adv. Math. 470 (2025) 110247]","authors":"Maria Manuel Clementino ,&nbsp;Dirk Hofmann ,&nbsp;Walter Tholen","doi":"10.1016/j.aim.2025.110337","DOIUrl":"10.1016/j.aim.2025.110337","url":null,"abstract":"<div><div>We correct a faulty argument given in the justification of Corollary 7.5 of “Cauchy convergence in <span><math><mi>V</mi></math></span>-normed categories” and clarify that it does not have any impact on the validity of other results stated in the paper either.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110337"},"PeriodicalIF":1.5,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936094","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":"Measure and dimension theory of permeable sets and its applications to fractals","authors":"Gunther Leobacher ,&nbsp;Tapio Rajala ,&nbsp;Alexander Steinicke ,&nbsp;Jörg Thuswaldner","doi":"10.1016/j.aim.2025.110316","DOIUrl":"10.1016/j.aim.2025.110316","url":null,"abstract":"<div><div>We study <em>permeable</em> sets. These are sets <span><math><mi>Θ</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> which have the property that each two points <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> can be connected by a short path <em>γ</em> which has small (or even empty, apart from the end points of <em>γ</em>) intersection with Θ. We investigate relations between permeability and Lebesgue measure and establish theorems on the relation of permeability with several notions of dimension. It turns out that for most notions of dimension each subset of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> of dimension less than <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span> is permeable. We use our permeability result on the Nagata dimension to characterize permeability properties of self-similar sets with certain finiteness properties.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110316"},"PeriodicalIF":1.5,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923441","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":"Alternating sign pentagons and Magog pentagons","authors":"Moritz Gangl","doi":"10.1016/j.aim.2025.110315","DOIUrl":"10.1016/j.aim.2025.110315","url":null,"abstract":"<div><div>Alternating sign triangles were introduced by Ayyer, Behrend and Fischer in 2016 and it was proven that there is the same number of alternating sign triangles with <em>n</em> rows as there is of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> alternating sign matrices. Later on Fischer gave a refined enumeration of alternating sign triangles with respect to a statistic <em>ρ</em>, which has the same distribution as the position of the unique 1 in the top row of an alternating sign matrix, by connecting alternating sign triangles to <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-Magog trapezoids for which such a statistic is known. We introduce two more statistics counting the all 0-columns on the left and right in an alternating sign triangle yielding objects we call alternating sign pentagons. We then show the equinumeracy of these alternating sign pentagons with Magog pentagons of a certain shape taking into account the statistic <em>ρ</em>. Furthermore we deduce a generating function of these alternating sign pentagons with respect to the statistic <em>ρ</em> in terms of a Pfaffian and consider the implications of our new results for some open conjectures. In particular we conjecture a refined equinumerosity between our Magog pentagons and Gog pentagons of a certain shape.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110315"},"PeriodicalIF":1.5,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923529","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":"Colored sl(N) homology and SU(N) representations","authors":"Joshua Wang","doi":"10.1016/j.aim.2025.110314","DOIUrl":"10.1016/j.aim.2025.110314","url":null,"abstract":"<div><div>We provide the first complete computations of colored <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math></span> homology for a nontrivial knot. In doing so, we show that the colored <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math></span> homology of the trefoil labeled by an exterior power of the defining representation is isomorphic to the cohomology of a closed manifold naturally associated to the trefoil. This manifold is the set of homomorphisms from the fundamental group of the complement of the trefoil to <span><math><mi>SU</mi><mo>(</mo><mi>N</mi><mo>)</mo></math></span> that send meridians to a particular conjugacy class depending on the label. We also provide complete computations and analogous isomorphisms for the first nontrivial link, the Hopf link.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110314"},"PeriodicalIF":1.5,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923442","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}