Pub Date : 2025-12-22DOI: 10.1016/j.aim.2025.110732
Carsten Peterson
<div><div>Let <span><math><mi>p</mi><mo>⊂</mo><mi>V</mi></math></span> be a polytope and <span><math><mi>ξ</mi><mo>∈</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. We obtain an expression for <span><math><mi>I</mi><mo>(</mo><mi>p</mi><mo>;</mo><mi>α</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>α</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow></msup><mi>d</mi><mi>x</mi></math></span> as a sum of meromorphic functions in <span><math><mi>α</mi><mo>∈</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> parametrized by the faces <span><math><mi>f</mi></math></span> of <span><math><mi>p</mi></math></span> on which <span><math><mo>〈</mo><mi>ξ</mi><mo>,</mo><mi>x</mi><mo>〉</mo></math></span> is constant. Each term only depends on the local geometry of <span><math><mi>p</mi></math></span> near <span><math><mi>f</mi></math></span> (and on <em>ξ</em>) and is holomorphic at <span><math><mi>α</mi><mo>=</mo><mi>ξ</mi></math></span>. When <span><math><mo>〈</mo><mi>ξ</mi><mo>,</mo><mo>⋅</mo><mo>〉</mo></math></span> is only constant on the vertices of <span><math><mi>p</mi></math></span> our formula reduces to Brion's formula.</div><div>Suppose <span><math><mi>p</mi></math></span> is a rational polytope with respect to a lattice Λ. We obtain an expression for <span><math><mi>S</mi><mo>(</mo><mi>p</mi><mo>;</mo><mi>α</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>λ</mi><mo>∈</mo><mi>p</mi><mo>∩</mo><mi>Λ</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>α</mi><mo>,</mo><mi>λ</mi><mo>〉</mo></mrow></msup></math></span> as a sum of meromorphic functions parametrized by the faces <span><math><mi>f</mi></math></span> on which <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>ξ</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow></msup><mo>=</mo><mn>1</mn></math></span> on a finite index sublattice of <span><math><mtext>lin</mtext><mo>(</mo><mi>f</mi><mo>)</mo><mo>∩</mo><mi>Λ</mi></math></span>. Each term only depends on the local geometry of <span><math><mi>p</mi></math></span> near <span><math><mi>f</mi></math></span> (and on <em>ξ</em> and Λ) and is holomorphic at <span><math><mi>α</mi><mo>=</mo><mi>ξ</mi></math></span>. When <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>ξ</mi><mo>,</mo><mo>⋅</mo><mo>〉</mo></mrow></msup><mo>≠</mo><mn>1</mn></math></span> at any non-zero lattice point on a line through the origin parallel to an edge of <span><math><mi>p</mi></math></span>, our formula reduces to Brion's formula, and when <span><math><mi>ξ</mi><mo>=</mo><mn>0</mn></math></span>, it reduces to the Ehrhart quasi-polynomial.</div><div>Our formulas are particularly useful for understanding how <span><math><mi>I</mi><mo>(</mo><mi>p</mi><mo>(</mo><mi>h</mi><mo>)</mo><mo>;</mo><mi>ξ</mi><mo>)</mo></math></span> and <
{"title":"A degenerate version of Brion's formula","authors":"Carsten Peterson","doi":"10.1016/j.aim.2025.110732","DOIUrl":"10.1016/j.aim.2025.110732","url":null,"abstract":"<div><div>Let <span><math><mi>p</mi><mo>⊂</mo><mi>V</mi></math></span> be a polytope and <span><math><mi>ξ</mi><mo>∈</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. We obtain an expression for <span><math><mi>I</mi><mo>(</mo><mi>p</mi><mo>;</mo><mi>α</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>α</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow></msup><mi>d</mi><mi>x</mi></math></span> as a sum of meromorphic functions in <span><math><mi>α</mi><mo>∈</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> parametrized by the faces <span><math><mi>f</mi></math></span> of <span><math><mi>p</mi></math></span> on which <span><math><mo>〈</mo><mi>ξ</mi><mo>,</mo><mi>x</mi><mo>〉</mo></math></span> is constant. Each term only depends on the local geometry of <span><math><mi>p</mi></math></span> near <span><math><mi>f</mi></math></span> (and on <em>ξ</em>) and is holomorphic at <span><math><mi>α</mi><mo>=</mo><mi>ξ</mi></math></span>. When <span><math><mo>〈</mo><mi>ξ</mi><mo>,</mo><mo>⋅</mo><mo>〉</mo></math></span> is only constant on the vertices of <span><math><mi>p</mi></math></span> our formula reduces to Brion's formula.</div><div>Suppose <span><math><mi>p</mi></math></span> is a rational polytope with respect to a lattice Λ. We obtain an expression for <span><math><mi>S</mi><mo>(</mo><mi>p</mi><mo>;</mo><mi>α</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>λ</mi><mo>∈</mo><mi>p</mi><mo>∩</mo><mi>Λ</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>α</mi><mo>,</mo><mi>λ</mi><mo>〉</mo></mrow></msup></math></span> as a sum of meromorphic functions parametrized by the faces <span><math><mi>f</mi></math></span> on which <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>ξ</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow></msup><mo>=</mo><mn>1</mn></math></span> on a finite index sublattice of <span><math><mtext>lin</mtext><mo>(</mo><mi>f</mi><mo>)</mo><mo>∩</mo><mi>Λ</mi></math></span>. Each term only depends on the local geometry of <span><math><mi>p</mi></math></span> near <span><math><mi>f</mi></math></span> (and on <em>ξ</em> and Λ) and is holomorphic at <span><math><mi>α</mi><mo>=</mo><mi>ξ</mi></math></span>. When <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>ξ</mi><mo>,</mo><mo>⋅</mo><mo>〉</mo></mrow></msup><mo>≠</mo><mn>1</mn></math></span> at any non-zero lattice point on a line through the origin parallel to an edge of <span><math><mi>p</mi></math></span>, our formula reduces to Brion's formula, and when <span><math><mi>ξ</mi><mo>=</mo><mn>0</mn></math></span>, it reduces to the Ehrhart quasi-polynomial.</div><div>Our formulas are particularly useful for understanding how <span><math><mi>I</mi><mo>(</mo><mi>p</mi><mo>(</mo><mi>h</mi><mo>)</mo><mo>;</mo><mi>ξ</mi><mo>)</mo></math></span> and <","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110732"},"PeriodicalIF":1.5,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841833","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 : 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":"2025-12-19","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 : 2025-12-19DOI: 10.1016/j.aim.2025.110744
Spyridon Kakaroumpas , Zoe Nieraeth
In this work we fully characterize the classes of matrix weights for which multilinear Calderón–Zygmund operators extend to bounded operators on matrix weighted Lebesgue spaces. To this end, we develop the theory of multilinear singular integrals taking values in tensor products of finite dimensional Hilbert spaces. We establish quantitative bounds in terms of multilinear Muckenhoupt matrix weight characteristics and scalar Fujii–Wilson conditions of a tensor product analogue of the convex body sparse operator, of a convex-set valued tensor product analogue of the Hardy–Littlewood maximal operator, and of a multilinear analogue of the Christ–Goldberg maximal operator. These bounds recover the sharpest known bounds in the linear case. Moreover, we define a notion of directional nondegeneracy for multilinear Calderón–Zygmund operators, which is new even in the scalar case. The noncommutativity of matrix multiplication, the absence of duality, and the natural presence of quasinorms in the multilinear setting present several new difficulties in comparison to previous works in the scalar or in the linear case. To overcome them, we use techniques inspired from convex combinatorics and differential geometry.
{"title":"Multilinear matrix weights","authors":"Spyridon Kakaroumpas , Zoe Nieraeth","doi":"10.1016/j.aim.2025.110744","DOIUrl":"10.1016/j.aim.2025.110744","url":null,"abstract":"<div><div>In this work we fully characterize the classes of matrix weights for which multilinear Calderón–Zygmund operators extend to bounded operators on matrix weighted Lebesgue spaces. To this end, we develop the theory of multilinear singular integrals taking values in tensor products of finite dimensional Hilbert spaces. We establish quantitative bounds in terms of multilinear Muckenhoupt matrix weight characteristics and scalar Fujii–Wilson conditions of a tensor product analogue of the convex body sparse operator, of a convex-set valued tensor product analogue of the Hardy–Littlewood maximal operator, and of a multilinear analogue of the Christ–Goldberg maximal operator. These bounds recover the sharpest known bounds in the linear case. Moreover, we define a notion of directional nondegeneracy for multilinear Calderón–Zygmund operators, which is new even in the scalar case. The noncommutativity of matrix multiplication, the absence of duality, and the natural presence of quasinorms in the multilinear setting present several new difficulties in comparison to previous works in the scalar or in the linear case. To overcome them, we use techniques inspired from convex combinatorics and differential geometry.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110744"},"PeriodicalIF":1.5,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792176","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 : 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":"2025-12-17","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 : 2025-12-17DOI: 10.1016/j.aim.2025.110728
Ian Hambleton , John Nicholson
Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any , there exist a family of k closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent.
{"title":"Four-manifolds, two-complexes and the quadratic bias invariant","authors":"Ian Hambleton , John Nicholson","doi":"10.1016/j.aim.2025.110728","DOIUrl":"10.1016/j.aim.2025.110728","url":null,"abstract":"<div><div>Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there exist a family of <em>k</em> closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110728"},"PeriodicalIF":1.5,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760671","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 : 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":"2025-12-16","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 : 2025-12-16DOI: 10.1016/j.aim.2025.110736
Giorgio Mangioni, Alessandro Sisto
We prove that most Artin groups of large and hyperbolic type are Hopfian, meaning that every self-epimorphism is an isomorphism. The class covered by our result is generic, in the sense of Goldsborough-Vaskou. Moreover, assuming the residual finiteness of certain hyperbolic groups with an explicit presentation, we get that all large and hyperbolic type Artin groups are residually finite. We also show that “most” quotients of the five-holed sphere mapping class group are hierarchically hyperbolic, up to taking powers of the normal generators of the kernels.
The main tool we use to prove both results is a Dehn-filling-like procedure for short hierarchically hyperbolic groups (these also include e.g. non-geometric 3-manifolds, and triangle- and square-free RAAGs).
{"title":"Short hierarchically hyperbolic groups II: Quotients and the Hopf property for Artin groups","authors":"Giorgio Mangioni, Alessandro Sisto","doi":"10.1016/j.aim.2025.110736","DOIUrl":"10.1016/j.aim.2025.110736","url":null,"abstract":"<div><div>We prove that most Artin groups of large and hyperbolic type are Hopfian, meaning that every self-epimorphism is an isomorphism. The class covered by our result is generic, in the sense of Goldsborough-Vaskou. Moreover, assuming the residual finiteness of certain hyperbolic groups with an explicit presentation, we get that all large and hyperbolic type Artin groups are residually finite. We also show that “most” quotients of the five-holed sphere mapping class group are hierarchically hyperbolic, up to taking powers of the normal generators of the kernels.</div><div>The main tool we use to prove both results is a Dehn-filling-like procedure for short hierarchically hyperbolic groups (these also include e.g. non-geometric 3-manifolds, and triangle- and square-free RAAGs).</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110736"},"PeriodicalIF":1.5,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760668","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 : 2025-12-16DOI: 10.1016/j.aim.2025.110734
Fernando Ballesta-Yagüe, María J. Carro
We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering -measures on the boundary. More specifically, we study the -solvability for the Dirichlet problem, complementing results of [25] and [10]. Then, we study -solvability of the Neumann problem, obtaining a range of solvability which is empty in some cases, a clear difference with the arc-length case. When it is not empty, it is an interval, and we consider solvability at its endpoints, establishing conditions for Lorentz space solvability when and atomic Hardy space solvability when . Solving the Lorentz endpoint leads us to a two-weight Sawyer-type inequality, for which we give a sufficient condition. Finally, we show how to adapt to the Regularity problem the results for the Neumann problem.
{"title":"Boundary value problems in graph Lipschitz domains in the plane with A∞-measures on the boundary","authors":"Fernando Ballesta-Yagüe, María J. Carro","doi":"10.1016/j.aim.2025.110734","DOIUrl":"10.1016/j.aim.2025.110734","url":null,"abstract":"<div><div>We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-measures on the boundary. More specifically, we study the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></math></span>-solvability for the Dirichlet problem, complementing results of <span><span>[25]</span></span> and <span><span>[10]</span></span>. Then, we study <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-solvability of the Neumann problem, obtaining a range of solvability which is empty in some cases, a clear difference with the arc-length case. When it is not empty, it is an interval, and we consider solvability at its endpoints, establishing conditions for Lorentz space solvability when <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and atomic Hardy space solvability when <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span>. Solving the Lorentz endpoint leads us to a two-weight Sawyer-type inequality, for which we give a sufficient condition. Finally, we show how to adapt to the Regularity problem the results for the Neumann problem.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110734"},"PeriodicalIF":1.5,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760670","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 : 2025-12-16DOI: 10.1016/j.aim.2025.110738
Jeremy B. Hume
We compute the K-theory of the three -algebras associated to a rational function R acting on the Riemann sphere, its Fatou set, and its Julia set. The latter -algebra is a unital UCT Kirchberg algebra and is thus classified by its K-theory. The K-theory in all three cases is shown to depend only on the degree of R, the critical points of R, and the Fatou cycles of R. Our results yield new dynamical invariants for rational functions and a -algebraic interpretation of the Density of Hyperbolicity Conjecture for quadratic polynomials. These calculations are possible due to new exact sequences in K-theory we induce from morphisms of -correspondences.
我们计算了作用于Riemann球上的一个有理函数R及其Fatou集和Julia集的三个C -代数的k理论。后一种C -代数是一个统一的UCT Kirchberg代数，因此可以用它的k理论来分类。这三种情况下的k理论只依赖于R的度、R的临界点和R的Fatou环。我们的结果给出了有理函数的新的动态不变量和二次多项式的双曲猜想密度的C - C -代数解释。这些计算是可能的，因为我们从C -对应的态射中推导出了k理论中的新的精确序列。
{"title":"The K-theory of the C*-algebras associated to rational functions","authors":"Jeremy B. Hume","doi":"10.1016/j.aim.2025.110738","DOIUrl":"10.1016/j.aim.2025.110738","url":null,"abstract":"<div><div>We compute the <em>K</em>-theory of the three <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras associated to a rational function <em>R</em> acting on the Riemann sphere, its Fatou set, and its Julia set. The latter <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra is a unital UCT Kirchberg algebra and is thus classified by its <em>K</em>-theory. The <em>K</em>-theory in all three cases is shown to depend only on the degree of <em>R</em>, the critical points of <em>R</em>, and the Fatou cycles of <em>R</em>. Our results yield new dynamical invariants for rational functions and a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebraic interpretation of the Density of Hyperbolicity Conjecture for quadratic polynomials. These calculations are possible due to new exact sequences in <em>K</em>-theory we induce from morphisms of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-correspondences.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110738"},"PeriodicalIF":1.5,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760667","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 : 2025-12-16DOI: 10.1016/j.aim.2025.110735
Stavros Garoufalidis , Tao Yu
We define a quantum trace map from the skein module of a 3-manifold with torus boundary components to a module (left and right quotient of a quantum torus) constructed from an ideal triangulation. Our map is a 3-dimensional version of the well-known quantum trace map on surfaces introduced by Bonahon and Wong and further developed by Lê.
{"title":"A quantum trace map for 3-manifolds","authors":"Stavros Garoufalidis , Tao Yu","doi":"10.1016/j.aim.2025.110735","DOIUrl":"10.1016/j.aim.2025.110735","url":null,"abstract":"<div><div>We define a quantum trace map from the skein module of a 3-manifold with torus boundary components to a module (left and right quotient of a quantum torus) constructed from an ideal triangulation. Our map is a 3-dimensional version of the well-known quantum trace map on surfaces introduced by Bonahon and Wong and further developed by Lê.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"486 ","pages":"Article 110735"},"PeriodicalIF":1.5,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760669","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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