Advances in Mathematics最新文献

英文中文

Lax functorialities of the comma construction for ω-categories ω-范畴的逗号结构的松弛功能

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-15 DOI: 10.1016/j.aim.2025.110762

Dimitri Ara , Léonard Guetta

Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict ω-categories. To state the most general functorialities, we use the language of Gray ω-categories, that is, categories enriched in the category of strict ω-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict ω-categories defines a Gray ω-functor, that is, a morphism of Gray ω-categories. To makes sense of this statement, we prove that slices of Gray ω-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict ω-categories defines a Gray ω-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray ω-functors, which we plan to investigate in future work.

在Grothendieck结构的启发下，我们研究了严格ω-范畴的逗号结构的泛函性。为了描述最一般的功能，我们使用Gray ω-范畴的语言，即在赋予oplax Gray张量积的严格ω-范畴中丰富的范畴。我们的主要结果是严格ω-范畴的逗号构造定义了一个Gray ω-函子，即Gray ω-范畴的态射。为了使这句话有意义，我们证明了Gray ω-范畴片的存在。回到格罗滕狄克构造，我们提出了一个关于逗号构造的定义，作为结果，我们得到严格ω-范畴的格罗滕狄克构造定义了一个Gray ω-函子。最后，作为一个副产品，我们得到了Gray ω-函子的Grothendieck构造的概念，我们计划在未来的工作中对此进行研究。

引用次数: 0

Fractionally Calabi–Yau lattices that tilt to higher Auslander algebras of type A 向A型高等Auslander代数倾斜的分数Calabi-Yau格

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-15 DOI: 10.1016/j.aim.2026.110785

Tal Gottesman

We prove that the bounded derived category of the lattice of order ideals of the product of two ordered chains is fractionally Calabi–Yau. We also show that these lattices are derived equivalent to higher Auslander algebras of type A. The proofs involve the study of intervals of the poset that have resolutions described with antichains having rigid properties. These two results combined corroborate a conjecture by Chapoton linking posets to Fukaya–Seidel Categories.

证明了两有序链积的有序理想格的有界派生范畴是分数Calabi-Yau。我们还证明了这些格的推导等价于a型的更高的Auslander代数。证明涉及到用具有刚性性质的反链描述分辨率的偏序集的区间的研究。这两个结果结合起来证实了Chapoton将偏置集与Fukaya-Seidel范畴联系起来的猜想。

引用次数: 0

Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature 非负Ricci曲率流形的体积增长与渐近锥

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-15 DOI: 10.1016/j.aim.2026.110786

Zhu Ye

Let M be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of M is always greater than or equal to the dimension of some (or every) asymptotic cone of M.

Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of M equals k, then there exists an asymptotic cone of M whose upper box dimension is at most k. In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature.

In the subsequent part of the paper, we extend or partially extend Sormani's results concerning M 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

Ric \geq 0

and linear volume growth.

Finally, we construct an example of an open n-manifold M with

\sec_{M} \geq 0

whose volume growth order oscillates between 1 and n.

设M是一个具有非负里奇曲率的开（即完全非紧）流形。在本文中,我们研究M的销量增长顺序是否总是大于或等于的维度(或所有)的渐近锥面M.Our第一主要结果断言,在圆锥在无穷远处条件下,如果销量增长的下确界的M = k,然后有一个渐近锥面的M上盒子尺寸是最多k。特别是,这个收益率完全肯定的回答我们的问题在负的截面曲率的设置。在本文的后续部分，我们将Sormani关于线性体积增长的M的结果推广或部分推广到更宽松的体积增长条件。我们的方法也允许我们对Ric≥0和线性体积增长的开放流形给出Sormani的次线性直径增长定理的一个新的证明。最后，构造了secM≥0且体积增长阶在1和n之间振荡的开放n流形M的一个例子。

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引用次数: 0

On the large time asymptotics of Schrödinger type equations with general data 一般数据下Schrödinger型方程的大时间渐近性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-14 DOI: 10.1016/j.aim.2025.110774

Avy Soffer , Xiaoxu Wu

For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in an adapted way the Free Channel Wave Operator, and further tools from the recent works [21], [22], [35]. This work generalizes the results of the first part of [21], [22] to arbitrary dimension, and non-radial data.

对于具有一般相互作用项的Schrödinger方程，它可以是线性的，也可以是非线性的，时间相关的，并包含电荷转移势，我们证明了它的整体解是由自由波和弱局域部分的和渐近给出的。该证明是基于以一种适应的方式构造自由通道波算子，以及来自最近工作[21]，[22]，[35]的进一步工具。本工作将[21]、[22]第一部分的结果推广到任意维度和非径向数据。

引用次数: 0

The Lyubashenko modular functor for Drinfeld centers via non-semisimple string-nets 非半单弦网上Drinfeld中心的Lyubashenko模函子

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-14 DOI: 10.1016/j.aim.2025.110770

Lukas Müller , Christoph Schweigert , Lukas Woike , Yang Yang

The Levin-Wen string-nets of a spherical fusion category

C

describe, by results of Kirillov and Bartlett, the representations of mapping class groups of closed surfaces obtained from the Turaev-Viro construction applied to

C

. 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

Z (C)

球面融合范畴C的Levin-Wen弦网，通过Kirillov和Bartlett的结果，描述了应用于C的Turaev-Viro构造得到的闭曲面映射类群的表示。我们将这一陈述推广到任意关键有限张量范畴，包括非半单质或非球面。我们证明了由关键有限张量范畴的射影对象构造的有限共完备弦网模函子等价于由Drinfeld中心Z(C)构造的Lyubashenko模函子。

引用次数: 0

Nuclear dimension and virtually polycyclic groups 核维度和几乎多环基团

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.aim.2025.110768

Caleb Eckhardt , Jianchao Wu

We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on

C_{0} (X)

-algebras.

证明了虚多环群的（扭曲）群C*-代数的核维数是有限的。这促使我们对群C*-代数的有限核维数和有限Hirsch长度提出了一个猜想，然后我们对一类超越虚多环的初等可调群进行了验证。特别地，我们给出了有限核维有限生成的非剩余有限群的第一个例子。给出了有限分解秩的一个平行猜想，并得到了类似的结果。我们的方法很大程度上依赖于Hirshberg和第二位作者最近关于C0(X)-代数上的虚幂零群作用的研究。

引用次数: 0

Weighted Ehrhart theory via equivariant toric geometry 基于等变环几何的加权Ehrhart理论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.aim.2025.110771

Laurenţiu Maxim , Jörg Schürmann

<div><div>We give a K-theoretic and geometric interpretation for a generalized weighted Ehrhart theory of a full-dimensional lattice polytope P, depending on a given homogeneous polynomial function φ on P, and with Laurent polynomial weights <math><msub><mrow><mi>f</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math> associated to the faces <math><mi>Q</mi><mo>⪯</mo><mi>P</mi></math> of the polytope. For this purpose, we calculate equivariant K-theoretic Hodge–Chern classes of a torus-equivariant mixed Hodge module <math><mi>M</mi></math> on the toric variety <math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math> associated to P (defined via an equivariant embedding of <math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math> into an ambient smooth variety). For any integer ℓ, we introduce a corresponding equivariant Hodge <math><msub><mrow><mi>χ</mi></mrow><mrow><mi>y</mi></mrow></msub></math>-polynomial <math><msub><mrow><mi>χ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>,</mo><mi>ℓ</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>;</mo><mo>[</mo><mi>M</mi><mo>]</mo><mo>)</mo></math>, with <math><msub><mrow><mi>D</mi></mrow><mrow><mi>P</mi></mrow></msub></math> the corresponding ample Cartier divisor on <math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math> (defined by the facet presentation of P). Motivic properties of the Hodge–Chern classes are used to express this equivariant Hodge polynomial in terms of weighted character sums fitting with a generalized weighted Ehrhart theory. The equivariant Hodge polynomials are shown to satisfy a reciprocity and purity formula fitting with the duality for equivariant mixed Hodge modules, and implying the corresponding properties for the generalized weighted Ehrhart polynomials. In the special case of the equivariant intersection cohomology mixed Hodge module, with the weight function corresponding to Stanley's g-function of the polar polytope of P, we recover in geometric terms a recent combinatorial formula of Beck–Gunnells–Materov. More generally, motivated by the analogy to the Kazhdan–Lusztig theory, we introduce a duality involution on the free <math><mi>Z</mi><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math>-module of weight functions corresponding to the duality of equivariant mixed Hodge modules, and prove a new reciprocity formula in terms of this duality. This unifies and generalizes the classical reciprocity formula of Brion–Vergne in Ehrhart theory as well as t

我们给出了全维晶格多面体P的广义加权Ehrhart理论的k理论和几何解释，这取决于P上给定的齐次多项式函数φ，以及多面体的面Q⪯P相关的Laurent多项式权值fQ(y)∈Z[y±1]。为此，我们计算了与P相关的环-等变混合Hodge模M在环型变量XP上的等变k -理论Hodge - chern类（通过将XP等变嵌入到环境光滑变量中来定义）。对于任意整数，我们引入一个对应的等变Hodge χy-多项式χy(XP， P;[M])，其中DP是XP上对应的样例Cartier因子（由P的面表示定义）。利用Hodge - chern类的动机性质，用广义加权Ehrhart理论拟合的加权特征和来表示该等变Hodge多项式。证明了等变混合Hodge模的等变Hodge多项式满足与对偶拟合的互易性和纯度公式，并给出了广义加权Ehrhart多项式的相应性质。在等变交上同调混合Hodge模的特殊情况下，利用P的极多面体Stanley的g函数对应的权函数，用几何形式恢复了最近的Beck-Gunnells-Materov组合公式。更一般地说，在与Kazhdan-Lusztig理论类比的基础上，我们引入了与等变混合Hodge模对偶性相对应的自由权重函数Z[y±1]模的对偶对合，并利用该对偶性证明了一个新的互易公式。这统一和推广了Ehrhart理论中经典的Brion-Vergne互易公式以及上述最近的beck - gunnell - materov组合公式。

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引用次数: 0

Borel graphable equivalence relations 可图等价关系

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.aim.2025.110765

Tyler Arant , Alexander S. Kechris , Patrick Lutz

This paper is devoted to the study of analytic equivalence relations which are Borel graphable, 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

ZFC

) 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

S_{\infty}

, 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 graphic groups. We show that besides

S_{\infty}

, 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.

本文研究了可Borel图的解析等价关系，即可实现为Borel图的连通关系。我们的主要焦点是哪些解析等价关系是Borel可图的问题。首先，我们研究了由可数容许序数理论引起的等价关系，并证明了当且仅当存在不可构造实数时，该等价关系是Borel可图的。作为证明的一个推论，我们构造了一个（可证明在ZFC中）不可Borel图的解析等价关系和一个可Borel图但不可有效Borel图的有效解析等价关系。其次，我们研究了由同构关系给出的一类可数结构的解析等价关系。我们证明了所有这些等价关系都是Borel可图的，这意味着对于S∞上的每一个Borel作用，相关的轨道等价关系都是Borel可图的。这导致我们研究一类波兰群，它们的Borel作用总是产生Borel可图轨道等价关系；我们把这样的群称为图形群。证明了除S∞外，图群类包含了所有连通的波兰群，并且在可数积下是闭的。最后，我们研究了Borel可图解析等价关系类的结构性质，并考虑了Borel可图性的两种变体：用超图代替图的泛化和在可计算可数等价关系设置下对Borel可图性的模拟。

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引用次数: 0

Powers of ghost ideals 幽灵理想的力量

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.aim.2025.110777

S. Estrada , X.H. Fu , I. Herzog , S. Odabaşı

A cocomplete additive category

A

may be equipped with the transfinite filtration of inductive ordinal powers of an ideal

I ◃ A

. If

S \subseteq A

is a subcategory of an exact category, the

S

-ghost morphisms, i.e., the morphisms right Ext-orthogonal to

S

, 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

S^{⊥}

of right Ext-perpendicular objects. The question of convergence for this filtration yields a transfinite formulation of the Generating Hypothesis. For an ordinal λ, the Generalized λ-Generating Hypothesis is the proposition that the λ-th power of the ideal of

S

-ghost morphisms is the (object) ideal of morphisms that factor through an object in

S^{⊥}

. It is shown to hold when the category

A

is a locally λ-presentable Grothendieck category and

S

is a set of λ-presentable objects.

Two cases of interest are treated: when the exact category is the category

C (R)

of chain complexes of left R-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 ω-Generating Hypothesis is shown to hold; when the exact category is the module category R-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.

协完全加性范畴A可以具有理想I / A的归纳序幂的超限滤除。如果S⊆是一个精确的类别的子类别,在S-ghost射,即正确射Ext-orthogonal年代,形成这样一个理想和相应的超限过滤被射的理想有下界的因素通过对象的子类年代⊥Ext-perpendicular对象。这种过滤的收敛性问题产生了生成假设的超限公式。对于一个序数λ，广义λ生成假设是这样一个命题：S-鬼态射的理想的λ次幂是通过S⊥中的一个对象进行因子化的态射的（对象）理想。当范畴A是一个局部λ可呈现的Grothendieck范畴，而S是一组λ可呈现的客体时，它成立。处理了两种有趣的情况：当左R模的链复形的精确范畴是C(R)范畴时，则在同调上平凡的态射的理想是Cartan-Eilenberg投影子范畴的残影，并证明了广义的生成假设成立；当左纯射影模在可拓下闭合的环的精确范畴为模范畴R-Mod时，分析了由fp -鬼理想引起的超限滤波，证明了每一个左fp -射影模都是纯射影。

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引用次数: 0

Gromov width of the disk cotangent bundle of spheres of revolution 公转球体的圆盘共切束的格罗莫夫宽度

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-08 DOI: 10.1016/j.aim.2025.110761

Brayan Ferreira , Vinicius G.B. Ramos , Alejandro Vicente

Inspired by work of the first and second authors, this paper studies the Gromov width of the disk cotangent bundle of spheroids and Zoll spheres of revolution. This is achieved with the use of techniques from integrable systems and embedded contact homology capacities.

受第一和第二作者工作的启发，本文研究了椭球和Zoll旋转球的盘共切束的Gromov宽度。这是通过使用可积系统和嵌入式接触同源能力的技术来实现的。

引用次数: 0

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