Advances in Mathematics最新文献

英文中文

On the Farrell–Jones conjecture for localising invariants 关于定域不变量的Farrell-Jones猜想

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.aim.2026.110788

Ulrich Bunke , Daniel Kasprowski , Christoph Winges

We prove the Farrell–Jones conjecture for finitary localising invariants with coefficients in left-exact ∞-categories for finitely

F

-amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over

E_{1}

-ring spectra.

我们证明了有限f -可服从群和更一般的Dress-Farrell-Hsiang-Jones群在左正∞-范畴中有限带系数的局部不变量的Farrell-Jones猜想。我们的结果包含并统一了关于可加范畴和球群环的k理论的论证，并将其推广到e1环谱上的完美模范畴。

引用次数: 0

Hölder regularity of harmonic functions on metric measure spaces Hölder度量度量空间上调和函数的正则性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.aim.2026.110797

Jin Gao , Meng Yang

We introduce a Hölder regularity condition for harmonic functions on metric measure spaces and prove that, under a slow volume regular condition and an upper heat kernel estimate, the Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse Hölder inequality on the Sierpiński carpet cable system, resolving an open problem left by Devyver et al. (2023) [26]. Second, we prove that two-sided heat kernel estimates alone imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain Hölder (Lipschitz) estimates for the heat kernel on strongly recurrent metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds.

引入了度量度量空间上调和函数的一个Hölder正则性条件，证明了在慢体积正则条件和上热核估计下，Hölder正则性条件、弱Bakry-Émery非负曲率条件、热核（含或不含指数项）的Hölder连续性条件和热核的近对角下界是等价的。作为应用，首先，我们在Sierpiński地毯电缆系统上建立了所谓的广义反向Hölder不等式的有效性，解决了Devyver等人（2023）[26]留下的一个开放性问题。其次，我们证明了双面热核估计单独暗示了强循环分形索系统热核的梯度估计，改进了上述论文的主要结果。第三，我们得到了热核在强循环度量空间上的Hölder （Lipschitz）估计，扩展了黎曼流形上热核的经典Li-Yau梯度估计。

引用次数: 0

Symmetry in deformation quantization and geometric quantization 变形量化和几何量化中的对称性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.aim.2026.110804

Naichung Conan Leung , Qin Li , Ziming Nikolas Ma

In this paper, we explore the quantization of Kähler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form

f = f_{0} - \frac{ħ}{4 π} (Δ f_{0} + c)

for a certain smooth (non-formal) function

f_{0}

. If

f_{0}

is real-valued then

f_{0}

corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian G-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.

本文探讨了Kähler流形的量化问题，重点讨论了变形量化与几何量化的关系。我们在Berezin-Toeplitz变形量化中给出了1阶形式可量化函数的分类，建立了对于某光滑（非正式）函数f0，这些形式函数的形式为f=f0−ħ4π（Δf0+c）。如果f0是实值，那么f0对应于哈密顿杀戮向量场。在hamilton g对称存在的情况下，我们讨论了通过量子矩映射进行变形量子化的无穷小对称性与通过Berezin-Toeplitz量子化作用于全纯截面Hilbert空间的几何量子化的无穷小对称性之间的相容性。

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引用次数: 0

Corrigendum to “Notes on Plücker's relations in geometric algebra” [Advances in Mathematics 363 (2020) 106959] “几何代数中plencker关系注释”的勘误表[数学进展363 (2020)106959]

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-21 DOI: 10.1016/j.aim.2026.110801

Garret Sobczyk

引用次数: 0

The defect of the F-pure threshold F-pure阈值的缺陷

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-20 DOI: 10.1016/j.aim.2026.110792

Alessandro De Stefani , Luis Núñez-Betancourt , Ilya Smirnov

Introduced by Takagi and Watanabe, F-pure thresholds are invariants defined in terms of the Frobenius homomorphism. While they find applications in various settings, they are primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring

(R, m)

by setting

dfpt (R) = \dim (R) - fpt (m)

. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring.

由Takagi和Watanabe引入，f纯阈值是根据Frobenius同态定义的不变量。虽然它们在各种设置中都有应用，但它们主要用作局部不变量。本文的目的是通过打开其在方案上的行为研究来填补这一空白。为此，我们通过设dfpt(R)=dim (R)−fpt(m)来定义局部环（R,m）的f -纯阈值缺陷。结果表明，这个不变量定义了一个上半连续函数在一个方案上，并且满足bertini型定理。我们还研究了f -纯阈值在平面扩展和局部环极大理想爆破后的缺陷行为。

引用次数: 0

Categories of abstract and noncommutative measurable spaces 抽象和非交换可测空间的范畴

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-20 DOI: 10.1016/j.aim.2026.110793

Tobias Fritz , Antonio Lorenzin

Gelfand duality is a fundamental result that justifies thinking of general unital

C^{⁎}

-algebras as noncommutative versions of compact Hausdorff spaces. Inspired by this perspective, we investigate what noncommutative measurable spaces should be. This leads us to consider categories of monotone σ-complete

C^{⁎}

-algebras as well as categories of Boolean σ-algebras, which can be thought of as abstract measurable spaces.

Motivated by the search for a good notion of noncommutative measurable space, we provide a unified overview of these categories, alongside those of measurable spaces, and formalize their relationships through functors, adjunctions and equivalences. This includes an equivalence between Boolean σ-algebras and commutative monotone σ-complete

C^{⁎}

-algebras, as well as a Gelfand-type duality adjunction between the latter category and the category of measurable spaces. This duality restricts to two equivalences: one involving standard Borel spaces, which are widely used in probability theory, and another involving the more general Baire measurable spaces. Moreover, this result admits a probabilistic version, where the morphisms are σ-normal cpu maps and Markov kernels, respectively.

We hope that these developments can also contribute to the ongoing search for a well-behaved Markov category for measure-theoretic probability beyond the standard Borel setting — an open problem in the current state of the art.

Gelfand对偶性是证明将一般单位C -代数视为紧化Hausdorff空间的非交换版本的一个基本结果。受此启发，我们研究了什么是非交换可测空间。这导致我们考虑单调σ-完备C -代数的范畴以及布尔σ-代数的范畴，它们可以被认为是抽象的可测量空间。在寻找非交换可测空间的良好概念的激励下，我们提供了这些范畴的统一概述，以及可测空间的范畴，并通过函子、辅式和等价形式化了它们之间的关系。这包括布尔σ-代数与交换单调σ-完备C -代数之间的等价，以及后者与可测空间范畴之间的gelfand型对偶附加。这种对偶性限制了两个等价：一个涉及广泛用于概率论的标准Borel空间，另一个涉及更一般的Baire可测量空间。此外，该结果承认一个概率版本，其中态射分别是σ-正态cpu映射和马尔可夫核。我们希望这些发展也可以有助于正在进行的寻找超出标准Borel设置的测量理论概率的良好马尔可夫类别-这是当前艺术状态中的一个开放问题。

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引用次数: 0

Fukaya A∞-structure near infinity and the categorical formal completion 近无穷A∞结构与范畴形式补全

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-20 DOI: 10.1016/j.aim.2026.110795

Yuan Gao

For a stopped Liouville manifold arising from a Liouville sector, we construct a symplectic analogue of the formal neighborhood of the stop on the level of Fukaya categories. This geometric construction is performed via Floer-theoretic methods by allowing wrappings in the negative direction. On the other hand, inspired by homological mirror symmetry for pairs, where the mirror is the formal neighborhood of a divisor in an ambient projective variety, there is a different approach by taking a ‘categorical formal completion’ introduced by Efimov. Our main result establishes equivalence of these two approaches, confirms computability of this new type of Floer theory by categorical and algebraic means, and indicates contributions from and to computations in homological mirror symmetry.

对于由Liouville扇区产生的停止Liouville流形，我们在Fukaya范畴的水平上构造了该停止的形式邻域的辛模拟。这种几何结构通过允许在负方向上包裹的花理论方法来执行。另一方面，受对的同调镜像对称的启发，其中镜像是周围射影变化中的除数的形式邻域，有一种不同的方法，采用Efimov引入的“绝对形式补全”。我们的主要结果建立了这两种方法的等价性，用范畴和代数方法证实了这种新型Floer理论的可计算性，并指出了对同调镜像对称计算的贡献。

引用次数: 0

Dedekind's problem in the hypergrid 戴德金在超级电网中的问题

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-20 DOI: 10.1016/j.aim.2026.110796

Victor Falgas–Ravry , Eero Räty , István Tomon

<div><div>Consider the partially ordered set on <span><math><msup><mrow><mo>[</mo><mi>t</mi><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>:</mo><mo>=</mo><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> equipped with the natural coordinate-wise ordering, and let <span><math><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> denote the number of antichains of this poset. Determining <span><math><mi>A</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is the celebrated problem of Dedekind from 1897, and the general quantity <span><math><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> has a number of combinatorial interpretations: it is precisely the number of <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional partitions with entries from <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>t</mi><mo>}</mo></math></span>, and by a result of Moshkovitz and Shapira, <span><math><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> is equal to the <em>n</em>-color Ramsey number of monotone paths of length <em>t</em> in 3-uniform hypergraphs. This has led to significant interest in the growth rate of <span><math><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>.</div><div>Trivially, <span><math><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⁡</mo><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mi>α</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><mi>α</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is the size of a maximal antichain in <span><math><msup><mrow><mo>[</mo><mi>t</mi><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span>. In the present paper, we prove that this simple lower bound is close to optimal, in particular for every <span><math><mi>t</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>,<span><span><span><math><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⁡</mo><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>≤</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></mrow><mo>)</mo></mrow><mo>⋅</mo><mi>α</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>.</mo></math></span></span></span> This resolves a conjecture of Moshkovitz and Shapira, and gives the first bound that is close to optimal for growing <em>t</em>. Our proof is based on the graph container method, partly inspired by previous work of Pohoata and Zakharov.</div><div>One of our main contributions is a novel supersaturation result in <span><math><msup><mrow><

考虑在[t]n:={0，…，t−1}n上具有自然坐标有序的偏序集，设A（t,n）表示该偏序集的反链个数。确定A（2,n）是Dedekind自1897年以来的著名问题，一般量A（t,n）有许多组合解释：它精确地是（n−1）维分区的个数，条目为{0，…，t}，并且根据Moshkovitz和Shapira的结果，A(t,n)+1等于3-均匀超图中长度为t的单调路径的n色拉姆齐数。这引起了对A（t,n）增长率的极大兴趣。一般来说，log2 (A, t,n)≥α(t,n)，其中α(t,n)是[t]n中最大反链的大小。在本文中，我们证明了这个简单下界是接近最优的，特别是对于t，n≥2,log2 (A, t,n)≤(1+O((log (n)3n)))·α(t,n)。这解决了Moshkovitz和Shapira的一个猜想，并给出了t增长时接近最优的第一个界。我们的证明是基于图容器方法，部分灵感来自Pohoata和Zakharov之前的工作。我们的主要贡献之一是[t]n中的一个新的过饱和结果。我们证明了对于任意k∈Z+和δ>；0，任意大小至少为(k+δ)α(t,n)的集合A∧[t]n包含一个至少与A的Ωδ，k（(n/log (n))k）个其他元素相媲美的顶点，该界是最优的，直至对数因子。我们通过在[t]n的封面图上构造一个归一化匹配流来实现这一点，其中权重分布接近均匀，这是一个可能独立感兴趣的结果。

引用次数: 0

On the invariant surface area functionals in 3-dimensional CR geometry 三维CR几何中的不变表面积泛函

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.aim.2026.110790

Pak Tung Ho

Cheng, Yang, and Zhang have studied two invariant surface area functionals in 3-dimensional CR manifolds. They deduced the Euler–Lagrange equations of the associated energy functionals when the 3-dimensional CR manifold has constant Webster curvature and vanishing torsion. In this paper, we deduce the Euler–Lagrange equations of the energy functionals in a more general 3-dimensional CR manifold. Moreover, we study the invariant area functionals on the disk bundle, on the Rossi sphere, and on 3-dimensional tori. In particular, we show that the Clifford torus is a minimizer for

E_{1}

on the Rossi sphere

S_{t}^{3}

when

t = - 4 + \sqrt{15}

. Also, by computing the second variation formula, we show that the Clifford torus is not a minimizer for

E_{1}

on the Rossi sphere

S_{t}^{3}

when

t > - 4 + \sqrt{15}

Cheng， Yang和Zhang研究了三维CR流形中的两个不变表面积泛函。当三维CR流形具有恒定的韦氏曲率和消失的扭转时，他们推导出了相关能量泛函的欧拉-拉格朗日方程。本文推导了一般三维CR流形中能量泛函的欧拉-拉格朗日方程。此外，我们还研究了盘束、罗西球和三维环面上的不变面积泛函。特别地，我们证明了当t= - 4+15时，Clifford环面是罗西球St3上E1的最小化器。此外，通过计算第二变分公式，我们证明了当t>；−4+15时，Clifford环面不是罗西球St3上E1的最小值。

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引用次数: 0

Expander graphs are globally synchronizing 扩展器图是全局同步的

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.aim.2025.110773

Pedro Abdalla , Afonso S. Bandeira , Martin Kassabov , Victor Souza , Steven H. Strogatz , Alex Townsend

The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any

ε > 0

and

p ⩾ (1 + ε) (\log n) / n

, the homogeneous Kuramoto model on the Erdős–Rényi random graph

G (n, p)

is globally synchronizing with probability tending to one as n goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the Kuramoto model is globally synchronizing on any d-regular Ramanujan graph, and on typical d-regular graphs, for

d ⩾ 600

Kuramoto模型是同步研究的基础。它由一组由网络给出的具有相互作用的振子组成，我们分别用图的顶点和边来识别这些振子。在本文中，我们证明了具有充分展开的图必须是全局同步的，这意味着在这样的图上具有相同振子的齐次Kuramoto模型将收敛到所有振子具有相同相位的完全同步状态，对于每个初始状态直到一组测度零。特别是，我们表明，对于任何ε>；0和p小于（1+ε）(log log n)/n， Erdős-Rényi随机图G（n,p）上的齐次Kuramoto模型与当n趋于无穷时趋向于1的概率在全局同步。这改进了Kassabov、Strogatz和Townsend先前的结果，并解决了Ling、Xu和Bandeira的一个猜想。我们还表明，Kuramoto模型在任何d规则Ramanujan图上，以及在d大于或等于600的典型d规则图上，是全局同步的。

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