Advances in Mathematics最新文献

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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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引用次数: 0

The weighted ambient metric for manifolds with density 带密度流形的加权环境度规

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.aim.2026.110787

Ayush Khaitan

We prove the existence and uniqueness of a weighted analogue of the Fefferman-Graham ambient metric for manifolds with density. We then show that this ambient metric forms the natural geometric framework for the Ricci flow by constructing infinite families of fully non-linear analogues of Perelman's

F

and

W

functionals. We extend Perelman's monotonicity result to these two families of functionals under several conditions, including for shrinking solitons and Einstein manifolds. We do so by constructing a “Ricci flow vector field” in the ambient space, which may be of independent research interest. We also prove that the weighted GJMS operators associated with the weighted ambient metric are formally self-adjoint, and that the associated weighted renormalized volume coefficients are variational.

我们证明了具有密度流形的Fefferman-Graham环境度量的一个加权模拟的存在唯一性。然后，我们通过构造Perelman的F和W泛函的无限族的完全非线性类似物，证明了这个环境度量形成了Ricci流的自然几何框架。我们将Perelman的单调性结果推广到这两类泛函的若干条件下，包括缩孤子和爱因斯坦流形。我们通过在环境空间中构建一个“里奇流向量场”来实现这一点，这可能是一个独立的研究兴趣。我们还证明了与加权环境度量相关的加权GJMS算子在形式上是自伴随的，并且相关的加权重归一化体积系数是变分的。

引用次数: 0

Discrete fractals: Dimensions, quasi-isometric invariance and self-similarity 离散分形：维数、准等距不变性和自相似性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.aim.2026.110791

Kan Jiang , Junjie Miao , Lifeng Xi

It is well-known that fractal dimensions are invariant under bi-Lipschitz mappings on Euclidean spaces, and therefore, bi-Lipschitz mappings are important in the classification of fractal sets. On locally finite discrete metric spaces, bi-Lipschitz mappings are a class of special quasi-isometries which constitute a fundamental concept in geometric group theory.

In this paper, we extend discrete fractal dimensions to locally finite discrete metric spaces, establishing their quasi-isometric invariance. For discrete self-similar sets with integer digits, we prove a complete classification of bi-Lipschitz and quasi-isometric equivalences, providing a discrete analogue to Falconer and Marsh's seminal results on Lipschitz equivalence of self-similar fractals. Our main theorem shows that two non-trivial such sets are quasi-isometric if and only if the logarithm of their scaling ratios and digit set cardinalities are rationally proportional. Furthermore, the bi-Lipschitz equivalence of these structures is strictly determined by the inclusion of zero in their digit sets, distinguishing them from standard self-similar fractals.

众所周知，欧几里得空间上的双lipschitz映射下分形维数是不变的，因此，双lipschitz映射在分形集的分类中具有重要的意义。在局部有限离散度量空间上，bi-Lipschitz映射是一类特殊的拟等距，是几何群论中的一个基本概念。本文将离散分形维推广到局部有限离散度量空间，建立了它们的拟等距不变性。对于具有整数位数的离散自相似集，我们证明了双Lipschitz和准等距等价的完全分类，提供了Falconer和Marsh关于自相似分形的Lipschitz等价的开创性结果的离散类比。我们的主要定理表明，两个非平凡的这样的集合是拟等距的，当且仅当它们的比例比和数字集基数的对数是合理比例的。此外，这些结构的双lipschitz等价性是由它们的数字集中包含0严格确定的，从而与标准的自相似分形区分开来。

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引用次数: 0

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