Advances in Mathematics最新文献

英文中文

Generalized rank functions and quilts of alternating sign matrices 交替符号矩阵的广义秩函数和矩阵

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-24 DOI: 10.1016/j.aim.2026.110863

Sara C. Billey , Matjaž Konvalinka

In this paper, we present new objects, quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with respect to two Boolean lattices. When the two posets are chains, a quilt is equivalent to an alternating sign matrix and its corresponding corner sum matrix. Quilts also generalize the monotone Boolean functions counted by the Dedekind numbers. Quilts form a distributive lattice with many beautiful properties and contain many classical and well-known sublattices, such as the lattice of matroids of a given rank and ground set. While enumerating quilts is hard in general, we prove two major enumerative results, when one of the posets is an antichain and when one of them is a chain. We also give some bounds for the number of quilts when one poset is the Boolean lattice.

在本文中，我们给出了关于两个给定序集的交替符号矩阵的新对象。Quilts概括了数学中几个常用的概念。例如，一个矩阵的子矩阵上的秩函数产生关于两个布尔格的被子。当两个序集为链时，被子等价于一个交变符号矩阵及其对应的角和矩阵。Quilts还推广了由Dedekind数计数的单调布尔函数。被子构成了一个具有许多美丽性质的分布格，它包含了许多经典的和众所周知的子格，如给定秩和基集的拟阵格。虽然一般来说，列举被子是困难的，但我们证明了两个主要的列举结果，即当其中一个偏序集是反链和其中一个偏序集是链时。当一个偏置集是布尔格时，我们还给出了被子的个数的界限。

引用次数: 0

Minimal slopes and bubbling for complex Hessian equations 复Hessian方程的最小斜率和冒泡

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-18 DOI: 10.1016/j.aim.2026.110865

Ved Datar , Ramesh Mete , Jian Song

The existence of smooth solutions to a broad class of complex Hessian equations is related to nonlinear Nakai type criteria on intersection numbers on Kähler manifolds. Such a Nakai criteria can be interpreted as a slope stability condition analogous to the slope stability for Hermitian vector bundles over Kähler manifolds. In the present work, we initiate a program to find canonical solutions to such equations in the unstable case when the Nakai criteria fails. Conjecturally such solutions should arise as limits of natural parabolic flows and should be minimisers of the corresponding moment-map energy functionals. We implement our approach for the J-equation and the deformed Hermitian Yang-Mills equation on surfaces and some examples with symmetry. We prove that there always exist unique canonical solutions to these two equations on Kähler surfaces in the unstable cases. Such canonical solutions with singularities are also shown to be the limits of the corresponding J-flow and the cotangent flow on certain projective bundles. We further present the bubbling phenomena for the J-equation by constructing minimizing sequences of the moment-map energy functionals, whose Gromov-Hausdorff limits are singular algebraic spaces.

广义复Hessian方程光滑解的存在性与Kähler流形上相交数的非线性Nakai型判据有关。这样的Nakai准则可以解释为类似于Kähler流形上厄米向量束的边坡稳定性的边坡稳定性条件。在目前的工作中，我们启动了一个程序，以在Nakai准则失效的不稳定情况下找到这些方程的正则解。据推测，这样的解应该作为自然抛物流的极限出现，并且应该是相应的矩图能量泛函的最小值。我们在曲面上实现了j方程和变形厄米杨-米尔斯方程的方法，并给出了一些具有对称性的例子。证明了在不稳定情况下，这两个方程在Kähler表面上总是存在唯一正则解。这些具有奇异性的正则解也被证明是相应的j流和余切流在某些射影束上的极限。通过构造具有奇异代数空间Gromov-Hausdorff极限的矩映射能量泛函的最小化序列，进一步给出了j方程的冒泡现象。

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引用次数: 0

Nerves of generalized multicategories 广义多范畴神经

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-17 DOI: 10.1016/j.aim.2026.110862

Soichiro Fujii , Stephen Lack

For any category

E

and monad T thereon, we introduce the notion of T-simplicial object in

E

. Any T-category in the sense of Burroni induces a T-simplicial object as its nerve. This nerve construction defines a fully faithful functor from the category

{Cat}_{T} (E)

of T-categories to the category

s_{T} E

of T-simplicial objects, whose essential image is characterized by a simple condition. We show that the category

s_{T} E

is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on

{Cat}_{T} (E)

. We also study enriched limits and colimits in

s_{T} E

and

{Cat}_{T} (E)

, and show that if

E

is locally finitely presentable and T is finitary, then

{Cat}_{T} (E)

is locally finitely presentable as a 2-category and

s_{T} E

is locally finitely presentable as a simplicially-enriched category.

对于任意范畴E及其上的单子T，我们在E中引入单纯T对象的概念。任何Burroni意义上的T范畴都诱导一个单纯T对象作为其神经。这种神经结构定义了一个完全忠实的函子，从t -范畴的范畴CatT(E)到t -简单对象的范畴sTE，其本质象以一个简单条件为特征。我们证明了范畴sTE在简单集合范畴上是丰富的，并且这在CatT(E)上推导出通常的2范畴结构。我们还研究了sTE和CatT(E)中的丰富极限和极限，并证明了如果E是局部有限可表示的，T是有限的，那么CatT(E)是局部有限可表示为2范畴的，而sTE是局部有限可表示为简单丰富范畴的。

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引用次数: 0

Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications 秩一正交群的酉对偶的显式Hilbert空间及其应用

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-18 DOI: 10.1016/j.aim.2026.110868

Christian Arends, Frederik Bang-Jensen, Jan Frahm

We realize all irreducible unitary representations of the group

{SO}_{0} (n + 1, 1)

on explicit Hilbert spaces of vector-valued

L^{2}

-functions on

R^{n} ∖ {0}

. The key ingredient in our construction is an explicit expression for the standard Knapp–Stein intertwining operators between arbitrary principal series representations in terms of the Euclidean Fourier transform on a maximal unipotent subgroup isomorphic to

R^{n}

As an application, we describe the space of Whittaker vectors on all irreducible Casselman–Wallach representations. Moreover, the new realizations of the irreducible unitary representations immediately reveal their decomposition into irreducible representations of a parabolic subgroup, thus providing a simple proof of a recent result of Liu–Oshima–Yu.

我们在Rn∈{0}上的向量值l2函数的显式Hilbert空间上实现了群SO0（n+1,1）的所有不可约酉表示。我们构造的关键成分是任意主级数表示之间的标准Knapp-Stein交织算子在与Rn同构的极大单幂子群上的欧氏傅里叶变换的显式表达式。作为应用，我们描述了所有不可约Casselman-Wallach表示上的Whittaker向量空间。此外，不可约酉表示的新实现直接揭示了它们分解为抛物子群的不可约表示，从而提供了Liu-Oshima-Yu最近结果的简单证明。

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引用次数: 0

A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds 环流形log Landau-Ginzburg镜原始形式的微扰构造

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-24 DOI: 10.1016/j.aim.2026.110867

Kwokwai Chan , Ziming Nikolas Ma , Hao Wen

We introduce the notion of a logarithmic Landau-Ginzburg (log LG) model, which is essentially given by equipping the central degenerate fiber of the family of Landau-Ginzburg (LG) models mirror to a projective toric manifold with a natural log structure. We show that the state space of the mirror log LG model is naturally isomorphic to that of the original toric manifold. Following [32], [33], we give a perturbative construction of primitive forms by studying the deformation theory of such a log LG model, which involves both smoothing of the central degenerate fiber and unfolding of the superpotential. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. The primitive forms and flat coordinates we obtained are computable and closely related to the bulk-deformed Lagrangian Floer superpotential of a projective toric manifold, at least in the semi-Fano case.

我们引入了对数朗道-金兹堡（log LG）模型的概念，该模型本质上是通过将朗道-金兹堡（LG）模型族的中心简并纤维镜像到具有自然对数结构的射影环流形来给出的。我们证明了镜像logg模型的状态空间与原环流形的状态空间自然同构。在[32]，[33]之后，我们通过研究这种log LG模型的变形理论，给出了原始形式的微扰构造，其中涉及到中心简并纤维的平滑和超势的展开。这在普遍展开的基空间上得到一个对数Frobenius流形结构。我们得到的原始形式和平面坐标是可计算的，并且与射影环流形的体积变形拉格朗日花超势密切相关，至少在半法诺情况下是这样。

引用次数: 0

On the superadditivity of anticanonical Iitaka dimension 反正则Iitaka维的超可加性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-13 DOI: 10.1016/j.aim.2026.110860

Marta Benozzo , Iacopo Brivio , Chi-Kang Chang

Given a fibration

f : X \to Y

with normal general fibre

X_{y}

, over a field of any characteristic, we establish the Iitaka-type inequality

κ (X, - K_{X}) \leq κ (X_{y}, - K_{X_{y}}) + κ (Y, - K_{Y})

whenever the Q-linear series

| - K_{X} |_{Q}

has good singularities on

X_{y}

给定具有普通纤维Xy的纤维f:X→Y，在任意特征域上，只要Q-线性级数|−KX|Q在Xy上具有良好的奇异性，我们就建立了iitaka型不等式κ(X,−KX)≤κ(Xy,−KXy)+κ(Y,−KY)。

引用次数: 0

(Co)Minuscule Hecke categories (Co)小类别

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-18 DOI: 10.1016/j.aim.2026.110866

Joseph Baine

We determine the p-Kazhdan–Lusztig bases for antispherical (co)minuscule Hecke categories in all characteristics, and for spherical (co)minuscule Hecke categories in good characteristic. This is achieved using geometric and diagrammatic methods. The 2-Kazhdan–Lusztig bases of antispherical minuscule Hecke categories exhibit extremely pathological behaviour. The notions of p-small resolutions and p-tight elements are introduced and conjecturally explain this behaviour.

我们确定了具有所有特征的反球面（co）极小Hecke范畴和具有良好特征的球面（co）极小Hecke范畴的p-Kazhdan-Lusztig基。这是通过几何和图解方法实现的。反球面极小Hecke类的2-Kazhdan-Lusztig碱基表现出极其病态的行为。引入了p小分辨率和p紧元素的概念，并从理论上解释了这种行为。

引用次数: 0

Tropical thermodynamic formalism 热带热力学形式论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-02-19 DOI: 10.1016/j.aim.2026.110864

Zhiqiang Li , Yiqing Sun

We investigate the zero-temperature large deviation principle for equilibrium states in the context of distance-expanding maps. The logarithmic-type zero-temperature limit in the large deviation principle induces a tropical algebra structure, which motivates our study of the tropical adjoint Bousch operator

since the Bousch operator

L_{A}

is tropical linear and corresponds to the Ruelle operator

R_{A}

We extend tropical functional analysis, define the adjoint operator

as a tropical analog of the adjoint Ruelle operator

R_{A}^{⁎}

, and establish the existence and generic uniqueness of tropical eigendensities of

associated with the maximal eigenvalue. The Aubry set and the Mañé potential, both originating from weak KAM theory, serve as important tools in the representations of tropical eigendensities. With our notion of tropical completeness and our tropical Riesz representation theorem,

can also be seen as a version of the tropical Koopman operator.

我们研究了在距离扩展图的背景下平衡态的零温度大偏差原理。由于Bousch算子LA是热带线性的，对应于Ruelle算子RA，因此大偏差原理中的对数型零温度极限引出了热带代数结构，从而激发了我们对热带伴随Bousch算子的研究。我们扩展了热带泛函分析，将伴随算子定义为伴随Ruelle算子RA的热带模拟，并建立了与极大特征值相关的热带特征的存在性和一般唯一性。Aubry集和Mañé势都源自弱KAM理论，是表征热带特征的重要工具。有了热带完备性的概念和热带Riesz表示定理，也可以看作是热带Koopman算子的一个版本。

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引用次数: 0

The monodromy divisor of an exact algebraic Lagrangian 精确代数拉格朗日的单除数

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-04-01 Epub Date: 2026-02-12 DOI: 10.1016/j.aim.2026.110851

Christopher Dodd

We construct and study a new invariant, the monodromy divisor, attached to a smooth exact Lagrangian subvariety inside the cotangent bundle of a smooth affine variety. This invariant lives in a group of

Q / Z

-valued divisors and reflects subtle arithmetic properties of the Lagrangian.

构造并研究了光滑仿射变体的余切束内光滑精确拉格朗日子变体的一个新的不变量——单除数。这个不变量存在于一组Q/ z值因子中，反映了拉格朗日的微妙算术性质。

引用次数: 0

A simple proof of the Atkin-O'Brien partition Hecke congruence conjecture for powers of 13 13次幂的Atkin-O'Brien分合Hecke同余猜想的一个简单证明

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-04-01 Epub Date: 2026-01-30 DOI: 10.1016/j.aim.2026.110840

Frank Garvan, Zhumagali Shomanov

In 1967, Atkin and O'Brien conjectured congruences for the partition function involving Hecke operators modulo powers of 13. While they proved the conjecture modulo 13 and 13², a proof for all powers of 13 has remained open. In this paper we provide a simple and complete proof of the conjecture.

1967年，Atkin和O'Brien猜想了包含Hecke算子模为13的幂的配分函数的同余。虽然他们证明了模13和模132的猜想，但13的所有幂的证明仍然是开放的。本文给出了这个猜想的一个简单而完整的证明。

引用次数: 0

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