Advances in Mathematics最新文献

英文中文

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110038

Jonas Stelzig

We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting theory naturally accomodates higher operations involving double primitives. As applications, we obtain various refinements of the homotopy groups, sensitive to the complex structure. Under a simple connectedness assumption, one obtains minimal models which are unique up to isomorphism and allow for explicit computations of the new invariants.

我们为复流形上的形式建立了自由的、大等级的双微分代数模型，它尊重一个强的准同构概念，并与共轭对称性兼容。这回答了苏利文的一个问题。由此产生的理论自然可以容纳涉及双重基元的高级运算。作为应用，我们得到了对复杂结构敏感的同调群的各种细化。在一个简单的连通性假设下，我们可以得到最小模型，这些模型在同构时是唯一的，并且可以明确计算新的不变式。

引用次数: 0

On the isometric version of Whitney's strong embedding theorem 关于惠特尼强嵌入定理的等距版本

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110040

Wentao Cao , László Székelyhidi Jr.

We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any n-dimensional smooth compact manifold admits infinitely many global isometric embeddings into 2n-dimensional Euclidean space, of Hölder class

C^{1, θ}

with

θ < 1 / 3

for

n = 2

and

θ < {(n + 2)}^{- 1}

for

n \geq 3

. The proof is performed by Nash-Kuiper's convex integration construction and applying the gluing technique of the authors on short embeddings with small amplitude.

我们证明了惠特尼强嵌入定理的一个版本，即在纳什-柯伊伯 h 原则的一般环境中的等距嵌入。更确切地说，我们证明了任何 n 维光滑紧凑流形都有无穷多个全局等距嵌入到 2n 维欧几里得空间中，其中荷尔德类 C1,θ 在 n=2 时为 θ<1/3，在 n≥3 时为 θ<(n+2)-1。证明是通过纳什-柯伊伯的凸积分构造和作者对小振幅短嵌入的胶合技术进行的。

引用次数: 0

Orlov's theorem for dg-algebras 二元数组的奥洛夫定理

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110035

Michael K. Brown , Prashanth Sridhar

A landmark theorem of Orlov relates the singularity category of a graded Gorenstein algebra to the derived category of the associated noncommutative projective scheme. We generalize this theorem to the setting of differential graded algebras. As an application, we obtain new cases of the Lattice Conjecture in noncommutative Hodge theory.

奥洛夫（Orlov）的一个具有里程碑意义的定理将分级戈伦斯坦代数的奇点范畴与相关非交换投影方案的派生范畴联系起来。我们将这一定理推广到微分级数代数的环境中。作为应用，我们得到了非交换霍奇理论中的格点猜想的新案例。

引用次数: 0

Linearization of Virasoro symmetries associated with semisimple Frobenius manifolds 与半简单弗罗贝尼斯流形相关的维拉索罗对称性线性化

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110046

Si-Qi Liu , Zhe Wang , Youjin Zhang

For any semisimple Frobenius manifold, we prove that a tau-symmetric bihamiltonian deformation of its Principal Hierarchy admits an infinite family of linearizable Virasoro symmetries if and only if all the central invariants of the corresponding deformation of the bihamiltonian structure are equal to

\frac{1}{24}

. As an important application of this result, we prove that the Dubrovin-Zhang hierarchy associated with the semisimple Frobenius manifold possesses a bihamiltonian structure which can be represented in terms of differential polynomials.

对于任何半简单弗罗本尼乌斯流形，我们证明了其主层次结构的头对称双哈密顿变形具有无限的可线性化维拉索罗对称性族，当且仅当相应的双哈密顿结构变形的所有中心不变式等于 124。作为这一结果的重要应用，我们证明了与半简单弗罗本尼乌斯流形相关的杜布罗文-张层次结构具有可以用微分多项式表示的双哈密顿结构。

引用次数: 0

Prime orthogeodesics, concave cores and families of identities on hyperbolic surfaces 双曲面上的质点正交、凹核和同素族

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110026

Ara Basmajian , Hugo Parlier , Ser Peow Tan

We prove and explore a family of identities relating lengths of curves and orthogeodesics of hyperbolic surfaces. These identities hold over a large space of metrics including ones with hyperbolic cone points, and in particular, show how to extend a result of the first author to surfaces with cusps. One of the main ingredients in the approach is a partition of the set of orthogeodesics into sets depending on their dynamical behavior, which can be understood geometrically by relating them to geodesics on orbifold surfaces. These orbifold surfaces turn out to be exactly on the boundary of the space in which the underlying identity holds.

我们证明并探讨了与双曲面的曲线长度和正交曲线有关的一系列等式。这些等式在很大的度量空间中都成立，包括具有双曲锥点的度量空间，特别是展示了如何将第一作者的一个结果扩展到具有尖点的曲面。该方法的主要内容之一是根据正交线的动力学行为将其划分为若干集合，通过将它们与球面上的正交线联系起来，可以从几何学角度理解这些集合。事实证明，这些球面恰好位于基本特征成立的空间边界上。

引用次数: 0

Critical well-posedness for the 2D Peskin problem with general tension 具有一般张力的二维佩斯金问题的临界好求解性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110047

Eduardo García-Juárez , Susanna V. Haziot

In this paper, we study the two dimensional Peskin problem with general elasticity law. Specifically, we prove global regularity for small perturbations, in suitable critical spaces, of the circle solution, possibly containing corners. For such initial data we prove asymptotic stability in the sense that as

t \to \infty

, the solution converges to a translated and rotated disk.

本文研究了具有一般弹性规律的二维佩斯金问题。具体来说，我们证明了在合适的临界空间中，圆解（可能包含角）的小扰动的全局正则性。对于这样的初始数据，我们证明了渐近稳定性，即随着 t→∞，解收敛于平移和旋转的圆盘。

引用次数: 0

Paralinearization and extended lifespan for solutions of the α-SQG sharp front equation α-SQG锐前方程解的副线性化和扩展寿命

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110034

Massimiliano Berti , Scipio Cuccagna , Francisco Gancedo , Stefano Scrobogna

In this paper we paralinearize the contour dynamics equation for sharp-fronts of α-SQG, for any

α \in (0, 1) \cup (1, 2)

, close to a circular vortex. This turns out to be a quasi-linear Hamiltonian PDE. The key idea relies on a novel desingularization of the Hamiltonian vector field which is a convolution integral operator with nonlinear singular kernel. After deriving the asymptotic expansion of the linear frequencies of oscillations at the vortex disk and verifying the absence of three wave interactions, we prove that, in the most singular cases

α \in (1, 2)

, any initial vortex patch which is ε-close to the disk exists for a time interval of size at least

\sim ε^{- 2}

. This quadratic lifespan result relies on a paradifferential Birkhoff normal form reduction and exploits cancellations arising from the Hamiltonian nature of the equation. This is the first normal form long time existence result of sharp fronts.

在本文中，我们对α∈(0,1)∪(1,2)的α-SQG尖锋的等值线动力学方程进行了拟线性化处理，使其接近于一个圆形旋涡。这原来是一个准线性哈密顿 PDE。哈密顿矢量场是一个具有非线性奇异内核的卷积积分算子。在推导出涡旋盘线性振荡频率的渐近展开并验证了不存在三波相互作用之后，我们证明了在最奇异的情况下α∈(1,2)，任何ε-接近盘的初始涡旋斑块都会存在至少∼ε-2大小的时间间隔。这个二次生命期结果依赖于范差伯克霍夫正态式还原，并利用了方程的哈密顿性质所产生的抵消。这是尖锐前沿的第一个正态长时间存在结果。

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引用次数: 0

Recurrence rates for shifts of finite type 有限类型转移的递推率

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110039

Demi Allen , Simon Baker , Balázs Bárány

Let

Σ_{A}

be a topologically mixing shift of finite type, let

σ : Σ_{A} \to Σ_{A}

be the usual left-shift, and let μ be the Gibbs measure for a Hölder continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system

(Σ_{A}, σ)

that hold μ-almost surely. In particular, given a function

ψ : N \to N

we are interested in the following set

R_{ψ} = {i \in Σ_{A} : i_{n + 1} \dots i_{n + ψ (n) + 1} = i_{1} \dots i_{ψ (n)} for infinitely many n \in N} .

We provide sufficient conditions for

μ (R_{ψ}) = 1

and sufficient conditions for

μ (R_{ψ}) = 0

. As a corollary of these results, we discover a new critical threshold where the measure of

R_{ψ}

transitions from zero to one. This threshold was previously unknown even in the special case of a non-uniform Bernoulli measure defined on the full shift. The proofs of our results combine ideas from Probability Theory and Thermodynamic Formalism. In our final section we apply our results to the study of dynamics on self-similar sets.

设ΣA 是有限类型的拓扑混合位移，σ:ΣA→ΣA 是通常的左移，μ 是霍尔德连续势的吉布斯度量，而霍尔德连续势不是与常数同源的。在本文中，我们将研究μ几乎肯定成立的动力系统（ΣA,σ）的递推率。特别是，给定函数ψ:N→N，我们感兴趣的是以下集合Rψ={i∈ΣA:in+1...in+ψ(n)+1=i1...iψ(n)for infinitely manyn∈N}。我们提供了 μ(Rψ)=1 的充分条件和 μ(Rψ)=0 的充分条件。作为这些结果的推论，我们发现了一个新的临界阈值，在这个阈值上，Rψ的度量从零过渡到一。即使是在定义于全移的非均匀伯努利度量的特殊情况下，这个临界值以前也是未知的。我们的结果证明结合了概率论和热力学形式主义的思想。在最后一节，我们将我们的结果应用于自相似集上的动力学研究。

引用次数: 0

Empirical approximation of the gaussian distribution in Rd Rd 中高斯分布的经验近似值

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110041

Daniel Bartl , Shahar Mendelson

<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be independent copies of the standard gaussian random vector in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We show that there is an absolute constant <em>c</em> such that for any <span><math><mi>A</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, with probability at least <span><math><mn>1</mn><mo>−</mo><mn>2</mn><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mi>c</mi><mi>Δ</mi><mi>m</mi><mo>)</mo></math></span>, for every <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>,<span><span><span><math><munder><mi>sup</mi><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></munder><mo>⁡</mo><mrow><mo>|</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub><mo>−</mo><mi>P</mi><mo>(</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>)</mo><mo>|</mo></mrow><mo>≤</mo><mi>Δ</mi><mo>+</mo><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo><msqrt><mrow><mi>Δ</mi></mrow></msqrt><mo>.</mo></math></span></span></span> Here <span><math><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is the variance of <span><math><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub></math></span> and <span><math><mi>Δ</mi><mo>≥</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is determined by an unexpected complexity parameter of <em>A</em> that captures the set's geometry (Talagrand's <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> functional). The bound, the probability estimate, and the value of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are all (almost) optimal.</div><div>We use this fact to show that if <span><math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the random matrix that has <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> as its rows, then the structure of <span><math><mi>Γ</mi><mo>(</mo><mi>A</mi><mo>)</

假设 G1,...,Gm 是 Rd 中标准高斯随机向量的独立副本。我们证明，对于任何 A⊂Sd-1，对于每 t∈R，supx∈A|1m∑i=1m1{〈Gi,x〉≤t}-P(〈G,x〉≤t)|≤Δ+σ(t)Δ，概率至少为 1-2exp(-cΔm)。这里，σ(t) 是 1{〈G,x〉≤t}的方差，Δ≥Δ0，其中Δ0 由 A 的意外复杂度参数决定，该参数捕捉了集合的几何形状（塔拉格兰德的 γ1 函数）。我们利用这一事实证明，如果Γ=∑i=1m〈Gi,x〉ei 是以 G1,...,Gm 为行的随机矩阵，那么Γ(A)={Γx:x∈A} 的结构远比之前预期的要严格和规范。

引用次数: 0

Three dimensional topological quantum field theory from Uq(gl(1|1)) and U(1|1) Chern–Simons theory 来自 Uq(gl(1|1)) 和 U(1|1) Chern-Simons 理论的三维拓扑量子场论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110044

Nathan Geer, Matthew B. Young

We introduce an unrolled quantization

U_{q}^{E} (gl (1 | 1))

of the complex Lie superalgebra

gl (1 | 1)

and use its categories of weight modules to construct and study new three dimensional non-semisimple topological quantum field theories. These theories are defined on categories of cobordisms which are decorated by ribbon graphs and cohomology classes and take values in categories of graded super vector spaces. Computations in these theories are enabled by a detailed study of the representation theory of

U_{q}^{E} (gl (1 | 1))

. We argue that by restricting to subcategories of integral weight modules we obtain topological quantum field theories which are mathematical models of Chern–Simons theories with gauge supergroups

psl (1 | 1)

and

U (1 | 1)

coupled to background flat

C^{\times}

-connections, as studied in the physics literature by Rozansky–Saleur and Mikhaylov. In particular, we match Verlinde formulae and mapping class group actions on state spaces of non-generic tori with results in the physics literature. We also obtain explicit descriptions of state spaces of generic surfaces, including their graded dimensions, which go beyond results in the physics literature.

我们引入了复李超代数 gl(1|1) 的未卷积量子化 UqE(gl(1|1))，并利用其权重模块类别来构建和研究新的三维非半简单拓扑量子场论。这些理论定义在由带状图和同调类装饰的共线范畴上，并在分级超向量空间范畴中取值。通过对 UqE(gl(1|1))表示理论的详细研究，可以在这些理论中进行计算。我们认为，通过对积分权重模块子类的限制，我们可以得到拓扑量子场论，它是具有规超群 psl(1|1) 和 U(1|1) 并与背景平面 C× 连接耦合的切尔-西蒙斯理论的数学模型，正如罗赞斯基-萨利尔和米哈伊洛夫在物理学文献中所研究的那样。特别是，我们将非一般环的状态空间上的韦林德公式和映射类群作用与物理学文献中的结果相匹配。我们还获得了对一般曲面状态空间的明确描述，包括它们的梯度维数，这些都超越了物理学文献中的结果。

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引用次数: 0

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