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Relative perversity 相对任性
Pub Date : 2023-08-18 DOI: 10.1090/cams/21
David Hansen, P. Scholze

We define and study a relative perverse t t -structure associated with any finitely presented morphism of schemes f : X S f: Xto S , with relative perversity equivalent to perversity of the restrictions to all geometric fibres of f f . The existence of this t t -structure is closely related to perverse t t -exactness properties of nearby cycles. This t t -structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category P e r v U

我们定义并研究了与任意有限表示的格式f: X→S f: Xto S相关的相对反常t -结构,其相对反常等价于对f的所有几何纤维的限制反常。这种t - t结构的存在与邻近环的反常t - t -精确性质密切相关。这个t - t -结构保留了普遍的局部无环轴,得到了一个阿贝尔范畴Perv UL a (X/S) mathrm {Perv}^{mathrm {ULA}}(X/S),它具有许多在绝对环境中熟悉的性质(例如,诺etherian, artinian,与Verdier对偶相容)。对于S S连通且具有一般点η eta的几何单分支,函子Perv ULA (X/S)→Perv (X η) mathrm {Perv}^{mathrm {ULA}}(X/S)到mathrm {Perv}(X_eta)是精确的、完全忠实的,其本质象在传递到子商时是稳定的。这就产生了一个“良好减少”的概念。
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引用次数: 0
The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers 𝑚=2振幅面体和超单纯形:符号、簇、平铺、欧拉数
Pub Date : 2023-07-10 DOI: 10.1090/cams/23
M. Parisi, M. Sherman-Bennett, Lauren Williams

The hypersimplex Δ k + 1 , n Delta _{k+1,n} is the image of the positive Grassmannian G r k + 1 , n 0 Gr^{geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n 1 n-1 in R n mathbb {R}^n . Meanwhile, the amplituhedron A n , k , 2

《hypersimplex Δ k + 1 , n 三角洲的{k + 1, n}是阳性Grassmannian形象》 G r k + 1 , n ≥ 0 Gr ^ { geq 0} {k + 1, n}下的《地图的时刻。这是a的polytope维度 n−1 n-1 in R n mathbb {R) ^ n。Meanwhile, the amplituhedron A n , k , 2 ( Z ) mathcal {A} {n, k, 2}的投射》(Z)是阳性Grassmannian G r k , n ≥ 0 Gr ^ { geq 0} {k, n}》的Grassmannian G r k ,k + 2 Gr_ {k, k + 2下的a地图 Z ~ 蒂尔德{Z} induced by a阳性矩阵 公元Z∈a t n , k + 2 > 0 Z 在Mat_ {n, k + 2) ^{> 0}。在散射振幅的背景下进行介绍,它不是一个多边形,里面有一个2k的尺寸在Gr r k,k+2 G {k,k+2}。Nevertheless,那里似乎成为非凡的connections这些通过T-duality两个物体之间,美国conjectured由Łukowski,帕里和威廉姆斯(Int)。数学。Res音符。(2023)。在这篇文章中,我们用的是来自东方matroid理论的概念,全正常性,以及高度对称和波利托对偶的几何知识。我们那个节目《不平等卡特房positroid polytopes——images of positroid细胞of G r k + 1 , n ≥ 0 Gr ^ { geq 0} {k + 1, n}下的《地图的时刻——翻译进入签约条件characterizing T-dual Grasstopes——images of positroid细胞of G r k ,n ≥ 0 Gr ^ { geq 0} {k, n}下的 Z ~ 蒂尔德{Z}。更重要的是,我们把振幅移植到钱伯斯,就像超硬化剂可以移植到钱伯斯,由长老会的数字修剪。我们用这些属性来证明
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引用次数: 2
Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules 输入𝐼𝐼的量子子群:𝔩_{𝔑}。局部模块的对称性
Pub Date : 2023-05-01 DOI: 10.1090/cams/19
Cain EDIE-MICHELL

This paper is the first of a pair that aims to classify a large number of the type I I II quantum subgroups of the categories C ( s l r + 1 , k ) mathcal {C}(mathfrak {sl}_{r+1}, k) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I I quantum subgroups of C ( s l r + 1 , k ) mathcal {C}(mathfrak {sl}_{r+1}, k) . We find that the symmetries are all non-exceptional e

本文是对范畴C (sl r+1, k) mathcal {C}(mathfrak {sl}_{r+1}, k)的大量II II型量子子群进行分类的第一个对。本文对C (sl r+1, k) mathcal {C}(mathfrak {sl}_{r+1}, k)的所有已知I I型量子子群的局部模范畴的编织自等价进行了分类。我们发现除了四种情况(直到水平-秩对偶)外,对称性都是非例外的。这些例外情况是轨道C (s 1,2, 16) Rep (Z 2) 0 mathcal {C}(mathfrak {sl}_{2}, 16)^0_{operatorname {Rep}(mathbb {Z}_{2})},C (sl 3,9) Rep (z3) 0 mathcal {C}(mathfrak {sl}_{3}, 9)^0_{operatorname {Rep}(mathbb {Z}_{3})},C (sl 1,8) Rep (Z 4) 0 mathcal {C}(mathfrak {sl}_{4}, 8)^0_{operatorname {Rep}(mathbb {Z}_{4})},和C (s 1,5,5) Rep (z5) 0 mathcal {C}(mathfrak {sl}_{5}, 5)^0_{operatorname {Rep}(mathbb {Z}_{5})}我们在这项工作中开发了几个技术工具。本文给出了
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引用次数: 0
Self-Bäcklund curves in centroaffine geometry and Lamé’s equation Self-Bäcklund仿心几何中的曲线和lam<s:1>方程
Pub Date : 2022-08-24 DOI: 10.1090/cams/9
M. Bialy, Gil Bor, S. Tabachnikov
Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centroaffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centroaffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves.Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam’s problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics.We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.
25年前,U. Pinkall发现Korteweg-de Vries方程可以用仿心几何曲线的演化来实现。从那时起,许多作者解释了KdV的各种性质及其在中心仿射几何方面的推广。特别是,Korteweg-de Vries方程的Bäcklund变换可以看作是中仿射曲线之间的关系。我们的论文关注self-Bäcklund中仿射曲线。我们描述了这些曲线的一般性质,并用椭圆函数对它们进行了详细的描述。我们的工作是F. Wegner在欧几里得几何中对一个类似问题的研究的仿心对应,该问题与Ulam描述在所有位置平衡漂浮的(二维)物体的问题以及自行车运动学有关。我们还考虑了用多边形代替曲线的离散化问题。这与理想多边形上KdV的离散化和交叉比动力学有关。
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引用次数: 3
Tumor growth with nutrients: Regularity and stability 肿瘤生长与营养:规律性和稳定性
Pub Date : 2022-04-15 DOI: 10.1090/cams/20
M. Jacobs, Inwon C. Kim, Jiajun Tong
In this paper, we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the incompressibility of the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics thanks to an unexpected comparison principle. Using the comparison principle, we provide quantitative L 1 L^1 -contraction estimates and establish the C 1 , α C^{1,alpha } -boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient n 0 n_0 either lies entirely above or entirely below the critical value n 0 = 1 n_0=1 , we are able to give a complete characterization of the long-time behavior of the system. When n 0 n_0 is constant, we can even describe the dynamics of the full system in terms of some simpler nutrient-free and parameter-free model problems. These results are in sharp contrast to the observed behavior of the models either with nutrient diffusion or with death rate in tumor cells.
本文研究了一种有营养物的肿瘤生长模型。由于肿瘤细胞的不可压缩性,该模型提出了动态补丁解。我们表明,当营养物质不扩散和细胞不死亡时,由于意想不到的比较原理,肿瘤密度呈现正则化动态。利用比较原理,我们给出了定量的l1l ^1 -收缩估计,并建立了肿瘤斑块的c1, α C^{1, α} -边界正则性。此外,只要初始营养物n 0 n_0完全高于或完全低于临界值n 0=1 n_0=1,我们就能够给出系统长期行为的完整表征。当n0 n_0为常数时,我们甚至可以用一些更简单的无营养和无参数模型问题来描述整个系统的动力学。这些结果与观察到的模型在肿瘤细胞中营养物质扩散或死亡率方面的行为形成鲜明对比。
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引用次数: 9
Ribbon concordance of knots is a partial ordering 结的缎带一致性是一种部分排序
Pub Date : 2022-01-10 DOI: 10.1090/cams/15
I. Agol
In this note we show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon [Math. Ann. 257 (1981), pp. 157–170, Conjecture 1.1]. The proof makes use of representation varieties of the knot groups to S O ( N ) SO(N) and the subquotient relation between them induced by a ribbon concordance.
在这篇笔记中,我们证明了丝带的一致性在结的集合上形成了偏序,回答了Gordon[数学]的一个问题。Ann. 257 (1981), pp 157-170,猜想1.1]。证明利用代表品种的结组S O (N) (N)和subquotient引起一个丝带和谐之间的关系。
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引用次数: 13
Smooth local rigidity for hyperbolic toral automorphisms 双曲型全自同构的光滑局部刚性
Pub Date : 2021-11-02 DOI: 10.1090/cams/22
B. Kalinin, V. Sadovskaya, Zhenqi Wang

We study the regularity of a conjugacy H H between a hyperbolic toral automorphism A A and its smooth perturbation f f . We show that if H H is weakly differentiable then it is C 1 + H"older C^{1+text {H"older}} and, if A A is also weakly irreducible, then H H is C C^infty . As a part of the proof, we establish results of independent interest on Hölder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, w

研究了双曲总自同构a a与其光滑摄动f之间共轭H H的正则性。我们证明了如果H H是弱可微的,那么它是c1 + Hölder C^{1+text Hölder{,如果A A也是弱不可约的,那么H H是C∞C^ }}infty。作为证明的一部分,我们建立了关于双曲系统上线性环间可测共轭的Hölder连续性的独立有趣的结果。作为一个推论,我们改进了先前局部刚度结果中C∞C^ infty共轭的正则性。
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引用次数: 1
Asymptotic vanishing of syzygies of algebraic varieties 代数变种合子的渐近消失
Pub Date : 2021-10-24 DOI: 10.1090/cams/7
Jinhyung Park
The purpose of this paper is to prove Ein–Lazarsfeld’s conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein–Lazarsfeld’s asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.
本文的目的是证明Ein-Lazarsfeld关于代数变量合子渐近消失的猜想。该结果与Ein-Lazarsfeld的渐近不消失定理一起,描述了随着线束正性的增长,线束梯度截面环在射影变化上的最小自由分辨率渐近行为的全貌。之前,Raicu将问题简化为三个射影空间积的情况,我们在这里解决了这种情况。
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引用次数: 3
Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere 高半加性Grothendieck-Witt理论与𝐾(1)局部球
Pub Date : 2021-09-24 DOI: 10.1090/cams/17
Shachar Carmeli, Allen Yuan

We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the K ( 1 ) K(1) -local sphere S K ( 1 ) mathbb {S}_{K(1)} at the prime 2 2 , in particular realizing the non- 2 2 -adic rational element 1 + ε π 0 S K ( 1 ) 1+varepsilon in pi _0mathbb {S}_{K(1)} as a “semiadditive cardinality.” As a fur

我们发展了Grothendieck-Witt理论的高半加性版本。然后,我们将该理论应用于有限域的情况下,研究了K(1) K(1) -局部球S K(1) mathbb S_K{(1)在素数22处}的高半加性结构。特别是实现了非- 2 2进有理数元素1+ ε∈π 0 S K(1) 1+ {}varepsiloninpi _0 mathbb S_K{(}1)作为{“半加性的cardinality”。作为进一步的应用,我们计算并阐明了π 0 S_K(1) }pi _0 mathbb S_K{(}1)中的一些幂运算。{}
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引用次数: 5
Maximum spread of graphs and bipartite graphs 图与二部图的最大扩展
Pub Date : 2021-09-07 DOI: 10.1090/cams/14
Jane Breen, Alexander W. N. Riasanovsky, Michael Tait, John C. Urschel

Given any graph G G , the spread of G G is the maximum difference between any two eigenvalues of the adjacency matrix of G G . In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n n , the n n -vertex graph G G that maximizes spread is the join of a clique and an independent set, with 2 n / 3 lfloor 2n/3 rfloor and n

给定任意图G G, G G的扩展是G G的邻接矩阵的任意两个特征值之差的最大值。本文解决了Gregory, Hershkowitz和Kirkland关于图的传播的一对20年猜想。第一个表述了对于所有正整数n n,扩展最大的n n顶点图G G是一个团和一个独立集合的连接,其顶点分别为⌊2n/3⌋lfloor 2n/3 rfloor和⌊n/3 lceil n/3 rceil。利用图极限理论和数值分析的技术,我们证明了所有n n足够大的情况下的这一说法。作为中间步骤,我们证明了l2 [0,1] mathscr L{^2[0,1]上Hilbert空间中算子族的一个类似结果。第二个猜想表明,对于任意固定的m≤n 2/4 m }leq n^2/4,如果G在所有n个n顶点、m个边的图上展开最大,则G是二部的。我们证明了这个猜想的渐近版本。此外,我们构造了一个无限族的反例,这表明我们的渐近解对低阶误差项是紧绷的。
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引用次数: 5
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Communications of the American Mathematical Society
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