We define and study a relative perverse tt-structure associated with any finitely presented morphism of schemes f:X→Sf: Xto S, with relative perversity equivalent to perversity of the restrictions to all geometric fibres of ff. The existence of this tt-structure is closely related to perverse tt-exactness properties of nearby cycles. This tt-structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category PervU
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。
{"title":"Relative perversity","authors":"David Hansen, P. Scholze","doi":"10.1090/cams/21","DOIUrl":"https://doi.org/10.1090/cams/21","url":null,"abstract":"<p>We define and study a relative perverse <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-structure associated with any finitely presented morphism of schemes <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper X right-arrow upper S\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi>S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">f: Xto S</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, with relative perversity equivalent to perversity of the restrictions to all geometric fibres of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The existence of this <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-structure is closely related to perverse <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-exactness properties of nearby cycles. This <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal e normal r normal v Superscript normal upper U normal upper L normal upper A Baseline left-parenthesis upper X slash upper S right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">P</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">e</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">r</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">v</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">U</mml:mi>\u0000 <mml:mi ","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"47 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120992600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The hypersimplex Δk+1,nDelta _{k+1,n} is the image of the positive Grassmannian Grk+1,n≥0Gr^{geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n−1n-1 in Rnmathbb {R}^n. Meanwhile, the amplituhedron An,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
{"title":"The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers","authors":"M. Parisi, M. Sherman-Bennett, Lauren Williams","doi":"10.1090/cams/23","DOIUrl":"https://doi.org/10.1090/cams/23","url":null,"abstract":"<p>The hypersimplex <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta Subscript k plus 1 comma n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">Delta _{k+1,n}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the image of the positive Grassmannian <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G r Subscript k plus 1 comma n Superscript greater-than-or-equal-to 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:msubsup>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Gr^{geq 0}_{k+1,n}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> under the moment map. It is a polytope of dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n minus 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n-1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript n\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Meanwhile, the amplituhedron <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A Subscript n comma k comma 2 Baseline left-parenthesis upper Z right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126165091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is the first of a pair that aims to classify a large number of the type IIII quantum subgroups of the categories C(slr+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 II quantum subgroups of C(slr+1,k)mathcal {C}(mathfrak {sl}_{r+1}, k). We find that the symmetries are all non-exceptional e
{"title":"Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules","authors":"Cain EDIE-MICHELL","doi":"10.1090/cams/19","DOIUrl":"https://doi.org/10.1090/cams/19","url":null,"abstract":"<p>This paper is the first of a pair that aims to classify a large number of the type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I upper I\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mi>I</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">II</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> quantum subgroups of the categories <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript r plus 1 Baseline comma k right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">s</mml:mi>\u0000 <mml:mi mathvariant=\"fraktur\">l</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {C}(mathfrak {sl}_{r+1}, k)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this work we classify the braided auto-equivalences of the categories of local modules for all known type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\u0000 <mml:semantics>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> quantum subgroups of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript r plus 1 Baseline comma k right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">s</mml:mi>\u0000 <mml:mi mathvariant=\"fraktur\">l</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {C}(mathfrak {sl}_{r+1}, k)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We find that the symmetries are all non-exceptional e","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127256285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Self-Bäcklund curves in centroaffine geometry and Lamé’s equation","authors":"M. Bialy, Gil Bor, S. Tabachnikov","doi":"10.1090/cams/9","DOIUrl":"https://doi.org/10.1090/cams/9","url":null,"abstract":"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.\u0000\u0000Our 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.\u0000\u0000We 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.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116757699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Tumor growth with nutrients: Regularity and stability","authors":"M. Jacobs, Inwon C. Kim, Jiajun Tong","doi":"10.1090/cams/20","DOIUrl":"https://doi.org/10.1090/cams/20","url":null,"abstract":"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 \u0000\u0000 \u0000 \u0000 L\u0000 1\u0000 \u0000 L^1\u0000 \u0000\u0000-contraction estimates and establish the \u0000\u0000 \u0000 \u0000 C\u0000 \u0000 1\u0000 ,\u0000 α\u0000 \u0000 \u0000 C^{1,alpha }\u0000 \u0000\u0000-boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient \u0000\u0000 \u0000 \u0000 n\u0000 0\u0000 \u0000 n_0\u0000 \u0000\u0000 either lies entirely above or entirely below the critical value \u0000\u0000 \u0000 \u0000 \u0000 n\u0000 0\u0000 \u0000 =\u0000 1\u0000 \u0000 n_0=1\u0000 \u0000\u0000, we are able to give a complete characterization of the long-time behavior of the system. When \u0000\u0000 \u0000 \u0000 n\u0000 0\u0000 \u0000 n_0\u0000 \u0000\u0000 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.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126947183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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。
{"title":"Ribbon concordance of knots is a partial ordering","authors":"I. Agol","doi":"10.1090/cams/15","DOIUrl":"https://doi.org/10.1090/cams/15","url":null,"abstract":"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 \u0000\u0000 \u0000 \u0000 S\u0000 O\u0000 (\u0000 N\u0000 )\u0000 \u0000 SO(N)\u0000 \u0000\u0000 and the subquotient relation between them induced by a ribbon concordance.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"64 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120919741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the regularity of a conjugacy HH between a hyperbolic toral automorphism AA and its smooth perturbation ff. We show that if HH is weakly differentiable then it is C1+H"olderC^{1+text {H"older}} and, if AA is also weakly irreducible, then HH 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
{"title":"Smooth local rigidity for hyperbolic toral automorphisms","authors":"B. Kalinin, V. Sadovskaya, Zhenqi Wang","doi":"10.1090/cams/22","DOIUrl":"https://doi.org/10.1090/cams/22","url":null,"abstract":"<p>We study the regularity of a conjugacy <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> between a hyperbolic toral automorphism <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and its smooth perturbation <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We show that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is weakly differentiable then it is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript 1 plus upper H reverse-solidus quotation-mark older\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mtext>H\"older</mml:mtext>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^{1+text {H\"older}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and, if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is also weakly irreducible, then <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. 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","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129741365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Asymptotic vanishing of syzygies of algebraic varieties","authors":"Jinhyung Park","doi":"10.1090/cams/7","DOIUrl":"https://doi.org/10.1090/cams/7","url":null,"abstract":"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.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114231278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 SK(1)mathbb {S}_{K(1)} at the prime 22, in particular realizing the non-22-adic rational element 1+ε∈π0SK(1)1+varepsilon in pi _0mathbb {S}_{K(1)} as a “semiadditive cardinality.” As a fur
{"title":"Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere","authors":"Shachar Carmeli, Allen Yuan","doi":"10.1090/cams/17","DOIUrl":"https://doi.org/10.1090/cams/17","url":null,"abstract":"<p>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 <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K left-parenthesis 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K(1)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-local sphere <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {S}_{K(1)}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> at the prime <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, in particular realizing the non-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic rational element <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 plus epsilon element-of pi 0 double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">1+varepsilon in pi _0mathbb {S}_{K(1)}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> as a “semiadditive cardinality.” As a fur","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134144285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jane Breen, Alexander W. N. Riasanovsky, Michael Tait, John C. Urschel
Given any graph GG, the spread of GG is the maximum difference between any two eigenvalues of the adjacency matrix of GG. 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 nn, the nn-vertex graph GG that maximizes spread is the join of a clique and an independent set, with ⌊2n/3⌋lfloor 2n/3 rfloor and ⌈n
{"title":"Maximum spread of graphs and bipartite graphs","authors":"Jane Breen, Alexander W. N. Riasanovsky, Michael Tait, John C. Urschel","doi":"10.1090/cams/14","DOIUrl":"https://doi.org/10.1090/cams/14","url":null,"abstract":"<p>Given any graph <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the spread of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the maximum difference between any two eigenvalues of the adjacency matrix of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. 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 <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-vertex graph <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that maximizes spread is the join of a clique and an independent set, with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left floor 2 n slash 3 right floor\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⌊<!-- ⌊ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⌋<!-- ⌋ --></mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">lfloor 2n/3 rfloor</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left ceiling n slash 3 right ceiling\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⌈<!-- ⌈ --></mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125845519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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