In this paper, we construct unipotent representations for the real orthagonal groups and the metaplectic groups in the sense of Vogan. Our construction is based on quantum induction which involves the compositions of even number of theta correspondences. In particular, our results imply that there are irreducible unitary representations attached to each special nilpotent orbit.
{"title":"Unipotent Representations, Theta Correspondences, and Quantum Induction","authors":"Hongyu He","doi":"10.1090/memo/1496","DOIUrl":"https://doi.org/10.1090/memo/1496","url":null,"abstract":"In this paper, we construct unipotent representations for the real orthagonal groups and the metaplectic groups in the sense of Vogan. Our construction is based on quantum induction which involves the compositions of even number of theta correspondences. In particular, our results imply that there are irreducible unitary representations attached to each special nilpotent orbit.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141699406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct p p -adic L L -functions associated with p p -refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in p p -adic families, and does not require any small slope or non-criticality assumptions on the p p -refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group, and a smoothness theorem for certain eigenvarieties at critically refined points.
在一个温和的假设条件下，我们构建了与任何全实数域上 p p 精化同调尖顶希尔伯特模形式相关的 p p -adic L L 函数。我们的构造是典型的，在 p p -adic 族中自然变化，并且不需要任何小斜率或 p p - 精化的非临界假设。主要的新成分是一个从过敛同调到相关伽罗瓦群上局部解析分布空间的规范映射的自洽定义，以及在临界细化点上某些特征变量的平滑性定理。
{"title":"On 𝑝-adic 𝐿-functions for Hilbert modular forms","authors":"John Bergdall, David Hansen","doi":"10.1090/memo/1489","DOIUrl":"https://doi.org/10.1090/memo/1489","url":null,"abstract":"We construct \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-functions associated with \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic families, and does not require any small slope or non-criticality assumptions on the \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group, and a smoothness theorem for certain eigenvarieties at critically refined points.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141408679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this memoir, for quasi-split classical groups over pp-adic fields, we determine the LL-packets consisting of simple supercuspidal representations and their corresponding LL-parameters, under the assumption that pp is not equal to 22. The key is an explicit computation of characters of simple supercuspidal representations and the endoscopic character relation, which is a characterization of the local Langlands correspondence for quasi-split classical groups.
在这篇回忆录中，对于 p p -adic 场上的准分裂经典群，我们在 p p 不等于 2 2 的假设下，确定了由简单超pidal 表示及其相应 L L -参数组成的 L L -包。关键在于对简单超pidal 表示的特征和内视特征关系的明确计算，而内视特征关系是对准分裂经典群的局部朗兰兹对应关系的描述。
{"title":"Simple Supercuspidal 𝐿-Packets of Quasi-Split Classical Groups","authors":"Masao Oi","doi":"10.1090/memo/1483","DOIUrl":"https://doi.org/10.1090/memo/1483","url":null,"abstract":"<p>In this memoir, for quasi-split classical groups over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic fields, we determine the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-packets consisting of simple supercuspidal representations and their corresponding <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-parameters, under the assumption that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is not equal to <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>. The key is an explicit computation of characters of simple supercuspidal representations and the endoscopic character relation, which is a characterization of the local Langlands correspondence for quasi-split classical groups.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141034417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Brandon Doherty, Krzysztof Kapulkin, Zachery Lindsey, Christian Sattler
We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DK-equivalences.
{"title":"Cubical Models of (∞,1)-Categories","authors":"Brandon Doherty, Krzysztof Kapulkin, Zachery Lindsey, Christian Sattler","doi":"10.1090/memo/1484","DOIUrl":"https://doi.org/10.1090/memo/1484","url":null,"abstract":"We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DK-equivalences.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141033393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We continue the study of multiple cluster structures in the rings of regular functions on G L n GL_n , S L n SL_n and M a t n Mat_n that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group G mathcal {G} corresponds to a cluster structure in O ( G ) mathcal {O}(mathcal {G}) . Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of A n A_n type, which includes all the previously known examples. Namely, we subdivide all possible A n A_n type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on S L n SL_n compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of S L n SL_n equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications.
我们继续研究 G L n GL_n 、S L n SL_n 和 M a t n Mat_n 上与泊松里结构和泊松均质结构相容的正则函数环中的多重簇结构。根据我们最初的猜想，半简单复群 G mathcal {G} 上 Poisson-Lie 结构的 Belavin-Drinfeld 分类中的每一类都对应于 O ( G ) mathcal {O}(mathcal {G}) 中的一个簇结构。在这里，我们为 A n A_n 类型的贝拉文-德林费尔德（BD）数据的一个大子集证明了这一猜想，其中包括所有之前已知的例子。也就是说，我们将所有可能的 A n A_n 类型 BD 数据细分为定向和非定向两种。我们进一步挑选出满足特定组合条件（我们称之为非周期性）的 BD 数据，并证明对于任何此类定向 BD 数据，都存在与相应的泊松李括号兼容的规则簇结构。事实上，我们将非周期性条件扩展到了成对的定向 BD 数据，并证明了一个更一般的结果，即在 S L n SL_n 上存在一个与泊松括号相容的正则簇结构，该泊松括号与 S L n SL_n 的两个副本的左右作用同质，而这两个副本配备了两个不同的泊松-李括号。类似的结果也适用于非周期性的无取向 BD 数据，但相应的规则簇结构的分析更为复杂，在此不再赘述。如果不满足非周期性条件，则必须用广义簇结构来替代兼容簇结构。我们将在今后的出版物中讨论这些情况。
{"title":"A Plethora of Cluster Structures on 𝐺𝐿_{𝑛}","authors":"M. Gekhtman, M. Shapiro, A. Vainshtein","doi":"10.1090/memo/1486","DOIUrl":"https://doi.org/10.1090/memo/1486","url":null,"abstract":"We continue the study of multiple cluster structures in the rings of regular functions on \u0000\u0000 \u0000 \u0000 G\u0000 \u0000 L\u0000 n\u0000 \u0000 \u0000 GL_n\u0000 \u0000\u0000, \u0000\u0000 \u0000 \u0000 S\u0000 \u0000 L\u0000 n\u0000 \u0000 \u0000 SL_n\u0000 \u0000\u0000 and \u0000\u0000 \u0000 \u0000 M\u0000 a\u0000 \u0000 t\u0000 n\u0000 \u0000 \u0000 Mat_n\u0000 \u0000\u0000 that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group \u0000\u0000 \u0000 \u0000 G\u0000 \u0000 mathcal {G}\u0000 \u0000\u0000 corresponds to a cluster structure in \u0000\u0000 \u0000 \u0000 \u0000 O\u0000 \u0000 (\u0000 \u0000 G\u0000 \u0000 )\u0000 \u0000 mathcal {O}(mathcal {G})\u0000 \u0000\u0000. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of \u0000\u0000 \u0000 \u0000 A\u0000 n\u0000 \u0000 A_n\u0000 \u0000\u0000 type, which includes all the previously known examples. Namely, we subdivide all possible \u0000\u0000 \u0000 \u0000 A\u0000 n\u0000 \u0000 A_n\u0000 \u0000\u0000 type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on \u0000\u0000 \u0000 \u0000 S\u0000 \u0000 L\u0000 n\u0000 \u0000 \u0000 SL_n\u0000 \u0000\u0000 compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of \u0000\u0000 \u0000 \u0000 S\u0000 \u0000 L\u0000 n\u0000 \u0000 \u0000 SL_n\u0000 \u0000\u0000 equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141036038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study nonlinear stability of the 3D plane Couette flow (y,0,0)(y,0,0) at high Reynolds number Re{Re} in a finite channel T×[−1,1]×Tmathbb {T}times [-1,1]times mathbb {T}. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity v0v_0 satisfies
本文研究了三维平面 Couette 流 ( y , 0 , 0 ) (y,0,0) 在高雷诺数 R e {Re} 下在有限通道 T × [ - 1 , 1 ] × T mathbb {T} times [-1,1]times mathbb {T} 中的非线性稳定性。众所周知，平面库埃特流在任何雷诺数下都是线性稳定的。然而，在高雷诺数下，对于微小但有限的扰动，它可能变得非线性不稳定并过渡到湍流。这就是所谓的 Sommerfeld 悖论。解决这一悖论的方法之一是研究过渡阈值问题，即多少扰动会导致流动不稳定以及扰动与雷诺数的关系。这项工作表明，如果初速度 v 0 v_0 满足 ‖ v 0 - ( y , 0 , 0 ) ‖ H 2 ≤ c 0 R e - 1 |v_0-(y,0、0)|_{H^2}le c_0{Re}^{-1} 对于与 R e Re 无关的某个 c 0 > 0 c_0>0，则三维纳维-斯托克斯方程的解在时间上是全局的，不会偏离 L ∞ L^infty 意义上的 Couette 流，并且由于混合增强的耗散效应，在 t ≫ R e 1 3 tgg Re^{frac 13} 时迅速收敛到条纹解。这一结果证实了查普曼通过渐近分析得到的阈值结果（JFM 2002）。证明的最关键部分是围绕流动的全线性化三维纳维-斯托克斯系统的解析量估计（ V ( y , z ) , 0 , 0 ) (V(y,z), 0,0) ，其中 V ( y , z ) V(y,z) 是 y y 的小扰动（但与 R e Re 无关）。
{"title":"Transition Threshold for the 3D Couette Flow in a Finite Channel","authors":"Qi Chen, Dongyi Wei, Zhifei Zhang","doi":"10.1090/memo/1478","DOIUrl":"https://doi.org/10.1090/memo/1478","url":null,"abstract":"<p>In this paper, we study nonlinear stability of the 3D plane Couette flow <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis y comma 0 comma 0 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(y,0,0)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> at high Reynolds number <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R e\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>R</mml:mi>\u0000 <mml:mi>e</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{Re}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in a finite channel <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper T times left-bracket negative 1 comma 1 right-bracket times double-struck upper T\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">T</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">T</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {T}times [-1,1]times mathbb {T}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v 0\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>v</mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">v_0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> satisfies <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-vertical-bar v 0 minus left-parenthesis y comma 0 comma 0 right-parent","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140777519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the first-order theories of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if H H is elementarily embedded in a torsion-free hyperbolic group G G , then G G is a tower over H H relative to H H (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous. The converse to Sela and Perin’s results just mentioned is true. This follows from the solution to Tarski’s problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper. We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fraïssé’s method, we associate to every torsion-free hyperbolic group H H a unique homogeneous countable group M {mathcal {M}} in which any hyperbolic group H ′ H’ elementarily equivalent to H H has an elementary embedding. In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group H H are H H -limit groups.
本文专门讨论无扭双曲群的一阶理论。本文的目的之一是回顾一些结果，并提供精确、正确的陈述和定义，以及一些证明和新结果。一个关键概念是塔（Sela）或 NTQ 系统（Kharlampovich-Myasnikov）。我们阐述并证明了一个新的一般定理，它统一了文献中的几个结果：元素等价的无扭双曲群具有同构核（塞拉）；如果 H H 元素嵌入无扭双曲群 G G 中，那么相对于 H H，G G 是 H H 上的塔（佩林）；自由群（佩林-斯克利诺斯，乌尔德-侯辛），以及更一般的原型和自由群的自由积，都是同质的。刚才提到的塞拉和佩林结果的反面是真的。我们提出了许多例子和反例，并证明了一些新的模型理论结果。我们描述了无扭双曲群中的质模型和无元素群中的最小模型。利用弗雷泽的方法，我们给每个无扭双曲群 H H 关联了一个唯一的同质可数群 M {mathcal {M}}，在这个群中，任何与 H H 元素等价的双曲群 H ′ H' 都有一个基本嵌入。在附录中，我们完整地证明了塞拉提出的事实，即无扭双曲群 H H 上的塔是 H H 极限群。
{"title":"Towers and the First-order Theories of Hyperbolic Groups","authors":"Vincent Guirardel, Gilbert Levitt, R. Sklinos","doi":"10.1090/memo/1477","DOIUrl":"https://doi.org/10.1090/memo/1477","url":null,"abstract":"This paper is devoted to the first-order theories of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results.\u0000\u0000A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly.\u0000\u0000We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if \u0000\u0000 \u0000 H\u0000 H\u0000 \u0000\u0000 is elementarily embedded in a torsion-free hyperbolic group \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000, then \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000 is a tower over \u0000\u0000 \u0000 H\u0000 H\u0000 \u0000\u0000 relative to \u0000\u0000 \u0000 H\u0000 H\u0000 \u0000\u0000 (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous.\u0000\u0000The converse to Sela and Perin’s results just mentioned is true. This follows from the solution to Tarski’s problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper.\u0000\u0000We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fraïssé’s method, we associate to every torsion-free hyperbolic group \u0000\u0000 \u0000 H\u0000 H\u0000 \u0000\u0000 a unique homogeneous countable group \u0000\u0000 \u0000 \u0000 \u0000 M\u0000 \u0000 \u0000 {mathcal {M}}\u0000 \u0000\u0000 in which any hyperbolic group \u0000\u0000 \u0000 \u0000 H\u0000 ′\u0000 \u0000 H’\u0000 \u0000\u0000 elementarily equivalent to \u0000\u0000 \u0000 H\u0000 H\u0000 \u0000\u0000 has an elementary embedding.\u0000\u0000In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group \u0000\u0000 \u0000 H\u0000 H\u0000 \u0000\u0000 are \u0000\u0000 \u0000 H\u0000 H\u0000 \u0000\u0000-limit groups.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140776116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eva Elduque, C. Geske, Moisés Herradón Cueto, L. Maxim, Botong Wang
Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let UU be a smooth connected complex algebraic variety and let f:U→C∗fcolon Uto mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C∗mathbb {C}^* by ff gives rise to an infinite cyclic cover UfU^f of UU. The action of the deck group
受超曲面奇点胚芽的米尔诺纤维上的极限混合霍奇结构的启发，我们在光滑连通复代数簇的亚历山大模块的扭转部分上构造了一个自然的混合霍奇结构。更确切地说，设 U U 是光滑连通复代数簇，设 f : U → C ∗ fcolon Uto mathbb {C}^* 是基本群中诱导外变形的代数映射。通过 f f 对 C ∗ mathbb {C}^* 的普遍盖的拉回，可以得到 U U 的无限循环盖 U f U^f。甲板群 Z 在 U f U^f 上的作用在 H ∗ ( U f ; Q ) H_*(U^f;mathbb {Q}) 上诱导出一个 Q [ t ± 1 ] mathbb {Q}[t^{pm 1}] 模块结构。我们证明亚历山大模块 H ∗ ( U f ; Q ) H_*(U^f;mathbb {Q}) 的扭转部分 A ∗ ( U f ; Q ) A_*(U^f;mathbb {Q}) 带有典型的 Q mathbb {Q} -mixed Hodge 结构。 -混合霍奇结构。我们还证明了覆盖映射 U f → U U^f to U 在亚历山大模块的扭转部分上诱导了混合霍奇结构变形。作为应用，我们研究了 A ∗ ( U f ; Q ) A_*(U^f;mathbb {Q}) 的半简单性，以及所构造的混合霍奇结构的可能权重。最后，在 f : U → C ∗ fcolon Uto mathbb {C}^* 是适当的情况下，我们证明了
{"title":"Mixed Hodge Structures on Alexander Modules","authors":"Eva Elduque, C. Geske, Moisés Herradón Cueto, L. Maxim, Botong Wang","doi":"10.1090/memo/1479","DOIUrl":"https://doi.org/10.1090/memo/1479","url":null,"abstract":"<p>Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\">\u0000 <mml:semantics>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">U</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a smooth connected complex algebraic variety and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper U right-arrow double-struck upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:<!-- : --></mml:mo>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">fcolon Uto mathbb {C}^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {C}^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> by <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> gives rise to an infinite cyclic cover <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U Superscript f\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mi>f</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">U^f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\">\u0000 <mml:semantics>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">U</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The action of the deck group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z\">\u0000 <mm","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140795898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known “Benjamin-Feir resonances”. We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.
{"title":"Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity","authors":"Filippo Giuliani, R. Feola","doi":"10.1090/memo/1471","DOIUrl":"https://doi.org/10.1090/memo/1471","url":null,"abstract":"We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations.\u0000\u0000The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known “Benjamin-Feir resonances”. We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140085786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I present a proof of Kirchberg’s classification theorem: two separable, nuclear, O∞mathcal O_infty-stable C∗C^ast-algebras are stably isomorphic if and only if they are ideal-related KKKK-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.
我提出了基希贝格分类定理的一个证明：当且仅当两个可分离的、核的、O ∞ mathcal O_infty -stable C ∗ C^ast -gealbras 是理想相关的 K KK -equivalent 时，它们是稳定同构的。特别是，这为基希贝格-菲利普斯定理提供了一个更基本的证明，该定理在论文中被单独列出，以增加这一重要特例的可读性。
{"title":"Classification of 𝒪_{∞}-Stable 𝒞*-Algebras","authors":"James Gabe","doi":"10.1090/memo/1461","DOIUrl":"https://doi.org/10.1090/memo/1461","url":null,"abstract":"<p>I present a proof of Kirchberg’s classification theorem: two separable, nuclear, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal O_infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-stable <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^ast</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-algebras are stably isomorphic if and only if they are ideal-related <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K upper K\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>K</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">KK</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139391793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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