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Stabilization distance between surfaces 面间稳定距离
L’Enseignement Mathématique Pub Date : 2019-08-19 DOI: 10.4171/LEM/65-3/4-4
Allison N. Miller, Mark Powell
{"title":"Stabilization distance between surfaces","authors":"Allison N. Miller, Mark Powell","doi":"10.4171/LEM/65-3/4-4","DOIUrl":"https://doi.org/10.4171/LEM/65-3/4-4","url":null,"abstract":"Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer $m$ we find a pair of 2-knots in the 4-sphere whose stabilization distance equals $m$. Next, using a generalized stabilization distance that counts connected sum with arbitrary 2-knots as distance zero, for every nonnegative integer $m$ we exhibit a knot $J_m$ in the 3-sphere with two slice discs in the 4-ball whose generalized stabilization distance equals $m$. We show this using homology of cyclic covers. Finally, we use metabelian twisted homology to show that for each $m$ there exists a knot and pair of slice discs with generalized stabilization distance at least $m$, with the additional property that abelian invariants associated to cyclic covering spaces coincide. This detects different choices of slicing discs corresponding to a fixed metabolising link on a Seifert surface.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115083968","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}
引用次数: 10
On extensions of algebraic groups 代数群的扩展
L’Enseignement Mathématique Pub Date : 2019-07-29 DOI: 10.4171/lem/65-3/4-5
M. Florence, G. Arteche
{"title":"On extensions of algebraic groups","authors":"M. Florence, G. Arteche","doi":"10.4171/lem/65-3/4-5","DOIUrl":"https://doi.org/10.4171/lem/65-3/4-5","url":null,"abstract":"We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of extensions of $G$ by the center $Z$ of $H$. The proof should be easily generalizable to other contexts. We also study the subset of classes of split extensions and give a quick application by proving a finiteness result on these sets over a finite field.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114211794","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}
引用次数: 0
Commission Internationale de l’Enseignement Mathématique. Discussion document twenty-fifth ICMI study 国际宇航组织委员会。讨论文件25 ICMI研究
L’Enseignement Mathématique Pub Date : 2019-07-23 DOI: 10.4171/LEM/64-3/4-14
H. Borko, D. Potari
{"title":"Commission Internationale de l’Enseignement Mathématique. Discussion document twenty-fifth ICMI study","authors":"H. Borko, D. Potari","doi":"10.4171/LEM/64-3/4-14","DOIUrl":"https://doi.org/10.4171/LEM/64-3/4-14","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128480499","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}
引用次数: 0
The critical Ising model on amenable graphs of exponential growth 可服从指数增长图的临界伊辛模型
L’Enseignement Mathématique Pub Date : 2019-07-23 DOI: 10.4171/LEM/64-3/4-4
Aran Raoufi
{"title":"The critical Ising model on amenable graphs of exponential growth","authors":"Aran Raoufi","doi":"10.4171/LEM/64-3/4-4","DOIUrl":"https://doi.org/10.4171/LEM/64-3/4-4","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128798368","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}
引用次数: 1
Commission Internationale de l’Enseignement Mathématique. ICME 14 in 2020 in Shanghai 国际数学教育委员会。2020年上海ICME 14展
L’Enseignement Mathématique Pub Date : 2019-07-23 DOI: 10.4171/lem/64-3/4-13
{"title":"Commission Internationale de l’Enseignement Mathématique. ICME 14 in 2020 in Shanghai","authors":"","doi":"10.4171/lem/64-3/4-13","DOIUrl":"https://doi.org/10.4171/lem/64-3/4-13","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122342498","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}
引用次数: 0
Basic matrix perturbation theory 基本矩阵摄动理论
L’Enseignement Mathématique Pub Date : 2019-07-23 DOI: 10.4171/LEM/64-3/4-1
B. Texier
{"title":"Basic matrix perturbation theory","authors":"B. Texier","doi":"10.4171/LEM/64-3/4-1","DOIUrl":"https://doi.org/10.4171/LEM/64-3/4-1","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114252272","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}
引用次数: 6
Siegel modular forms of weight 13 and the Leech lattice 权重13和Leech格的西格尔模形式
L’Enseignement Mathématique Pub Date : 2019-07-20 DOI: 10.4171/lem/1021
Gaëtan Chenevier, O. Taibi
{"title":"Siegel modular forms of weight 13 and the Leech lattice","authors":"Gaëtan Chenevier, O. Taibi","doi":"10.4171/lem/1021","DOIUrl":"https://doi.org/10.4171/lem/1021","url":null,"abstract":"For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the ${rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${rm Leech}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${rm Sp}_{2g}(mathbb{Z})$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard ${rm L}$-functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular forms of weight $13$ for ${rm Sp}_{2n}(mathbb{Z})$, for any $ngeq 1$.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122424494","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}
引用次数: 2
Equivariant discretizations of diffusions, random walks, and harmonic functions 扩散、随机游走和调和函数的等变离散化
L’Enseignement Mathématique Pub Date : 2019-06-27 DOI: 10.4171/lem/1011
W. Ballmann, Panagiotis Polymerakis
{"title":"Equivariant discretizations of diffusions, random walks, and harmonic functions","authors":"W. Ballmann, Panagiotis Polymerakis","doi":"10.4171/lem/1011","DOIUrl":"https://doi.org/10.4171/lem/1011","url":null,"abstract":"For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127018188","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}
引用次数: 4
Complément to the Thurston 3D-geometrization picture 对瑟斯顿3d几何图的赞美
L’Enseignement Mathématique Pub Date : 2019-06-26 DOI: 10.4171/lem/1034
Alice Kwon, D. Sullivan
{"title":"Complément to the Thurston 3D-geometrization picture","authors":"Alice Kwon, D. Sullivan","doi":"10.4171/lem/1034","DOIUrl":"https://doi.org/10.4171/lem/1034","url":null,"abstract":"Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four particular Thurston geometries\"(see Introduction and Figure 1). These geometric components have finite volume for the hyperbolic geometries (the H labeled vertices). They also have finite volume for each of the two geometries appearing as Seifert fibrations (the S labeled vertices). The remaining pieces (the I labeled vertices) have Euclidean geometries of linear volume growth. Then these vertex geometries are combined topologically to recover the original manifold. This, by cutting off the toroidal ends and then gluing the torus boundaries by affine mappings (indicated by the labeled edges in Figure 1). The point of this work is to make the affine gluing respect an interpretation of the metric geometry in terms of a new notion of `` regional Lie generated geometry\". The vertex regions use four geometries in Lie form combined in the overlap edge regions via affine geometry. The Theorem solves, using Geometrization, a 45 year old question/approach to the Poincar'{e} Conjecture. This was described in a '76 Princeton Math dept. preprint and finally documented in the 1983 reference by Thurston and the second author.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134027361","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}
引用次数: 1
Spines for amoebas of rational curves 有理曲线变形虫的棘
L’Enseignement Mathématique Pub Date : 2019-06-11 DOI: 10.4171/LEM/65-3/4-3
G. Mikhalkin, Johannes Rau
{"title":"Spines for amoebas of rational curves","authors":"G. Mikhalkin, Johannes Rau","doi":"10.4171/LEM/65-3/4-3","DOIUrl":"https://doi.org/10.4171/LEM/65-3/4-3","url":null,"abstract":"To every rational complex curve $C subset (mathbf{C}^times)^n$ we associate a rational tropical curve $Gamma subset mathbf{R}^n$ so that the amoeba $mathcal{A}(C) subset mathbf{R}^n$ of $C$ is within a bounded distance from $Gamma$. In accordance with the terminology introduced by Passare and Rullgard, we call $Gamma$ the spine of $mathcal{A}(C)$. We use spines to describe tropical limits of sequences of rational complex curves.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121948587","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}
引用次数: 2
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