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From octonions to composition superalgebras via tensor categories 通过张量范畴从八元数到复合超代数
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-05-13 DOI: 10.4171/rmi/1408
Alberto Daza-Garcia, A. Elduque, Umut Sayın
{"title":"From octonions to composition superalgebras via tensor categories","authors":"Alberto Daza-Garcia, A. Elduque, Umut Sayın","doi":"10.4171/rmi/1408","DOIUrl":"https://doi.org/10.4171/rmi/1408","url":null,"abstract":". The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal Magic Square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42792284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Non-isotopic transverse tori in Engel manifolds Engel流形中的非同位素横向tori
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-05-10 DOI: 10.4171/rmi/1413
M. Kegel
{"title":"Non-isotopic transverse tori in Engel manifolds","authors":"M. Kegel","doi":"10.4171/rmi/1413","DOIUrl":"https://doi.org/10.4171/rmi/1413","url":null,"abstract":". In every Engel manifold we construct an infinite family of pairwise non-isotopic transverse tori that are all smoothly isotopic. To distinguish the transverse tori in the family we introduce a homological invariant of transverse tori that is similar to the self-linking number for transverse knots in contact 3-manifolds. Analogous results are presented for Legendrian tori in even contact 4-manifolds.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46471623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Tangent ray foliations and their associated outer billiards 切线射线叶理及其相关的外部台球
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-05-09 DOI: 10.4171/rmi/1434
Yamile Godoy, Michael C. Harrison, M. Salvai
{"title":"Tangent ray foliations and their associated outer billiards","authors":"Yamile Godoy, Michael C. Harrison, M. Salvai","doi":"10.4171/rmi/1434","DOIUrl":"https://doi.org/10.4171/rmi/1434","url":null,"abstract":"Let $v$ be a unit vector field on a complete, umbilic (but not totally geodesic) hypersurface $N$ in a space form; for example on the unit sphere $S^{2k-1} subset mathbb{R}^{2k}$, or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on $v$ for the rays with initial velocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find and explore relationships among these vector fields, geodesic vector fields, and contact structures on $N$. When the rays corresponding to each of $pm v$ foliate $U$, $v$ induces an outer billiard map whose billiard table is $U$. We describe the unit vector fields on $N$ whose associated outer billiard map is volume preserving. Also we study a particular example in detail, namely, when $N simeq mathbb{R}^3$ is a horosphere of the four-dimensional hyperbolic space and $v$ is the unit vector field on $N$ obtained by normalizing the stereographic projection of a Hopf vector field on $S^{3}$. In the corresponding outer billiard map we find explicit periodic orbits, unbounded orbits, and bounded nonperiodic orbits. We conclude with several questions regarding the topology and geometry of bifoliating vector fields and the dynamics of their associated outer billiards.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46396385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Horizontally quasiconvex envelope in the Heisenberg group Heisenberg群中的水平拟凸包络
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-05-04 DOI: 10.4171/rmi/1417
Antoni Kijowski, Qing Liu, Xiaodan Zhou
{"title":"Horizontally quasiconvex envelope in the Heisenberg group","authors":"Antoni Kijowski, Qing Liu, Xiaodan Zhou","doi":"10.4171/rmi/1417","DOIUrl":"https://doi.org/10.4171/rmi/1417","url":null,"abstract":"This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex for short) envelope of a given continuous function in the Heisenberg group. We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problems for the nonlocal Hamilton-Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45615346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Infinite dimensional spaces in the set of strongly norm-attaining Lipschitz maps 强范数达到的Lipschitz映射集合中的无限维空间
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-04-26 DOI: 10.4171/rmi/1425
Antonio Avil'es, Gonzalo Mart'inez-Cervantes, A. R. Zoca, P. Tradacete
{"title":"Infinite dimensional spaces in the set of strongly norm-attaining Lipschitz maps","authors":"Antonio Avil'es, Gonzalo Mart'inez-Cervantes, A. R. Zoca, P. Tradacete","doi":"10.4171/rmi/1425","DOIUrl":"https://doi.org/10.4171/rmi/1425","url":null,"abstract":"We prove that if $M$ is an infinite complete metric space then the set of strongly norm-attaining Lipschitz functions $SA(M)$ contains a linear subspace isomorphic to $c_0$. This solves an open question posed by V. Kadets and O. Rold'an.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48998473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Real Kaehler submanifolds in codimension up to four 四维以下的实Kaehler子流形
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-04-24 DOI: 10.4171/rmi/1427
S. Chión, M. Dajczer
{"title":"Real Kaehler submanifolds in codimension up to four","authors":"S. Chión, M. Dajczer","doi":"10.4171/rmi/1427","DOIUrl":"https://doi.org/10.4171/rmi/1427","url":null,"abstract":"Let $fcolon M^{2n}tomathbb{R}^{2n+4}$ be an isometric immersion of a Kaehler manifold of complex dimension $ngeq 5$ into Euclidean space with complex rank at least $5$ everywhere. Our main result is that, along each connected component of an open dense subset of $M^{2n}$, either $f$ is holomorphic in $mathbb{R}^{2n+4}congmathbb{C}^{n+2}$ or it is in a unique way a composition $f=Fcirc h$ of isometric immersions. In the latter case, we have that $hcolon M^{2n}to N^{2n+2}$ is holomorphic and $Fcolon N^{2n+2}tomathbb{R}^{2n+4}$ belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifold in codimension two. Moreover, the submanifold $F$ is minimal if and only if $f$ is minimal.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46136827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On the sequence $n! bmod p$ 序列$n! bmod p $
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-04-03 DOI: 10.4171/rmi/1422
A. Grebénnikov, A. Sagdeev, A. Semchankau, A. Vasilevskii
{"title":"On the sequence $n! bmod p$","authors":"A. Grebénnikov, A. Sagdeev, A. Semchankau, A. Vasilevskii","doi":"10.4171/rmi/1422","DOIUrl":"https://doi.org/10.4171/rmi/1422","url":null,"abstract":"We prove, that the sequence $1!, 2!, 3!, dots$ produces at least $(sqrt{2} + o(1))sqrt{p}$ distinct residues modulo prime $p$. Moreover, factorials on an interval $mathcal{I} subseteq {0, 1, dots, p - 1}$ of length $N>p^{7/8 + varepsilon}$ produce at least $(1 + o(1))sqrt{p}$ distinct residues modulo $p$. As a corollary, we prove that every non-zero residue class can be expressed as a product of seven factorials $n_1! dots n_7!$ modulo $p$, where $n_i = O(p^{6/7+varepsilon})$ for all $i=1,dots,7$, which provides a polynomial improvement upon the preceding results.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48490006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the topology of leaves of singular Riemannian foliations 奇异黎曼叶理的叶拓扑
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-03-29 DOI: 10.4171/rmi/1435
M. Radeschi, E. K. Samani
{"title":"On the topology of leaves of singular Riemannian foliations","authors":"M. Radeschi, E. K. Samani","doi":"10.4171/rmi/1435","DOIUrl":"https://doi.org/10.4171/rmi/1435","url":null,"abstract":"In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation $(M,fol)$. If $M$ is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If $M$ has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44542036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Carleson perturbations for the regularity problem 正则问题的Carleson摄动
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-03-15 DOI: 10.4171/rmi/1401
Zanbing Dai, J. Feneuil, S. Mayboroda
{"title":"Carleson perturbations for the regularity problem","authors":"Zanbing Dai, J. Feneuil, S. Mayboroda","doi":"10.4171/rmi/1401","DOIUrl":"https://doi.org/10.4171/rmi/1401","url":null,"abstract":". We prove that the solvability of the regularity problem in L q ( ∂ Ω) is stable under Carleson perturbations. If the perturbation is small, then the solvability is preserved in the same L q , and if the perturbation is large, the regularity problem is solvable in L r for some other r ∈ (1 , ∞ ). We extend an earlier result from Kenig and Pipher to very general unbounded domains, possibly with lower dimensional boundaries as in the theory developed by Guy David and the last two authors. To be precise, we only need the domain to have non-tangential access to its Ahlfors regular boundary, together with a notion of gradient on the boundary.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43146697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Non-transversal multilinear duality and joints 非横向多重线性对偶与关节
IF 1.2 2区 数学
Revista Matematica Iberoamericana Pub Date : 2022-03-04 DOI: 10.4171/rmi/1402
A. Carbery, M. Tang
{"title":"Non-transversal multilinear duality and joints","authors":"A. Carbery, M. Tang","doi":"10.4171/rmi/1402","DOIUrl":"https://doi.org/10.4171/rmi/1402","url":null,"abstract":". We develop a framework for a duality theory for general multilin- ear operators which extends that for transversal multilinear operators which has been established in [4]. We apply it to the setting of joints and multijoints, and obtain a “factorisation” theorem which provides an analogue in the discrete setting of results of Bourgain and Guth ([7] and [2]) from the Euclidean setting.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42368682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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