. 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.
{"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}
. 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.
{"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}
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.
{"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}
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.
{"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}
Antonio Avil'es, Gonzalo Mart'inez-Cervantes, A. R. Zoca, P. Tradacete
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.
{"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}
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.
{"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}
A. Grebénnikov, A. Sagdeev, A. Semchankau, A. Vasilevskii
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.
{"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}
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.
{"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}
. 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.
{"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}
. 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.
{"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}
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