A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich features. This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces.
{"title":"Translation surfaces: Dynamics and Hodge theory","authors":"Simion Filip","doi":"10.4171/emss/78","DOIUrl":"https://doi.org/10.4171/emss/78","url":null,"abstract":"A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich features. This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141021438","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}
{"title":"On the Whitehead nightmare and some related topics","authors":"Valentin Poénaru","doi":"10.4171/emss/73","DOIUrl":"https://doi.org/10.4171/emss/73","url":null,"abstract":"","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139272950","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}
{"title":"Rational homotopy via Sullivan models and enriched Lie algebras","authors":"Y. Félix, S. Halperin","doi":"10.4171/emss/67","DOIUrl":"https://doi.org/10.4171/emss/67","url":null,"abstract":"","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139272299","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}
{"title":"String topology in three flavors","authors":"Florian Naef, Manuel Rivera, Nathalie Wahl","doi":"10.4171/emss/72","DOIUrl":"https://doi.org/10.4171/emss/72","url":null,"abstract":"","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139274963","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}
{"title":"Singularity formation in the incompressible Euler equation in finite and infinite time","authors":"Theodore D. Drivas, T. Elgindi","doi":"10.4171/emss/66","DOIUrl":"https://doi.org/10.4171/emss/66","url":null,"abstract":"","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139275110","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 survey article we discuss certain homotopy coherent enhancements of the coalgebra structure on cellular chains defined by an approximation to the diagonal. Over the rational numbers, $C_infty$-coalgebra structures control the $mathbb Q$-complete homotopy theory of spaces, and over the integers, $E_infty$-coalgebras provide an appropriate setting to model the full homotopy category. Effective constructions of these structures, the focus of this work, carry geometric and combinatorial information which has found applications in various fields including deformation theory, higher category theory, and condensed matter physics.
{"title":"The diagonal of cellular spaces and effective algebro-homotopical constructions","authors":"Anibal M. Medina-Mardones","doi":"10.4171/emss/71","DOIUrl":"https://doi.org/10.4171/emss/71","url":null,"abstract":"In this survey article we discuss certain homotopy coherent enhancements of the coalgebra structure on cellular chains defined by an approximation to the diagonal. Over the rational numbers, $C_infty$-coalgebra structures control the $mathbb Q$-complete homotopy theory of spaces, and over the integers, $E_infty$-coalgebras provide an appropriate setting to model the full homotopy category. Effective constructions of these structures, the focus of this work, carry geometric and combinatorial information which has found applications in various fields including deformation theory, higher category theory, and condensed matter physics.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227560","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}
Guruswami and Xing introduced subspace designs in 2013 to give the first construction of positive rate rank metric codes list-decodable beyond half the distance. In this paper we provide bounds involving the parameters of a subspace design, showing they are tight via explicit constructions. We point out a connection with sum-rank metric codes, dealing with optimal codes and minimal codes with respect to this metric. Applications to two-intersection sets with respect to hyperplanes, two-weight codes, cutting blocking sets and lossless dimension expanders are also provided.
{"title":"On subspace designs","authors":"Paolo Santonastaso, Ferdinando Zullo","doi":"10.4171/emss/77","DOIUrl":"https://doi.org/10.4171/emss/77","url":null,"abstract":"Guruswami and Xing introduced subspace designs in 2013 to give the first construction of positive rate rank metric codes list-decodable beyond half the distance. In this paper we provide bounds involving the parameters of a subspace design, showing they are tight via explicit constructions. We point out a connection with sum-rank metric codes, dealing with optimal codes and minimal codes with respect to this metric. Applications to two-intersection sets with respect to hyperplanes, two-weight codes, cutting blocking sets and lossless dimension expanders are also provided.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432403","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 show that the additive category of chain complexes parametrized by a finite simplicial complex $K$ forms a category with chain duality. This fact, never fully proven in the original reference (Ranicki, 1992), is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincaré duality to global Poincaré duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes.
{"title":"Chain duality for categories over complexes","authors":"James F. Davis, Carmen Rovi","doi":"10.4171/emss/65","DOIUrl":"https://doi.org/10.4171/emss/65","url":null,"abstract":"We show that the additive category of chain complexes parametrized by a finite simplicial complex $K$ forms a category with chain duality. This fact, never fully proven in the original reference (Ranicki, 1992), is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincaré duality to global Poincaré duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135265680","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 survey, we first give a summary of characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or $C^{1+alpha}$) in terms of Beurling–Ahlfors extension, Douady–Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism. Then we provide a brief account of characterizations of the elements of the tangent spaces of these sub-Teichmüller spaces at the base point in the universal Teichmüuller space. We also investigate the regularity of the Beurling–Ahlfors extension $BA(h)$ of a $C^{1+mathrm{Zygmund}}$ orientation-preserving diffeomorphism $h$ of the real line, and show that the Beltrami coefficient $mu (BA(h))(x+iy)$ vanishes as $O(y)$ uniformly on $x$ near the boundary of the upper half plane if and only if $h$ is $C^{1+mathrmtarted with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.{Lipschitz}}$. Finally, we show this criterion is indeed true when $h$ is started with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.
在本文中，我们首先从Beurling-Ahlfors扩展、doudy - earle扩展和Thurston的保向圆同胚的地震表示三个方面总结了不同规律圆同胚(拟对称、对称或$C^{1+alpha}$)的特征。然后，我们简要地描述了这些子teichm ller空间的切空间的元素在泛teichm uller空间的基点处的特征。我们还研究了实线的$C^{1+mathrm{Zygmund}}$保向微分同态$h$的Beurling-Ahlfors扩展$BA(h)$的规律性，并证明了当且仅当$h$为$C^{1+mathrmtarted with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.{Lipschitz}}$时，Beltrami系数$mu (BA(h))(x+iy)$在靠近上半平面边界的$x$上均匀消失为$O(y)$。最后，我们证明了当$h$以保持方向的圆同胚的提升映射的实线的任何同胚开始时，这个判据确实成立。
{"title":"Characterizations of circle homeomorphisms of different regularities in the universal Teichmüller space","authors":"Jun Hu","doi":"10.4171/emss/60","DOIUrl":"https://doi.org/10.4171/emss/60","url":null,"abstract":"In this survey, we first give a summary of characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or $C^{1+alpha}$) in terms of Beurling–Ahlfors extension, Douady–Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism. Then we provide a brief account of characterizations of the elements of the tangent spaces of these sub-Teichmüller spaces at the base point in the universal Teichmüuller space. We also investigate the regularity of the Beurling–Ahlfors extension $BA(h)$ of a $C^{1+mathrm{Zygmund}}$ orientation-preserving diffeomorphism $h$ of the real line, and show that the Beltrami coefficient $mu (BA(h))(x+iy)$ vanishes as $O(y)$ uniformly on $x$ near the boundary of the upper half plane if and only if $h$ is $C^{1+mathrmtarted with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.{Lipschitz}}$. Finally, we show this criterion is indeed true when $h$ is started with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135266286","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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