{"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":"43 10","pages":""},"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":"4 1","pages":""},"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":"54 2","pages":""},"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":"19 1","pages":""},"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":"9 11","pages":"0"},"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":"33 15‐16","pages":"0"},"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":"30 1","pages":"0"},"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":"11 4","pages":"0"},"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}
Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor–Wood inequality is for circle bundles over surfaces, whereas the Mather–Thurston theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern–Weil theory and other smooth obstructions such as the Bott classes and the Godbillion–Vey invariant. Contradiction is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g., $operatorname{PSL}(2,mathbb{R})$ versus $operatorname{U}(1)$ in the former case and $C^1$ versus $C^2$ in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather–Thurston cobordism to a semi-$s$-cobordism (ssc), and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one. The motivation is to lay mathematical foundations for a physical program. The philosophy is that living in the IR we cannot expect to know, for a given bundle, if it has curvature or is flat, because we cannot resolve the fine scale topology which may be present in the base, introduced by an ssc, nor minute symmetry violating distortions of the fiber. Small scale, UV, “distortions“ of the base topology and structure group allow flat connections to simulate curvature at larger scales. The goal is to find a duality under which curvature terms, such as Maxwell's $F wedge F^ast$ and Hilbert's $int R, dmathrm{vol}$ are replaced by an action which measures such “distortions“. In this view, curvature results from renormalizing a discrete, group theoretic, structure.
米尔诺、伍德、马瑟和瑟斯顿的经典结果在令人惊讶的地方产生了平连接。米尔诺-伍德不等式适用于曲面上的圆束，而马瑟-瑟斯顿定理是关于将一般流形束与允许平面连接的流形束结合在一起的。这个惊喜来自于与chen - weil理论中的障碍物和其他平滑障碍物(如Bott类和Godbillion-Vey不变量)的近距离接触。避免了矛盾，因为正结果的结构组比障碍所需的结构组大，例如，前一种情况下$operatorname{PSL}(2，mathbb{R})$与$operatorname{U}(1)$，后一种情况下$C^1$与$C^2$。本文增加了两种类型的控制来加强积极的结果:在许多情况下，我们能够(1)将Mather-Thurston协同改进为半s -协同(ssc)，并且(2)提供关于过渡函数如何以及在多大程度上必须从初始的小结构群漂移到更大的结构群的细节。其动机是为物理程序奠定数学基础。哲学是，生活在红外中，我们不能期望知道，对于一个给定的束，它是否有曲率或平坦，因为我们不能解决可能存在于基础中的精细尺度拓扑结构，由ssc引入，也不能解决纤维的微小对称性违反扭曲。小尺度，UV，基础拓扑和结构组的“扭曲”允许平面连接在更大尺度上模拟曲率。目标是找到一种对偶性，在这种对偶性下，曲率项，如麦克斯韦的$F wedge F^ast$和希尔伯特的$int R, d mathm {vol}$被测量这种“扭曲”的作用所取代。在这个观点中，曲率是由一个离散的、群论的结构的重整而来的。
{"title":"Controlled Mather–Thurston theorems","authors":"Michael Freedman","doi":"10.4171/emss/63","DOIUrl":"https://doi.org/10.4171/emss/63","url":null,"abstract":"Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor–Wood inequality is for circle bundles over surfaces, whereas the Mather–Thurston theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern–Weil theory and other smooth obstructions such as the Bott classes and the Godbillion–Vey invariant. Contradiction is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g., $operatorname{PSL}(2,mathbb{R})$ versus $operatorname{U}(1)$ in the former case and $C^1$ versus $C^2$ in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather–Thurston cobordism to a semi-$s$-cobordism (ssc), and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one. The motivation is to lay mathematical foundations for a physical program. The philosophy is that living in the IR we cannot expect to know, for a given bundle, if it has curvature or is flat, because we cannot resolve the fine scale topology which may be present in the base, introduced by an ssc, nor minute symmetry violating distortions of the fiber. Small scale, UV, “distortions“ of the base topology and structure group allow flat connections to simulate curvature at larger scales. The goal is to find a duality under which curvature terms, such as Maxwell's $F wedge F^ast$ and Hilbert's $int R, dmathrm{vol}$ are replaced by an action which measures such “distortions“. In this view, curvature results from renormalizing a discrete, group theoretic, structure.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":"BME-30 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135218337","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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