{"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}
{"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}
{"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}
{"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}
{"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":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":"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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