Forum of Mathematics Sigma最新文献

{"title":"Persistence and the Sheaf-Function Correspondence","authors":"Nicolas Berkouk","doi":"10.1017/fms.2023.115","DOIUrl":"https://doi.org/10.1017/fms.2023.115","url":null,"abstract":"<p>The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold <span>M</span> with the Grothendieck group of constructible sheaves on <span>M</span>. When <span>M</span> is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215022810799-0847:S2050509423001159:S2050509423001159_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {k}$</span></span></img></span></span>-vector spaces on <span>M</span>. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138717222","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}

{"title":"E-Polynomials of Generic -Character Varieties: Branched Case","authors":"Cheng Shu","doi":"10.1017/fms.2023.119","DOIUrl":"https://doi.org/10.1017/fms.2023.119","url":null,"abstract":"<p>For any branched double covering of compact Riemann surfaces, we consider the associated character varieties that are unitary in the global sense, which we call <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_nrtimes !&lt;!sigma {&gt;}$</span></span></img></span></span>-character varieties. We restrict the monodromies around the branch points to generic semi-simple conjugacy classes contained in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_nsigma $</span></span></img></span></span> and compute the E-polynomials of these character varieties using the character table of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_n(q)rtimes !&lt;!sigma !&gt;!$</span></span></img></span></span>. The result is expressed as the inner product of certain symmetric functions associated to the wreath product <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$(mathbb {Z}/2mathbb {Z})^Nrtimes mathfrak {S}_N$</span></span></img></span></span>. We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138717524","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}

{"title":"Counting geodesics of given commutator length","authors":"Viveka Erlandsson, Juan Souto","doi":"10.1017/fms.2023.114","DOIUrl":"https://doi.org/10.1017/fms.2023.114","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001147_inline1.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a closed hyperbolic surface. We study, for fixed <jats:italic>g</jats:italic>, the asymptotics of the number of those periodic geodesics in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001147_inline2.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> having at most length <jats:italic>L</jats:italic> and which can be written as the product of <jats:italic>g</jats:italic> commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001147_inline3.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138685932","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}

{"title":"Two-variable fibrations, factorisation systems and -categories of spans","authors":"Rune Haugseng, Fabian Hebestreit, Sil Linskens, Joost Nuiten","doi":"10.1017/fms.2023.107","DOIUrl":"https://doi.org/10.1017/fms.2023.107","url":null,"abstract":"We prove a universal property for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline2.png\" /> <jats:tex-math> $infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose. As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline3.png\" /> <jats:tex-math> $(infty ,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline4.png\" /> <jats:tex-math> $infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-category of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline5.png\" /> <jats:tex-math> $infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen).","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579507","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}

{"title":"A question of Frohardt on -groups, skew translation quadrangles of even order and cyclic STGQs","authors":"Koen Thas","doi":"10.1017/fms.2023.105","DOIUrl":"https://doi.org/10.1017/fms.2023.105","url":null,"abstract":"Abstract We solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite \u0000$2$\u0000 -groups with Kantor familes, by showing that finite groups K with a Kantor family \u0000$(mathcal {F},mathcal {F}^*)$\u0000 having distinct members \u0000$A, B in mathcal {F}$\u0000 such that \u0000$A^* cap B^*$\u0000 is a central subgroup of K and the quotient \u0000$K/(A^* cap B^*)$\u0000 is abelian cannot exist if the center of K has exponent \u0000$4$\u0000 and the members of \u0000$mathcal {F}$\u0000 are elementary abelian. Then we give a short geometrical proof of a recent result of Ott which says that finite skew translation quadrangles of even order \u0000$(t,t)$\u0000 (where t is not a square) are always translation generalized quadrangles. This is a consequence of a complete classification of finite cyclic skew translation quadrangles of order \u0000$(t,t)$\u0000 that we carry out in the present paper.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529171","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}

{"title":"The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems","authors":"Chokri Manai, Simone Warzel","doi":"10.1017/fms.2023.111","DOIUrl":"https://doi.org/10.1017/fms.2023.111","url":null,"abstract":"A semiclassical analysis based on spin-coherent states is used to establish a classification and novel simple formulae for the spectral gap of mean-field spin Hamiltonians. For gapped systems, we provide a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian, hence justifying fluctuation theory at zero temperature for this case. We also point out a shift caused by the spherical geometry in these second-order approximations.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529165","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}

{"title":"Gluing approximable triangulated categories","authors":"Jesse Burke, Amnon Neeman, Bregje Pauwels","doi":"10.1017/fms.2023.97","DOIUrl":"https://doi.org/10.1017/fms.2023.97","url":null,"abstract":"Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is <jats:italic>approximable</jats:italic> if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300097X_inline1.png\" /> <jats:tex-math> $mathsf {D}_{mathsf {qc}}( X )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is approximable when <jats:italic>X</jats:italic> is a quasi compact, separated scheme led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article, we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which, in conjunction with the groundwork laid in [16], has interesting applications in existing and forthcoming work by the authors.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529172","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}

{"title":"Categorical and K-theoretic Donaldson–Thomas theory of (part II)","authors":"Tudor Pădurariu, Yukinobu Toda","doi":"10.1017/fms.2023.103","DOIUrl":"https://doi.org/10.1017/fms.2023.103","url":null,"abstract":"Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory. We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight. We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529193","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}

{"title":"Thermalization in Kitaev’s quantum double models via tensor network techniques","authors":"Angelo Lucia, David Pérez-García, Antonio Pérez-Hernández","doi":"10.1017/fms.2023.98","DOIUrl":"https://doi.org/10.1017/fms.2023.98","url":null,"abstract":"We show that every ergodic Davies generator associated to <jats:italic>any</jats:italic> 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529164","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}

{"title":"On the nilpotent orbit theorem of complex variations of Hodge structure","authors":"Ya Deng","doi":"10.1017/fms.2023.109","DOIUrl":"https://doi.org/10.1017/fms.2023.109","url":null,"abstract":"<p>We prove some results on the nilpotent orbit theorem for complex variations of Hodge structure.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529173","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}