Forum of Mathematics Sigma最新文献

{"title":"Exact solutions to the Erdős-Rothschild problem","authors":"Oleg Pikhurko, Katherine Staden","doi":"10.1017/fms.2023.117","DOIUrl":"https://doi.org/10.1017/fms.2023.117","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$boldsymbol {k} := (k_1,ldots ,k_s)$</span></span></img></span></span> be a sequence of natural numbers. For a graph <span>G</span>, let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$F(G;boldsymbol {k})$</span></span></img></span></span> denote the number of colourings of the edges of <span>G</span> with colours <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$1,dots ,s$</span></span></img></span></span> such that, for every <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$c in {1,dots ,s}$</span></span></img></span></span>, the edges of colour <span>c</span> contain no clique of order <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$k_c$</span></span></img></span></span>. Write <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$F(n;boldsymbol {k})$</span></span></img></span></span> to denote the maximum of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$F(G;boldsymbol {k})$</span></span></img></span></span> over all graphs <span>G</span> on <span>n</span> vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105145234572-0397:S2050509423001172:S2050509423001172_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$n to infty $</span></span></img></span></span>: </p><ol><li><p><span>(i)</span> A sufficient condition on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139397609","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":"Positivity of Schur forms for strongly decomposably positive vector bundles","authors":"Xueyuan Wan","doi":"10.1017/fms.2023.125","DOIUrl":"https://doi.org/10.1017/fms.2023.125","url":null,"abstract":"<p>In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139397998","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":"Hypercontractivity on the symmetric group","authors":"Yuval Filmus, Guy Kindler, Noam Lifshitz, Dor Minzer","doi":"10.1017/fms.2023.118","DOIUrl":"https://doi.org/10.1017/fms.2023.118","url":null,"abstract":"<p>The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains.</p><p>We consider the symmetric group, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121623186-0467:S2050509423001184:S2050509423001184_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$S_n$</span></span></img></span></span>, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of <span>global functions</span> on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121623186-0467:S2050509423001184:S2050509423001184_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$S_n$</span></span></img></span></span>, which are functions whose <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121623186-0467:S2050509423001184:S2050509423001184_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>-norm remains small when restricting <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121623186-0467:S2050509423001184:S2050509423001184_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$O(1)$</span></span></img></span></span> coordinates of the input, and assert that low-degree, global functions have small <span>q</span>-norms, for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121623186-0467:S2050509423001184:S2050509423001184_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$q&gt;2$</span></span></img></span></span>.</p><p>As applications, we show the following: </p><ol><li><p><span>1.</span> An analog of the level-<span>d</span> inequality on the hypercube, asserting that the mass of a global function on low degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121623186-0467:S2050509423001184:S2050509423001184_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$A_n$</span></span></img></span></span>.</p></li><li><p><span>2.</span> Isoperimetric inequalities on the transposition Cayley graph of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.or","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139397937","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":"Tropical Fock–Goncharov coordinates for -webs on surfaces I: construction","authors":"Daniel C. Douglas, Zhe Sun","doi":"10.1017/fms.2023.120","DOIUrl":"https://doi.org/10.1017/fms.2023.120","url":null,"abstract":"<p>For a finite-type surface <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104142456266-0825:S2050509423001202:S2050509423001202_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathfrak {S}$</span></span></img></span></span>, we study a preferred basis for the commutative algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104142456266-0825:S2050509423001202:S2050509423001202_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {C}[mathscr {R}_{mathrm {SL}_3(mathbb {C})}(mathfrak {S})]$</span></span></img></span></span> of regular functions on the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104142456266-0825:S2050509423001202:S2050509423001202_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {SL}_3(mathbb {C})$</span></span></img></span></span>-character variety, introduced by Sikora–Westbury. These basis elements come from the trace functions associated to certain trivalent graphs embedded in the surface <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104142456266-0825:S2050509423001202:S2050509423001202_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathfrak {S}$</span></span></img></span></span>. We show that this basis can be naturally indexed by nonnegative integer coordinates, defined by Knutson–Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105069","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":"Every complex Hénon map is exponentially mixing of all orders and satisfies the CLT","authors":"Fabrizio Bianchi, Tien-Cuong Dinh","doi":"10.1017/fms.2023.110","DOIUrl":"https://doi.org/10.1017/fms.2023.110","url":null,"abstract":"<p>We show that the measure of maximal entropy of every complex Hénon map is exponentially mixing of all orders for Hölder observables. As a consequence, the Central Limit Theorem holds for all Hölder observables.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105070","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":"Modularity of trianguline Galois representations","authors":"Rebecca Bellovin","doi":"10.1017/fms.2023.116","DOIUrl":"https://doi.org/10.1017/fms.2023.116","url":null,"abstract":"<p>We use the theory of trianguline <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104083513942-0464:S2050509423001160:S2050509423001160_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(varphi ,Gamma )$</span></span></img></span></span>-modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at <span>p</span>, including those with characteristic <span>p</span> coefficients. The use of pseudorigid spaces lets us construct integral models of the trianguline varieties of [BHS17], [Che13] after bounding the slope, and we carry out a Taylor–Wiles patching argument for families of overconvergent modular forms. This permits us to construct a patched quaternionic eigenvariety and deduce our modularity results.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105118","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":"Chern classes in equivariant bordism","authors":"Stefan Schwede","doi":"10.1017/fms.2023.124","DOIUrl":"https://doi.org/10.1017/fms.2023.124","url":null,"abstract":"<p>We introduce Chern classes in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$U(m)$</span></span></img></span></span>-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {MU}$</span></span></img></span></span>-cohomology of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$B U(m)$</span></span></img></span></span>. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {MU}$</span></span></img></span></span>-completion theorem of Greenlees–May and La Vecchia.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105112","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 stable cohomology of self-equivalences of connected sums of products of spheres","authors":"Robin Stoll","doi":"10.1017/fms.2023.113","DOIUrl":"https://doi.org/10.1017/fms.2023.113","url":null,"abstract":"<p>We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121641127-0961:S2050509423001135:S2050509423001135_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {S}^{k}} times {mathrm {S}^{l}}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105121641127-0961:S2050509423001135:S2050509423001135_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$3 le k &lt; l le 2k - 2$</span></span></img></span></span>. The result is expressed in terms of Lie graph complex homology.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105428","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":"Base sizes of primitive groups of diagonal type","authors":"Hong Yi Huang","doi":"10.1017/fms.2023.121","DOIUrl":"https://doi.org/10.1017/fms.2023.121","url":null,"abstract":"<p>Let <span>G</span> be a permutation group on a finite set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104065504114-0233:S2050509423001214:S2050509423001214_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Omega $</span></span></img></span></span>. The base size of <span>G</span> is the minimal size of a subset of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104065504114-0233:S2050509423001214:S2050509423001214_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Omega $</span></span></img></span></span> with trivial pointwise stabiliser in <span>G</span>. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094095","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":"Lim Ulrich sequences and Boij-Söderberg cones","authors":"Srikanth B. Iyengar, Linquan Ma, Mark E. Walker","doi":"10.1017/fms.2023.108","DOIUrl":"https://doi.org/10.1017/fms.2023.108","url":null,"abstract":"<p>This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings.</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":"138717512","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}