The Swiss-Cheese operads, which encode actions of algebras over the little n-cubes operad on algebras over the little $(n-1)$-cubes operad, comes in several variants. We prove that the variant in which open operations must have at least one open input is not formal in characteristic zero. This is slightly stronger than earlier results of Livernet and Willwacher. The obstruction to formality that we find lies in arity $(2, 2^n)$, rather than $(2, 0)$ (Livernet) or $(4, 0)$ (Willwacher).
{"title":"Non-formality of Voronov’s Swiss-Cheese operads","authors":"Najib Idrissi, Renato Vasconcellos Vieira","doi":"10.1093/qmath/haad041","DOIUrl":"https://doi.org/10.1093/qmath/haad041","url":null,"abstract":"The Swiss-Cheese operads, which encode actions of algebras over the little n-cubes operad on algebras over the little $(n-1)$-cubes operad, comes in several variants. We prove that the variant in which open operations must have at least one open input is not formal in characteristic zero. This is slightly stronger than earlier results of Livernet and Willwacher. The obstruction to formality that we find lies in arity $(2, 2^n)$, rather than $(2, 0)$ (Livernet) or $(4, 0)$ (Willwacher).","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We give a version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices. As applications, we obtain sufficient conditions for the reflexivity of ${mathcal P}^r(^nE;F)$, the space of regular n-homogeneous polynomials from a Banach lattice E to a Banach lattice F, and sufficient conditions for the positive Grothendieck property of $hat{otimes}_{n,s,|pi|}E$, the n-fold positive projective symmetric tensor product of a Banach lattice E. Moreover, we also prove that these sufficient conditions are also necessary under the bounded regular approximation property.
{"title":"A version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices","authors":"Qingying Bu","doi":"10.1093/qmath/haad040","DOIUrl":"https://doi.org/10.1093/qmath/haad040","url":null,"abstract":"Abstract We give a version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices. As applications, we obtain sufficient conditions for the reflexivity of ${mathcal P}^r(^nE;F)$, the space of regular n-homogeneous polynomials from a Banach lattice E to a Banach lattice F, and sufficient conditions for the positive Grothendieck property of $hat{otimes}_{n,s,|pi|}E$, the n-fold positive projective symmetric tensor product of a Banach lattice E. Moreover, we also prove that these sufficient conditions are also necessary under the bounded regular approximation property.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136158461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For $mge 2$, consider K the m-fold Cartesian product of the limit set of an iterated function system (IFS) of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and K does not degenerate to singleton, we construct vectors in K that lie within the ‘folklore set’ as defined by Beresnevich et al., meaning that they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within K; however, we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.
{"title":"Dirichlet is not just bad and singular in many rational IFS fractals","authors":"Johannes Schleischitz","doi":"10.1093/qmath/haad039","DOIUrl":"https://doi.org/10.1093/qmath/haad039","url":null,"abstract":"Abstract For $mge 2$, consider K the m-fold Cartesian product of the limit set of an iterated function system (IFS) of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and K does not degenerate to singleton, we construct vectors in K that lie within the ‘folklore set’ as defined by Beresnevich et al., meaning that they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within K; however, we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we improve our bounds on the Rankin–Selberg problem. That is, we obtain a smaller error term of the second moment of Fourier coefficients of a GL(2) cusp form (both holomorphic and Maass).
{"title":"On the Rankin–Selberg problem, II","authors":"Bingrong Huang","doi":"10.1093/qmath/haad037","DOIUrl":"https://doi.org/10.1093/qmath/haad037","url":null,"abstract":"Abstract In this paper, we improve our bounds on the Rankin–Selberg problem. That is, we obtain a smaller error term of the second moment of Fourier coefficients of a GL(2) cusp form (both holomorphic and Maass).","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136375734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The paper deals with a class of periods, Frobenius constants, which describe monodromy of Frobenius solutions of differential equations arising in algebraic geometry. We represent Frobenius constants related to families of elliptic curves as iterated integrals of modular forms. Using the theory of periods of modular forms, we then witness some of these constants in terms of zeta values.
{"title":"Frobenius constants for families of elliptic curves","authors":"Bidisha Roy, Masha Vlasenko","doi":"10.1093/qmath/haad034","DOIUrl":"https://doi.org/10.1093/qmath/haad034","url":null,"abstract":"Abstract The paper deals with a class of periods, Frobenius constants, which describe monodromy of Frobenius solutions of differential equations arising in algebraic geometry. We represent Frobenius constants related to families of elliptic curves as iterated integrals of modular forms. Using the theory of periods of modular forms, we then witness some of these constants in terms of zeta values.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135519419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let χ4 be the non-principal Dirichlet character mod 4 and $L(s,chi_4)$ be the Dirichlet L-function associated with χ4 and put $R(s):= s 4^{s} L(s+1,chi_4) + pi L(s-1,chi_4)$. In the present paper, we show that the function R(s) has the Riemann’s functional equation and its zeros only at the non-positive even integers and complex numbers with real part $1/2$. We also give other L-functions that have the same property.
设χ4为非主狄利克雷特征模4,$L(s,chi_4)$为与χ4相关的狄利克雷l函数，并代入$R(s):= s 4^{s} L(s+1,chi_4) + pi L(s-1,chi_4)$。本文证明了函数R(s)只有在非正偶数和带实部的复数$1/2$处才存在黎曼函数方程及其零点。我们也给出了其他具有相同性质的l函数。
{"title":"<i>L</i>-functions with Riemann’s functional equation and the Riemann hypothesis","authors":"Takashi Nakamura","doi":"10.1093/qmath/haad032","DOIUrl":"https://doi.org/10.1093/qmath/haad032","url":null,"abstract":"Abstract Let χ4 be the non-principal Dirichlet character mod 4 and $L(s,chi_4)$ be the Dirichlet L-function associated with χ4 and put $R(s):= s 4^{s} L(s+1,chi_4) + pi L(s-1,chi_4)$. In the present paper, we show that the function R(s) has the Riemann’s functional equation and its zeros only at the non-positive even integers and complex numbers with real part $1/2$. We also give other L-functions that have the same property.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134919941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We establish sharp relations between several regularity coefficients for flows generated by non-autonomous differential equations. We verify that these relations are sharp by showing that all possible values of the regularity coefficients satisfying these relations are attained by some linear equation with bounded piecewise-continuous coefficient matrix. In addition, we introduce two new regularity coefficients and we obtain sharp relations between these and other coefficients.
{"title":"Realization of regularity coefficients for flows","authors":"Luis Barreira, Claudia Valls","doi":"10.1093/qmath/haad020","DOIUrl":"https://doi.org/10.1093/qmath/haad020","url":null,"abstract":"Abstract We establish sharp relations between several regularity coefficients for flows generated by non-autonomous differential equations. We verify that these relations are sharp by showing that all possible values of the regularity coefficients satisfying these relations are attained by some linear equation with bounded piecewise-continuous coefficient matrix. In addition, we introduce two new regularity coefficients and we obtain sharp relations between these and other coefficients.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135752417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We construct infinitely many distinct irreducible smooth structures on $n(S^2,times,S^2)$, the connected sum of n copies of $S^2,times,S^2$, for every odd integer $ngeq 27$.
{"title":"Exotic smooth structures on connected sums of S2×S2","authors":"Anar Akhmedov, B Doug Park, Sümeyra Sakallı","doi":"10.1093/qmath/haad002","DOIUrl":"https://doi.org/10.1093/qmath/haad002","url":null,"abstract":"Abstract We construct infinitely many distinct irreducible smooth structures on $n(S^2,times,S^2)$, the connected sum of n copies of $S^2,times,S^2$, for every odd integer $ngeq 27$.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135539249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let M be a simply connected closed 4-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on M by homeomorphisms is an abelian group of rank at most two, when $b_{2}(M)gt2$. As applications, let $mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $mathrm{Aut}(F_{n})$ and $mathrm{GL}_{n}mathbb{Z}$, n > = 4, on $Mneq S^{4}$ factors through $mathbb{Z}/2$, if the group action is by homologically trivial homeomorphisms.
{"title":"Topological symmetries of simply connected 4-manifolds and actions of automorphism groups of free groups","authors":"Shengkui Ye","doi":"10.1093/qmath/haac042","DOIUrl":"https://doi.org/10.1093/qmath/haac042","url":null,"abstract":"Abstract Let M be a simply connected closed 4-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on M by homeomorphisms is an abelian group of rank at most two, when $b_{2}(M)gt2$. As applications, let $mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $mathrm{Aut}(F_{n})$ and $mathrm{GL}_{n}mathbb{Z}$, n &gt; = 4, on $Mneq S^{4}$ factors through $mathbb{Z}/2$, if the group action is by homologically trivial homeomorphisms.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134902887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell–Liberman theorem. As an application, this confirms a conjecture raised by Battaglia–Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds.
{"title":"Basic kirwan injectivity and its applications","authors":"Yi Lin, Xiangdong Yang","doi":"10.1093/qmath/haac038","DOIUrl":"https://doi.org/10.1093/qmath/haac038","url":null,"abstract":"Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell–Liberman theorem. As an application, this confirms a conjecture raised by Battaglia–Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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