Pub Date : 2020-07-19DOI: 10.1215/00127094-2023-0007
Corey Bregman, Ruth Charney, K. Vogtmann
For any right-angled Artin group $A_{Gamma}$ we construct a finite-dimensional space $mathcal{O}_{Gamma}$ on which the group $text{Out}(A_{Gamma})$ of outer automorphisms of $A_{Gamma}$ acts properly. We prove that $mathcal{O}_{Gamma}$ is contractible, so that the quotient is a rational classifying space for $text{Out}(A_{Gamma})$. The space $mathcal{O}_{Gamma}$ blends features of the symmetric space of lattices in $mathbb{R}^n$ with those of Outer space for the free group $F_n$. Points in $mathcal{O}_{Gamma}$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with $A_{Gamma}$.
{"title":"Outer space for RAAGs","authors":"Corey Bregman, Ruth Charney, K. Vogtmann","doi":"10.1215/00127094-2023-0007","DOIUrl":"https://doi.org/10.1215/00127094-2023-0007","url":null,"abstract":"For any right-angled Artin group $A_{Gamma}$ we construct a finite-dimensional space $mathcal{O}_{Gamma}$ on which the group $text{Out}(A_{Gamma})$ of outer automorphisms of $A_{Gamma}$ acts properly. We prove that $mathcal{O}_{Gamma}$ is contractible, so that the quotient is a rational classifying space for $text{Out}(A_{Gamma})$. The space $mathcal{O}_{Gamma}$ blends features of the symmetric space of lattices in $mathbb{R}^n$ with those of Outer space for the free group $F_n$. Points in $mathcal{O}_{Gamma}$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with $A_{Gamma}$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43758456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-15DOI: 10.1215/00127094-2019-0083
M. Drmota, C. Mauduit, J. Rivat
If q1 and q2 are two coprime bases, f (resp. g) a strongly q1-multiplicative (resp. strongly q2-multiplicative) function of modulus 1 and θ a real number, we estimate the sums ∑ n≤x Λ(n)f(n)g(n) exp(2iπθn) (and ∑ n≤x μ(n)f(n)g(n) exp(2iπθn)), where Λ denotes the von Mangoldt function (and μ the Möbius function). The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation and combinatorial arguments. We deduce from these estimates a Prime Number Theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.
{"title":"Prime numbers in two bases","authors":"M. Drmota, C. Mauduit, J. Rivat","doi":"10.1215/00127094-2019-0083","DOIUrl":"https://doi.org/10.1215/00127094-2019-0083","url":null,"abstract":"If q1 and q2 are two coprime bases, f (resp. g) a strongly q1-multiplicative (resp. strongly q2-multiplicative) function of modulus 1 and θ a real number, we estimate the sums ∑ n≤x Λ(n)f(n)g(n) exp(2iπθn) (and ∑ n≤x μ(n)f(n)g(n) exp(2iπθn)), where Λ denotes the von Mangoldt function (and μ the Möbius function). The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation and combinatorial arguments. We deduce from these estimates a Prime Number Theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47289272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-15DOI: 10.1215/00127094-2022-0011
S. Lynch
We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.
{"title":"Convexity estimates for hypersurfaces moving by concave curvature functions","authors":"S. Lynch","doi":"10.1215/00127094-2022-0011","DOIUrl":"https://doi.org/10.1215/00127094-2022-0011","url":null,"abstract":"We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42043778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-18DOI: 10.1215/00127094-2022-0076
Tsao-Hsien Chen, D. Nadler
We construct a stratified homeomorphism between the space of $ntimes n$ real matrices with real eigenvalues and the space of $ntimes n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual $GL_n(mathbb R)$-adjoint orbits and $O_n(mathbb C)$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kahler quotients of linear spaces. We discuss applications to the (generalized) Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.
{"title":"Real and symmetric matrices","authors":"Tsao-Hsien Chen, D. Nadler","doi":"10.1215/00127094-2022-0076","DOIUrl":"https://doi.org/10.1215/00127094-2022-0076","url":null,"abstract":"We construct a stratified homeomorphism between the space of $ntimes n$ real matrices with real eigenvalues and the space of $ntimes n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual $GL_n(mathbb R)$-adjoint orbits and $O_n(mathbb C)$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kahler quotients of linear spaces. We discuss applications to the (generalized) Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46569461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-15DOI: 10.1215/00127094-2019-0075
V. Pilloni
— We investigate the p-adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on the Siegel threefolds. 2000 Mathematics Subject Classification: 11F33, 11G18, 14G35
{"title":"Higher coherent cohomology and p -adic modular forms of singular weights","authors":"V. Pilloni","doi":"10.1215/00127094-2019-0075","DOIUrl":"https://doi.org/10.1215/00127094-2019-0075","url":null,"abstract":"— We investigate the p-adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on the Siegel threefolds. 2000 Mathematics Subject Classification: 11F33, 11G18, 14G35","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48500603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1215/00127094-2022-0036
R. Kenyon, I. Prause
We prove a new integrability principle for gradient variational problems in $mathbb{R}^2$, showing that solutions are explicitly parameterized by $kappa$-harmonic functions, that is, functions which are harmonic for the laplacian with varying conductivity $kappa$, where $kappa$ is the square root of the Hessian determinant of the surface tension.
{"title":"Gradient variational problems in R2","authors":"R. Kenyon, I. Prause","doi":"10.1215/00127094-2022-0036","DOIUrl":"https://doi.org/10.1215/00127094-2022-0036","url":null,"abstract":"We prove a new integrability principle for gradient variational problems in $mathbb{R}^2$, showing that solutions are explicitly parameterized by $kappa$-harmonic functions, that is, functions which are harmonic for the laplacian with varying conductivity $kappa$, where $kappa$ is the square root of the Hessian determinant of the surface tension.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47543479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-28DOI: 10.1215/00127094-2022-0046
R. Oliver, F. Thorne
Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (log n)^2})$ for an explicit constant $c$.
{"title":"Upper bounds on number fields of given degree and bounded discriminant","authors":"R. Oliver, F. Thorne","doi":"10.1215/00127094-2022-0046","DOIUrl":"https://doi.org/10.1215/00127094-2022-0046","url":null,"abstract":"Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (log n)^2})$ for an explicit constant $c$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46491984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-26DOI: 10.1215/00127094-2023-0008
Nguyen Viet Dang IMJ-PRG, Iuf, Gabriel Rivière
On a negatively curved surface, we show that the Poincare series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincare series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesic arcs orthogonal to both geodesics determine the linking number of the two geodesics.
{"title":"Poincaré series and linking of Legendrian knots","authors":"Nguyen Viet Dang IMJ-PRG, Iuf, Gabriel Rivière","doi":"10.1215/00127094-2023-0008","DOIUrl":"https://doi.org/10.1215/00127094-2023-0008","url":null,"abstract":"On a negatively curved surface, we show that the Poincare series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincare series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesic arcs orthogonal to both geodesics determine the linking number of the two geodesics.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141202811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-19DOI: 10.1215/00127094-2022-0048
A. Sah
We improve the upper bound for diagonal Ramsey numbers to [R(k+1,k+1)leexp(-c(log k)^2)binom{2k}{k}] for $kge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal "effective quasirandomness" results about convergence of graphs. This optimality represents a natural barrier to improvement.
{"title":"Diagonal Ramsey via effective quasirandomness","authors":"A. Sah","doi":"10.1215/00127094-2022-0048","DOIUrl":"https://doi.org/10.1215/00127094-2022-0048","url":null,"abstract":"We improve the upper bound for diagonal Ramsey numbers to [R(k+1,k+1)leexp(-c(log k)^2)binom{2k}{k}] for $kge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal \"effective quasirandomness\" results about convergence of graphs. This optimality represents a natural barrier to improvement.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47237326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-15DOI: 10.1215/00127094-2019-0078
G. González‐diez
F. Catanese has recently asked if there exists an element of the absolute Galois group σ ∈ Gal(Q) for which there is a Kodaira fibration f : S → B defined over a number field such that the universal covers of S and its Galois conjugate surface S are not isomorphic. The main result of this article is that every element σ 6= Id. has this property.
f . Catanese最近提出了一个问题:在绝对伽罗瓦群σ∈Gal(Q)中是否存在一个元素，其在数域上定义了一个Kodaira颤振f: S→B，使得S的全称覆盖与其伽罗瓦共轭曲面S不同构。本文的主要结论是每个元素σ 6= Id。有这个性质。
{"title":"Galois action on universal covers of Kodaira fibrations","authors":"G. González‐diez","doi":"10.1215/00127094-2019-0078","DOIUrl":"https://doi.org/10.1215/00127094-2019-0078","url":null,"abstract":"F. Catanese has recently asked if there exists an element of the absolute Galois group σ ∈ Gal(Q) for which there is a Kodaira fibration f : S → B defined over a number field such that the universal covers of S and its Galois conjugate surface S are not isomorphic. The main result of this article is that every element σ 6= Id. has this property.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48659962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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