Given a function $f: A to{mathbb{R}}^{n}$ of a certain regularity defined on some open subset $A subseteq{mathbb{R}}^{m}$, it is a classical problem of analysis to investigate whether the function can be extended to all of ${mathbb{R}}^{m}$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on ${mathbb{R}}^{m}$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions that are anchored by an inextendible causal curve $gamma : [-1,0) to M$ in the sense that $gamma $ has limit points in both extensions must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions that are only Hölder continuous do in general not enjoy this local uniqueness result.
{"title":"Uniqueness and Non-Uniqueness Results for Spacetime Extensions","authors":"Jan Sbierski","doi":"10.1093/imrn/rnae194","DOIUrl":"https://doi.org/10.1093/imrn/rnae194","url":null,"abstract":"Given a function $f: A to{mathbb{R}}^{n}$ of a certain regularity defined on some open subset $A subseteq{mathbb{R}}^{m}$, it is a classical problem of analysis to investigate whether the function can be extended to all of ${mathbb{R}}^{m}$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on ${mathbb{R}}^{m}$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions that are anchored by an inextendible causal curve $gamma : [-1,0) to M$ in the sense that $gamma $ has limit points in both extensions must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions that are only Hölder continuous do in general not enjoy this local uniqueness result.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252538","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}
Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $ell ^{infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $infty $. This norm decays as $n^{-1/N}$ for some $Nge 3$. Furthermore, for every $Nge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $ntimes n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|det{T}|cdot |T^{-1}|cdot |T|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.
{"title":"On the Fourier Coefficients of Powers of a Finite Blaschke Product","authors":"Alexander Borichev, Karine Fouchet, Rachid Zarouf","doi":"10.1093/imrn/rnae199","DOIUrl":"https://doi.org/10.1093/imrn/rnae199","url":null,"abstract":"Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $ell ^{infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $infty $. This norm decays as $n^{-1/N}$ for some $Nge 3$. Furthermore, for every $Nge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $ntimes n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|det{T}|cdot |T^{-1}|cdot |T|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252537","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}
We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.
{"title":"The Prime Geodesic Theorem in Arithmetic Progressions","authors":"Dimitrios Chatzakos, Gergely Harcos, Ikuya Kaneko","doi":"10.1093/imrn/rnae198","DOIUrl":"https://doi.org/10.1093/imrn/rnae198","url":null,"abstract":"We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252535","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}
In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson $L$-class $L_{*}(X)$ and the Hirzebruch homology class $T_{1,*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest.
{"title":"The Brasselet–Schürmann–Yokura Conjecture on L-Classes of Projective Rational Homology Manifolds","authors":"Javier Fernández de Bobadilla, Irma Pallarés","doi":"10.1093/imrn/rnae193","DOIUrl":"https://doi.org/10.1093/imrn/rnae193","url":null,"abstract":"In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson $L$-class $L_{*}(X)$ and the Hirzebruch homology class $T_{1,*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226159","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}
For a finite dimensional algebra $A$, the notion of $g$-fan $Sigma (A)$ is defined from two-term silting complexes of $textsf{K}^{textrm{b}}(textsf{proj} A)$ in the real Grothendieck group $K_{0}(textsf{proj} A)_{mathbb{R}}$. In this paper, we discuss the theory of shards to $Sigma (A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of torsion classes of $textsf{mod}A$ and the set of shards of $Sigma (A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $textsf{mod}A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $textsf{mod}A$.
{"title":"Shard Theory for g-Fans","authors":"Yuya Mizuno","doi":"10.1093/imrn/rnae196","DOIUrl":"https://doi.org/10.1093/imrn/rnae196","url":null,"abstract":"For a finite dimensional algebra $A$, the notion of $g$-fan $Sigma (A)$ is defined from two-term silting complexes of $textsf{K}^{textrm{b}}(textsf{proj} A)$ in the real Grothendieck group $K_{0}(textsf{proj} A)_{mathbb{R}}$. In this paper, we discuss the theory of shards to $Sigma (A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of torsion classes of $textsf{mod}A$ and the set of shards of $Sigma (A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $textsf{mod}A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $textsf{mod}A$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226166","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}
This paper investigates the (semi)group action of $textrm{SL}_{n}({mathbb R})$ on ${mathbb P}({mathbb R}^{n})$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32], they obtain a dimension formula for the stationary measures on ${mathbb P}({mathbb R}^{3})$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $textrm{SL}_{3}({mathbb R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.
{"title":"On the Dimension of Limit Sets on ℙ(ℝ3) via Stationary Measures: Variational Principles and Applications","authors":"Yuxiang Jiao, Jialun Li, Wenyu Pan, Disheng Xu","doi":"10.1093/imrn/rnae190","DOIUrl":"https://doi.org/10.1093/imrn/rnae190","url":null,"abstract":"This paper investigates the (semi)group action of $textrm{SL}_{n}({mathbb R})$ on ${mathbb P}({mathbb R}^{n})$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32], they obtain a dimension formula for the stationary measures on ${mathbb P}({mathbb R}^{3})$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $textrm{SL}_{3}({mathbb R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203303","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}
Let ${mathcal{M}}_{operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${mathcal{M}}_{operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $operatorname{Tw}({mathcal{M}}_{operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $operatorname{Tw}({mathcal{M}}_{operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.
{"title":"The Dirac–Higgs Complex and Categorification of (BBB)-Branes","authors":"Emilio Franco, Robert Hanson","doi":"10.1093/imrn/rnae187","DOIUrl":"https://doi.org/10.1093/imrn/rnae187","url":null,"abstract":"Let ${mathcal{M}}_{operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${mathcal{M}}_{operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $operatorname{Tw}({mathcal{M}}_{operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $operatorname{Tw}({mathcal{M}}_{operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203307","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}
We introduce a new method to detect the zeros of the Riemann zeta function, which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few “half-isolated” zeros. By combining this with classical methods, we improve the Ingham–Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption.
{"title":"Half-Isolated Zeros and Zero-Density Estimates","authors":"James Maynard, Kyle Pratt","doi":"10.1093/imrn/rnae191","DOIUrl":"https://doi.org/10.1093/imrn/rnae191","url":null,"abstract":"We introduce a new method to detect the zeros of the Riemann zeta function, which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few “half-isolated” zeros. By combining this with classical methods, we improve the Ingham–Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203304","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}
We study the asymptotic behavior of the fluctuations of smooth and rough linear statistics for determinantal point processes on the sphere and on the Euclidean space. The main tool is the generalization of some norm representation results for functions in Sobolev spaces and in the space of functions of bounded variation.
{"title":"Linear Statistics of Determinantal Point Processes and Norm Representations","authors":"Matteo Levi, Jordi Marzo, Joaquim Ortega-Cerdà","doi":"10.1093/imrn/rnae182","DOIUrl":"https://doi.org/10.1093/imrn/rnae182","url":null,"abstract":"We study the asymptotic behavior of the fluctuations of smooth and rough linear statistics for determinantal point processes on the sphere and on the Euclidean space. The main tool is the generalization of some norm representation results for functions in Sobolev spaces and in the space of functions of bounded variation.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203305","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}
Let $mathcal{F}_{g}$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over ${mathbb{Q}}$. Poonen and Stoll have shown that for every $g geq 3$, a positive proportion of curves in $mathcal{F}_{g}$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $ggeq 4$, a positive proportion of curves in $mathcal{F}_{g}$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $mathbb{P}^{1}$.
{"title":"A Positive Proportion of Monic Odd-Degree Hyperelliptic Curves of Genus g ≥ 4 Have no Unexpected Quadratic Points","authors":"Jef Laga, Ashvin A Swaminathan","doi":"10.1093/imrn/rnae184","DOIUrl":"https://doi.org/10.1093/imrn/rnae184","url":null,"abstract":"Let $mathcal{F}_{g}$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over ${mathbb{Q}}$. Poonen and Stoll have shown that for every $g geq 3$, a positive proportion of curves in $mathcal{F}_{g}$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $ggeq 4$, a positive proportion of curves in $mathcal{F}_{g}$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $mathbb{P}^{1}$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203309","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}
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