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On $L^infty$ estimates for fully non-linear partial differential equations 关于全非线性偏微分方程的 $L^infty$ 估计数
IF 5.7 1区 数学
Annals of Mathematics Pub Date : 2024-07-01 DOI: 10.4007/annals.2024.200.1.6
Bin Guo, Duong H. Phong
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引用次数: 0
Eldan’s stochastic localization and the KLS conjecture: Isoperimetry, concentration and mixing | Annals of Mathematics 埃尔丹的随机局部化和KLS猜想：等时性、浓度和混合 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-05-01 DOI: 10.4007/annals.2024.199.3.2
Yin Tat Lee, Santosh S. Vempala
{"title":"Eldan’s stochastic localization and the KLS conjecture: Isoperimetry, concentration and mixing | Annals of Mathematics","authors":"Yin Tat Lee, Santosh S. Vempala","doi":"10.4007/annals.2024.199.3.2","DOIUrl":"https://doi.org/10.4007/annals.2024.199.3.2","url":null,"abstract":"<p>We analyze the Poincaré and Log-Sobolev constants of logconcave densities in $mathbb{R}^{n}$. For the Poincaré constant, we give an improved estimate of $O(sqrt{n})$ for any isotropic logconcave density. For the Log-Sobolev constant, we prove a bound of $Omega (1/D)$, where $D$ is the diameter of the support of the density, and show that this is asymptotically the best possible bound, resolving a question posed by Frieze and Kannan in 1997. These bounds have several interesting consequences. Improved bounds on the thin-shell and Cheeger/KLS constants are immediate. The ball walk to sample an isotropic logconcave density in $mathbb{R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start, and the speedy version of the ball walk, studied by Kannan, Lov’aasz and Simonovits mixes in $O^{*}(n^{2}D)$ proper steps from any start, also a tight bound. As another consequence, we obtain a unified and improved large deviation inequality for the concentration of any $L$-Lipshitz function over an isotropic logconcave density (studied by many), generalizing bounds of Paouris and Guedon-E. Milman. Our proof technique is a development of stochastic localization, first introduced by Eldan.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140830906","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}
引用次数: 0
On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes | Annals of Mathematics 论塔拉格朗关于选择过程的猜想和关于正经验过程的结果 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-05-01 DOI: 10.4007/annals.2024.199.3.6
Jinyoung Park, Huy Tuan Pham
{"title":"On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes | Annals of Mathematics","authors":"Jinyoung Park, Huy Tuan Pham","doi":"10.4007/annals.2024.199.3.6","DOIUrl":"https://doi.org/10.4007/annals.2024.199.3.6","url":null,"abstract":"<p>For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831279","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}
引用次数: 0
Nonlinear inviscid damping for a class of monotone shear flows in a finite channel | Annals of Mathematics 有限通道中一类单调剪切流的非线性不粘性阻尼 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-05-01 DOI: 10.4007/annals.2024.199.3.3
Nader Masmoudi, Weiren Zhao
{"title":"Nonlinear inviscid damping for a class of monotone shear flows in a finite channel | Annals of Mathematics","authors":"Nader Masmoudi, Weiren Zhao","doi":"10.4007/annals.2024.199.3.3","DOIUrl":"https://doi.org/10.4007/annals.2024.199.3.3","url":null,"abstract":"<p>We prove the nonlinear inviscid damping for a class of monotone shear flows in $mathbb{T}times [0,1]$ for initial perturbation in Gevrey-$frac{1}{s}$ class ($1lt frac{1}{s}&lt;2$) with compact support. The main new idea of the proof is to construct and use the wave operator of a slightly modified Rayleigh operator in a well-chosen coordinate system.\u0000</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140830884","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}
引用次数: 0
Torsion-free abelian groups are Borel complete | Annals of Mathematics 无扭无边群是伯尔完全群 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-05-01 DOI: 10.4007/annals.2024.199.3.4
Gianluca Paolini, Saharon Shelah
{"title":"Torsion-free abelian groups are Borel complete | Annals of Mathematics","authors":"Gianluca Paolini, Saharon Shelah","doi":"10.4007/annals.2024.199.3.4","DOIUrl":"https://doi.org/10.4007/annals.2024.199.3.4","url":null,"abstract":"<p>We prove that the Borel space of torsion-free abelian groups with domain $omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140830899","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}
引用次数: 0
Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves | Annals of Mathematics 稳定曲线模空间上的多项式点数和奇数同调消失 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-05-01 DOI: 10.4007/annals.2024.199.3.7
Jonas Bergström, Carel Faber, Sam Payne
{"title":"Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves | Annals of Mathematics","authors":"Jonas Bergström, Carel Faber, Sam Payne","doi":"10.4007/annals.2024.199.3.7","DOIUrl":"https://doi.org/10.4007/annals.2024.199.3.7","url":null,"abstract":"<p>We compute the number of $mathbb{F}_q$-points on $overline{mathcal{M}}_{4,n}$ for $n leq 3$ and show that it is a polynomial in $q$, using a sieve based on Hasse–Weil zeta functions. As an application, we prove that the rational singular cohomology group $H^k (overline{mathcal{M}}_{g,n})$ vanishes for all odd $k leq 9$. Both results confirm predictions of the Langlands program, via the conjectural correspondence with polarized algebraic cuspidal automorphic representations of conductor $1$, which are classified in low weight. Our vanishing result for odd cohomology resolves a problem posed by Arbarello and Cornalba in the 1990s.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831105","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}
引用次数: 0
Exponential mixing implies Bernoulli | Annals of Mathematics 指数混合意味着伯努利 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-05-01 DOI: 10.4007/annals.2024.199.3.5
Dmitry Dolgopyat, Adam Kanigowski, Federico Rodriguez Hertz
{"title":"Exponential mixing implies Bernoulli | Annals of Mathematics","authors":"Dmitry Dolgopyat, Adam Kanigowski, Federico Rodriguez Hertz","doi":"10.4007/annals.2024.199.3.5","DOIUrl":"https://doi.org/10.4007/annals.2024.199.3.5","url":null,"abstract":"<p>Let $f$ be a $C^{1+alpha}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $mu$. We show that if $fcolon (M,mu)to (M,mu)$ is exponentially mixing, then it is Bernoulli.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140841835","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}
引用次数: 0
Microlocal Morse theory of wrapped Fukaya categories | Annals of Mathematics 缠绕 Fukaya 类别的微局域莫尔斯理论 | 数学年刊
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-05-01 DOI: 10.4007/annals.2024.199.3.1
Sheel Ganatra, John Pardon, Vivek Shende
{"title":"Microlocal Morse theory of wrapped Fukaya categories | Annals of Mathematics","authors":"Sheel Ganatra, John Pardon, Vivek Shende","doi":"10.4007/annals.2024.199.3.1","DOIUrl":"https://doi.org/10.4007/annals.2024.199.3.1","url":null,"abstract":"<p>The Nadler–Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms “at infinity,” given on the Floer side by Reeb trajectories (also known as “wrapping”) and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry. par More precisely, given a real analytic manifold $M$ and a subanalytic isotropic subset $Lambda$ of its co-sphere bundle $S^*M$, we show that the partially wrapped Fukaya category of $T^*M$ stopped at $Lambda$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $Lambda$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831403","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}
引用次数: 0
Pointwise convergence of the non-linear Fourier transform | Annals of Mathematics 非线性傅立叶变换的点收敛性 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-03-05 DOI: 10.4007/annals.2024.199.2.4
A. Poltoratski
{"title":"Pointwise convergence of the non-linear Fourier transform | Annals of Mathematics","authors":"A. Poltoratski","doi":"10.4007/annals.2024.199.2.4","DOIUrl":"https://doi.org/10.4007/annals.2024.199.2.4","url":null,"abstract":"<p>We prove pointwise convergence for the scattering data of a Dirac system of differential equations. Equivalently, we prove an analog of Carleson’s theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. Our proofs are based on the study of resonances of Dirac systems using families of meromorphic inner functions, generated by a Riccati equation corresponding to the system.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045624","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}
引用次数: 0
Prime number theorem for analytic skew products | Annals of Mathematics 解析偏斜积的素数定理 | 数学年鉴
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2024-03-05 DOI: 10.4007/annals.2024.199.2.2
Adam Kanigowski, Mariusz Lemańczyk, Maksym Radziwiłł
{"title":"Prime number theorem for analytic skew products | Annals of Mathematics","authors":"Adam Kanigowski, Mariusz Lemańczyk, Maksym Radziwiłł","doi":"10.4007/annals.2024.199.2.2","DOIUrl":"https://doi.org/10.4007/annals.2024.199.2.2","url":null,"abstract":"<p>We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $mathbb{T}^2$. More precisely, for every irrational $alpha$ and every $1$-periodic real analytic $g:mathbb{R}tomathbb{R}$ of zero mean, let $T_{alpha,g} : mathbb{T}^2 rightarrow mathbb{T}^2$ be defined by $(x,y) mapsto (x+alpha,y+g(x))$. We prove that if $T_{alpha, g}$ is uniquely ergodic then, for every $(x,y) in mathbb{T}^2$, the sequence ${T_{alpha, g}^p(x,y)}$ is equidistributed on $mathbb{T}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if $g$ is only continuous on $mathbb{T}$.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045620","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}
引用次数: 0
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