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The Fuglede conjecture for convex domains is true in all dimensions 凸域的Fuglede猜想在所有维度上都是正确的
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2019-04-28 DOI: 10.4310/ACTA.2022.v228.n2.a3
Nir Lev, M. Matolcsi
{"title":"The Fuglede conjecture for convex domains is true in all dimensions","authors":"Nir Lev, M. Matolcsi","doi":"10.4310/ACTA.2022.v228.n2.a3","DOIUrl":"https://doi.org/10.4310/ACTA.2022.v228.n2.a3","url":null,"abstract":"A set $Omega subset mathbb{R}^d$ is said to be spectral if the space $L^2(Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $Omega subset mathbb{R}^d$ the \"tiling implies spectral\" part of the conjecture is in fact true. \u0000To the contrary, the \"spectral implies tiling\" direction of the conjecture for convex bodies was proved only in $mathbb{R}^2$, and also in $mathbb{R}^3$ under the a priori assumption that $Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $Omega$ is a polytope) and could not be treated using the previously developed techniques. \u0000In this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $Omega subset mathbb{R}^d$ is a spectral set then it can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric \"weak tiling\" condition necessary for a set $Omega subset mathbb{R}^d$ to be spectral.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47687210","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: 44
Correction to “On topological cyclic homology” 对“On拓扑循环同调”的更正
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2019-03-01 DOI: 10.4310/ACTA.2019.V222.N1.A2
T. Nikolaus, P. Scholze
{"title":"Correction to “On topological cyclic homology”","authors":"T. Nikolaus, P. Scholze","doi":"10.4310/ACTA.2019.V222.N1.A2","DOIUrl":"https://doi.org/10.4310/ACTA.2019.V222.N1.A2","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45207672","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: 2
Quotients of higher-dimensional Cremona groups 高维Cremona群的群
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2019-01-14 DOI: 10.4310/ACTA.2021.v226.n2.a1
J. Blanc, St'ephane Lamy, Susanna Zimmermann
{"title":"Quotients of higher-dimensional Cremona groups","authors":"J. Blanc, St'ephane Lamy, Susanna Zimmermann","doi":"10.4310/ACTA.2021.v226.n2.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2021.v226.n2.a1","url":null,"abstract":"We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space, in which case Bir(X) is the Cremona group of rank n, or when X is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect and thus not simple. As a consequence we also obtain that the Cremona group of rank n at least 3 is not generated by linear and Jonqui`eres elements.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41893558","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: 26
The directed landscape 定向景观
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2018-12-02 DOI: 10.4310/acta.2022.v229.n2.a1
Duncan Dauvergne, Janosch Ortmann, B'alint Vir'ag
{"title":"The directed landscape","authors":"Duncan Dauvergne, Janosch Ortmann, B'alint Vir'ag","doi":"10.4310/acta.2022.v229.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2022.v229.n2.a1","url":null,"abstract":"The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Holder-$2/3^-$ continuous paths. This work completes the construction of the central object in the Kardar-Parisi-Zhang universality class, the directed landscape.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41458245","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: 116
Ancient solutions to the Ricci flow in dimension $3$ 利玛窦流的古代解法（3美元）$
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2018-11-06 DOI: 10.4310/acta.2020.v225.n1.a1
S. Brendle
{"title":"Ancient solutions to the Ricci flow in dimension $3$","authors":"S. Brendle","doi":"10.4310/acta.2020.v225.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2020.v225.n1.a1","url":null,"abstract":"It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $kappa$-solution. \u0000We prove that the every noncompact ancient $kappa$-solution in dimension $3$ is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45410223","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: 60
Systems of holomorphic multivalued projections on complex manifolds 复流形上的全纯多值投影系统
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2018-11-01 DOI: 10.4467/20843828am.18.001.9717
Kamil Drzyzga
{"title":"Systems of holomorphic multivalued projections on complex manifolds","authors":"Kamil Drzyzga","doi":"10.4467/20843828am.18.001.9717","DOIUrl":"https://doi.org/10.4467/20843828am.18.001.9717","url":null,"abstract":". Let M be a submanifold of a connected Stein manifold X . We construct a global system of holomorphic multivalued projections X −→ M . In particular, for every locally bounded family F ⊂ O ( M ) we get a continuous extension operator F −→ O ( X ).","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47468449","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
Irreducibility of random polynomials of large degree 大次随机多项式的不可约性
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2018-10-31 DOI: 10.4310/ACTA.2019.v223.n2.a1
E. Breuillard, P. P. Varj'u
{"title":"Irreducibility of random polynomials of large degree","authors":"E. Breuillard, P. P. Varj'u","doi":"10.4310/ACTA.2019.v223.n2.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2019.v223.n2.a1","url":null,"abstract":"We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48776717","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: 27
Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness 古代低熵流、均值凸邻域和唯一性
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2018-10-19 DOI: 10.4310/acta.2022.v228.n2.a1
K. Choi, Robert Haslhofer, Or Hershkovits
{"title":"Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness","authors":"K. Choi, Robert Haslhofer, Or Hershkovits","doi":"10.4310/acta.2022.v228.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2022.v228.n2.a1","url":null,"abstract":"In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $varepsilon=varepsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in $mathbb{R}^3$ with entropy at most $sqrt{2pi/e}+delta$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47680734","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: 53
The special fiber of the motivic deformation of the stable homotopy category is algebraic 稳定同伦范畴运动变形的特殊纤维是代数的
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2018-09-25 DOI: 10.4310/acta.2021.v226.n2.a2
Bogdan Gheorghe, Guozhen Wang, Zhouli Xu
{"title":"The special fiber of the motivic deformation of the stable homotopy category is algebraic","authors":"Bogdan Gheorghe, Guozhen Wang, Zhouli Xu","doi":"10.4310/acta.2021.v226.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2021.v226.n2.a2","url":null,"abstract":"For each prime $p$, we define a $t$-structure on the category $widehat{S^{0,0}}/tautext{-}mathbf{Mod}_{harm}^b$ of harmonic $mathbb{C}$-motivic left module spectra over $widehat{S^{0,0}}/tau$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $widehat{S^{0,0}}/tautext{-}mathbf{Mod}_{harm}^b$ is equivalent to $mathcal{D}^b({{BP}_*{BP}text{-}mathbf{Comod}}^{ev})$ as stable $infty$-categories equipped with $t$-structures. \u0000As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $widehat{S^{0,0}}/tau$, which converges to the motivic homotopy groups of $widehat{S^{0,0}}/tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams-Novikov $E_2$-page for the sphere spectrum $widehat{S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the 90-stem, with ongoing computations into even higher dimensions.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42083646","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: 32
On the structure of band edges of $2$-dimensional periodic elliptic operators $2$维周期椭圆算子的带边结构
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2018-09-01 DOI: 10.4310/ACTA.2018.V221.N1.A2
N. Filonov, I. Kachkovskiy
{"title":"On the structure of band edges of $2$-dimensional periodic elliptic operators","authors":"N. Filonov, I. Kachkovskiy","doi":"10.4310/ACTA.2018.V221.N1.A2","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V221.N1.A2","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"221 1","pages":"59-80"},"PeriodicalIF":3.7,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46641159","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: 16
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