Pub Date : 2019-04-28DOI: 10.4310/ACTA.2022.v228.n2.a3
Nir Lev, M. Matolcsi
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. To 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. In 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.
{"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}
Pub Date : 2019-01-14DOI: 10.4310/ACTA.2021.v226.n2.a1
J. Blanc, St'ephane Lamy, Susanna Zimmermann
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.
{"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}
Pub Date : 2018-12-02DOI: 10.4310/acta.2022.v229.n2.a1
Duncan Dauvergne, Janosch Ortmann, B'alint Vir'ag
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.
{"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}
Pub Date : 2018-11-06DOI: 10.4310/acta.2020.v225.n1.a1
S. Brendle
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. We 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.
{"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}
Pub Date : 2018-11-01DOI: 10.4467/20843828am.18.001.9717
Kamil Drzyzga
. 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 ).
{"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}
Pub Date : 2018-10-31DOI: 10.4310/ACTA.2019.v223.n2.a1
E. Breuillard, P. P. Varj'u
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.
{"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}
Pub Date : 2018-10-19DOI: 10.4310/acta.2022.v228.n2.a1
K. Choi, Robert Haslhofer, Or Hershkovits
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.
{"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}
Pub Date : 2018-09-25DOI: 10.4310/acta.2021.v226.n2.a2
Bogdan Gheorghe, Guozhen Wang, Zhouli Xu
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. As 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.
{"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}
Pub Date : 2018-09-01DOI: 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}
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