Acta MathematicaPub Date : 2017-10-27DOI: 10.4310/acta.2019.v223.n2.a2
L. Guth, J. Hickman, Marina Iliopoulou
{"title":"Sharp estimates for oscillatory integral operators via polynomial partitioning","authors":"L. Guth, J. Hickman, Marina Iliopoulou","doi":"10.4310/acta.2019.v223.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2019.v223.n2.a2","url":null,"abstract":"The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42141769","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}
Acta MathematicaPub Date : 2017-10-17DOI: 10.4310/acta.2021.v227.n2.a3
H. Hedenmalm, Aron Wennman
{"title":"Planar orthogonal polynomials and boundary universality in the random normal matrix model","authors":"H. Hedenmalm, Aron Wennman","doi":"10.4310/acta.2021.v227.n2.a3","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n2.a3","url":null,"abstract":"We prove that the planar normalized orthogonal polynomials $P_{n,m}(z)$ of degree $n$ with respect to an exponentially varying planar measure $e^{-2mQ(z)}mathrm{dA}(z)$ enjoy an asymptotic expansion [ P_{n,m}(z)sim m^{frac{1}{4}}sqrt{phi_tau'(z)}[phi_tau(z)]^n e^{mmathcal{Q}_tau(z)}left(mathcal{B}_{0,tau}(z) +frac{1}{m}mathcal{B}_{1,tau}(z)+frac{1}{m^2} mathcal{B}_{2,tau}(z)+ldotsright), ] as $n=mtautoinfty$. Here $mathcal{S}_tau$ denotes the droplet, the boundary of which is assumed to be a smooth, simple, closed curve, and $phi_tau$ is a conformal mapping $mathcal{S}_tau^ctomathbb{D}_e$. The functions $mathcal{Q}_tau$ and $mathcal{B}_{j,tau}(z)$ are bounded holomorphic functions which may be computed in terms of $Q$ and $mathcal{S}_tau$. We apply these results to prove universality at the boundary for regular droplets in the random normal matrix model, i.e., that the limiting rescaled process is the random process with correlation kernel $$ mathrm{k}(xi,eta)= e^{xibareta,-frac12(lvertxirvert^2+lvert etarvert^2)} operatorname{erf}(xi+bar{eta}). $$ A key ingredient in the proof of the asymptotic expansion is the construction of an {orthogonal foliation} -- a smooth flow of closed curves near $partialmathcal{S}_tau$, on each of which $P_{n,m}$ is orthogonal to lower order polynomials, with respect to an induced measure. To compute the coefficients, we develop an algorithm which determines $mathcal{B}_{j,tau}$ up to any desired order in terms of inhomogeneous Toeplitz kernel conditions.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42156274","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}
Acta MathematicaPub Date : 2017-09-13DOI: 10.4310/acta.2022.v228.n1.a1
R. Bamler, B. Kleiner
{"title":"Uniqueness and stability of Ricci flow through singularities","authors":"R. Bamler, B. Kleiner","doi":"10.4310/acta.2022.v228.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2022.v228.n1.a1","url":null,"abstract":"We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. \u0000Our results have applications to the study of diffeomorphism groups of three manifolds --- in particular to the Generalized Smale Conjecture --- which will appear in a subsequent paper.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44402472","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}
Acta MathematicaPub Date : 2017-07-10DOI: 10.4310/ACTA.2022.v229.n2.a2
Benjamin Gammage, V. Shende
{"title":"Mirror symmetry for very affine hypersurfaces","authors":"Benjamin Gammage, V. Shende","doi":"10.4310/ACTA.2022.v229.n2.a2","DOIUrl":"https://doi.org/10.4310/ACTA.2022.v229.n2.a2","url":null,"abstract":"We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective DM toric stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. \u0000Our proof has the following ingredients. Using Mikhalkin-Viro patchworking, we compute the skeleton of the hypersurface. The result matches the [FLTZ] skeleton and is naturally realized as a Legendrian in the cosphere bundle of a torus. By [GPS1, GPS2, GPS3], we trade wrapped Fukaya categories for microlocal sheaf theory. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41520437","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}
Acta MathematicaPub Date : 2017-01-09DOI: 10.4310/acta.2021.v227.n2.a1
P. Berger
{"title":"Generic family displaying robustly a fast growth of the number of periodic points","authors":"P. Berger","doi":"10.4310/acta.2021.v227.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n2.a1","url":null,"abstract":"For any $2le rle infty$, $nge 2$, we prove the existence of an open set $U$ of $C^r$-self-mappings of any $n$-manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $N$-periodic points grows as fast as asked. This complements the works of Martens-de Melo-van Strien, Gochenko-Shil'nikov-Turaev, Kaloshin, Bonatti-Diaz-Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. \u0000Furthermore for any $2le r<infty$ and any $kge 0$, we prove the existence of an open set $hat U$ of $k$-parameter families in $U$ so that for a generic $(f_a)_ain hat U$, for every $|a|le 1$, the map $f_a$ displays a fast growth of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45288215","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}
Acta MathematicaPub Date : 2016-12-30DOI: 10.4310/ACTA.2021.v227.n1.a3
K. Matetski, J. Quastel, Daniel Remenik
{"title":"The KPZ fixed point","authors":"K. Matetski, J. Quastel, Daniel Remenik","doi":"10.4310/ACTA.2021.v227.n1.a3","DOIUrl":"https://doi.org/10.4310/ACTA.2021.v227.n1.a3","url":null,"abstract":"An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process with arbitrary initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [Sas05; BFPS07]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the Airy$_1$ and Airy$_2$ processes. The invariant Markov process takes values in real valued functions which look locally like Brownian motion, and is H\"older $1/3-$ in time.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71153243","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}
Acta MathematicaPub Date : 2016-11-16DOI: 10.1007/S11511-016-0140-6
Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang
{"title":"Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems","authors":"Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang","doi":"10.1007/S11511-016-0140-6","DOIUrl":"https://doi.org/10.1007/S11511-016-0140-6","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S11511-016-0140-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53064695","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}
Acta MathematicaPub Date : 2016-11-16DOI: 10.4310/ACTA.2018.V221.N1.A5
A. Lytchak, S. Wenger
{"title":"Isoperimetric characterization of upper curvature bounds","authors":"A. Lytchak, S. Wenger","doi":"10.4310/ACTA.2018.V221.N1.A5","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V221.N1.A5","url":null,"abstract":"We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71152818","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}
Acta MathematicaPub Date : 2016-09-12DOI: 10.4310/acta.2019.v223.n1.a1
Gebhard Bockle, M. Harris, Chandrashekhar B. Khare, J. Thorne
{"title":"$hat{G}$-local systems on smooth projective curves are potentially automorphic","authors":"Gebhard Bockle, M. Harris, Chandrashekhar B. Khare, J. Thorne","doi":"10.4310/acta.2019.v223.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2019.v223.n1.a1","url":null,"abstract":"Let $X$ be a smooth, projective, geometrically connected curve over a finite field $mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $mathbb{F}_q$. Its dual group $widehat{G}$ is a split reductive group over $mathbb{Z}$. Conjecturally, any $l$-adic $widehat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $pi_1(X) to widehat{G}(overline{mathbb{Q}}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$. \u0000We show that for any homomorphism $pi_1(X) to widehat{G}(overline{mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y to X$ over which the associated local system becomes automorphic.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71152943","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}
Acta MathematicaPub Date : 2016-09-01DOI: 10.1007/s11511-016-0141-5
P. Bernard, V. Kaloshin, Ke Zhang
{"title":"Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders","authors":"P. Bernard, V. Kaloshin, Ke Zhang","doi":"10.1007/s11511-016-0141-5","DOIUrl":"https://doi.org/10.1007/s11511-016-0141-5","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11511-016-0141-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53064704","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}