Acta Mathematica最新文献

{"title":"Convergence and divergence of formal CR mappings","authors":"B. Lamel, N. Mir","doi":"10.4310/ACTA.2018.V220.N2.A5","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V220.N2.A5","url":null,"abstract":"A formal holomorphic map H: (M,p)!M ′ from a germ of a real-analytic submanifold M⊂C at p∈M into a real-analytic subset M ′⊂CN ′ is an N ′-tuple of formal holomorphic power series H=(H1, ...,HN ′) satisfying H(p)∈M ′ with the property that, for any germ of a real-analytic function δ(w, w) at H(p)∈C ′ which vanishes on M ′, the formal power series δ(H(z), H(z)) vanishes on M . There is an abundance of examples showing that formal maps may diverge: After the trivial example of self-maps of a complex submanifold, possibly the simplest non-trivial example is given by the formal maps of (R, 0) into R which are just given by the formal power series in z∈C with real coefficients, that is, by elements of R[[z]]. It is a surprising fact at first that, for formal mappings between real submanifolds in complex spaces, if one assumes that the trivial examples above are excluded in a suitable sense, the situation is fundamentally different. The first result of this kind was encountered by Chern and Moser in [CM], where—as a byproduct of the convergence of their normal form—it follows that every formal holomorphic invertible map between Levinon-degenerate hypersurfaces in C necessarily converges. The convergence problem, that is, deciding whether formal maps, as described above, are in fact convergent, has been studied intensively in different contexts, both for CR manifolds and for manifolds with CR singularities, for which we refer the reader to the papers [Rot], [MMZ2], [LM1], [HY1], [HY2], [HY3], [Sto], [GS] and the references therein. Solutions to the convergence problem have important applications, for example, to the biholomorphic equivalence","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42862607","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}

{"title":"Algebraic actions of discrete groups: the $p$-adic method","authors":"Serge Cantat, Junyi Xie","doi":"10.4310/ACTA.2018.V220.N2.A2","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V220.N2.A2","url":null,"abstract":"We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on p-adic analysis, the second makes use of isoperimetric inequalities and LangWeil estimates. For instance, we show that if SL n(Z) acts faithfully on a complex quasi-projective variety X by birational transformations, then dim(X) ≥ n−1 and X is rational if dim(X) = n−1.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48028557","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}

{"title":"The orbit method and analysis of automorphic forms","authors":"Paul D. Nelson, Akshay Venkatesh","doi":"10.4310/ACTA.2021.v226.n1.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2021.v226.n1.a1","url":null,"abstract":"We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. \u0000Our main global application is an asymptotic formula for averages of Gan--Gross--Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding $L$-functions. Ratner's results on measure classification provide an important input to the proof. \u0000Our local results include asymptotic expansions for certain special functions arising from representations of higher rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino--Ikeda conjecture.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2018-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48408084","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}

{"title":"Maximization of the second non-trivial Neumann eigenvalue","authors":"D. Bucur, A. Henrot","doi":"10.4310/ACTA.2019.V222.N2.A2","DOIUrl":"https://doi.org/10.4310/ACTA.2019.V222.N2.A2","url":null,"abstract":"In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{'o}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2018-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41510317","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}

{"title":"Bogoliubov theory in the Gross–Pitaevskii limit","authors":"Chiara Boccato, C. Brennecke, S. Cenatiempo, B. Schlein","doi":"10.4310/ACTA.2019.V222.N2.A1","DOIUrl":"https://doi.org/10.4310/ACTA.2019.V222.N2.A1","url":null,"abstract":"We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order $N^{-1}$(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N to infty$. Our results confirm Bogoliubov's predictions.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2018-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45255097","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}

{"title":"On topological cyclic homology","authors":"Thomas Nikolaus, Peter Scholze","doi":"10.4310/acta.2018.v221.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2018.v221.n2.a1","url":null,"abstract":"Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $varphi_p: Xto X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=mathrm{cofib}(mathrm{Nm}: X_{hC_p}to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology. In order to construct the maps $varphi_p: Xto X^{tC_p}$ in the example of topological Hochschild homology we introduce and study Tate diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular we prove a version of the Segal conjecture for the Tate diagonals and relate these Frobenius homomorphisms to power operations.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494786","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}

{"title":"Counterexamples to Strassen’s direct sum conjecture","authors":"Y. Shitov","doi":"10.4310/ACTA.2019.V222.N2.A3","DOIUrl":"https://doi.org/10.4310/ACTA.2019.V222.N2.A3","url":null,"abstract":"The multiplicative complexity of systems of bilinear forms (and, in particular, the famous question of fast matrix multiplication) is an important area of research in modern theory of computation. One of the foundational papers on the topic is Strassen’s work [20], which contains an O(n 7/ ln ) algorithm for the multiplication of two n×n matrices. In his subsequent paper [21] published in 1973, Strassen asked whether the multiplicative complexity of the union of two bilinear systems depending on different variables is equal to the sum of the multiplicative complexities of both systems. A stronger version of this problem was proposed in the 1981 paper [10] by Feig and Winograd, who asked whether any optimal algorithm that computes such a pair of bilinear systems must compute each system separately. These questions became known as the direct sum conjecture and strong direct sum conjecture, respectively, and they were attracting a notable amount of attention during the four decades. As Feig and Winograd wrote, ‘either a proof of, or a counterexample to, the direct sum conjecture will be a major step forward in our understanding of complexity of systems of bilinear forms.’ The modern formulation of this conjecture is based on a natural representation of a bilinear system as a three-dimensional tensor, that is, an array of elements T (i|j|k) taken from a field F , where the triples (i, j, k) run over the Cartesian product of finite indexing sets I, J,K. A tensor T is called decomposable if T = a⊗b⊗c (which should be read as T (i|j|k) = aibjck), for some vectors a ∈ FI , b ∈ FJ , c ∈ FK . The rank of a tensor T , or the multiplicative complexity of the corresponding bilinear system, is the smallest r for which T can be written as a sum of r decomposable tensors with entries in F . We denote this quantity by rankF T , and we note that the rank of a tensor may change if one allows to take the entries of decomposable tensors as above from an extension of F , see [3]. Taking the union of two bilinear systems depending on disjoint sets of variables corresponds to the direct sum operation on tensors. More precisely, if T and T ′ are tensors with disjoint indexing sets I, I , J, J ,K,K , then we can define the direct sum T⊕T ′ as a tensor with indexing sets I ∪ I , J ∪ J , K ∪ K ′ such that the (I|J |K) block equals T and (I ′|J ′|K ) block equals T , and all entries outside of these blocks are zero. In other words, direct sums of tensors are a multidimensional analogue of block-diagonal matrices; a basic result of linear algebra says that the ranks of such matrices are equal to the sums of the ranks of their diagonal blocks. Strassen’s direct sum conjecture is a three-dimensional analogue of this statement.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46675899","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}

{"title":"Purely unrectifiable metric spaces and perturbations of Lipschitz functions","authors":"David Bate","doi":"10.4310/ACTA.2020.v224.n1.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2020.v224.n1.a1","url":null,"abstract":"We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $fcolon X to mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $mathcal H^n(f(S))=0$ is residual. Conversely, if $Esubset X$ is $n$-rectifiable with $mathcal H^n(E)>0$, the set of all $f$ with $mathcal H^n(f(E))>0$ is residual. \u0000These results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46725174","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}

{"title":"On the complex Monge-Ampère operator in unbounded domains","authors":"P. Åhag, R. Czyż","doi":"10.4467/20843828am.17.001.7077","DOIUrl":"https://doi.org/10.4467/20843828am.17.001.7077","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48900290","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}

{"title":"Strong property (T) for higher rank lattices","authors":"M. Salle","doi":"10.4310/acta.2019.v223.n1.a3","DOIUrl":"https://doi.org/10.4310/acta.2019.v223.n1.a3","url":null,"abstract":"We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue's strong property (T). Over non-archimedean local fields, we also prove that they have strong Banach proerty (T) with respect to all Banach spaces with nontrivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-cocompact lattices, such as $mathrm{SL}_n(Z)$ for $n geq 3$. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher rank groups have this property and that this property passes to undistorted lattices.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46328008","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}