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Acta Mathematica, 2012
ISSN:
0001-5962 (Print)
1871-2509 (Online)
In this year (16 articles)
Volume 211, Issue 1
[1] Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory
Bob Oliver Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539 du CNRS, 99, Av. J-B Clément, Villetaneuse, France
K-Theory and Homology Rings and Algebras mathscidoc:1701.20002
Acta Mathematica, 211, (1), 141-175, 2011.8
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We present a version of a proof by Andy Chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems. This proof differs from the one in [Ch2] in that it is based on the computation of derived functors of certain inverse limits. This leads to a much shorter proof, but one which is aimed mostly at researchers familiar with homological algebra.
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[2] Analyticity of the Stokes semigroup in spaces of bounded functions
Ken Abe Graduate School of Mathematical Sciences, University of Tokyo Yoshikazu Giga Graduate School of Mathematical Sciences, University of Tokyo
Functional Analysis Spectral Theory and Operator Algebra mathscidoc:1701.12004
Acta Mathematica, 211, (1), 1-46, 2011.7
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The analyticity of the Stokes semigroup with the Dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the Helmholtz decomposition. The proof is based on a blow-up argument. This is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years.
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[3] Fusion systems and localities
Andrew Chermak Mathematics Department, Kansas State University
Group Theory and Lie Theory mathscidoc:1701.17001
Acta Mathematica, 211, (1), 47-139, 2011.8
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We introduce$objective partial groups$, of which the linking systems and$p$-local finite groups of Broto, Levi, and Oliver, the transporter systems of Oliver and Ventura, and the $${\mathcal{F}}$$ -localities of Puig are examples, as are groups in the ordinary sense. As an application we show that if $${\mathcal{F}}$$ is a saturated fusion system over a finite$p$-group then there exists a centric linking system $${\mathcal{L}}$$ having $${\mathcal{F}}$$ as its fusion system, and that $${\mathcal{L}}$$ is unique up to isomorphism. The proof relies on the classification of the finite simple groups in an indirect and—for that reason—perhaps ultimately removable way.
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Volume 211, Issue 2
[1] Embedded minimal tori in$S$^{3}and the Lawson conjecture
Simon Brendle Department of Mathematics, Stanford University
Analysis of PDEs Differential Geometry mathscidoc:1701.03010
Acta Mathematica, 211, (2), 177-190, 2012.4
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[2] The energy density in the planar Ising model
Clément Hongler Department of Mathematics, Columbia University Stanislav Smirnov Section de Mathématiques, Université de Genève
Dynamical Systems Probability mathscidoc:1701.11005
Acta Mathematica, 211, (2), 191-225, 2011.7
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We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.
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[3] Isotropy of unitary involutions
Nikita Karpenko Department of Mathematical and Statistical Sciences, University of Alberta Maksim Zhykhovich Johannes Gutenberg-Universität, Institut für Mathematik
mathscidoc:1701.01005
Acta Mathematica, 211, (2), 227-253, 2011.6
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We prove the so-called$unitary isotropy theorem$, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.
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[4] Proof of the BMV conjecture
Herbert R. Stahl Beuth Hochschule/FB II, Luxemburger Str. 10, Berlin, Germany
Rings and Algebras mathscidoc:1701.31001
Acta Mathematica, 211, (2), 255-290, 2012.8
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We prove the BMV (Bessis, Moussa, Villani, [1]) conjecture, which states that the function $${t \mapsto \mathop{\rm Tr}\exp(A-tB)}$$ , $${t \geqslant 0}$$ , is the Laplace transform of a positive measure on [0,∞) if$A$and$B$are $${n \times n}$$ Hermitian matrices and$B$is positive semidefinite. A semi-explicit representation for this measure is given.
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[5] On divisors of Lucas and Lehmer numbers
Cameron L. Stewart Department of Pure Mathematics, University of Waterloo
Number Theory mathscidoc:1701.24002
Acta Mathematica, 211, (2), 291-314, 2012.2
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Let$u$_{$n$}be the$n$th term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of$u$_{$n$}which is of the form$n$exp(log$n$/104 log log$n$). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.
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[6] $p$-adic logarithmic forms and a problem of Erdős
Kunrui Yu Department of Mathematics, Hong Kong University of Science and Technology
Number Theory mathscidoc:1701.24003
Acta Mathematica, 211, (2), 315-382, 2011.6
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For any natural number$m$(>1) let$P$($m$) denote the greatest prime divisor of$m$. By the problem of Erdős in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that $$\frac{P(2^n-1)}{n} \to \infty \quad {\rm as}\, n \to \infty$$ (see Erdős [10]) and its far-reaching generalization to Lucas and Lehmer numbers. In the present paper the author delivers three refinements upon Yu [40] required by C. L. Stewart for solving completely the problem of Erdős (see Stewart [25]). The author gives also some remarks on the solution of this problem, aiming to be more streamlined with respect to the$p$-adic theory of logarithmic forms.
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Volume 212, Issue 1
[1] A sharp lower bound for the log canonical threshold
Jean-Pierre Demailly Université de Grenoble I, Département de Mathématiques, Institut Fourier, Saint-Martin d’Hères, France Hoàng Hiệp Phạm Department of Mathematics, Hanoi National University of Education
Analysis of PDEs mathscidoc:1701.03011
Acta Mathematica, 212, (1), 1-9, 2012.1
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In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $${\varphi}$$ with an isolated singularity at 0 in an open subset of $${\mathbb{C}^n}$$ . This threshold is defined as the supremum of constants$c$> 0 such that $${e^{-2c\varphi}}$$ is integrable on a neighborhood of 0. We relate $${c(\varphi)}$$ to the intermediate multiplicity numbers $${e_j(\varphi)}$$ , defined as the Lelong numbers of $${(dd^c\varphi)^j}$$ at 0 (so that in particular $${e_0(\varphi)=1}$$ ). Our main result is that $${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$$ . This inequality is shown to be sharp; it simultaneously improves the classical result $${c(\varphi)\geqslant 1/e_1(\varphi)}$$ due to Skoda, as well as the lower estimate $${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.
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[2] Geometric quantization for proper moment maps: the Vergne conjecture
Xiaonan Ma Institut de Mathématiques de Jussieu, Université Paris Diderot – Paris 7 Weiping Zhang Chern Institute of Mathematics& LPMC, Nankai University
Differential Geometry mathscidoc:1701.10005
Acta Mathematica, 212, (1), 11-57, 2012.6
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We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.
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[3] Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton
Yvan Martel Université de Versailles St-Quentin and Institut Universitaire de France, LMV CNRS UMR8100, Versailles Cedex, France Frank Merle Université de Cergy Pontoise and Institut des Hautes Études Scientifiques, AGM CNRS UMR8088, Paris, France Pierre Raphaël Université Paul Sabatier and Institut Universitaire de France, IMT CNRS UMR 5219, Toulouse, France
Analysis of PDEs mathscidoc:1701.03012
Acta Mathematica, 212, (1), 59-140, 2012.2
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We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in$H$^{1}close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].
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[4] Unique Cartan decomposition for II_{1}factors arising from arbitrary actions of free groups
Sorin Popa Mathematics Department, University of California, Los Angeles Stefaan Vaes Department of Mathematics, KU Leuven
Functional Analysis Group Theory and Lie Theory Spectral Theory and Operator Algebra mathscidoc:1701.12005
Acta Mathematica, 212, (1), 141-198, 2012.6
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We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$ of a free group on$n$generators $${\mathbb{F}_n, 2\leq n \leq \infty}$$ , the associated group measure space II_{1}factor $${L^\infty (X)\rtimes \mathbb{F}_n}$$ has$L$^{∞}($X$) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_{1}factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_{1}factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.
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Volume 212, Issue 2
[1] The thick-thin decomposition and the bilipschitz classification of normal surface singularities
Lev Birbrair Departamento de Matemática, Universidade Federal do Ceará Walter D. Neumann Department of Mathematics, Barnard College, Columbia University Anne Pichon Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France
Complex Variables and Complex Analysis mathscidoc:1701.01006
Acta Mathematica, 212, (2), 199-256, 2012.1
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We describe a natural decomposition of a normal complex surface singularity ($X$, 0) into its “thick” and “thin” parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts.
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[2] Stable moduli spaces of high-dimensional manifolds
Søren Galatius Department of Mathematics, Stanford University Oscar Randal-Williams Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
Geometric Analysis and Geometric Topology mathscidoc:1701.15001
Acta Mathematica, 212, (2), 257-377, 2012.9
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We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of$g$copies of$S$^{$n$}×$S$^{$n$}, in the limit $${g \to \infty}$$ . Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture.
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[3] Proof of an entropy conjecture for Bloch coherent spin states and its generalizations
Elliott H. Lieb Departments of Physics and Mathematics, Jadwin Hall, Princeton University Jan Philip Solovej Department of Mathematics, University of Copenhagen
Mathematical Physics mathscidoc:1701.22002
Acta Mathematica, 212, (2), 379-398, 2012.9
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Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum$J$. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to $${K=J+\frac{1}{2}, J+1, ... ,}$$ with $${K=\infty}$$ corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each$J$and $${J < K \leqslant \infty}$$ we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.
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Impact Factor
3.719
Available
1882 - 2016
Volumes
228
Issues
301
Articles
2167