Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.\n¶ This paper provides the details of the results announced in Christensen et al. Proc. Natl. Acad. Sci. USA 107 (2010), 587–591.

If X is a closed subset of the real line, denote by GX the supremum of the size of the gap in the Fourier spectrum of a measure, taken over all non-trivial finite complex measures supported on X. In this paper we attempt to find GX in terms of X.

Let $ \\gamma = \\frac{1}{2}\\left( {1 + \\sqrt {5} } \\right) $ denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant c > 0 with the property that, for arbitrarily large values of X, the inequalities$ \\left| {{x_0}} \\right| \\leqslant X,\\,\\,\\,\\left| {{x_0}\\xi - {x_1}} \\right| \\leqslant c{X^{{{{ - 1}} \\left/ {\\gamma } \\right.}}}\\,\\,\\,{\\text{and}}\\,\\,\\,\\left| {{x_0}{\\xi^2} - {x_2}} \\right| \\leqslant c{X^{{{{ - 1}} \\left/ {\\gamma } \\right.}}} $admit no non-zero solution $ \\left( {{x_0},{x_1},{x_2}} \\right) \\in {\\mathbb{Z}^3} $. Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X ≥ 1. Such extremal numbers are transcendental and their set is stable under the action of $ {\\text{G}}{{\\text{L}}_2}\\left( \\mathbb{Z} \\right) $ on $ \\mathbb{R}\\backslash \\mathbb{Q} $ by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf<sub>q→∞</sub>q||qξ|| is $ \\frac{1}{3} $, the largest possible value for a non-quadratic number, and that there is a natural bijection between the $ {\\text{G}}{{\\text{L}}_2}\\left( \\mathbb{Z} \\right) $-equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.

We consider Stokes' conjecture concerning the shape of the extremal two-dimensional water wave. By new geometric methods including a nonlinear frequency formula, we prove Stokes' conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types ${E_{n}^{(1)}}$ , n=6, 7, 8) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases).

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in Rn × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].

We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.

Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W<sup>1, p</sup>(R<sup>p+1</sup>, N) be the Sobolev space of measurable maps from R<sup>p+1</sup> into N whose gradients are in L<sup>p</sup>. The restriction of u to almost every p-dimensional sphere S in R<sup>p+1</sup> is in W<sup>1, p</sup>(S, N) and defines an homotopy class in π<sub>p</sub>(N) (White 1988). Evaluating a fixed element z of Hom(π<sub>p</sub>(N), R) on this homotopy class thus gives a real number Φ<sub>z, u</sub>(S). The main result of the paper is that any W<sup>1, p</sup>-weakly convergent limit u of a sequence of smooth maps in C<sup>∞</sup>(R<sup>p+1</sup>, N), Φ<sub>z, u</sub> has a rectifiable Poincaré dual$ {\\left( {\\Gamma ,{\\overrightarrow{\\Gamma }} ,\\theta } \\right)} $. Here Γ is a a countable union of C<sup>1</sup> curves in R<sup>p+1</sup> with Hausdorff $ {\\user1{\\mathcal{H}}}^{1} $-measurable orientation $ {\\overrightarrow{\\Gamma }} :\\Gamma \\to S^{p} $ and density function θ: Γ→R. The intersection number between $ {\\left( {\\Gamma ,{\\overrightarrow{\\Gamma }} ,\\theta } \\right)} $ and S evaluates Φ<sub>z, u</sub>(S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n<sub>z</sub>, depending only on homotopy operation z, such that $ {\\int_\\Gamma {{\\left| \\theta \\right|}^{{p \\mathord{\\left/ {\\vphantom {p {{\\left( {p + n_{z} } \\right)}}}} \\right. \\kern-\\nulldelimiterspace} {{\\left( {p + n_{z} } \\right)}}}} d{\\user1{\\mathcal{H}}}^{1} < \\infty } } $ even though the mass $ {\\int_\\Gamma {{\\left| \\theta \\right|}d{\\user1{\\mathcal{H}}}^{1} } } $ may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n<sub>z</sub>) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to R<sup>m</sup> for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.

Par des méthodes de nature probabiliste, nous démontrons des nouveaux résultats de rigidité pour des groupes et des pseudo-groupes de difféomorphismes de variétés unidimensionnelles dont la classe de différentiabilité est intermédiaire (i.e. entre C1 et C2). En particulier, nous prouvons des généralisations du théorème de Denjoy et d'un lemme classique de Kopell pour des groupes abéliens. Ensuite, nous appliquons les techniques introduites à l'étude des feuilletages de codimension 1 dont la régularité transverse est intermédiaire. Nous obtenons notamment des versions généralisées du théorème de Sacksteder en classe C1. Nous finissons par quelques remarques à propos de la mesure stationnaire.