Mostrando recursos 1 - 20 de 449

  1. Non-quasi-projective moduli spaces

    Kollár, János
    We show that every smooth toric variety (and many other algebraic spaces as well) can be realized as a moduli space for smooth, projective, polarized varieties. Some of these are not quasi-projective. This contradicts a recent paper (Quasi-projectivity of moduli spaces of polarized varieties, Ann. of Math. 159 (2004) 597-639.).

  2. Proofs without syntax

    Hughes, Dominic J. D.
    Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a combinatorial proof of a proposition $\phi$ as a graph homomorphism $h:C\to \graphof{\phi}$, where $\graphof{\phi}$ is a graph associated with $\phi\mkern2mu$ and $C$ is a coloured graph. The main theorem is soundness and completeness: $\,\phi$ is true if and only if there exists a combinatorial proof $h:C\to \graphof{\phi}$.

  3. Analytic representation of functions and a new quasi-analyticity threshold

    Kozma, Gady; Olevski?, Alexander
    We characterize precisely the possible rate of decay of the anti-analytic half of a trigonometric series converging to zero almost everywhere.

  4. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing

    Hairer, Martin; Mattingly, Jonathan C.
    The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in $\L^2_0(\TT^2)$. Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when...

  5. A Mass Transference Principle and the Duffin-Schaeffer conjecture for Hausdorff measures

    Beresnevich, Victor; Velani, Sanju
    A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for $\limsup$ subsets of $\R^k$ to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of $\limsup $ sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

  6. Isometric actions of simple Lie groups on pseudoRiemannian manifolds

    Quiroga-Barranco, Raul
    Let $M$ be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group $G$. If $m_0, n_0$ are the dimensions of the maximal lightlike subspaces tangent to $M$ and $G$, respectively, where $G$ carries any bi-invariant metric, then we have $n_0 \leq m_0$. We study $G$-actions that satisfy the condition $n_0 = m_0$. With no rank restrictions on $G$, we prove that $M$ has a finite covering $\widehat{M}$ to which the $G$-action lifts so that $\widehat{M}$ is $G$-equivariantly diffeomorphic to an action on a double coset $K\backslash L/\Gamma$, as considered in Zimmer's...

  7. Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles

    Avila, Artur; Krikorian, Raphaël
    We show that for almost every frequency $\alpha \in \R \setminus \Q$, for every $C^\omega$ potential $v:\R/\Z \to \R$, and for almost every energy $E$ the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrödinger operator, and allows us to complete the proof of the Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator.

  8. A Paley-Wiener theorem for reductive symmetric spaces

    van den Ban, E. P.; Schlichtkrull, H.
    Let $X=G/H$ be a reductive symmetric space and $K$ a maximal compact subgroup of $G$. The image under the Fourier transform of the space of $K$-finite compactly supported smooth functions on $X$ is characterized.

  9. Orbit equivalence rigidity and bounded cohomology

    Monod, Nicolas; Shalom, Yehuda
    We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type $II_1$ equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic.

  10. Global hyperbolicity of renormalization for $C^r$ unimodal mappings

    de Faria, Edson; de Melo, Welington; Pinto, Alberto
    In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of $C^r$ unimodal maps with quadratic critical point. We show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided $r \ge 2+\alpha$ with $\alpha$ close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are $C^1$...

  11. Deligne's integrality theorem in unequal characteristic and rational points over finite fields

    Esnault, Hélène
    If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its étale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective regular model carries a rational point. As a consequence, if the Chow group of 0-cycles of $V$ over a large algebraically closed field is trivial, then the mod $p$ reduction of a projective regular model carries a rational point.

  12. The $\dbar_b$-complex on decoupled boundaries in $\C^n$

    Nagel, Alexander; Stein, Elias M.

  13. Combinatorics of random processes and sections of convex bodies

    Rudelson, M.; Vershynin, R.
    We find a sharp combinatorial bound for the metric entropy of sets in $\R^n$ and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in $\R^n$. In the operator theory, this essentially proves for all normed spaces the restricted invertibility principle of Bourgain and Tzafriri.

  14. Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products

    Costello, Kevin
    I prove a formula expressing the descendent genus $g$ Gromov-Witten invariants of a projective variety $X$ in terms of genus $0$ invariants of its symmetric product stack $S^{g+1}(X)$. When $X$ is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the Gromov-Witten invariants of a point.

  15. Invariant measures and the set of exceptions to Littlewood's conjecture

    Ensiedler, Manfred; Katok, Anatole; Lindenstrauss, Elon
    We classify the measures on $\SL (k,\Bbb R) / \SL (k, \ZZ)$ which are invariant and ergodic under the action of the group $A$ of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.

  16. Schubert induction

    Vakil, Ravi
    We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule of "A geometric Littlewood-Richardson Rule." As applications, we show that all Schubert problems for all Grassmannians are enumerative over the real numbers, and sufficiently large finite fields. We prove a generic smoothness theorem as a substitute for the Kleiman-Bertini theorem in positive characteristic. We compute the monodromy groups of many Schubert problems, and give some surprising examples where the monodromy group is much smaller than the full symmetric group.

  17. Automorphic distributions, $L$-functions, and Voronoi summation for ${\rm GL}(3)$

    Miller, Stephen D.; Schmid, Wilfried

  18. A geometric Littlewood-Richardson Rule

    Vakil, Ravi
    We describe a geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are $1$; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has straightforward bijections to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. Geometric consequences are described here. For example, the rule also has an...

  19. Higher-order tangents and Fefferman's paper on Whitney's extension problem

    Bierstone, Edward; Milman, Pierre D.; Paw?ucki, Wies?aw
    Whitney proved that a function defined on a closed subset of $\IR$ is the restriction of a $\cC^m$ function if the limiting values of all $m^{\rm th}$ divided differences form a continuous function. We show that Fefferman's solution of Whitney's problem for $\IR^n$ is equivalent to a variant of our conjecture in our article, "Differentiable functions defined in closed sets, A problem of Whitney," giving a criterion for $\cC^m$ extension in terms of iterated limits of finite differences.

  20. Whitney's extension problem for $C^m$

    Fefferman, Charles
    Let $f$ be a real-valued function on a compact set in $\mathbb{R}^n$, and let $m$ be a positive integer. We show how to decide whether $f$ extends to a $\mathbb{C}^m$ function on $\mathbb{R}^n$

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