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作者：Monod, Nicolas; Shalom, Yehuda
摘要：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 ...
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作者：Andersen, Jorgen Ellegaard
摘要：We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)-Verlinde bundles over Teichmuller space, is asymptotically faithful. That is, the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least 3. For the genus 2 case, this intersection is exactly the order 2 subgroup, generated by the hyper-elliptic involution, in the case of e...
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作者：Darmon, Henri; Dasgupta, Samit
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作者：Popa, Sorin
摘要：We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A subset of M, the Betti numbers of the standard eqilivalence relation associated with A subset of M ([G2]), are in fact isomorphism invariants for HT the factors M, beta n(HT) (M), n >= 0. The class HT is closed under amplifications and tensor product...
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作者：Szemeredi, E.; Vu, V. H.
摘要：We prove that if A is a subset of at least cn(1/2) elements of {1,..., n}, where c is a sufficiently large constant, then the collection of subset sums of A contains an arithmetic progression of length n. As an application, we confirm a long standing conjecture of Erdos and Folkman on complete sequences.
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作者：Miller, Stephen D.; Schmid, Wilfried
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作者：Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott
摘要：We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is-the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of th...
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作者：Quiroga-Barranco, Raul
摘要：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) <= 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 (M) over cap to which the G-action lifts s...
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作者：Rudelson, M.; Vershynin, R.
摘要：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 bo...
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作者：Avila, Artur; Krikorian, Raphael
摘要：We show that for almost every frequency alpha epsilon R\Q, for every C-omega potential nu: R/Z -> R, and for almost every energy E the corresponding quasiperiodic Schrodinger 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 Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Opera...
