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作者：Gowers, W. T.
摘要：We prove analogues for hypergraphs of Szemeredi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemeredi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, Rodl, Schacht and Skokan.
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作者：Beresnevich, Victor; Dickinson, Detta; Velani, Sanju; Vaughan, R. C.
摘要：Let C be a nondegenerate planar curve and for a real, positive decreasing function psi let C(psi) denote the set of simultaneously psi-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(psi) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results...
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作者：McMullen, Curtis T.
摘要：This paper classifies orbit closures and invariant measures for the natural action of SL2(R) on Omega M-2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.
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作者：Adamczewski, Boris; Bugeaud, Yann
摘要：Let b > 2 be an integer. We prove that the b-ary expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.
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作者：Bruedern, Joerg; Wooley, Trevor D.
摘要：By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables.
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作者：Epstein, Charles L.
摘要：We assume that the manifold with boundary, X, has a Spin(C)-structure with spinor bundle $. Along the boundary, this structure agrees with the structure defined by an infinite order, integrable, almost complex structure and the metric is Kahler. In this case the Spin(C)-Dirac operator a agrees with (partial derivative) over bar +(partial derivative) over bar* along the boundary. The induced CR-structure on bX is integrable and either strictly pseudoconvex or strictly pseudoconcave. We assume t...
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作者：Gursky, Matthew J.; Viaclovsky, Jeff A.
摘要：We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
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作者：Kronheimer, P.; Mrowka, T.; Ozsvath, P.; Szabo, Z.
摘要：Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do ...
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作者：Fefferman, Charles
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作者：Mazur, Barry; Rubin, Karl
摘要：We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K- denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(K-/K) by inversion). We prove (under mild hypotheses on p) that if th...
