American Journal of Mathematics

American Journal of Mathematics 122.3, June 2000

Contents

    Hrushovski, Ehud.
    Pillay, Anand.
  • Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties
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    Abstract:
      Let A be a semi-abelian variety, and X a subvariety of A, both defined over a number field. Assume that X does not contain X1 + X2 for any positive-dimensional subvarieties X1, X2 of A. Let G be a subgroup of A(C) of finite rational rank. We give doubly exponential bounds for the size of . Among the ingredients is a uniform bound, doubly exponential in the data, on finite sets which are quantifier-free definable in differentially closed fields. We also give uniform bounds on in the case where X contains no translate of any semi-abelian subvariety of A and G is a subgroup of A(C) of finite rational rank which has trivial intersection with . (Here A is assumed to be defined over a number field, but X need not be.) (Pages 439-450.) Abstract in TeX
    Tao, Terence.
  • Ill-posedness for one-dimensional wave maps at the critical regularity
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    Abstract:
      We show that the wave map equation in R1+1 is in general ill posed in the critical space , and the Besov space . The problem is attributed to the bad behavior of the one-dimensional bilinear expression D-1(fDg) in these spaces. (Pages 451-463.) Abstract in TeX
    Bridgeman, Martin.
    Taylor, Edward C.
  • Length distortion and the Hausdorff dimension of limit sets
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    Abstract:
      Let G be a convex co-compact quasi-Fuchsian Kleinian group. We define the distortion function along geodesic rays lying on the boundary of the convex hull of the limit set, where each ray is pointing in a randomly chosen direction. The distortion function measures the ratio of the intrinsic to extrinsic metrics, and is defined asymptotically as the length of the ray goes to infinity. Our main result is that the distortion function is both almost everywhere constant and bounded above by the Hausdorff dimension of the limit set of G. As a consequence, we are able to provide a geometric proof of the following result of Bowen: If the limit set of G is not a round circle, then the Hausdorff dimension of the limit set is strictly greater than one. The proofs are developed from results in Patterson-Sullivan theory and ergodic theory. (Pages 465-482.) Abstract in TeX
    Schlag, W.
  • On minima of the absolute value of certain random exponential sums
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    Abstract:
      Let where + stands for a random choice of sign with equal probability. It is shown here that with high probability provided n is large and s < 1/12. Similar results are proved for other powers than squares. The problem of determining the optimal s is open. For the case , where d=2,3,... is fixed and with standard normal rj we show that the minima are typically on the order of n-d+1/2 with high probability and for large n. (Pages 483-514.) Abstract in TeX
    Demailly, Jean-Pierre.
    El Goul, Jawher.
  • Hyperbolicity of generic surfaces of high degree in projective 3-space
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    Abstract:
      The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map to the surface is constant. In 1970, Kobayashi conjectured more generally that a (very) generic hypersurface of sufficiently high degree in projective space is hyperbolic. Our technique follows the stream of ideas initiated by Green and Griffiths in 1979, which consists of considering jet differentials and their associated base loci. However, a key ingredient is the use of a different kind of jet bundle, namely the "Semple jet bundles" previously studied by the first named author. The base locus calculation is achieved through a sequence of Riemann-Roch formulas combined with a suitable generic vanishing theorem for order 2-jets. Our method covers the case of surfaces of general type with Picard group and , where q2 is the "2-jet threshold" (bounded below by -1/6 for surfaces in ). The final conclusion is obtained by using recent results of McQuillan on holomorphic foliations. (Pages 515-546.) Abstract in TeX
    Broto, Carlos.
    Levi, Ran.
  • Loop structures on homotopy fibers of self maps of spheres
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    Abstract:
      Let S2n-1{k} denote the fiber of the degree k map on the sphere S2n-1. If k=pr, where p is an odd prime and n divides p-1, then S2n-1{k} is known to be a loop space. It is also known that S3{2r} is a loop space for r > 3. In this paper we study the possible loop structures on this family of spaces for all primes p. In particular we show that S3{4} is not a loop space. Our main result is that whenever S2n-1{pr} is a loop space, the loop structure is unique up to homotopy. (Pages 547-580.) Abstract in TeX
    Dadarlat, Marius.
  • Nonnuclear subalgebras of AF algebras
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    Abstract:
      We show that any non-type I separable unital AF algebra B can be modeled from inside by a nonnuclear C*-algebra and from outside by a nonexact C*-algebra. More precisely there exist unital separable quasidiagonal C*-algebras of real rank zero, stable rank one, such that A is nonnuclear, C is nonexact, and both A and C are asymptotically homotopy equivalent to B. In particular A, B and C have the same ordered K-theory groups, hence isomorphic ideal lattices, and, A and B have (affinely) homeomorphic trace spaces. (Pages 581-597.) Abstract in TeX
    Bendersky, Martin.
    Thompson, Robert D.
  • The Bousfield-Kan spectral sequence for periodic homology theories
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    Abstract:
      We construct the Bousfield-Kan (unstable Adams) spectral sequence based on certain nonconnective periodic homology theories E such as complex periodic K-theory, and define an E-completion of a space X. For X = S2n+1 and E=K we calculate the E2-term and show that the spectral sequence converges to the homotopy groups of the K-completion of the sphere. This also determines all of the homotopy groups of the (unstable) K-theory localization of S2n+1 including three divisible groups in negative stems. (Pages 599-635.) Abstract in TeX
    Kirchberg, Eberhard.
    Rørdam, Mikael.
  • Non-simple purely infinite C*-algebras
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    Abstract:
      A C*-algebra A is defined to be purely infinite if there are no characters on A, and if for every pair of positive elements a,b in A, such that b lies in the closed two-sided ideal generated by a, there exists a sequence {rn} in A such that . This definition agrees with the usual definition by J. Cuntz when A is simple. It is shown that the property of being purely infinite is preserved under extensions, Morita equivalence, inductive limits, and it passes to quotients, and to hereditary sub-C*-algebras. It is shown that is purely infinite for every C*-algebra A. Purely infinite C*-algebras admit no traces, and, conversely, an approximately divisible exact C*-algebra is purely infinite if it admits no nonzero trace. (Pages 637-666.) Abstract in TeX



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