American Journal of Mathematics 122.3, June 2000
Contents
Hrushovski, Ehud.
Pillay, Anand.

Effective bounds for the number of transcendental points on subvarieties
of semiabelian varieties
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Abstract:
Let A be a semiabelian variety, and X a subvariety of
A, both defined over a number field. Assume that X does not contain X_{1} + X_{2} for any
positivedimensional subvarieties X_{1}, X_{2} 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 quantifierfree definable in differentially closed fields.
We also give uniform bounds on _{} in the case where X contains no translate of any
semiabelian 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 439450.)
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Tao, Terence.

Illposedness for onedimensional wave maps at the critical regularity
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Abstract:
We show that the wave map equation in R^{1+1} is in general ill posed
in the critical space _{}, and the Besov space
_{}. The problem is attributed to the bad behavior
of the onedimensional bilinear expression D^{1}(fDg) in
these spaces. (Pages 451463.)
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Bridgeman, Martin.
Taylor, Edward C.

Length distortion and the Hausdorff dimension of limit sets
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Abstract:
Let G be a convex cocompact quasiFuchsian 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
PattersonSullivan theory and ergodic theory.
(Pages 465482.)
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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 r_{j} we show that the minima are typically on the order
of n^{d+1/2} with high probability and for large n.
(Pages 483514.)
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Demailly, JeanPierre.
El Goul, Jawher.

Hyperbolicity of generic surfaces of high degree in projective 3space
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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 3dimensional 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 RiemannRoch formulas combined
with a suitable generic vanishing theorem for order 2jets. Our method covers the case
of surfaces of general type with Picard group _{} and _{}, where q_{2} is the "2jet threshold"
(bounded below by 1/6 for surfaces in _{}). The final
conclusion is obtained by using recent results of McQuillan on holomorphic
foliations. (Pages 515546.)
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Broto, Carlos.
Levi, Ran.

Loop structures on homotopy fibers of self maps of spheres
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Abstract:
Let S^{2n1}{k} denote the fiber of the degree k map on the
sphere S^{2n1}. If k=p^{r}, where p is an odd prime and n
divides p1, then S^{2n1}{k} is known to be a loop space. It is
also known that S^{3}{2^{r}} 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 S^{3}{4} is
not a loop space. Our main result is that whenever S^{2n1}{p^{r}} is
a loop space, the loop structure is unique up to homotopy. (Pages 547580.)
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Dadarlat, Marius.

Nonnuclear subalgebras of AF algebras
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Abstract:
We show that any nontype 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 Ktheory groups, hence
isomorphic
ideal lattices, and, A and B have (affinely) homeomorphic trace
spaces. (Pages 581597.)
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Bendersky, Martin.
Thompson, Robert D.

The BousfieldKan spectral sequence for periodic homology theories
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Abstract:
We construct the BousfieldKan (unstable Adams) spectral sequence
based on certain nonconnective periodic homology theories E such
as complex periodic Ktheory, and define an Ecompletion of a
space X. For X = S^{2n+1} and E=K we calculate the
E_{2}term and show that the spectral sequence converges to
the homotopy groups of the Kcompletion of the sphere. This also
determines all of the homotopy groups of the (unstable) Ktheory
localization of S^{2n+1} including three divisible groups in
negative stems. (Pages 599635.)
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Kirchberg, Eberhard.
Rørdam, Mikael.

Nonsimple 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 twosided ideal generated by a, there exists a sequence
{r_{n}} 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 subC*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 637666.)
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