Volume 2005, Issue 589

Let (ℝ n   , || ⋅ ||) be the space ℝ N equipped with a norm || ⋅ || whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N   ×  n matrix with N   >  n  , whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n  -dimensional Euclidean space (ℝ N  , | ⋅ |) onto its image in (ℝ N  , || ⋅ ||) , i.e. there exist α ,  β  > 0 such that for all  x  ∈ ℝ N , . This solves a conjecture of Schechtman on random embeddings of ℓ 2 n into ℓ 1 N  .

The 1-covering property of omega limit sets is established for monotone and uniformly stable skew-product semiflows with the componentwise separating property of bounded and ordered full orbits. Then these results are applied to study the asymptotic almost periodicity of solutions to almost periodic reaction-diffusion equations and differential systems with time delays. The earlier convergence results for autonomous and periodic monotone systems are generalized to the almost periodic case without the strong monotonicity assumption.

We consider relationships between families of K3 surfaces. Donagi and Pantev [ R. Donagi and T. Pantev , Torus fibrations, gerbes, and duality, arXiv: math.AG/0306123] have exhibited a duality between derived categories of twisted sheaves on certain genus one fibred K3 surfaces. We investigate partial extensions of this result to the more general case of non fibred K3s. We also provide an explicit construction of a quasiuniversal sheaf associated to the classical Mukai correspondence [ S. Mukai , Moduli of vector bundles on K3 surfaces, and symplectic manifolds, Sugaku Expos. 1 (1988), no. 2, 139–174, Sugaku 39 (1987), 216–235].

The classical Kloosterman sums give rise to a Galois representation of the function field unramified outside 0 and ∞. We study the local monodromy of this representation at ∞ using l  -adic method based on the work of Deligne and Katz. As an application, we determine the degrees and the bad factors of the L -functions of the symmetric products of the above representation. Our results generalize some results of Robba obtained through p  -adic method.

Consider a Hamiltonian action of a compact Lie group K  on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map κ K . We start with the case when K  is a torus T  : we determine the kernel of the equivariant Kirwan map (defined by Goldin in [ R. F. Goldin , An effective algorithm for the cohomology ring of symplectic reductions, Geom. Func. Anal. 12 (2002), 567–583]) corresponding to a generic circle S   ⊂  T , and show how to recover from this the kernel of κ T  , as described by Tolman and Weitsman in [ S. Tolman and J. Weitsman , The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [ L. C. Jeffrey , The residue formula and the Tolman-Weitsman theorem, J. reine angew. Math. 562 (2003), 51–58], [ L. C. Jeffrey and A.-L. Mare , The kernel of the equivariant Kirwan map and the residue formula, Quart. J. Math. Oxford 54 (2004), 435–444].) For a compact nonabelian Lie group K  we will use the ‘‘non-abelian localization formula’’ of [ L. C. Jeffrey and F. C. Kirwan , Localization for nonabelian group actions, Topology 34 (1995), 291–327] and [ L. C. Jeffrey and F. C. Kirwan , Localization and the quantization conjecture, Topology 36 (1995), 647–693] to establish relationships—some of them obtained by Tolman and Weitsman in [ S. Tolman and J. Weitsman , The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]—between Ker( κ K  ) and Ker( κ T  ), where T   ⊂  K  is a maximal torus. In the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of μ T  .

We study the moduli space of framed holomorphic bundles of any rank, over the blow-up of ℂℙ 2 at q  points. For c 2  = 1, 2 we introduce an open cover of the moduli space and describe its nerve. In the limit when r → ∞ we use this result to obtain the homotopy type of the moduli spaces. In particular, we compute the cohomology of the moduli spaces.

For a function field F |  over a finite field of cardinality  ℓ  , denote by g  ( F  ) (resp. N  ( F  )) the genus (resp. the number of rational places) of F |  . In this paper we present an explicit tower of function fields  over for  ℓ   =  q   3 , such that lim r → ∞   N  ( F r   )  | g  ( F r   ) ≧ 2( q   2  − 1)  |  ( q  + 2).

An action of a finite group on a smooth projective curve over an algebraically closed field of positive characteristic is called weakly ramified , if all second ramification groups are trivial (e.g., every group action on an ordinary curve is weakly ramified). When the ramification indices satisfy certain numerical criteria, we construct a degenerating equivariant quasi-projective family to which the given curve belongs, and which in a sense is the unique building block for all such weakly ramified equivariant families that ramify above a fixed set of points. The result is used to inductively study automorphisms of ordinary curves.