Volume 2007, Issue 611

This paper gives a geometric method for uniformly bounding some classes of real oscillatory integrals with phase λ 1 P 1 + λ 2 P 2 , P 1 , P 2 real analytic or polynomial functions. A nontrivial decay rate uniform in (λ 1 , λ 2 ) is shown to exist and characterized geometrically when the mapping x → ( P 1 ( x ), P 2 ( x )) can be “uniformized”, an analog for maps of local uniformization for varieties.

We consider a nonlocal perturbation of an isoperimetric variational problem. The problem may be viewed as a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation associated with competing short and long-range interactions. In particular, it arises as a Γ-limit of a model for microphase separation of diblock copolymers. In this article, we establish precise conditions for criticality and stability (i.e. we explicitly compute the first and second variations). We also present some applications.

Suppose that is a Lucas sequence, and suppose that l 1 , … , l t are primes. We show that the equation has only finitely many solutions. Moreover, we explain a practical method of solving these equations. For example, if is the Fibonacci sequence, then we solve the equation under the restrictions: p is prime and m < p .

For any measure preserving system ( X ,, μ , T ) and A ∈ with μ ( A ) > 0, we show that there exist infinitely many primes p such that (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 ( μ ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p .

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E ⊂ℂ N by a regular polynomial map F , expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E . We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F .

We compute rationally the topological (complex) K -theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW -model for its classifying space for proper G -actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.

This paper is the first response to a question of Igusa that asked for a multivariable extension of his adelic Poisson formula in one variable. We prove a Poisson summation formula over the adèles between two distributions, depending upon two variables. Each distribution is defined by a mapping P : ℚ n → ℚ 2 whose components are homogeneous polynomials, and whose critical locus is a union of lines.

We study the hyperbolicity of the log variety (ℙ n , X ), where X is a very general hypersurface of degree d ≧ 2 n + 1 (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of (ℙ n , X ) is of log-general type, give a new proof of the algebraic hyperbolicity of (ℙ n , X ), and exclude the existence of maximal rank families of entire curves in the complement of the universal degree d hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.