On the representation of integers as linear combinations of consecutive values of a polynomial

Let be a cyclotomic field with ring of integers and let be a polynomial whose values on belong to . If the ideal of generated by the values of on is itself, then every algebraic integer of may be written in the following form:

for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where

     

Hardy space of exact forms on

We show that the Hardy space of divergence-free vector fields on has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of . Using the duality result we prove a ``div-curl" type theorem: for in , is equivalent to a -type norm of , where the supremum is taken over all with This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on , study their atomic decompositions and dual spaces, and establish ``div-curl" type theorems on .

     

Maximal holonomy of infra-nilmanifolds with -dimensional quaternionic Heisenberg geometry

Let be the quaternionic Heisenberg group of real dimension and let denote the maximal order of the holonomy groups of all infra-nilmanifolds with -geometry. We prove that . As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a -dimensional quaternionic hyperbolic manifold with cusps is at least

     

Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

Weighted estimates in for Laplace's equation on Lipschitz domains

Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

     

Lack of natural weighted estimates for some singular integral operators

We show that the classical Hörmander condition, or analogously the -Hörmander condition, for singular integral operators is not sufficient to derive Coifman's inequality

where , is the Hardy-Littlewood maximal operator, is any weight and is a constant depending upon and the constant of . This estimate is well known to hold when is a Calderón-Zygmund operator.

As a consequence we deduce that the following estimate does not hold:

where and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator.

One of the main ingredients of the proof is a very general extrapolation theorem for weights.

     

Cremer fixed points and small cycles

Let , , and let denote the sequence of convergents to the regular continued fraction of . Let be a function holomorphic at the origin, with a power series of the form . We assume that for infinitely many we simultaneously have (i) , (ii) the coefficients stay outside two small disks, and (iii) the series is lacunary, with for . We then prove that has infinitely many periodic orbits in every neighborhood of the origin.

     

A generalization of half-plane mappings to the ball in

Let be a normalized (, ) biholomorphic mapping of the unit ball onto a convex domain that is the union of lines parallel to some unit vector . We consider the situation in which there is one infinite singularity of on . In one case with a simple change-of-variables, we classify all convex mappings of that are half-plane mappings in the first coordinate. In the more complicated case, when is not in the span of the infinite singularity, we derive a form of the mappings in dimension .

     

Hyperpolygon spaces and their cores

Given an -tuple of positive real numbers , Konno (2000) defines the hyperpolygon space , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the core. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

     

Dimension of families of determinantal schemes

A scheme of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers and we denote by (resp. ) the locus of good (resp. standard) determinantal schemes of codimension defined by the maximal minors of a matrix where is a homogeneous polynomial of degree .

In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. ) in terms of and , (2) whether the closure of is an irreducible component of , and (3) when is generically smooth along . Concerning question (1) we give an upper bound for the dimension of (resp. ) which works for all integers and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions.