Sets of minimal Hausdorff dimension for quasiconformal maps

For any , there is a compact set of (Hausdorff) dimension whose dimension cannot be lowered by any quasiconformal map . We conjecture that no such set exists in the case . More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.

     

Minkowski's conjecture, well-rounded lattices and topological dimension

Let be the diagonal subgroup, and identify with the space of unimodular lattices in . In this paper we show that the closure of any bounded orbit

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of and .

     

Radix representations, self-affine tiles, and multivariable wavelets

We investigate the connection between radix representations for and self-affine tilings of . We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficiently large.

     

Geometric inequalities for a class of exponential measures

Using -ellipsoids we prove versions of the inverse Santaló inequality and the inverse Brunn-Minkowski inequality for a general class of measures replacing the usual volume on . This class contains in particular the Gaussian measure on .

     

Every real ellipsoid in admits CR umbilical points

We prove that every real ellipsoid admits at least four umbilical points, which can be compared to the result of Webster that a generic real ellipsoid in with does not admit any umbilical point.

     

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on

The local Phragmén-Lindelöf condition for analytic subvarieties of  at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in  . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on  .

     

Legendrian contact homology in

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form , where is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of and, more generally, invariants of self transverse immersions into up to restricted regular homotopies. When , this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.

     

Diophantine approximation by algebraic hypersurfaces and varieties

Questions on rational approximations to a real number can be generalized in two directions. On the one hand, we may ask about ``approximation' to a point in by hyperplanes defined over the rationals. That is, we seek hyperplanes with small distance from the given point. On the other hand, following Wirsing, we may ask about approximation to a real number by real algebraic numbers of degree at most .

The present paper deals with a common generalization of both directions, namely with approximation to a point in by algebraic hypersurfaces, or more generally algebraic varieties defined over the rationals.

     

A priori estimates for second order operators with symplectic characteristic manifold

We prove Fefferman's SAK Principle for a class of classical pseudodifferential operators on with symplectic characteristic manifold.

     

Manifolds with an -action on the tangent bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over with the normal bundle of each section isomorphic to .

     

Real -flats tangent to quadrics in

Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.

     

Interpolation inequalities in Besov spaces

In this paper we present an interpolation inequality in the homogeneous Besov spaces on , which reduces to a number of well-known inequalities in special cases.

     

Enumerative tropical algebraic geometry in

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space and holomorphic curves with certain piecewise-linear graphs there.

     

A modified Brauer algebra as centralizer algebra of the unitary group

The centralizer algebra of the action of on the real tensor powers of its natural module, , is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for and with the decomposition of into irreducible submodules is considered.

     

The size of the Dini subdifferential

Given a lower semicontinuous function , we prove that the points of , where the lower Dini subdifferential contains more than one element, lie in a countable union of sets which are isomorphic to graphs of some Lipschitzian functions defined on . Consequently, the set of all these points has a null Lebesgue measure.

     

Some properties of the gamma and psi functions, with applications

In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume of the unit ball , the surface area of the unit sphere , and some related constants.

     

Lines tangent to

We show that for there are complex common tangent lines to general spheres in and that there is a choice of spheres with all common tangents real.

     

A note on periodic points of order preserving subhomogeneous maps

Let be the standard closed positive cone in and let be the set of integers for which there exists a continuous, order preserving, subhomogeneous map , which has a periodic point with period . It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that is contained in the set consisting of those for which there exist integers and such that , , and for some . This note shows that for all .

     

A moment approach to analyze zeros of triangular polynomial sets

Let be a zero-dimensional ideal of such that its associated set of polynomial equations for all is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in . We also provide a necessary and sufficient (numerical) condition for all the zeros of to be in a given set , without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the 's for to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set . In the proof technique, we use a deep result of Curto and Fialkow (2000) on the -moment problem, and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the 's via the Newton sums of . In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.

     

On commutators of fractional integrals

Let be the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds, and let be the fractional integrals of for . For a BMO function on , we show boundedness of the commutators from to , where . Our result of this boundedness still holds when is replaced by a Lipschitz domain of with infinite measure. We give applications to large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form.