Links in an open book decomposition and in the standard contact structure

We study a relationship between arc presentations of links in and Legendrian links in with the standard tight contact structure. We determine the arc indices of torus knots.

     

Toric hypersymplectic quotients

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space by the action of a compact Abelian group. These -dimensional quotients carry a multi-Hamilitonian action of an -torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in . We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on and metrics on complements of hypersurfaces in compact manifolds.

     

A new result on the Pompeiu problem

A nonempty bounded open set () is said to have the Pompeiu property if and only if the only continuous function on for which the integral of over is zero for all rigid motions of is . We consider a nonempty bounded open set with Lipschitz boundary and we assume that the complement of is connected. We show that the failure of the Pompeiu property for implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in , , has the Pompeiu property. So far the result was proved only for solid tori in . We also examine the case of planar domains. Finally we extend the example of solid tori to domains in bounded by hypersurfaces of revolution.

     

Branch structure of -holomorphic curves near periodic orbits of a contact manifold

Let be a three-dimensional contact manifold, and a finite-energy pseudoholomorphic map from the punctured disc in that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into , where denotes a rational ellipsoid (contact structure induced by the standard complex structure on ), as well as contact structures arising from non-standard circle-fibrations of the three-sphere.

     

Conformal actions of on Lorentz manifolds

We prove that, for , a locally faithful action of or of by conformal transformations of a connected Lorentz manifold must be a proper action.

     

Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation

We establish that the initial value problem for the quadratic non-linear Schrödinger equation

where , is locally well-posed in when . The critical exponent for this problem is , and previous work by Colliander, Delort, Kenig and Staffilani, 2001, established local well-posedness for .

     

Quantum cohomology of Hilb and enumerative applications

We compute the small quantum cohomology of Hilb and determine recursively most of the big quantum cohomology. We prove a relationship between the invariants so obtained and the enumerative geometry of hyperelliptic curves in . This extends the results obtained by Graber (2001) for Hilb and hyperelliptic curves in .

     

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.

     

Generalized stationary random fields with linear regressions - an operator approach

Existence, -stationarity and linearity of conditional expectations of square integrable random sequences satisfying

for a real sequence is examined. The analysis is reliant upon the use of Laurent and Toeplitz operator techniques.

     

Complete minimal hypersurfaces in the hyperbolic space with vanishing Gauss-Kronecker curvature

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, a nowhere vanishing second fundamental form and a scalar curvature bounded from below.

     

On the biharmonic and harmonic indices of the Hopf map

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map and modify it into a nonharmonic biharmonic map . We show to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

     

Limiting weak-type behavior for singular integral and maximal operators

The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator in : for , ,

For the maximal operator , the corresponding result is

     

Spherical classes and the Lambda algebra

Let be Singer's invariant-theoretic model of the dual of the lambda algebra with , where denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, , into is a chain-level representation of the Lannes-Zarati dual homomorphism

The Lannes-Zarati homomorphisms themselves, , correspond to an associated graded of the Hurewicz map

Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism. One of these algebraic conjectures predicts that every Dickson element, i.e. element in , of positive degree represents the homology class in for 2$">.

We also show that factors through , where denotes the differential of . Therefore, the problem of determining should be of interest.

     

On symplectic fillings of lens spaces

Let be the contact structure naturally induced on the lens space by the standard contact structure on the three-sphere . We obtain a complete classification of the symplectic fillings of up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.

     

Average size of -Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.

     

Whitney's extension problem for multivariate -functions

We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that

The constant is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.

     

On the Castelnuovo-Mumford regularity of connected curves

In this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1. We also investigate the structure of connected curves for which this bound is optimal. In particular, we construct connected curves of arbitrarily high degree in having maximal regularity, but no extremal secants. We also show that any connected curve in of degree at least 5 with maximal regularity and no linear components has an extremal secant.

     

Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians and by two equal symplectic balls.

     

Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds

We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of , a subject which is of interest on its own.

     

Characterizations of regular almost periodicity in compact minimal abelian flows

Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of -dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations for the case when the acting group is . We extend Egawa's results to the case of an arbitrary abelian acting group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications (characterization of regularly almost periodic functions on arbitrary abelian topological groups, classification of uniformly regularly almost periodic compact minimal - and -flows, conditions equivalent with uniform regular almost periodicity, etc.).