an approximation procedure for the zakharov shabat scattering problem for real single humped potentials a procedure is presented for finding simple approximations of the discrete eigenvalues of the zakharov shabat scattering problem corresponding to the nonlinear schrodinger equation the approximation is in the form of an interpolation formula which combines results for small eigenvalues obtained by a direct variational approach and for large eigenvalues obtained by the bohr sommerfeld quantization rule

half flat nilmanifolds we introduce a double complex that can be associated to certain lie algebras and show that its cohomology determines an obstruction to the existence of a half flat su 3 structure we obtain a classification of the 6 dimensional nilmanifolds carrying an invariant half flat structure

kahler reduction of metrics with holonomy g2 a torsion free g 2 structure admitting an infinitesimal isometry such that the quotient is a kahler manifold is shown to give rise to a 4 manifold equipped with a complex symplectic structure and a 1 parameter family of functions and 2 forms linked by second order equations reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to g 2

variational approach to the zakharov shabat scattering problem it is shown that the zakharov shabat scattering problem associated with the inverse scattering transform technique for solving the nonlinear schr odinger equation can be reformulated as a variational problem this reformulation makes it possible to use direct variational methods for finding the eigenvalues of the scattering problem which determines the speed and amplitude of the solitons emerging from a given initial condition the approach is illustrated by an application to sech shaped initial conditions

quasi minimizing surfaces in hyperbolic space let h 3 g denote 3 dimensional hyperbolic space with constant curvature 1 we are interested in surfaces f in h 3 with the following quasi minimality property if it f h 3 0 1 which does not depend on the homotopy if this condition is satisfied with q 1 then f is called homotopically minimizing a surface f c h 3 with this property is not necessarily area minimizing in contrast to the situation in higher dimensions cf 13 our main result is

geometric model for complex non kahler manifolds with su 3 structure for a given complex n fold m we present an explicit construction of all complex n 1 folds which are principal holomorphic t2 fibrations over m for physical applications we consider the case of m being a calabi yau 2 fold we show that for such m there is a subclass of the 3 folds that we construct which has natural families of non kahler su 3 structures satisfying the conditions for open image in new window supersymmetry in the heterotic string theory compactified on the 3 folds we present examples in the aforementioned subclass with m being a k3 surface and a 4 torus

harmonicity of sections of sphere bundles we consider the energy functional on the space of sections of a sphere bundle over a riemannian manifold m langle cdot cdot rangle equipped with the sasaki metric and discuss the characterising condition for critical points furthermore we provide a useful method for computing the tension field in some particular situations such a method is shown to be adequate for many tensor fields defined on manifolds m equipped with a g structure compatible with langle cdot cdot rangle this leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from m langle cdot cdot rangle into its sphere bundle

conformally parallel g2 structures on a class of solvmanifolds starting from a 6 dimensional nilpotent lie group n endowed with an invariant su 3 structure we construct a homogeneous conformally parallel g 2 metric on an associated solvmanifold we classify all half flat su 3 structures that endow the rank one solvable extension of n with a conformally parallel g 2 structure by suitably deforming the su 3 structures obtained we are able to describe the corresponding non homogeneous ricci flat metrics with holonomy contained in g 2 in the process we also find a new metric with exceptional holonomy

half flat structures and special holonomy it was proven by hitchin that any solution of his evolution equations for a half flat su 3 structure on a compact six manifold m defines an extension of m to a seven manifold with holonomy in g2 we give a new proof which does not require the compactness of m more generally we prove that the evolution of any half flat g structure on a six manifold m defines an extension of m to a ricci flat seven manifold n for any real form g of sl 3 c if g is noncompact then the holonomy group of n is a subgroup of the noncompact form gof g c similar results are obtained for the extension of nearly half flat structures by nearly parallel g2 or g structures as well as for the extension of cocalibrated g2 and g structures by parallel spin 7 and spin0 3 4 structures respectively as an application we obtain that any six dimensional homogeneous manifold with an invariant half flat structure admits a canonical extension to a seven manifold with a parallel g2 or g structure for the group h3 h3 where h3 is the three dimensional heisenberg group we describe all left invariant half flat structures and develop a method to explicitly determine the resulting parallel g2 or g structure without integrating in particular we construct three eight parameter families of metrics with holonomy equal to g2 and g moreover we obtain a strong rigidity result for the metrics induced by a half flat structure on h3 h3 satisfying z z 0 where z denotes the centre finally we describe the special geometry of the space of stable three forms satisfying a reality condition considering all possible reality conditions we find four different special kahler manifolds and one special para kahler manifold

multi moment maps we introduce a notion of moment map adapted to actions of lie groups that preserve a closed three form we show existence of our multi moment maps in many circumstances including mild topological assumptions on the underlying manifold such maps are also shown to exist for all groups whose second and third lie algebra betti numbers are zero we show that these form a special class of solvable lie groups and provide a structural characterisation we provide many examples of multi moment maps for different geometries and use them to describe manifolds with holonomy contained in g2 preserved by a two torus symmetry in terms of tri symplectic geometry of four manifolds

variational principle of the zakharov shabat equations abstract we put forth a variational principle for the zakharov shabat zs equations which is the basis of the inverse scattering transform of a number of important nonlinear pde s using the variational representation of the zs equations we develop an approximate analytical technique for finding discrete eigenvalues of the complex spectral parameter in the zs equations for a given pulse shaped potential which is equivalent to the physically important problem of finding the soliton content of the given initial pulse we apply the technique to several particular shapes of the pulse and demonstrate that the simplest version of the variational approximation based on trial functions with one or two free parameters proves to be fully analytically tractable and it yields threshold conditions for the appearance of the first soliton or of the first soliton pair which are in a fairly good agreement with available numerical results however more free parameters are necessary to allow prediction of additional solitons produced by the given pulse

fractional sobolev spaces and topology abstract consider the sobolev class w s p m n where m and n are compact manifolds and p 1 s 0 1 1 p we present a necessary and sufficient condition for two maps u and v in w s p m n to be continuously connected in w s p m n we also discuss the problem of connecting a map u w s p m n to a smooth map f c m n

some mathematical challenges in materials science four challenges to mathematics research posed by the field of materials science are given plus an additional challenge purely from the field of geometric measure theory the problems all concern the effects of surface and grain boundary free energy motion by weighted mean curvature and or surface diffusion and or other kinetics proofs of minimality of soap bubble clusters and their anisotropic analogs shapes of crystals in a gravitational or other field incorporating data from simulations into mathematics and understanding flat chains modulo the figures are selected copies of transparencies presented at the lecture on which this paper is based materials science is both much older and much younger than mathematics ancient humans discovered various elemental metals like gold and copper at some point they discovered that if you mix a little tin into copper you get something quite different from either namely bronze there is a period of human history labeled the bronze age i don t think there is any labeled the arithmetic age on the other hand the term materials science only came into general use within the last 30 years or so as the subjects of metallurgy ceramics and polymers were collected into one discipline the materials research society was founded in 1973 and the international congress of materials research societies in 1991 and along with the old problems of making better structural materials such as steel came new issues unimaginable a century ago some of these include the ability to see and manipulate individual atoms with an atomic force microscope the study of carbon nanotubes the fabrication of electronic chips and the making of thin films for cd s and other storage media what are mathematicians doing in such a field first of all as rodney nichols said at the new york academy of sciences on june 5 2000 much more than a few decades ago there are mathematicians everywhere and that is for the good secondly there has been an effort by the national science foundation to encourage this interdisciplinary research a search of the nsf database under the division of mathematical sciences performed in the year 2000 revealed 85 grants since january 1 1997 with the words materials science in their abstracts there have been several multi million dollar awards co sponsored by darpa for such cooperative efforts targeted at specific problems part of the result is that the collection of mathematicians working on problems from materials science has grown from isolated individuals prior to the mid 1980 s to a small community and now received by the editors november 6 2000 and in revised form february 21 2002 2000 mathematics subject classification primary 74n20 secondary 49q20 49q15 35r99

on the geometry of closed g 2 structures we give an answer to a question posed in physics by cvetic et al 9 and recently in mathematics by bryant 3 namely we show that a compact 7 dimensional manifold equipped with a g2 structure with closed fundamental form is einstein if and only if the riemannian holonomy of the induced metric is contained in g2 this could be considered to be a g2 analogue of the goldberg conjecture in almost kahler geometry and was indicated by cvetic et al in 9 the result was generalized by bryant to closed g2 structures with too tightly pinched ricci tensor we extend it in another direction proving that a compact g2 manifold with closed fundamental form and divergence free weyl tensor is a g2 manifold with parallel fundamental form we introduce a second symmetric ricci type tensor and show that einstein conditions applied to the two ricci tensors on a closed g2 structure again imply that the induced metric has holonomy group contained in g2

homological and homotopical dehn functions are different the homological and homotopical dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space the homological dehn function measures fillings of cycles by chains whereas the homotopical dehn function measures fillings of curves by disks because the two definitions involve different sorts of boundaries and fillings there is no a priori relationship between the two functions however before this work there were no known examples of finitely presented groups for which the two functions differ this paper gives such examples constructed by amalgamating a free by cyclic group with several bestvina brady groups

mirror symmetric su 3 structure manifolds with ns fluxes when string theory is compactified on a six dimensional manifold with a nontrivial ns flux turned on mirror symmetry exchanges the flux with a purely geometrical composite ns form associated with lack of integrability of the complex structure on the mirror side considering a general class of t3 fibered geometries admitting su 3 structure we find an exchange of pure spinors e ij and in dual geometries under fiberwise t duality and study the transformations of the ns flux and the components of intrinsic torsion a complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with ns flux

einstein locally conformal calibrated g_2 structures we study locally conformal calibrated g 2 structures whose underlying riemannian metric is einstein showing that in the compact case the scalar curvature cannot be positive as a consequence a compact homogeneous 7 manifold cannot admit an invariant einstein locally conformal calibrated g 2 structure unless the underlying metric is flat in contrast to the compact case we provide a non compact example of homogeneous manifold endowed with a locally conformal calibrated g 2 structure whose associated riemannian metric is einstein and non ricci flat the homogeneous einstein metric is a rank one extension of a ricci soliton on the 3 dimensional complex heisenberg group endowed with a left invariant coupled mathrm su 3 structure omega psi i e such that d omega c mathrm re psi with c in mathbb r 0 nilpotent lie algebras admitting a coupled mathrm su 3 structure are also classified

computation of the direct scattering transform for the nonlinear schroedinger equation abstract the cubic nonlinear schroedinger equation nls describes the space time evolution of narrow banded wave trains in one space and one time 1 1 dimensions the richness of nonlinear wave motions described by nls is exemplified by the fully nonlinear envelope soliton and breather solutions which are fully understood only in terms of the general solution of the equation as described by the inverse scattering transform 1st the method may be viewed as a nonlinear generalization of the linear fourier transform herein we develop a numerical algorithm for determining the scattering transform spectrum of a nonlinear wave train described by the nls equation the analysis of space or time series data obtained from computer simulations of nonlinear narrow band wave trains or from experimental measurements is thus a central point of discussion in particular we develop a numerical algorithm for computing the direct scattering transform dst which may be interpreted as the nonlinear fourier spectrum of the complex envelope function of a wave train the fact that the nonlinear fourier modes are constants of the motion for all time provides a physical basis for the analysis of data while the nonlinear fourier method is specifically applied to the nls equation the approach is easily generalized to include the class of spectral problems due to ablowitz kaup newell and segur akns this class includes several other nonlinear wave equations of physical interest in 1 1 including the kortewegde vries kdv modified kdv sine gordon and sinh gordon equations