model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

lie powers of modules for groups of prime order let l v be the free lie algebra on a finite dimensional vector space v over a field k with homogeneous components l n v for n geq 1 if g is a group and v is a kg module the action of g extends naturally to l v and the l n v become finite dimensional kg modules called the lie powers of v in the decomposition problem the aim is to identify the isomorphism types of indecomposable kg modules with their multiplicities in unrefinable direct decompositions of the lie powers this paper is concerned with the case where g has prime order p and k has characteristic p as is well known there are p indecomposables denoted here by j 1 dots j p where j r has dimension r a theory is developed which provides information about the overall module structure of l v and gives a recursive method for finding the multiplicities of j 1 dots j p in the lie powers l n v for example the theory yields decompositions of l v as a direct sum of modules isomorphic either to j 1 or to an infinite sum of the form j r oplus j p 1 oplus j p 1 oplus ldots with r geq 2 closed formulae are obtained for the multiplicities of j 1 dots j p in l n j p and l n j p 1 for r p 1 the indecomposables which occur with non zero multiplicity in l n j r are identified for all sufficiently large n

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

global well posedness for an inhomogeneous lsw model in unbounded domains the lsw theory of domain coarsening describes the evolution of the size distribution of a system of particles evolving by diffusional mass exchange to reduce their total surface area we prove existence and uniqueness of solutions to an inhomogeneous extension of the lsw model in unbounded domains this model arises naturally as a homogenization limit of the underlying free boundary problem in the case of a system of particles for which the screening length the effective range of particle interactions is smaller than the system size the crucial ingredients in the analysis are first to establish the screening property by showing that the corresponding green s function decreases exponentially over the relevant distances second we have to control the mass fluxes between different regions to prevent aggregation of mass in few particles which would result in blow up of the solution

ostwald ripening for dilute systems under quasistationary dynamics we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying size evolving according to quasistationary dynamics under the scaling hypotheses that 1 size distance 2 deviation from sphericity size are both initially small we show robustness of the almost spherical shape by associating centers and radii to each particle and then derive rigorously a set of o d e s for the radii which we relate to the lifschitz slyozov wagner theory of coarsening and also establish an estimate for the speed of the centers

on an elaboration of m kac s theorem concerning eigenvalues of the laplacian in a region with randomly distributed small obstacles we removem balls of centersw 1 w m with the same radius m from a bounded domain inr 3 with smooth boundary let k m w m denote thek th eigenvalue of the laplacian in m balls under the dirichlet condition we consider k m w m as a random variable on a probability space w 1 w m and we examine a precise behaviour of k m w m asm we give an elaboration of m kac s theorem

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

the lower central series of a fiber type arrangement for a certain class fiber type of arrangements including the supersolvable ones of jambu and terao 3 we prove a formula relating the poincare polynomial of the complement with the ranks of successive quotients in the lower central series of the fundamental group such a formula was proved by kohno 5 for the single family of examplesa l

ostwald ripening in two dimensions the rigorous derivation of the equations from the mullins sekerka dynamics we consider a dilute mixture of a finite number of particles and we are interested in the coarsening of the spatial distribution in two space dimensions under mullins sekerka dynamics under the appropriate scaling hypotheses we associate radii and centers to each particle and derive equations for the whole evolution

ia automorphisms and lie algebras related to the mccool group abstract we investigate a subgroup i n of the mccool group m n we show that i n has solvable conjugacy problem next we investigate its lie algebra gr i n and we find a presentation for it finally we show that gr i n is naturally embedded into the andreadakis johnson lie algebra of the ia automorphisms of the free group f n

s modules and symmetric spectra we study a symmetric monoidal adjoint functor pair between the category of s modules and the category of symmetric spectra the functors induce equivalences between the respective homotopy categories of spectra module spectra and ring spectra then at approximately the same time two categories of spectra with nice smash products were discov ered elmendorf kriz mandell and may constructed the category of s modules ekmm and je smith introduced symmetric spectra hss both categories are quillen model categories and have associated notions of ring and module spectra however these two categories arise in completely dierent ways and even though the homotopy categories are equivalent it is not a priori clear if both frameworks give rise to the same homotopy theory of rings and modules both categories have their merits described in detail in the introductions of ekmm and hss and it is desirable to be able to translate results obtained in one category into conclusions valid in the other the present paper describes an easy mechanism which facilitates such comparisons below we dene a lax symmetric monoidal functor ms sp from the category of s modules to the category of symmetric spectra the functor preserves homotopy groups and has a strong symmetric monoidal left adjoint we show that the two functors induce inverse equivalences of the homotopy categories of spectra ring spectra commutative ring spectra and module spectra main theorem the functor from the category ms of s modules to the category sp of symmetric spectra passes to a symmetric monoidal equivalence of homotopy categories

the effect of the geometry of the particle distribution in ostwald ripening based on 1 we derive equations for the radii and the centers that we relate to the lifshitz slyozov wagner theory

on the effect of stochastic fluctuations in the dynamics of the lifshitz slyozov wagner model in this paper the dynamics of a system of spherical particles that fill a small volume fraction of the space and that evolves in a concentration field is discussed corrections to the lifshitz slyozov wagner lsw model that take into account the stochastic character of the problem are computed it is proved under suitable smallness assumptions for the volume fraction filled by the particles that the effect of these corrections does not modify much the dynamics of the self similar solutions of the lsw system of equations

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

on first order corrections to the lsw theory ii finite systems we consider first order corrections to the classical theory by lifshitz slyozov and wagner lsw for systems with a finite number of particles numerical simulations in v e fradkov etal phys rev e 53 3925 3932 1996 show a cross over in the scaling of the correction term from 1 3 to 1 2 is the volume fraction of particles when the system size becomes larger than the screening length we rigorously derive this cross over for the rate of change of the energy starting from the monopole approximation the proof exploits the fact that the rate of change of energy has a variational characterization

self similar behaviour for noncompactly supported solutions of the lsw model in this paper a method of approximating the classical lsw model near self similar solutions for initial data with infinite support is developed the resulting problem is an integrodifferential equation having two time scales that can be studied using multiple scale methods the analysis provides a detailed description of the precise manner in which the characteristics leak through the critical radius associated to the self similar solutions the analysis in this paper makes precise the meaning of the iterated logarithmic asymptotics in the dynamics of the lsw model that were already obtained in the original lifshitz slyozov paper examples of noncompactly supported solutions of the lsw model that do not behave in a self similar manner are also given

the method of reflections homogenization and screening for poisson and stokes equations in perforated domains we study the convergence of the method of reflections for the dirichlet problem of the poisson and the stokes equations in perforated domains which exist in the exterior of balls we prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small if the capacity density is too large or the balls extend to the whole space the method diverges but we provide a suitable modification of the method that converges to the solution of the dirichlet problem also in this case we give new proofs of classical homogenization results for the dirichlet problem of the poisson and the stokes equations in perforated domains using the modified method of reflections

screening in interacting particle systems we consider the green s function of the laplace operator in domains with spherical holes particles under natural assumptions on the distribution of particles we show that the green s function decays exponentially over distances larger than the screening length this result is fundamental for example when deriving effective equations for coarsening systems in unbounded domains