twistor spaces of hypercomplex manifolds are balanced abstract a hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations if in addition there exists a metric on the manifold which is hermitian with respect to the three structures and such that the corresponding hermitian forms are closed the manifold is said to be hyperkahler in the paper non hermitian yang mills connections 13 kaledin and verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric these were first studied in the article on the existence of special metrics in complex geometry 17 by michelsohn in the present article we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced

betti numbers of holomorphic symplectic quotients via arithmetic fourier transform a fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field this technique in turn gives information on betti numbers of holomorphic symplectic quotients as a consequence simple unified proofs are obtained for formulas of poincare polynomials of toric hyperkahler varieties recovering results of bielawski dancer and hausel sturmfels poincare polynomials of hilbert schemes of points and twisted atiyah drinfeld hitchin manin adhm spaces of instantons on 2 recovering results of nakajima yoshioka and poincare polynomials of all nakajima quiver varieties as an application a proof of a conjecture of kac on the number of absolutely indecomposable representations of a quiver is announced

moment maps scalar curvature and quantization of kahler manifolds building on donaldson s work on constant scalar curvature metrics we study the space of regular kahler metrics e i e those for which deformation quantization has been defined by cahen gutt and rawnsley after giving in sects 2 and 3 a review of donaldson s moment map approach we study the essential uniqueness of balanced basis i e of coherent states in a more general setting theorem 2 5 we then study the space e in sect 4 and we show in sect 5 how all the tools needed can be defined also in the case of non compact manifolds

modularity quaternion kahler spaces and mirror symmetry we provide an explicit twistorial construction of quaternion kahler manifolds ob tained by deformation of c map spaces and carrying an isometric action of the modular group sl 2 z the deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing ns5 brane instanton effects in string compactifications with n 2 supersymmetry in this context the modular invariant parametrization of twistor lines found in this work yields the complete non perturbative mirror map between type iia and type iib physical fields

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

hyper kahler metrics and harmonic superspace the most general unconstrained superfield action for self interactingn 2 matter hypermultiplets in analyticn 2 superspace is argued to produce a most generaln 2 hyper kahler model after eliminating an infinite set of auxiliary fields this suggests a new possibility of classifying hyper kahler metrics according to then 2 analytic superfield self interactions and provides an effective tool to compute these metrics explicitly as the simplest example the u 2 invariant quartic self coupling of a singleq hypermultiplet is analyzed and is shown to yield the familiar taub nut metric to see the geometric pattern directly in terms ofn 2 superfields we introduce a new on shell representation ofq hypermultiplets inn 2 harmonic superspace similar to the description ofn 2 gauge theories for the u 2 example this formulation is checked to coincide with that by sierra and townsend

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

manifolds of g2 holonomy from the n 4 sigma model using the two dimensional 2d n 4 sigma model with u 1 r gauge symmetry and introducing the ade cartan matrices as gauge matrix charges we build toric hyper kahler eight real dimensional manifolds x8 dividing by one toric geometry circle action of x8 manifolds we present examples describing quotients x7 x8 u 1 of g2 holonomy in particular for the ar cartan matrix the quotient space is a cone on an s2 bundle over r intersecting wcp2 1 2 1 weighted projective spaces according to the ar dynkin diagram

hyperkahler manifolds and nonlinear supermultiplets we present a new construction of hyperkahler metrics that derives from the 3 dimensionaln 4 nonlinear supermultiplet further we give a detailed description of the nonlinear multiplet inn 2 and 4 superspace

preconditioning heterogeneous boldsymbol h mathrm div problems by additive schur complement approximation and applications in this paper we propose and analyze a preconditioner for a system arising from a mixed finite element approximation of second order elliptic problems describing processes in highly heterogeneous media our approach uses the technique of multilevel methods see e g p vassilevski multilevel block factorization preconditioners matrix based analysis and algorithms for solving finite element equations springer new york 2008 and the recently proposed preconditioner based on additive schur complement approximation by j kraus siam j sci comput 34 2012 pp a2872 a2895 the main results are the design study and numerical justification of iterative algorithms for these problems that are robust with respect to the contrast of the media defined as the ratio between the maximum and minimum values of the coefficient of the problem numerical tests provide experimental evidence for the high quality of the preconditioner and its desired robustness with respect to the material contrast such resu

calabi yau manifolds motivations and constructions the possible ways of compacitification of thee8 e8 superstring theory to four dimensions are reviewed the phenomenological need forn 1 supersymmetry is argued on quite general grounds to favour the choice of a calabi yau manifold for the compact internal manifold the massless spectrum after compactification is derived in full detail revealing beside the usual particles others that may have great phenomenological impact the technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given a big family of such constructions giving rise to many new calabi yau manifolds is presented and its relevance to the search of a phenomenologically acceptable solution is discussed

subsolution theorem and the dirichlet problem for the quaternionic monge ampere equation in this paper we obtain the existence of solution to the dirichlet problem for quaternionic monge ampere equation after showing a stability theorem we prove the subsolution theorem for quaternionic monge ampere equation by combining our stability theorem and kolodziej s approach for the complex case

analysis and convergence of the mac scheme i the linear problem the mac marker and cell discretization of fluid flow is analysed for the stationary stokes equations it is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations estimates using mesh dependent norms analogous to the standard bf h 1 and l 2 norms are given for the velocity and pressure respectively

robust multilevel solvers for high contrast anisotropic multiscale problems a robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented the method which belongs to the class of nonlinear algebraic multilevel iterations amlis uses an abstract framework for general symmetric positive definite bilinear forms previously presented in another publication by the author the crucial ingredient for obtaining robustness with respect to the variations in the coefficients and the anisotropies is the design of a nested sequence of spaces based on local generalized eigenvalue problems a discussion on how to achieve large coarsening factors in this sequence of spaces which is desirable in terms of computational complexity is included particular emphasis is put on how to handle the situation when the computed generating sets of the nested spaces are not minimal i e do not constitute bases several numerical examples are provided verifying the theoretically established robustness results

robust multilevel methods for general symmetric positive definite operators a general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed the term robust refers to the convergence rate of the method being independent of discretization parameters i e the problem size and problem parameters important instances of such problem parameters are in particular highly varying coefficients the method belongs to the class of nonlinear algebraic multilevel iterations the crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems the method is analyzed in a rather general setting applying to the scalar elliptic equation the equations of linear elasticity and equations arising in the solution of maxwell s equations numerical results for the scalar elliptic equation are presented showing its robust convergence behavior

homogenization of the navier stokes equations in open sets perforated with tiny holes i abstract framework a volume distribution of holes this paper treats the homogenization of the stokes or navier stokes equations with a dirichlet boundary condition in a domain containing many tiny solid obstacles periodically distributed in each direction of the axes for example in the three dimensional case the obstacles have a size of e3 and are located at the nodes of a regular mesh of size e a suitable extension of the pressure is used to prove the convergence of the homogenization process to a brinkman type law in which a linear zero order term for the velocity is added to a stokes or navier stokes equation

geometric multigrid for darcy and brinkman models of flows in highly heterogeneous porous media we apply geometric multigrid methods for the finite element approximation of flow problems governed by darcy and brinkman systems used in modeling highly heterogeneous porous media the method is based on divergence conforming discontinuous galerkin methods and overlapping patch based domain decomposition smoothers we show in benchmark experiments that the method is robust with respect to mesh size and contrast of permeability for highly heterogeneous media

hyperkahler quotient construction of bps monopole moduli spaces we use the hyperkahler quotient of flat space to obtain some monopole moduli space metrics in explicit form using this new description we discuss their topology completeness and isometries we construct the moduli space metrics in the limit when some monopoles become massless which corresponds to non maximal symmetry breaking of the gauge group we also introduce a new family of hyperkahler metrics which depending on the mass parameter being positive or negative tend to either the asymptotic metric on the moduli space of many su 2 monopoles or to previously unknown metrics these new metrics are singular or complete depending on the particular choice of the level set of the moment map the singular metrics are of relevance to the moduli spaces of vacua of three dimensional gauge theories for higher rank gauge groups finally we make a few comments concerning the existence of closed or bound orbits on some of these manifolds and the integrability of the geodesic flow

abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations pdes however it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the pde coefficients especially for systems of pdes here we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities we achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence we prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous pdes and systems of pdes that confirm this property

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

some properties on the tensor product of graphs obtained by monogenic semigroups abstract in das et al 2013 8 a new graph s m on monogenic semigroups s m with zero having elements 0 x x 2 x 3 x n has been recently defined the vertices are the non zero elements x x 2 x 3 x n and for 1 i j n any two distinct vertices x i and x j are adjacent if x i x j 0 in s m as a continuing study in akgunes et al 2014 3 it has been investigated some well known indices first zagreb index second zagreb index randic index geometric arithmetic index atom bond connectivity index wiener index harary index first and second zagreb eccentricity indices eccentric connectivity index the degree distance over s m in the light of above references our main aim in this paper is to extend these studies over s m to the tensor product in detail we will investigate the diameter radius girth maximum and minimum degree chromatic number clique number and domination number for the tensor product of any two not necessarily different graphs s m 1 and s m 2

n 2 super yang mills theory in projective superspace we constructn 2 yang mills theory in projective superspace by exploiting the analogy to ward s twistor construction of self dual yang mills fields

integrating over higgs branches we develop some useful techniques for integrating over higgs branches in supersymmetric theories with 4 and 8 supercharges in particular we define a regularized volume for hyperkahler quotients we evaluate this volume for certain ale and alf spaces in terms of the hyperkahler periods we also reduce these volumes for a large class of hyperkahler quotients to simpler integrals these quotients include complex coadjoint orbits instanton moduli spaces on 4 and ale manifolds hitchin spaces and moduli spaces of parabolic higgs bundles on riemann surfaces in the case of hitchin spaces the evaluation of the volume reduces to a summation over solutions of bethe ansatz equations for the non linear schrodinger system we discuss some applications of our results

a note on the uniform inf sup condition for the brinkman problem in highly heterogeneous media abstract in this short note we study the interpolation spaces based uniform inf sup condition for brinkman equation in highly heterogeneous domains the result is known to hold in the case of a uniform constant permeability our numerical construction and experiments show that it is not true for the taylor hood and the mini elements applied to problems with highly heterogeneous coefficients

reflection functors for quiver varieties and weyl group actions we introduce reflectionfunctors on quiver varieties they are hyper kahler isometries between quiver varieties with different parameters related by elements in the weyl group the definition is motivated by the origial reflection functor given by bernstein gelfand ponomarev 1 but they behave much nicely they are isomorphisms and satisfy the weyl group relations as an application we define weyl group representations of homology groups of quiver varieties they are analogues of slodowy s construction of springer representations of the weyl group

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

complex symplectic geometry and compact locally hyper kahlerian manifolds complex symplectic geometry is the study of complex manifolds admitting a global closed nondegenerate holomorphic two form compact simply connected complex symplectic manifolds are of interest for various reasons for example they always admit a ricci flat metric in this work the close relationship between the complex geometry of such manifolds and their riemannian structures is exploited in order to obtain results in both directions riemannian techniques are used to obtain results on the complex automorphism group and the problem of constructing examples of compact locally hyper kahlerian manifolds with prescribed holonomy groups is discussed

gauging discrete symmetries recent developments in quantum gravity theory have led to the suggestion that various discrete symmetries in particular charge parity cp should be gauged that is interpreted as elements of some connected lie group as the parity operator is related to a space time isometry however it is far from clear that this suggestion has any real meaning we give a simple geometric construction in terms of which it is meaningful to gauge discrete symmetries provided that certain nontrivial conditions are satisfied by the space time manifold by the gauge group and by the discrete symmetry itself

the spectral radius of bicyclic graphs with prescribed degree sequences restricted to the bicyclic graphs with prescribed degree sequences we determine the unique graph with the largest spectral radius with respect to the adjacency matrix

a construction of non kahler calabi yau manifolds and new solutions to the strominger system abstract we propose a new construction of compact non kahler calabi yau manifolds with balanced metrics and study the strominger system on them in particular we obtain explicit solutions to the strominger system with degeneracies on g t 4 where g is an immersed minimal surface of genus g 3 in flat t 3

hyperkahler structures and infinite dimensional grassmannians abstract in this paper we describe an example of a hyperkahler quotient of a banach manifold by a banach lie group although the initial manifold is not diffeomorphic to a hilbert manifold not even to a manifold modelled on a reflexive banach space the quotient space obtained is a hilbert manifold which can be furthermore identified either with the cotangent space of a connected component gr res j j z of the restricted grassmannian or with a natural complexification of this connected component thus proving that these two manifolds are isomorphic hyperkahler manifolds moreover kahler potentials associated with the natural complex structure of the cotangent space of gr res j and with the natural complex structure of the complexification of gr res j are computed using kostant souriau s theory of prequantization

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

a new bound of radius with irregularity index in this paper we use a technique introduced in the paper p dankelmann r c entringer average distance minimum degree and spanning trees j graph theory 33 2000 1 13 to obtain a strengthening of an old classical theorem by erdos et al p erdos j pach r pollack z tuza radius diameter and minimum degree j combin theory b 47 1989 73 79 on radius and minimum degree to be more detailed we will prove that if g is a connected graph of order n with the minimum degree d then the radius g does not email protected 1 1 where t is the irregularity index that is the number of distinct terms of the degree sequence of g which has been recently defined in the paper s mukwembi a note on diameter and the degree sequence of a graph appl math lett 25 2012 175 178 we claim that our result represent the tightest bound that ever been obtained until now

tangentially positive isometric actions and conjugate points let m g be a complete riemannian manifold with no conjugate points and f m g b g b a principal g bundle where g is a lie group acting by isometries and b the smooth quotient with g b the riemannian submersion metric we obtain a characterization of conjugate point free quotients b g b in terms of symplectic reduction and a canonical pseudo riemannian metric on the tangent bundle tm from which we then derive necessary conditions involving g and m for the quotient metric to be conjugate point free particularly for m a reducible riemannian manifold let g tm with the lie algebra of g be the moment map of the tangential g action on tm and let gp be the canonical pseudo riemannian metric on tm defined by the symplectic form d and the map f tm m x m f z exp z exp z first we prove a theorem stating that if gp is not positive definite on the action vector fields for the tangential action along g 1 0 then b g b acquires conjugate points we proved the converse result in 2005 then we characterize self parallel vector fields on m in terms of the positivity of the gp length of their tangential lifts along certain canonical subsets of tm we use this to derive some necessary conditions on g and m for actions to be tangentially positive on relevant subsets of tm which we then apply to isometric actions on complete conjugate point free reducible riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions

linear perturbations of quaternionic metrics we extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds 1 to the case of quaternionic kahler manifolds via swann s construction deformations of a 4d dimensional quaternionic kahler manifold mathcal m are in one to one correspondence with deformations of its 4d 4 dimensional hyperkahler cone mathcal s the latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space mathcal z mathcal s with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved equivalently we show that the deformations of mathcal m can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space mathcal z mathcal m of mathcal m by passing the swann bundle and its twistor space we specialize these general results to the case of quaternionic kahler metrics with d 1 commuting isometries obtainable by the legendre transform method and linear deformations thereof we illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree and one loop level

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

a strongly conservative finite element method for the coupling of stokes and darcy flow we consider a model of coupled free and porous media flow governed by stokes and darcy equations with the beavers joseph saffman interface condition this model is discretized using divergence conforming finite elements for the velocities in the whole domain discontinuous galerkin techniques and mixed methods are used in the stokes and darcy subdomains respectively this discretization is strongly conservative in h d i v w and we show convergence numerical results validate our findings and indicate optimal convergence orders

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

energy of sections of the deligne hitchin twistor space we study a natural functional on the space of holomorphic sections of the deligne hitchin moduli space of a compact riemann surface generalizing the energy of equivariant harmonic maps corresponding to twistor lines we show that the energy is the residue of the pull back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by hitchin as a byproduct we show the existence of a hyper kahler potentials for new components of real holomorphic sections of twistor spaces of hyper kahler manifolds with rotating s 1 action additionally we prove that for a certain class of real holomorphic sections of the deligne hitchin moduli space the energy functional is basically given by the willmore energy of corresponding equivariant conformal map to the 3 sphere as an application we use the functional to distinguish new components of real holomorphic sections of the deligne hitchin moduli space from the space of twistor lines

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

gauge theory in witten s approach to the generation problem in witten s topological theory of the generation problem gauge groups are identified with thee 8 centraliser of the holonomy group of the internal manifold here we show that this amounts to interpreting gauge groups as generalised symmetry groups of the internal levi civita connection we then give techniques for computing centralisers in exceptional groups taking into account the fact that holonomy groups are frequently disconnected these techniques allow us to deal with compact locally irreducible ricci flat riemannian manifolds of all holonomy types and dimensions