generalized kahler manifolds commuting complex structures and split tangent bundles we study generalized kahler manifolds for which the corresponding complex structures commute and classify completely the compact four dimensional ones

holomorphic differentials thermostats and anosov flows we introduce a new family of thermostat flows on the unit tangent bundle of an oriented riemannian two manifold suitably reparametrised these flows include the geodesic flow of metrics of negative gauss curvature and the geodesic flow induced by the hilbert metric on the quotient surface of divisible convex sets we show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties in particular we prove that they admit a dominated splitting and we identify special cases in which the flows are anosov in the latter case we study when they admit an invariant measure in the lebesgue class and the regularity of the weak foliations

extremal solutions of p laplacian problems in d1 p rn via wolff potential estimates abstract we prove the existence of smallest positive and greatest negative solutions in d 1 p r n of the p laplacian problem of logistic type in r n with 1 p n u d 1 p r n p u a x u p 2 u g u and consider its extended version with a harvesting term h as well unlike papers on the topic here we only assume a superlinear growth for the nonlinearity g r r at zero and at infinity i e even supercritical growth is allowed and moreover no monotonicity for s g s s p 1 is required we prove a vazquez type maximum principle for nonnegative d 1 p r n solutions of some p laplacian problem and establish a sub supersolution method in d 1 p r n which along with wolff potential estimates for superharmonic functions will be used as our main tools

existence of solution for a class of quasilinear schrodinger equation in rn with zero mass abstract in this paper we use variational methods to establish a berestycki lions type result for two classes of quasilinear schrodinger equations in r n in the zero mass situation

the g generalized hitchin systems and prym varieties in this article we consider the generalized hitchin systems introduced by bottacin and markman for an arbitrary reductive complex group g they have the structure of lagrangian fibrations pr u by generalized prym varieties over sets u parametrizing families of weyl invariant curves the g generalized hitchin systems satisfy a rank two condition and one can find invariant varieties x which distinguish these systems it is then shown that there is a correspondence between these integrable systems and the varieties x k d h which are equipped with an appropriate two form with values in the cartan subalgebra h

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

positive solutions to logistic type equations with harvesting abstract we use comparison principles variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in r n and in a bounded domain r n with n 3 when the carrying capacity of the environment is not constant by relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved the solution of this new equation is then used to produce a positive solution of our original problem

on surfaces of class vii 0 with numerically anticanonical divisor we consider minimal compact complex surfaces s with betti numbers b 1 and n bi 0 a theorem of donaldson gives n exceptional line bundles we prove that if in a deformation these line bundles have sections s is a degeneration of blown up hopf surfaces besides if there exists an integer m 1 and a flat line bundle f such that h s mk 0 these surfaces admit no nonconstant mero morphic functions the major problem in classification of non kahlerian surfaces is to achieve the classification of surfaces s of class viij all known surfaces of this class contain global spherical shells gss i e admit a biholomorphic map ip u v from a neighbourhood u c c2 0 of the sphere 53 db2 onto an open set v such that i f s3 does not disconnect s are there other surfaces in first section we investigate the general situation a theorem of donaldson 13 gives a z base of 2 s z such that b of s these line bundles form families we propose the following conjecture which can be easily checked for surfaces with gss

topological field theories and integrable models we show that the classical non abelian pure chern simons action is related to nonrelativistic models in 2 1 dimensions via reductions of the gauge connection in hermitian symmetric spaces in such models the matter fields are coupled to gauge chern simons fields associated with the isotropy subgroup of the considered symmetric space thus a relation between the chern simons theory and the davey stewartson hierarchy is established in a natural way the backlund transformations are interpreted in terms of chern simons constraints moreover certain nonintegrable heisenberg models can be embedded into the pure chern simons theory the main classical and quantum properties of these systems are discussed

the classification of real projective structures on compact surfaces choi gratefully acknowledges partial support from garc kosef goldman gratefully acknowledges partial support from the national science foundation the alfred p sloan foundation and the institute for advanced computer studies at the university of maryland

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

bihermitian structures on complex surfaces proc london math soc 3 79 1999 414 428 we correct the statements of theorem 1 and corollary 2 of the above mentioned paper and provide a new argument for corollary 2 we also update some parts of the paper in the light of recent major developments in the theory of bihermitian structures due to m gualtieri and n hitchin

symplectic structures of moduli space of higgs bundles over a curve and hilbert scheme of points on the canonical bundle the moduli space of triples of the form e s are considered where e is a higgs bundle on a fixed riemann surface x and s is a nonzero holomorphic section of e such a moduli space admits a natural map to the moduli space of higgs bundles simply by forgetting s if y l is the spectral data for the higgs bundle e then s defines a section of the line bundle l over y the divisor of this section gives a point of a hilbert scheme parametrizing 0 dimensional subschemes of the total space of the canonical bundle k x since y is a curve on k x the main result says that the pullback of the symplectic form on the moduli space of higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the hilbert scheme using the map that sends any triple e s to the divisor of the corresponding section of the line bundle on the spectral curve

fractal dimension of random attractors for stochastic non autonomous reaction diffusion equations in this paper we first give some conditions for bounding the fractal dimension of a random invariant set for a non autonomous random dynamical system on a separable banach space then we apply these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic reaction diffusion equations with multiplicative white noise and additive white noise

orlicz sobolev versus holder local minimizer and multiplicity results for quasilinear elliptic equations national natural science foundation of china nsaf 10976026 national natural science foundation of china 11271305

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

on cyclic higgs bundles in this paper we derive a maximum principle for a type of elliptic systems and apply it to analyze the hitchin equation for cyclic higgs bundles we show several domination results on the pullback metric of the possibly branched minimal immersion f associated to cyclic higgs bundles also we obtain a lower and upper bound of the extrinsic curvature of the image of f as an application we give a complete picture for maximal sp 4 mathbb r representations in the 2g 3 gothen components and the hitchin components

longtime robustness of pullback random attractors for stochastic magneto hydrodynamics equations abstract we establish backward compactness of a pullback random attractor pra from a backward limit set compact cocycle we prove that backward compactness of a pra is a necessary and sufficient condition such that its time fibers are upper semi continuous at negative infinity and obtain the minimal compact limit set also we obtain the maximal limit set in the lower semi continuity sense these results greatly reinforce the upper semi continuity result at the finite time given by cui et al 2018 in the application part we show existence backward compactness and longtime robustness of a pra for the non autonomous stochastic magneto hydrodynamics equations perturbed by a backward tempered force moreover we prove that the pra does not change on different attracted universes this fact allows us to establish the finite time robustness of the pra if the force is tempered only

local c1 ω minimizers versus local w1 p ω minimizers of nonsmooth functionals abstract we study not necessarily differentiable functionals of the form j u 1 p u p d x 1 p u p d x j 1 x u d x j 2 x u d with 1 p involving locally lipschitz functions j 1 r r as well as j 2 r r we prove that local c 1 minimizers of j must be local w 1 p minimizers of j

random exponential attractor for stochastic reaction diffusion equation with multiplicative noise in r3 abstract in this paper we first improve the existing conditions for the existence of a random exponential attractor for a continuous cocycle on a separable banach space then we consider the existence of a random exponential attractor for stochastic non autonomous reaction diffusion equation with multiplicative noise defined in r 3 which implies the existence of a random attractor with finite fractal dimension the essential difficulty here is the continuity of the spectrum of the linear part of the equation which can be overcome by the tail estimation of solutions of equation and carefully decomposing the solution into a sum of three parts of whose one part is finite dimensional and other two parts are quickly decay in mean sense

mirror symmetry for moduli spaces of higgs bundles via p adic integration we prove the topological mirror symmetry conjecture by hausel thaddeus for smooth moduli spaces of higgs bundles of type sl n and pgl n more precisely we establish an equality of stringy hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties our proof utilises p adic integration relative to the fibres and interprets canonical gerbes present on these moduli spaces as characters on the hitchin fibres using tate duality furthermore we prove for d prime to n that the number of rank n higgs bundles of degree d over a fixed curve defined over a finite field is independent of d this proves a conjecture by mozgovoy schiffmann in the coprime case

pullback exponential attractors for nonautonomous equations part i semilinear parabolic problems a family of compact and positively invariant sets with uniformly bounded fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the process is constructed the existence of such a family called a pullback exponential attractor is proved for a nonautonomous semilinear abstract parabolic cauchy problem specific examples will be presented in the forthcoming part ii of this work

random exponential attractor for cocycle and application to non autonomous stochastic lattice systems with multiplicative white noise abstract we first establish some sufficient conditions for constructing a random exponential attractor for a continuous cocycle on a separable banach space and weighted spaces of infinite sequences then we apply our abstract result to study the existence of random exponential attractors for non autonomous first order dissipative lattice dynamical systems with multiplicative white noise

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

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation abstract in this paper we first establish a criteria for estimating the upper bound of fractal dimension of a random compact invariant set of a non autonomous random dynamical system on a separable banach space and give some special conditions which can be easily verified to some stochastic partial differential equations for bounding the fractal dimension of random attractors then we apply these conditions to obtain an upper bound of fractal dimension of random attractor for stochastic damped wave equation with additive white noise

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 symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

generalized pseudo kahler structures in this paper we consider pseudo bihermitian structures pairs of complex structures compatible with a pseudo riemannian metric we establish relations of these structures with generalized pseudo kahler geometry and holomorphic poisson structures similar to that in the positive definite case we provide a list of compact complex surfaces which could admit pseudo bihermitian structures and give examples of such structures on some of them we also consider a naturally defined null plane distribution on a generalized pseudo kahler 4 manifold and show that under a mild restriction it determines an engel structure

conformal limits and the bialynicki birula stratification of the space of λ connections abstract the bialynicki birula decomposition of the space of connections restricts to the morse stratification on the moduli space of higgs bundles and to the partial oper stratification on the de rham moduli space of holomorphic connections for both the morse and partial oper stratifications every stratum is a holomorphic lagrangian fibration over a component of the space of complex variations of hodge structure in this paper we generalize known results for the hitchin section and the space of opers to arbitrary strata these include the following a biholomorphic identification of the fibers of the two strata over a stable variation of hodge structure via the conformal limit of gaiotto a proof that the fibers of the morse and partial oper stratifications are transverse at the base point and an explicit parametrization of the fibers as half dimensional affine spaces

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

d1 2 rn versus c rn local minimizer and a hopf type maximum principle abstract we consider functionals of the form u 1 2 r n u 2 r n b x g u on d 1 2 r n n 3 whose critical points are the weak solutions of a corresponding elliptic equation in the whole r n we present a brezis nirenberg type result and a hopf type maximum principle in the context of the space d 1 2 r n more precisely we prove that a local minimizer of in the topology of the subspace v must be a local minimizer of in the d 1 2 r n topology where v is given by v v d 1 2 r n v c r n with sup x r n 1 x n 2 v x it is well known that the brezis nirenberg result has been proved a strong tool in the study of multiple solutions for elliptic boundary value problems in bounded domains we believe that the result obtained in this paper may play a similar role for elliptic problems in r n

regularity forward compactness and measurability of attractors for non autonomous stochastic lattice systems abstract this paper is concerned with the regularity forward compactness and measurability of pullback random attractors of non autonomous stochastic lattice systems with multiplicative white noise as well as random coefficients in weighted spaces the existence uniqueness and forward compactness of pullback random attractors in weighted space l 2 are proved for the systems when the random coefficient in the front of the discrete laplacian and the time dependent functions satisfy certain conditions in addition this forward compact l 2 l 2 attractor is proved to be a forward compact bi spatial l 2 l 2 l p attractor that is forward compact and measurable in l 2 l p and attracts all random subset of l 2 under the topology of l 2 l p the difficulty of establishing the measurability of the attractors in l 2 l p is overcome by proving the identity of the attractors on two different universes

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping in this article we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k dimensional integer set the well posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally lipschitz continuous the mean random dynamical system associated with the non autonomous system is shown to possess a unique tempered weak pullback mean random attractor in l2 f l2 l2 the existence of invariant measures for the autonomous system is also derived by using the krylov bogolyubov method the difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions

ends of the moduli space of higgs bundles we associate to each stable higgs pair a 0 phi 0 on a compact riemann surface x a singular limiting configuration a infinity phi infinity assuming that det phi has only simple zeroes we then prove a desingularization theorem by constructing a family of solutions a t t phi t to hitchin s equations which converge to this limiting configuration as t infinity this provides a new proof via gluing methods for elements in the ends of the higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space

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

extremal solutions of logistic type equations in exterior domain in r2 abstract let r 2 b 0 1 be the exterior of the closed unit ball we prove the existence of extremal constant sign solutions in d 0 1 2 of the following logistic type boundary value problem u a x u g u in u 0 on b 0 1 under the assumption that the nonnegative coefficient a decays like x 2 with 0 and the growth for the continuous nonlinearity g r r at zero and at infinity is superlinear which includes even exponential growth we are looking for solutions in the space d 0 1 2 which is the completion of c c with respect to the 2 norm for general unbounded domains in r 2 including the whole plane this completion may result in objects that do not belong to any function space this is one of the main reasons to consider the problem in the exterior of the unit ball instead in the whole plane unlike in the situation of r n with n 3 i e n p 2 the behavior of the solutions in the case n p 2 considered here is significantly different in particular it will be seen that the solutions are not decaying to zero at infinity and instead are bounded away from zero to prove our main results new tools have to be developed here such as for example a hopf type lemma and a sub supersolution method in d 0 1 2

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

bihermitian surfaces with odd first betti number compact bihermitian surfaces are considered that is compact oriented conformal four manifolds admitting two distinct compatible complex structures it is shown that if the first betti number is odd then with respect to either complex structure such a manifold belongs to class vii of the enriques kodaira classification moreover it must be either a special hopf or an inoue surface in the strongly bihermitian case or obtained by blowing up a minimal class vii surface with curves in the non strongly bihermitian case

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

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

separating coordinates for the generalized hitchin systems and the classical r matrices we exhibit natural darboux coordinates for the generalized hitchin systems studied by bottacin and markman these systems are defined on spaces of stable pairs consisting of a vector bundle and a form valued meromorphic endomorphism of the bundle in special cases genus zero genus one the bundles are rigid and one has the rational trigonometric and elliptic gaudin systems explicit formulae are given in these cases

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c

character varieties of free groups are gorenstein but not always factorial abstract fix a rank g free group f g and a connected reductive complex algebraic group g let x f g g be the g character variety of f g when the derived subgroup d g g is simply connected we show that x f g g is factorial which implies it is gorenstein and provide examples to show that when dg is not simply connected x f g g need not even be locally factorial despite the general failure of factoriality of these moduli spaces using different methods we show that x f g g is always gorenstein

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

characteristic exponents for a viscous fluid subjected to time dependent forces we consider a viscous incompressible fluid enclosed in a bounded region of 2 or 3 and subjected to time dependent forces using bound state estimates for the schrodinger operator we obtain rigorous bounds for the characteristic exponents entropy kolmogorov sinai invariant and hausdorff dimension of attracting sets our methods are of potential use for more general time evolutions described by nonlinear partial differential 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

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

limiting behavior of non autonomous stochastic reaction diffusion equations on unbounded thin domains this article is concerned with the limiting behavior of dynamics of a class of nonautonomous stochastic partial differential equations driven by multiplicative white noise on unbounded thin domains we first prove the existence of tempered pullback random attractors for the equations defined on n 1 dimensional unbounded thin domains then we show the upper semicontinuity of these attractors when the n 1 dimensional unbounded thin domains collapse onto the n dimensional space rn here the tail estimates are utilized to deal with the noncompactness of sobolev embeddings on unbounded domains

pathwise upper semi continuity of random pullback attractors along the time axis abstract the pullback attractor of a non autonomous random dynamical system is a time indexed family of random sets typically having the form a t t r with each a t a random set this paper is concerned with the nature of such time dependence it is shown that the upper semi continuity of the mapping t a t for each fixed has an equivalence relationship with the uniform compactness of the local union s i a s where i r is compact applied to a semi linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that in order to prove the above locally uniform compactness and upper semi continuity no additional conditions are required in which sense the two properties appear to be general properties satisfied by a large number of real models

constant sign and sign changing solutions for nonlinear eigenvalue problems abstract we prove the existence of multiple constant sign and sign changing solutions for a nonlinear elliptic eigenvalue problem under dirichlet boundary condition involving the p laplacian more precisely we establish the existence of a positive solution of a negative solution and of a nontrivial sign changing solution when the eigenvalue parameter is greater than the second eigenvalue 2 of the negative p laplacian extending results by ambrosetti lupo ambrosetti mancini and struwe our approach relies on a combined use of variational and topological tools such as e g critical points mountain pass theorem second deformation lemma variational characterization of the first and second eigenvalue of the p laplacian and comparison arguments for nonlinear differential inequalities in particular the existence of extremal nontrivial constant sign solutions plays an important role in the proof of sign changing solutions

pullback exponential attractors for nonautonomous equations part ii applications to reaction diffusion systems abstract the existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction diffusion equations with time dependent external forces including the case of the fitzhugh nagumo system

regularity and backward compactness of attractors for non autonomous lattice systems with random coefficients abstract we study longtime behavior for the non autonomous lattice model with multiplicative white noise and a random coefficient in the discrete laplace operator we first show existence of a bi spatial attractor when the initial space is the weighted square summation space and the terminal space is the weighted p times summation space for p 2 we then show backward compactness of the attractor in both initial and terminal spaces if the time indexed forces are backward tempered and backward null finally by proving identity of the attractors on the different universes of tempered or backward tempered sets we show measurability of the attractor in the initial space and in the terminal space respectively

the moduli of flat pgl 2 ℝ connections on riemann surfaces suppose x is a compact riemann surface with genus g 1 each class hom 1 x pgl 2 pgl 2 is associated with the first and second stiefel whitney classes w 1 and w 2 the set of representation classes with a fixed w 1 0 has two connected components these two connected components are characterized by w 2 being 0 or 1 for each fixed w 1 0 we prove that the component characterized by w 2 0 contains an open dense set diffeomorphic to the total space of a vector bundle of rank 2g 2 over a once punctured algebraic torus of dimension g 1 the other component characterized by w 2 1 contains an open dense set diffeomorphic to the total space of a vector bundle of rank 2g 2 over an algebraic torus of dimension g 1