flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

real holomorphic sections of the deligne hitchin twistor space we study the holomorphic sections of the deligne hitchin moduli space of a compact riemann surface especially the sections that are invariant under the natural anti holomorphic involutions of the moduli space their relationships with the harmonic maps are established as a by product a question of simpson on such sections posed in si4 is answered

hitchin s self duality equations on complete riemannian manifolds let m be a complete riemannian manifold let g be a real semisimple noncompact algebraic group and let k be its maximal compact subgroup let p be a principal g bundle over m fix a reduction of the structure group g of p to k and consider the space d of smooth g connections on p d is an affine space modeled on the vector space 21 ad p of 1 forms on m with values in the adjoint bundle associated to p for any connection d e d we have a decomposition d d 0 where d is a connection preserving the k structure and 0 is a 1 form on m with values in the orthogonal complement in ad p of ad pk here px is the k bundle defined by the reduction of structure group i f d is a flat connection we have

a quintic b spline finite elements scheme for the kdvb equation abstract an algorithm for solving the korteweg de vries burgers kdvb equation based on the collocation method with quintic b spline finite elements is set up to simulate the solutions of the kdv burgers and the kdvb equations the migration of solitary waves and temporal evolution of a maxwellian initial pulse are studied for the kdv equation burgers equation is solved for different values of reynolds number the time evaluation of the solutions of the kdvb equation with different values for the diffusion and dispersion coefficients is studied invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm a linear stability analysis shows the scheme to be unconditionally stable

matroid representations and free arrangements we show that terao s conjecture freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid holds over fields with at most four elements however an example demonstrates that the field characteristic has to be fixed for this

mathrm so p q so p q higgs bundles and higher teichmuller components some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics others are more alluring and unusual either because they are not detected by primary invariants or because they have special geometric significance or both in this paper we describe new examples of such exotic components in moduli spaces of mathrm so p q higgs bundles on closed riemann surfaces or equivalently moduli spaces of surface group representations into the lie group mathrm so p q furthermore we discuss how these exotic components are related to the notion of positive anosov representations recently developed by guichard and wienhard we also provide a complete count of the connected components of these moduli spaces except for mathrm so 2 q with q geqslant 4

a small time solutions for the korteweg de vries equation using spline approximation in this paper the one dimensional nonlinear korteweg de vries kdv equation is numerically solved using second order spline approximation the test problems concerning the propagation of a solution and two solution interaction are used to validate the proposal scheme and it is found to be both accurate and efficient at small times also it is shown that the second order spline approximation may be used effectively at small times when the exact solution of the kdv equation is not known

virtual classes of parabolic sl2 c character varieties abstract in this paper we compute the virtual classes in the grothendieck ring of algebraic varieties of sl 2 c character varieties over compact orientable surfaces with parabolic points of semi simple type when the parabolic punctures are chosen to be semi simple non generic we show that a new interaction phenomenon appears generating a recursive pattern

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

hitchin kobayashi correspondence quivers and vortices a twisted quiver bundle is a set of holomorphic vector bundles over a com plex manifold labelled by the vertices of a quiver linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles labelled by the arrows when the manifold is kahler quiver bundles admit natural gauge theoretic equations which unify many known equations for bundles with extra structure in this paper we prove a hitchin kobayashi correspondence for twisted quiver bundles over a compact kahler manifold relating the existence of solutions to the gauge equations to a stability criterion and con sider its application to a number of situations related to higgs bundles and dimensional reductions of the hermitian einstein equations

more on gauge theory and geometric langlands abstract the geometric langlands correspondence was described some years ago in terms of s duality of n 4 super yang mills theory some additional matters relevant to this story are described here the main goal is to explain directly why an a brane of a certain simple kind can be an eigenbrane for the action of t hooft operators to set the stage we review some facts about higgs bundles and the hitchin fibration we consider only the simplest examples in which many technical questions can be avoided

the galerkin method for the kdv equation using a new basis of smooth piecewise cubic polynomials abstract in this paper we introduce the galerkin method using a new basis of smooth piecewise cubic polynomials for the one dimensional nonlinear kdv equation the stability analysis shows that the present method is unconditionally stable in l 2 norm numerical experiments show that the proposed method preserves the conservation laws and is effective for simulating the motions and the interactions of solitary waves

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

linear shafarevich conjecture in this paper we settle armatively shafarevich s uniformization conjecture for varieties with linear fundamental groups we prove the strongest to date uniformization result the universal covering space of a complex projective manifold with a linear fundamental group is holomorphically convex the proof is based on both known and newly developed techniques in non abelian hodge theory

the moduli of flat pu p p structures with large toledo invariants for a compact riemann surface x of genus g 1 rm hom pi 1 x rm pu p q rm pu p q is the moduli space of flat rm pu p q connections on x there are two invariants the chern class c and the toledo invariant tau associated with each element in the moduli the toledo invariant is bounded in the range 2min p q g 1 le tau le 2min p q g 1 this paper shows that the component associated with a fixed tau 2 max p q 1 g 1 resp tau 2 max p q 1 g 1 and a fixed chern class c is connected the restriction on tau implies p q

moduli spaces of higgs bundles on degenerating riemann surfaces abstract we prove a gluing theorem for solutions a 0 0 of hitchin s self duality equations with logarithmic singularities on a rank 2 vector bundle over a noded riemann surface 0 representing a boundary point of teichmuller moduli space we show that every nearby smooth riemann surface 1 carries a smooth solution a 1 1 of the self duality equations which may be viewed as a desingularization of a 0 0

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

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

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

anosov flows surface groups and curves in projective space note that in 10 w goldman gives a complete description of these connected components in the case of finite covers of psl 2 r in the case of psl 2 r two homeomorphic components called teichmuller spaces play a central role these two components are well known to be homeomorphic to a ball of dimension 6g 6 n hitchin generalises this situation to psl n r indeed one of these components when n is odd and two when n is even has a very simple topology let us define an n fuchsian representation to be a representation which can be written as 0 where 0 is a cocompact representation with values in psl 2 r and is the irreducible representation of psl 2 r in psl n r we denote by reph 1 s psl n r a connected component that contains fuchsian representations and call it a hitchin component in fact when n is odd there is one hitchin component and when n is even two isomorphic ones n hitchin proves in 17

deformations of twisted harmonic maps and variation of the energy we study the deformations of twisted harmonic maps f with respect to the representation rho after constructing a continuous universal twisted harmonic map we give a construction of every first order deformation of f in terms of hodge theory we apply this result to the moduli space of reductive representations of a kahler group to show that the critical points of the energy functional e coincide with the monodromy representations of polarized complex variations of hodge structure we then proceed to second order deformations where obstructions arise we investigate the existence of such deformations and give a method for constructing them as well applying this to the energy functional as above we prove for every finitely presented group that the energy functional is a potential for the kahler form of the betti moduli space assuming furthermore that the group is kahler we study the eigenvalues of the hessian of e at critical points

a small time solutions for the korteweg de vries equation in this paper a heat balance integral hbi method is applied to the one dimensional non linear korteweg de vries kdv equation prescribed by appropriate homogenous boundary conditions and a set of initial conditions to obtain its approximate analytical solutions at small times it is shown that the hbi solutions obtained by the method may be used effectively at small times when the exact solution of the kdv equation is not known

why the characteristic polynomial factors we survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently the first technique uses geometric ideas and is is based on zaslavsky s theory of signed graphs the second approach is algebraic and employs results of saito and terao about free hyperplane arrangements finally we consider a purely combinatorial theorem of stanley about supersolvable lattices and its generalizations

classification of weil petersson isometries this paper contains two main results the first is the existence of an equivariant weil petersson geodesic in teichmuller space for any choice of pseudo anosov mapping class as a consequence one obtains a classification of the elements of the mapping class group as weil petersson isometries which is parallel to the thurston classification the second result concerns the asymptotic behavior of these geodesics it is shown that geodesics that are equivariant with respect to independent pseudo anosov s diverge it follows that subgroups of the mapping class group which contain independent pseudo anosov s act in a reductive manner with respect to the weil petersson geometry this implies an existence theorem for equivariant harmonic maps to the metric completion

local rigidity of discrete groups acting on complex hyperbolic space the superrigidity theorem of margulis see zimmer 17 classifies finite dimensional representations of lattices in semi simple lie groups of real rank strictly larger than 1 it is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank i semi simple lie groups that this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or in other words the existence of non trivial deformations in johnson millson 6 and kourouniotis 8 such deformations were constructed for certain representations of lattices in so n 1 based on a construction of thurston called bending the deformation space of the representation of a lattice f so n 1 obtained by restricting an inclusion so n 1 so n l 1 to f is of particular interest if n 2 the space of infinitesimal deformations is hi f r 1 where f acts on r 1 by the restriction of the standard action of so n 1 the space of infinitesimal deformations is non zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable in 6 it is also shown that some are not in this paper we study the complex analogue of the above example we let f be a cocompact torsion free lattice in su n 1 and consider the deformation space of the representation of f obtained by restricting an inclusion su n 1 su n i 1 if n l the space of infinitesimal deformations is h f r h f c 1 in the first summand f acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming f in u n 1 by a curve of homomorphisms into the center of u n 1 observe that the above inclusion factors as su n 1 u n 1 su n i 1 in the second summand f acts by the restriction of the standard action of su n 1 this summand is non zero for the standard arithmetic examples

moduli spaces of lambda modules on abelian varieties abstract we study the moduli space m x n of semistable modules of vanishing chern classes over an abelian variety x where belongs to a certain subclass of d algebras in particular for d x resp sym t x we obtain a description of the moduli spaces of flat connections resp higgs bundles we give a description of m x n in terms of a symmetric product of a certain fibre bundle over the dual abelian variety x we also give a moduli interpretation to the associated hilbert scheme as the classifying space of modules with extra structure finally we study the non abelian hodge theory associated to these new moduli spaces

maximal representations of surface groups in bounded symmetric domains let f be the fundamental group of a hyperbolic surface of genus g for 1 2 leave invariant a totally geodesic subspace of dp q isometric to dp p

a hybrid numerical method for the kdv equation by finite difference and sinc collocation method abstract in this paper we propose a hybrid numerical method for the kdv equation more precisely we discretize the temporal derivative of kdv equation by a weighted scheme and treat the implicitly nonlinear term with the combination of finite difference and sinc collocation method the stability analysis is presented and numerical experiments illustrate the efficiency and stability of the proposed hybrid method the motions and the interactions of solitary waves relying on particular initial boundary conditions are also simulated

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

a local discontinuous galerkin method for the korteweg de vries equation with boundary effect a local discontinuous galerkin method for solving korteweg de vries kdv type equations with non homogeneous boundary effect is developed we provide a criterion for imposing appropriate boundary conditions for general kdv type equations the discussion is then focused on the kdv equation posed on the negative half plane which arises in the modeling of transition dynamics in the plasma sheath formation h liu m slemrod kdv dynamics in the plasma sheath transition appl math lett 17 4 2004 401 410 the guiding principle for selecting inter cell fluxes and boundary fluxes is to ensure the l 2 stability and to incorporate given boundary conditions the local discontinuous galerkin method thus constructed is shown to be stable and efficient numerical examples are given to confirm the theoretical result and the capability of this method for capturing soliton wave phenomena and various boundary wave patterns

conservative numerical methods for solitary wave interactions the purpose of this paper is to show the advantages that represent the use of numerical methods that preserve invariant quantities in the study of solitary wave interactions for the regularized long wave equation it is shown that the so called conservative methods are more appropriate to study the phenomenon and provide a dynamic point of view that allows us to estimate the changes in the parameters of the solitary waves after the collision

volume invariant and maximal representations of discrete subgroups of lie groups let gamma be a lattice in a connected semisimple lie group g with trivial center and no compact factors we introduce a volume invariant for representations of gamma into g which generalizes the volume invariant for representations of uniform lattices introduced by goldman then we show that the maximality of this volume invariant exactly characterizes discrete faithful representations of gamma into g

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

parabolic higgs bundles and teichmuller spaces for punctured surfaces in this paper we study the relation between parabolic higgs vector bundles and irreducible representations of the fundamental group of punctured riemann surfaces established by simpson we generalize a result of hitchin identifying those parabolic higgs bundles that correspond to fuchsian representations we also study the higgs bundles that give representations whose image is contained after conjugation in sl k r we compute the real dimension of one of the components of this space of representations which in the absence of punctures is the generalized teichmuller space introduced by hitchin and which in the case of k 2 is the usual teichmuller space of the punctured surface

asymptotics of higgs bundles in the hitchin component in this paper we pursue a more geometric approach to compactification of the hitchin component our main motivation is wolf s harmonic map interpretation of thurston s compactification of teichmuller space with measured foliations using hitchin s parameterization of the hitchin component by holomorphic differentials we study asymptotics of certain rays of representations more precisely along these rays we solve the hitchin equations asymptotically and use the solution to study the asymptotics of the parallel transport operator of the associated flat connection the asymptotics of the corresponding family of equivariant harmonic maps to the symmetric space sl n r so n r proves a conjecture of katzarkov noll pandit and simpson 17 on the hitchin wkb problem in our setting

dimension and rigidity of quasi fuchsian representations let fo c so n 1 n 2 be a cocompact lattice and p fo f be an injective representation into a convex cocompact discrete isometric subgroup of a noncompact rank 1 symmetric space the hausdorff dimension 6 f of the limit set of f p fo satisfies 6 f 6 fo n 1 we prove that equality holds if and only if p is a fuchsian representation i e f preserves a totally geodesic copy of hri in hr this generalizes the result of 2 and settles a question raised by tukia 43 p 428 actually we prove a more general result in the context of variable negative curvature strikingly there are no quasi fuchsian representations at least for the lower codimensional case in complex hyperbolic geometry that is for a cocompact lattice fo c su n 1 n 2 and an injective representation p fo su m 1 n m 2n 1 with f p fo convex cocompact we prove that one always has 6 f 6 fo and moreover f must stabilize a totally geodesic copy of h in h this can be viewed as a global generalization of goldman and millson s local rigidity theorem see 20 another global generalization was obtained by k corlette 5 various other related rigidity results are also obtained

generators for the cohomology ring of the moduli space of rank 2 higgs bundles the moduli space of stable vector bundles on a riemann surface is smooth when the rank and degree are coprime and is diffeomorphic to the space of unitary connections of central constant curvature a classic result of newstead and atiyah and bott asserts that its rational cohomology ring is generated by the universal classes that is by the kunneth components of the chern classes of the universal bundle this paper studies the larger non compact moduli space of higgs bundles as introduced by hitchin and simpson with values in the canonical bundle k this is diffeomorphic to the space of all connections of central constant curvature whether unitary or not the main result of the paper is that in the rank 2 case the rational cohomology ring of this space is again generated by universal classes the spaces of higgs bundles with values in k n for n 0 turn out to be essential to the story indeed we show that their direct limit has the homotopy type of the classifying space of the gauge group and hence has cohomology generated by universal classes

an analytical numerical method for solving the korteweg de vries equation in this paper an analytical numerical method is applied to the one dimensional korteweg de vries equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times two test problem with known exact solutions are studied to demonstrate the accuracy of the present method the obtained results are compared with the exact solution of each problem and are found to be in good agreement with each other the numerical scheme is also compared with earlier work and shown to be accurate and efficient

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

arrangements of ideal type abstract in 2006 sommers and tymoczko defined so called arrangements of ideal type a i stemming from ideals i in the set of positive roots of a reduced root system they showed in a case by case argument that a i is free if the root system is of classical type or g 2 and conjectured that this is also the case for all types this was established only very recently in a uniform manner by abe barakat cuntz hoge and terao the set of non zero exponents of the free arrangement a i is given by the dual of the height partition of the roots in the complement of i in the set of positive roots generalizing the shapiro steinberg kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated weyl group our first aim in this paper is to investigate a stronger freeness property of the a i we show that all a i are inductively free with the possible exception of some cases in type e 8 in the same paper from 2006 sommers and tymoczko define a poincare polynomial i t associated with each ideal i which generalizes the poincare polynomial w t for the underlying weyl group w solomon showed that w t satisfies a product decomposition depending on the exponents of w for any coxeter group w sommers and tymoczko showed in a case by case analysis in types a n b n and c n and some small rank exceptional types that a similar factorization property holds for the poincare polynomials i t generalizing the formula of solomon for w t they conjectured that their multiplicative formula for i t holds in all types in our second aim to investigate this conjecture further the same inductive tools we develop to obtain inductive freeness of the a i are also employed here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions

error estimates of the semi discrete local discontinuous galerkin method for nonlinear convection diffusion and kdv equations in this paper we provide l 2 error estimates for the semi discrete local discontinuous galerkin methods for nonlinear convection diffusion equations and kdv equations with smooth solutions the main technical difficulty is the control of the inter element jump terms which arise because of the nonlinearity of the pdes and the discontinuous nature of the numerical method

maximal representations of uniform complex hyperbolic lattices let rho be a maximal representation of a uniform lattice gamma subset rm su n 1 n geq 2 in a classical lie group of hermitian type h we prove that necessarily h rm su p q with p geq qn and there exists a holomorphic or antiholomorphic rho equivariant map from complex hyperbolic space to the symmetric space associated to rm su p q this map is moreover a totally geodesic homothetic embedding in particular up to a representation in a compact subgroup of rm su p q the representation rho extends to a representation of rm su n 1 in rm su p q

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

cyclic surgery degrees of maps of character curves and volume rigidity for hyperbolic manifolds this paper proves a theorem about dehn surgery using a new theorem about psl2c character varieties confirming a conjecture of boyer and zhang this p aper shows that a small hyperbolic knot in a homotopy sphere having a non trivial cyclic slope r has an incompressible surface with non integer boundary slope strictly between r 1 and r 1 a corollary is that any small knot which has only integer boundary slopes has property p the proof uses connections between the topology of the complement of the knot m and the psl2c character variety of m that were discovered by culler and shalen the key lemma which should be of independent interest is that for certain components of the character variety of m the map on character varieties induced by m m is a birational isomorphism onto its image this in turn depends on a fancy version of mostow rigidity due to gromov thurston and goldman the proof of the theorem about dehn surgery also gives information about the diameter of the set of boundary slopes of m let me start with the theorem about dehn surgery let k be a knot in a compact closed 3manifold that is a tame embedding s 1 the complement of k m is minus an open regular neighborhood of k so m is a compact 3 manifold whose boundary is a torus suppose is a simple closed curve in m we can create a closed manifold m from m by taking a solid torus and gluing its boundary to m in such a way that bounds a disc in the solid torus m depends only on the isotopy class of the new manifold m is called a dehn filling of m or a dehn surgeryon k recently many people have studied what kinds of manifolds arise when you

moduli spaces of holomorphic triples over compact riemann surfaces a holomorphic triple over a compact riemann surface consists of two holomorphic vector bundles and a holomorphic map between them after fixing the topological types of the bundles and a real parameter there exist moduli spaces of stable holomorphic triples in this paper we study non emptiness irreducibility smoothness and birational descriptions of these moduli spaces for a certain range of the parameter our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary lie groups of indefinite signature 5 7 another application that we study in this paper is to the existence of stable bundles on the product of the surface by the complex projective line

involutions and higher order automorphisms of higgs bundle moduli spaces this research was partially supported by the spanish mineco under the icmat severo ochoa grant no sev 2011 0087 and grant no mtm2013 43963 p and by the european commission marie curie irses moduli programme pirses ga 2013 612534

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

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

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

numerical solution of korteweg de vries equation by galerkin b spline finite element method a finite element solution of the kdv equation is presented to demonstrate the efficiency of the method two test problems are considered the numerical solutions of the kdv equation are compared with both the exact solutions and other numerical solutions in the literature the numerical solutions are found to be in good agreement with the exact solutions

topology of hitchin systems and hodge theory of character varieties the case a1 for g gl2 pgl2 sl2 we prove that the perverse ltration associated with the hitchin map on the rational cohomology of the moduli space of twisted g higgs bundles on a compact riemann surface c agrees with the weight ltration on the rational cohomology of the twisted g character variety of c when the cohomologies are identied via non abelian hodge theory the proof is accomplished by means of a study of the topology of the hitchin map over the locus of integral spectral curves

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

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