generic smoothness of the moduli spaces of rank two stable vector bundles over algebraic surfaces we denote by m k the moduli space of all isomorphic classes of h stable rank 2 vector bundles with fixed determinant bundle d and second chern class k it is well known that m k is a quasi projective variety and is not empty if k is sufficiently large see g and m the local structure of m k is described precisely by the kuranishi deformation theorem in the following way suppose that v m k and let r h 1 endo v hz endo v be the kuranishi map then the stalk of the structure sheaf gm at v is naturally isomorphic to the germs of holomorphic functions at 0 h endo v divided by the ideal generated by the components of in particular if hz endo v o then m k is smooth at v and the tangent space t m k v is identified with h 1 endo v the hirzebruch riemann roch theorem gives the dimension formula

homeomorphic conjugates of fuchsian groups by a fuchsian group we mean a discrete subgroup of m it may contain also orientation reversing elements usually a fuchsian group is thought to act on i but this is not a problem since the action on s can be uniquely extended to an action on 1 and later we might also consider a fuchsian group to act on 1 depending on the situation we are here concerned with the problem when is a group g ofhomeornorphisms of s topologically conjugate to a fuchsian group we propose that this happens if and only if g is a convergence group of s which means that it satisfies the convergence property this means that whenever gi g are distinct we can pass to a subsequence in such a way that for some a b c s

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

actions of finite groups on homotopy 3 spheres it is conjectured that the action of a finite group of diffeomorphisms of the 3 sphere is equivariantly diffeomorphic to a linear action this conjecture is verified if both of the following conditions hold i each isotropy subgroup is dihedral or cyclic ii there is a point with cyclic isotropy subgroup of order not 1 2 3 or 5 0 introduction the generalized smith conjecture says that the effective action of a finite group g of orientation preserving diffeomorphisms on a homotopy 3 sphere i is essentially linear if y is diffeomorphic to s3 then essentially linear is equivalent to equivariantly diffeomorphic to a linear action work by thurston johannson jaco shalen bass meeks and yau established this conjecture when g is generated by a diffeomorphism with fixed points this is the classical smith conjecture davis and morgan improved this in certain cases by proving that if every isotropy subgroup of g is cyclic and if at least one of these subgroups has order greater than 5 then the action is essentially linear the group g ig finite and the action is differentiable and effective thus the isotropy subgroup of a point in e acts on the tangent space at the point and so is a finite subgroup of gl r therefore this group is cyclic dihedral tetrahedral octahedral or icosahedral in this paper the result of davis and morgan will be extended to include the case where one of the isotropy subgroups has order 4 we also show that the actions is essentially linear if every isotropy subgroup is either dihedral or cyclic and one of the cyclic isotropy subgroups has order not equal to 1 2 3 or 5 the results of this paper first appeared in the author s 1981 thesis at columbia university subsequently william thurston announced a theorem that includes these results as a special case for another proof of the davis morgan extension of the smith conjecture see culler and shalen 1 i would like to express my deepest gratitude to my thesis advisor john morgan i would also like to thank michael davis i preliminaries we start with pertinent definitions from orbifold theory and recall some results of davis and morgan 2 other results of that paper are extended to include the dihedral case for more on orbifolds see thurston 5 received by the editors march 9 1982 and in revised form june 20 1983 1980 mathematics subject classification primary 57s17 1984 american mathematical society 0002 9947 84 1 00 25 per page

fibration de hitchin et endoscopie we propose a geometric interpretation of the theory of elliptic endoscopy due to langlands and kottwitz in terms of the hitchin fibration as applications we prove a global analog of a purity conjecture due to goresky kottwitz and macpherson for unitary groups this global purity statement has been used in a joint work with g laumon to prove the fundamental lemma over a local fields of equal characteristics

twisted fourier mukai transforms for holomorphic symplectic four folds we apply the methods of c a ldararu to construct a twisted fourier mukai transform between a pair of holomorphic symplectic four folds which are fibred by lagrangian abelian surfaces more precisely we obtain an equivalence between the derived category of coherent sheaves on a certain lagrangian fibration and the derived category of twisted sheaves on its mirror partner as a corollary we extend the original fourier mukai transform to degenerations of abelian surfaces another consequence of the general theory is that the holomorphic symplectic four fold and its mirror are connected by a one parameter family of deformations through lagrangian fibrations

fourier mukai transforms for gorenstein schemes we extend to singular schemes with gorenstein singularities or fibered in schemes of that kind bondal and orlov s criterion for an integral functor to be fully faithful we also prove that the original condition of characteristic zero cannot be removed by providing a counterexample in positive characteristic we contemplate a criterion for equivalence as well in addition we prove that for locally projective gorenstein morphisms a relative integral functor is fully faithful if and only if its restriction to each fibre is also fully faithful these results imply the invertibility of the usual relative fourier mukai transform for an elliptic fibration as a direct corollary

d equivalence and k equivalence let x and y be smooth projective varieties over they are called d equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories and k equivalent if they are birationally equivalent and the pull backs of their canonical divisors to a common resolution coincide we expect that the two equivalences coincide for birationally equivalent varieties we shall provide a partial answer to the above problem in this paper

the local structure of compactified jacobians this paper studies the local geometry of compactified jacobians the main result is a presentation of the completed local ring of the compactified jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve the authors have investigated the geometric and combinatorial properties of these rings in previous work and consequences for compactified jacobians are presented in this paper similar results are given for the local structure of the universal compactified jacobian over the moduli space of stable curves

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

low complexity rls algorithms using dichotomous coordinate descent iterations in this paper we derive low complexity recursive least squares rls adaptive filtering algorithms we express the rls problem in terms of auxiliary normal equations with respect to increments of the filter weights and apply this approach to the exponentially weighted and sliding window cases to derive new rls techniques for solving the auxiliary equations line search methods are used we first consider conjugate gradient iterations with a complexity of operations per sample being the number of the filter weights to reduce the complexity and make the algorithms more suitable for finite precision implementation we propose a new dichotomous coordinate descent dcd algorithm and apply it to the auxiliary equations this results in a transversal rls adaptive filter with complexity as low as multiplications per sample which is only slightly higher than the complexity of the least mean squares lms algorithm multiplications simulations are used to compare the performance of the proposed algorithms against the classical rls and known advanced adaptive algorithms fixed point fpga implementation of the proposed dcd based rls algorithm is also discussed and results of such implementation are presented

strange actions of groups on spheres in fs we investigated certain topological analogs of schottky groups called admissible actions and their compatibility with various structures on spheres we constructed an action f 2 s 3 s 3 which was not topologically conjugate to a uniformly quasiconformal action also there was an example f r 2 s 3 s 3 r sufficiently large which was smooth and uniformly quasiconformal but not topologically conjugate to a conformal action and we gave examples of admissible actions on higher dimensional spheres but did not analyse the structures preserved

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights

ideals associated to deformations of singular plane curves we consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities these loci are the equisingular locus es which parametrizes equisingular or topologically trivial deformations the equigeneric locus eg which parametrizes deformations of constant geometric genus and the equiclassical locus ec which parametrizes deformations of constant geometric genus and class the class of a reduced plane curve is the degree of its dual it was previously known that the tangent space to es corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to eg corresponds to the conductor ideal we show that the support of the tangent cone to ec corresponds to an ideal which we call the equiclassical ideal by studying these ideals we are able to obtain information about the geometry and dimensions of es ec and eg this allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

a loop theorem for duality spaces and fibred ribbon knots in this paper we prove a version of the loop theorem for surfaces in the boundary of a 3 dimensional duality space i e a space which resembles a 3manifold only in that it satisfies the appropriate form of poincar6 lefschetz duality over some field of untwisted coefficients our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance and as the main application of our theorem we show that if a fibred knot in the 3 sphere is a ribbon knot then its monodromy extends over a handlebody we approach the loop theorem via the study of planar coverings of a surface as in the original paper of papakyriakopoulos 11 and the subsequent work of maskit 9 w contains a simple geometric treatment of these matters the main result of w is that a duality space actually satisfies duality with twisted coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal in w 4 the results of w167 2 and 3 together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring are used to prove the loop theorem for 3 dimensional duality spaces for the reader s convenience a proof of the algebraic temma is included as an appendix w contains the application to fibred ribbon knots mentioned above in w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4 manifolds in w we apply our methods to another aspect of knot concordance and show that for any concordance with a rationally anisotropic fibred knot see 7 at one end the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups for torus knots this question was raised by scharlemann 14

an inexact and nonmonotone proximal method for smooth unconstrained minimization an implementable proximal point algorithm is established for the smooth nonconvex unconstrained minimization problem at each iteration the algorithm minimizes approximately a general quadratic by a truncated strategy with step length control the main contributions are i a framework for updating the proximal parameter ii inexact criteria for approximately solving the subproblems iii a nonmonotone criterion for accepting the iterate the global convergence analysis is presented together with numerical results that validate and put into perspective the proposed approach

multivariate spectral gradient method for unconstrained optimization multivariate spectral gradient method is proposed for solving unconstrained optimization problems combined with some quasi newton property multivariate spectral gradient method allows an individual adaptive stepsize along each coordinate direction which guarantees that the method is finitely convergent for positive definite quadratics especially it converges no more than two steps for positive definite quadratics with diagonal hessian and quadratically for objective functions with positive definite diagonal hessian moreover based on a nonmonotone line search global convergence is established for multivariate spectral gradient algorithm at last numerical results are reported which show that this method is promising and deserves further discussing

a class of gradient unconstrained minimization algorithms with adaptive stepsize in this paper the development convergence theory and numerical testing of a class of gradient unconstrained minimization algorithms with adaptive stepsize are presented the proposed class comprises four algorithms the first two incorporate techniques for the adaptation of a common stepsize for all coordinate directions and the other two allow an individual adaptive stepsize along each coordinate direction all the algorithms are computationally efficient and possess interesting convergence properties utilizing estimates of the lipschitz constant that are obtained without additional function or gradient evaluations the algorithms have been implemented and tested on some well known test cases as well as on real life artificial neural network applications and the results have been very satisfactory

hyperbolic groups with planar boundaries we prove that the class of convex cocompact kleinian groups is quasi isometrically rigid we also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex cocompact kleinian group provided that its boundary has ahlfors regular conformal dimension strictly less than 2 or if it acts geometrically on a cat 0 cube complex

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

convergence of nonmonotone line search method in this paper we develop a new nonmonotone line search for general line search method and establish some global convergence theorems the new nonmonotone line search is a novel form of the nonmonotone armijo line search and allows one to choose a larger step size at each iteration which is available in constructing new line search methods and possibly reduces the function evaluations at each iteration moreover we analyze the convergence rate of some special line search methods with the new line search preliminary numerical results show that some line search methods with the new nonmonotone line search are available and efficient in practical computation

a nonmonotone line search technique and its application to unconstrained optimization a new nonmonotone line search algorithm is proposed and analyzed in our scheme we require that an average of the successive function values decreases while the traditional nonmonotone approach of grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 requires that a maximum of recent function values decreases we prove global convergence for nonconvex smooth functions and r linear convergence for strongly convex functions for the l bfgs method and the unconstrained optimization problems in the cute library the new nonmonotone line search algorithm used fewer function and gradient evaluations on average than either the monotone or the traditional nonmonotone scheme

the convergence of conjugate gradient method with nonmonotone line search abstract the conjugate gradient method is a useful and powerful approach for solving large scale minimization problems liu and storey developed a conjugate gradient method which has good numerical performance but no global convergence under traditional line searches such as armijo line search wolfe line search and goldstein line search in this paper we propose a new nonmonotone line search for liu storey conjugate gradient method ls in short the new nonmonotone line search can guarantee the global convergence of ls method and has a good numerical performance by estimating the lipschitz constant of the derivative of objective functions in the new nonmonotone line search we can find an adequate step size and substantially decrease the number of functional evaluations at each iteration numerical results show that the new approach is effective in practical computation

the compactified picard scheme of the compactified jacobian abstract let c be an integral projective curve in any characteristic given an invertible sheaf l on c of degree 1 form the corresponding abel map a l c j which maps c into its compactified jacobian and form its pullback map a l pic j 0 j which carries the connected component of 0 in the picard scheme back to the jacobian if c has at worst double points then a l is known to be an isomorphism we prove that a l always extends to a map between the natural compactifications pic j j and that the extended map is an isomorphism if c has at worst ordinary nodes and cusps

isometries of lorentz surfaces and convergence groups we study the isometry group of a globally hyperbolic spatially compact lorentz surface such a group acts on the circle and we show that when the isometry group acts non properly the subgroups of mathrm diff mathbb s 1 obtained are semi conjugate to subgroups of finite covers of mathrm psl 2 mathbb r by using convergence groups under an assumption on the conformal boundary we show that we have a conjugacy in mathrm homeo mathbb s 1

le lemme fondamental pondere ii enonces cohomologiques dans cet article on etudie la cohomologie de la fibration de hitchin tronquee introduite dans un article precedent on etend les principaux theoremes de ngo sur la cohomologie de la partie elliptique de la fibration de hitchin comme consequence on obtient une demonstration du lemme fondamental pondere d arthur in this paper we study the cohomology of the truncated hitchin fibration which was introduced in a previous paper we extend ngo s main theorems on the cohomology of the elliptic part of the hitchin fibration as a consequence we get a proof of arthur s weighted fundamental lemma

a new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization conjugate gradient methods have played a special role for solving large scale optimization problems due to the simplicity of their iteration convergence properties and their low memory requirements in this work we propose a new class of spectral conjugate gradient methods which ensures sufficient descent independent of the accuracy of the line search moreover an attractive property of our proposed methods is that they achieve a high order accuracy in approximating the second order curvature information of the objective function by utilizing the modified secant condition proposed by babaie kafaki et al s babaie kafaki r ghanbari n mahdavi amiri two new conjugate gradient methods based on modified secant equations journal of computational and applied mathematics 234 2010 1374 1386 further a global convergence result for general functions is established provided that the line search satisfies the wolfe conditions our numerical experiments indicate that our proposed methods are preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness

a new family of conjugate gradient methods in this paper we develop a new class of conjugate gradient methods for unconstrained optimization problems a new nonmonotone line search technique is proposed to guarantee the global convergence of these conjugate gradient methods under some mild conditions in particular polak ribiere polyak and liu storey conjugate gradient methods are special cases of the new class of conjugate gradient methods by estimating the local lipschitz constant of the derivative of objective functions we can find an adequate step size and substantially decrease the function evaluations at each iteration numerical results show that these new conjugate gradient methods are effective in minimizing large scale non convex non quadratic functions