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

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

on integrability of the yang baxter σ model we prove that the recently introduced yang baxter model can be considered as an integrable deformation of the principal chiral model we find also an explicit one to one map transforming every solution of the principal chiral model into a solution of the deformed model with the help of this map the standard procedure of the dressing of the principal chiral solutions can be directly transferred into the deformed yang baxter context

k means type algorithms a generalized convergence theorem and characterization of local optimality the k means algorithm is a commonly used technique in cluster analysis in this paper several questions about the algorithm are addressed the clustering problem is first cast as a nonconvex mathematical program then a rigorous proof of the finite convergence of the k means type algorithm is given for any metric it is shown that under certain conditions the algorithm may fail to converge to a local minimum and that it converges under differentiability conditions to a kuhn tucker point finally a method for obtaining a local minimum solution is given

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

surjectivity of galois representations associated with quadratic mathbb q curves we prove in this paper an uniform surjectivity result for galois representations associated with non cm mathbb q curves over imaginary quadratic fields using various tools for the proof such as mazur s method isogeny theorems runge s method and analytic estimates of sums of l functions

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions

on non integrability of classical strings in p brane backgrounds we investigate the question of possible integrability of classical string motion in curved p brane backgrounds for example the d3 brane metric interpolates between the flat and the ads5 s5 regions in which string propagation is integrable we find that while the point like string geodesic equations are integrable the equations describing an extended string in the complete d3 brane geometry are not the same conclusion is reached for similar brane intersection backgrounds interpolating between flat space and adsk sk we consider in particular the case of the ns 5 brane fundamental string background to demonstrate non integrability we make a special pulsating string ansatz for which the string equations reduce to an effective one dimensional system expanding near this simple solution leads to a linear differential equation for small fluctuations that cannot be solved in quadratures implying non integrability of the original set of string equations

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

surfaces and the sklyanin bracket we discuss the lie poisson group structures associated to splittings of the loop group lgl n due to sklyanin concentrating on the finite dimensional leaves of the associated poisson structure we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a invariant poisson structure in particular sklyanin s lie poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve the sklyanin structure is then equivalent to one considered by mukai tyurin and bottacin on a moduli space of sheaves on the poisson surface the abelianization procedure gives rise to natural darboux coordinates for these leaves as well as separation of variables for the integrable hamiltonian systems associated to invariant functions on the group

a data driven procedure for density estimation with some applications this paper deals with the probability density estimation using a kernel based approach where the window size of the kernel is found by a data driven procedure it is theoretically shown that under certain assumptions the estimated densities on bounded sets can be asymptotically unbiased when the width of window is obtained from the minimal spanning tree of the observed data the theoretical development initially carried out on r2 is applicable to higher dimensional spaces the results are experimentally verified on bounded sets with different types of distributions the behaviour of the estimator in the case of the unbounded set as in that for gaussian density is also experimentally seen to be good some applications of the proposed density estimation technique is demonstrated one application is the representative point detection algorithm which can be applied for data reduction and outlier rejection another application involves detection of border points of a dot pattern as well as finding a thinned version of the dot pattern

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

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

integrable coupled σ models a systematic procedure for constructing classical integrable field theories with arbitrarily many free parameters is outlined it is based on the recent interpretation of integrable field theories as realizations of affine gaudin models in this language one can associate integrable field theories with affine gaudin models having arbitrarily many sites we present the result of applying this general procedure to couple together an arbitrary number of principal chiral model fields on the same lie group each with a wess zumino term

nahm transform and spectral curves for doubly periodic instantons we present the nahm transform of the doubly periodic instantons i e instantons on t 2 2 converting them into certain meromorphic solutions of hitchin s equations over an elliptic curve we then show how to construct a triple consisting of an algebraic curve plus a line bundle with connection over it from a doubly periodic instanton and that such data coincides with the hitchin spectral data associated with the nahm transformed higgs bundle

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

selection of alpha for alpha hull in r2 abstract for finding the shape of a planar set edelsbrunner kirkpatrick and seidel introduced the concept of hulls as a natural generalization of convex hulls while the hull is elegant and efficient to compute it still suffers from a major drawback i e the single parameter namely must nevertheless be tuned this paper deals with finding a way to overcome this drawback i e we proposed here a selection criterion of for hulls corresponding to a point set in r 2 the selection criterion of is based on the concept of minimum spanning trees and certain existing results the effectiveness of the proposed selection criterion is demonstrated on some artificially generated data sets the convergence with sample size of the hull based on the proposed selection criterion for to the original pattern class has also been verified using symmetric difference the hausdorff metric and a similarity metric

generalized type iib supergravity equations and non abelian classical r matrices we study yang baxter deformations of the ads superstring with non abelian classical r matrices which satisfy the homogeneous classical yang baxter equation by performing a supercoset construction we can get deformed ads x s backgrounds while this is a new area of research the current understanding is that abelian classical r matrices give rise to solutions of type iib supergravity while non abelian classical r matrices lead to solutions of the generalized supergravity equations we examine here some examples of non abelian classical r matrices and derive the associated backgrounds explicitly all of the resulting backgrounds satisfy the generalized equations for some of them we derive t dualized backgrounds by adding a linear coordinate dependence to the dilaton and show that these satisfy the usual type iib supergravity equations remarkably some of the t dualized backgrounds are locally identical to undeformed ads x s after an appropriate coordinate transformation but this seems not to be generally the case

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

recent advances in the langlands program these are the notes for the lecture given by the author at the mathematical current events special session of the ams meeting in baltimore on january 17 2003 topics reviewed include the langlands correspondence for gl n in the function field case and its proof by v drinfeld and l lafforgue the geometric langlands correspondence for gl n and its proof by d gaitsgory k vilonen and the author and the work of a beilinson and v drinfeld on the quantization of the hitchin system and the langlands correspondence for an arbitrary semisimple algebraic group

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

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

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

superintegrability of geodesic motion on the sausage model reduction of the eta deformed sigma model on ads 5 x s 5 to the two dimensional squashed sphere s 2 eta can be viewed as a special case of the fateev sausage model where the coupling constant v i

serre s uniformity problem in the split cartan case we prove that there exists an integer p 0 such that x split p q is made of cusps and cm points for any prime p p 0 equivalently for any non cm elliptic curve e over q and any prime p p 0 the image of the galois representation induced by the galois action on the p division points of e is not contained in the normalizer of a split cartan subgroup this gives a partial answer to an old question of serre

hecke transformation and the generalized theta function we describe the behavior of the generalized theta function with respect to the hecke transformations let e be a holomorphic vector bundle over rank r and degree r g 1 over a compact riemann surface x of genus g let p e respectively p e be the space of all lines respectively hyperplanes in the fibers of e certain zariski open subsets of p e p e and p e p e p e p e parametrize holomorphic families of vector bundles over x of rank r and degree r g 1 we describe the generalized theta line bundle for these families

frobenius manifolds projective special geometry and hitchin systems we consider the construction of frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved in particular we prove that the underlying f manifold is canonical we then apply this construction to integrable systems of hitchin type

courant algebroids poisson lie t duality and type ii supergravities we reexamine the notions of generalized ricci tensor and scalar curvature on a general courant algebroid reformulate them using objects natural w r t pull backs and reductions and obtain them from the variation of a natural action functional this allows us to prove in a very general setup the compatibility of the poisson lie t duality with the renormalization group flow and with string background equations we thus extend the known results to a much wider class of dualities including the cases with gauging so called dressing cosets or equivariant poisson lie t duality as an illustration we use the formalism to provide new classes of solutions of modified supergravity equations on symmetric spaces

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

on the level of modular curves that give rise to isolated j invariants abstract we say a closed point x on a curve c is sporadic if c has only finitely many closed points of degree at most deg x and that x is isolated if it is not in a family of effective degree d divisors parametrized by p 1 or a positive rank abelian variety see section 4 for more precise definitions and a proof that sporadic points are isolated motivated by well known classification problems concerning rational torsion of elliptic curves we study sporadic and isolated points on the modular curves x 1 n in particular we show that any non cuspidal non cm sporadic respectively isolated point x x 1 n maps down to a sporadic respectively isolated point on a modular curve x 1 d where d is bounded by a constant depending only on j x conditionally we show that d is bounded by a constant depending only on the degree of q j x so in particular there are only finitely many j invariants of bounded degree that give rise to sporadic or isolated points

minimal surfaces in q deformed rm ad rm s _ 5 s 5 with poincare coordinates we study minimal surfaces in q deformed s5 with a new coordinate system introduced in the previous work kameyama and yoshida 2015 j phys a math theor 48 075401 in this paper we introduce poincare coordinates for the deformed theory we then construct minimal surfaces whose boundary shape is a circle the solution corresponds to a 1 2 bps circular wilson loop in the limit a remarkable point is that the classical euclidean action is not divergent unlike the original this finiteness indicates that the q deformation may be regarded as a uv regularization

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

deformed integrable σ models classical r matrices and classical exchange algebra on drinfel d doubles we describe a unifying framework for the systematic construction of integrable deformations of integrable models within the hamiltonian formalism it applies equally to both the yang baxter type as well as gauged wzw type deformations which were considered recently in the literature as a byproduct these two families of integrable deformations are shown to be poisson lie t dual of one another

on q deformed symmetries as poisson lie symmetries and application to yang baxter type models this is an author created un copyedited version of an article accepted for publication in journal of physics a mathematical and theoretical the publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it the version of record is available online at https doi org 1 1088 1751 8113 49 41 415402 2016 iop publishing ltd

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

some cases of serre s uniformity problem we show that if e mathbb q is an elliptic curve without complex multiplication and for which there is a prime q such that the image of bar rho e q is contained in the normaliser of a split cartan subgroup of mathrm gl 2 mathbb f q then bar rho e p surjects onto mathrm gl 2 mathbb f p for every prime p 37 this result complements a previous result by the author we also prove analogue results for certain families of mathbb q curves building on results of ellenberg am j math 126 4 763 787 2004 and le fourn math ann 365 1 2 173 214 2016

minimal spanning tree based clustering technique relationship with bayes classifier a minimal spanning tree mst based clustering technique along with its theoretical formulation is presented in this paper the proposed technique is compared with bayes classifier and it is shown theoretically that the clustering technique although an unsupervised one approaches the performance of bayes classifier under a condition as the number of sample points from each class increases experimental results with many synthetic data sets in 2 d and 3 d validate the theoretical prediction

elliptic calogero moser systems and isomonodromic deformations we show that various models of the elliptic calogero moser systems are accompanied with an isomonodromic system on a torus the isomonodromic partner is a nonautonomous hamiltonian system defined by the same hamiltonian the role of the time variable is played by the modulus of the base torus a suitably chosen lax pair with an elliptic spectral parameter of the elliptic calogero moser system turns out to give a lax representation of the nonautonomous system as well this lax representation ensures that the nonautonomous system describes isomonodromic deformations of a linear ordinary differential equation on the torus on which the spectral parameter of the lax pair is defined a particularly interesting example is the extended twisted bcl model recently introduced along with some other models by bordner and sasaki who remarked that this system is equivalent to inozemtsev s generalized elliptic calogero moser system we use the root type lax pair developed by bordner et al to formulate the associa

double coset construction of moduli space of holomorphic bundles and hitchin systems we present a description of the moduli space of holomorphic vector bundles over riemann curves as a double coset space which is differ from the standard loop group construction our approach is based on equivalent definitions of holomorphic bundles based on the transition maps or on the first order differential operators using this approach we present two independent derivations of the hitchin integrable systems we define a superfree upstairs system from which hitchin systems are obtained by three step hamiltonian reductions special attention is given to the schottky parameterization of curves

integrability of the eta deformed neumann rosochatius model an integrable deformation of the well known neumann rosochatius system is studied by considering generalised bosonic spinning solutions on the eta deformed ads 5 x s 5 background for this integra

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

r matrix structure of hitchin system in tyurin parameterization we present a classical r matrix for the hitchin system without marked points on an arbitrary non degenerate algebraic curve of genus g 2 using tyurin parameterization of holomorphic vector bundles