toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions in the present work some examples of toric hyperkahler metrics in eight dimensions are constructed first it is described how the calderbank pedersen metrics arise as a consequence of the joyce description of selfdual structures in four dimensions the jones tod correspondence and a result due to tod and przanowski it is also shown that any quaternionic kahler metric with t2 isometry is locally isometric to a calderbank pedersen one the swann construction of hyperkahler metrics in eight dimensions is applied to them to find hyperkahler examples with u 1 u 1 isometry the connection with the pedersen poon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation related to eigenfunctions of a certain linear equation these hyperkahler examples are lifted to solutions of the d 11 supergravity and type iia and iib backgrounds are found by use of dualities as before all the description is achieved in terms of a single eigenfunction f some explicit f are found together with the toda structure corresponding to the trajectories of the killing vectors of the calderbank pedersen bases

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

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

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

probability product kernels the advantages of discriminative learning algorithms and kernel machines are combined with generative modeling using a novel kernel between distributions in the probability product kernel data points in the input space are mapped to distributions over the sample space and a general inner product is then evaluated as the integral of the product of pairs of distributions the kernel is straightforward to evaluate for all exponential family models such as multinomials and gaussians and yields interesting nonlinear kernels furthermore the kernel is computable in closed form for latent distributions such as mixture models hidden markov models and linear dynamical systems for intractable models such as switching linear dynamical systems structured mean field approximations can be brought to bear on the kernel evaluation for general distributions even if an analytic expression for the kernel is not feasible we show a straightforward sampling method to evaluate it thus the kernel permits discriminative learning methods including support vector machines to exploit the properties metrics and invariances of the generative models we infer from each datum experiments are shown using multinomial models for text hidden markov models for biological data sets and linear dynamical systems for time series data

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

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

multivariate extension of matrix based renyi s alpha α order entropy functional the matrix based renyi s alpha order entropy functional was recently introduced using the normalized eigenspectrum of a hermitian matrix of the projected data in a reproducing kernel hilbert space rkhs however the current theory in the matrix based renyi s alpha order entropy functional only defines the entropy of a single variable or mutual information between two random variables in information theory and machine learning communities one is also frequently interested in multivariate information quantities such as the multivariate joint entropy and different interactive quantities among multiple variables in this paper we first define the matrix based renyi s alpha order joint entropy among multiple variables we then show how this definition can ease the estimation of various information quantities that measure the interactions among multiple variables such as interactive information and total correlation we finally present an application to feature selection to show how our definition provides a simple yet powerful way to estimate a widely acknowledged intractable quantity from data a real example on hyperspectral image hsi band selection is also provided

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

argyres seiberg duality and the higgs branch we demonstrate the agreement between the higgs branches of two mathcal n 2 theories proposed by argyres and seiberg to be s dual namely the su 3 gauge theory with six quarks and the su 2 gauge theory with one pair of quarks coupled to the superconformal theory with e 6 flavor symmetry in mathematical terms we demonstrate the equivalence between a hyperkahler quotient of a linear space and another hyperkahler quotient involving the minimal nilpotent orbit of e 6 modulo the identification of the twistor lines

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

reduction of courant algebroids and generalized complex structures we present a theory of reduction for courant algebroids as well as dirac structures generalized complex and generalized kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction the enhanced symmetry group of a courant algebroid leads us to define extended actions and a generalized notion of moment map key examples of generalized kahler reduced spaces include new explicit bi hermitian metrics on cp2

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

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

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

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

some integrable hierarchies in 2 1 dimensions and their twistor description the twistor space construction due to mason and sparling for solutions of the nonlinear schrodinger and korteweg devries hierarchies is generalized resulting in families of integrable models in 2 1 dimensions soliton solutions to one such hierarchy associated with the gauge group su 2 are constructed by adapting the methods used to construct instanton and monopole solutions these integrable models have the feature that their solutions depend on a number of arbitrary functions as well as arbitrary constants in contrast to the closely related davey stewartson hierachy

on the quantum renyi relative entropies and related capacity formulas following csiszar s approach in classical information theory it is shown that the quantum relative entropies with parameter 0 1 can be represented as generalized cutoff rates and hence a direct operational interpretation of the quantum relative entropies are provided it is also shown that various generalizations of the holevo capacity defined in terms of the relative entropies coincide for the parameter range 0 2 and an upper bound on the one shot e capacity of a classical quantum channel in terms of these capacities is given

realization of compact lie algebras in kahler manifolds the berezin quantization on a simply connected homogeneous kahler manifold which is considered as a phase space for a dynamical system enables a description of the quantal system in a finite dimensional hilbert space of holomorphic functions corresponding to generalized coherent states the lie algebra associated with the manifold symmetry group is given in terms of first order differential operators in the classical theory the lie algebra is represented by the momentum maps which are functions on the manifold and the lie product is the poisson bracket given by the kahler structure the kahler potentials are constructed for the manifolds related to all compact semi simple lie groups the complex coordinates are introduced by means of the borel method the kahler structure is obtained explicitly for any unitary group representation the cocycle functions for the lie algebra and the killing vector fields on the manifold are also obtained

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

variation of discrete spectra a formula see 1 below estimating collectively the variation of eigenvalues of a symmetric matrix under a perturbation is extended to the case of discrete eigenvalues of a selfadjoint operator in hilbert space under the assumption that the perturbation is compact for this purpose the notion of an extended enumeration of discrete eigenvalues is introduced

reduction of symplectic principal mathbb r bundles we describe a reduction process for symplectic principal bundles in the presence of a momentum map these types of structures play an important role in the geometric formulation of non autonomous hamiltonian systems we apply this procedure to the standard symplectic principal bundle associated with a fibration moreover we show a reduction process for non autonomous hamiltonian systems on symplectic principal bundles we apply these reduction processes to several examples

open gromov witten invariants on elliptic k3 surfaces and wall crossing in this paper we study holomorphic discs in k3 surfaces and defined the open gromov witten invariants using this new invariant we can establish a version of correspondence between tropical discs and holomorphic discs with non trivial invariants we give an example of wall crossing phenomenon of the invariant and expect it satisfies kontsevich soibelman wall crossing formula

a unified framework for quadratic measures of independence this paper proposes a unified framework for several available measures of independence by generalizing the concept of information theoretic learning itl the key component of itl is the use of inner product between two density functions as a measure of similarity between two random variables we show that by generalizing the inner product using a symmetric strictly positive definite kernel and by choosing appropriate kernels it is possible to reproduce a number of popular measures of independence this unified framework also allows the design of new strictly positive definite kernels and corresponding measures of independence following this framework we explore a new measure of independence and apply it in the context of linear independent component analysis ica an attractive property of the proposed method is that it does not involve any free parameter and we demonstrate that it performs equally well compared to the existing methods for ica

nonextensive information theoretic kernels on measures positive definite kernels on probability measures have been recently applied to classification problems involving text images and other types of structured data some of these kernels are related to classic information theoretic quantities such as shannon s mutual information and the jensen shannon js divergence meanwhile there have been recent advances in nonextensive generalizations of shannon s information theory this paper bridges these two trends by introducing nonextensive information theoretic kernels on probability measures based on new js type divergences these new divergences result from extending the the two building blocks of the classical js divergence convexity and shannon s entropy the notion of convexity is extended to the wider concept of q convexity for which we prove a jensen q inequality based on this inequality we introduce jensen tsallis jt q differences a nonextensive generalization of the js divergence and define a k th order jt q difference between stochastic processes we then define a new family of nonextensive mutual information kernels which allow weights to be assigned to their arguments and which includes the boolean js and linear kernels as particular cases nonextensive string kernels are also defined that generalize the p spectrum kernel we illustrate the performance of these kernels on text categorization tasks in which documents are modeled both as bags of words and as sequences of characters

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

nonzero scalar curvature generalizations of the ale hyperkahler metrics a quaternionic quotient construction of families of self dual einstein metrics with positive scalar curvature on compact riemannian orbifolds v manifolds is described the metrics are the positive scalar curvature quaternionic analogs of the asymptotically locally euclidean ale gravitational instantons constructed by kronheimer

cluster like coordinates in supersymmetric quantum field theory abstract recently it has become apparent that n 2 n 2 supersymmetric quantum field theory has something to do with cluster algebras i review one aspect of the connection supersymmetric quantum field theories have associated hyperkahler moduli spaces and these moduli spaces carry a structure that looks like an extension of the notion of cluster variety in particular one encounters the usual variables and mutations of the cluster story along with more exotic extra variables and generalized mutations i focus on a class of examples where the underlying cluster varieties are moduli spaces of flat connections on surfaces as considered by fock and goncharov fock v goncharov a 2006 publ math inst hautes etudes sci 103 1 211 the work reviewed here is largely joint with davide gaiotto and greg moore

the quotient construction for a class of compact einstein manifolds given any einstein manifoldm e one can obtain further examples of einstein manifolds by taking the quotientm e g by a freely acting properly discontinuous group of isometries we study this method in the case in whichm e is kahlerian m e g is compact and the ricci curvature is non negative in many cases the candidates forg can be completely classified

hyper hamiltonian dynamics we introduce an extension of hamiltonian dynamics defined on hyper kahler manifolds which we call hyper hamiltonian dynamics we show that this has many of the attractive features of standard hamiltonian dynamics we also discuss the prototypical integrable hyper hamiltonian systems i e quaternionic oscillators

teichmuller spaces and torelli theorems for hyperkahler manifolds kreck and yang su recently gave counterexamples to a version of the torelli theorem for hyperkahlerian manifolds as stated by verbitsky we extract the correct statement and give a short proof of it we also revisit a few of its consequences some of which are given new shorter proofs and ask some questions

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

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

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

measures of entropy from data using infinitely divisible kernels information theory provides principled ways to analyze different inference and learning problems such as hypothesis testing clustering dimensionality reduction classification and so forth however the use of information theoretic quantities as test statistics that is as quantities obtained from empirical data poses a challenging estimation problem that often leads to strong simplifications such as gaussian models or the use of plug in density estimators that are restricted to certain representation of the data in this paper a framework to nonparametrically obtain measures of entropy directly from data using operators in reproducing kernel hilbert spaces defined by infinitely divisible kernels is presented the entropy functionals which bear resemblance with quantum entropies are defined on positive definite matrices and satisfy similar axioms to those of renyi s definition of entropy convergence of the proposed estimators follows from concentration results on the difference between the ordered spectrum of the gram matrices and the integral operators associated to the population quantities in this way capitalizing on both the axiomatic definition of entropy and on the representation power of positive definite kernels the proposed measure of entropy avoids the estimation of the probability distribution underlying the data moreover estimators of kernel based conditional entropy and mutual information are also defined numerical experiments on independence tests compare favorably with state of the art

a reproducing kernel hilbert space framework for information theoretic learning this paper provides a functional analysis perspective of information theoretic learning itl by defining bottom up a reproducing kernel hilbert space rkhs uniquely determined by the symmetric nonnegative definite kernel function known as the cross information potential cip the cip as an integral of the product of two probability density functions characterizes similarity between two stochastic functions we prove the existence of a one to one congruence mapping between the itl rkhs and the hilbert space spanned by square integrable probability density functions therefore all the statistical descriptors in the original information theoretic learning formulation can be rewritten as algebraic computations on deterministic functional vectors in the itl rkhs instead of limiting the functional view to the estimators as is commonly done in kernel methods a connection between the itl rkhs and kernel approaches interested in quantifying the statistics of the projected data is also established

algorithms for learning kernels based on centered alignment this paper presents new and effective algorithms for learning kernels in particular as shown by our empirical results these algorithms consistently outperform the so called uniform combination solution that has proven to be difficult to improve upon in the past as well as other algorithms for learning kernels based on convex combinations of base kernels in both classification and regression our algorithms are based on the notion of centered alignment which is used as a similarity measure between kernels or kernel matrices we present a number of novel algorithmic theoretical and empirical results for learning kernels based on our notion of centered alignment in particular we describe efficient algorithms for learning a maximum alignment kernel by showing that the problem can be reduced to a simple qp and discuss a one stage algorithm for learning both a kernel and a hypothesis based on that kernel using an alignment based regularization our theoretical results include a novel concentration bound for centered alignment between kernel matrices the proof of the existence of effective predictors for kernels with high alignment both for classification and for regression and the proof of stability based generalization bounds for a broad family of algorithms for learning kernels based on centered alignment we also report the results of experiments with our centered alignment based algorithms in both classification and regression

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

geometry of hyper kahler connections with torsion the internal space of a n 4 supersymmetric model with wess zumino term has a connection with totally skew symmetric torsion and holonomy in sp n we study the mathematical background of this type of connection in particular we relate it to classical hermitian geometry construct homogeneous as well as inhomogeneous examples characterize it in terms of holomorphic data develop its potential theory and reduction theory

gravitational instantons of type d k and a generalization of the gibbons hawking ansatz we describe a quaternionic based ansatz generalizing the gibbons hawking ansatz to a class of hyperkahler metrics with hidden symmetries we then apply it to obtain explicit expressions for gravitational instanton metrics of type d k

hierarchy structure in integrable systems of gauge fields and underlying lie algebras an improved version of nakamura s self dual yang mills hierarchy is presentd and its symmetry contents are studied the new hierarchy as well as the previous one represents a set of commuting dynamical flows in an infinite dimensional manifolds of loop type but includes a large set of dependent variables because of new degrees of freedom the theory acquires a more symmetric form with richer structures for example it allows a large symmetry algebra of riemann hilbert type which is actually a direct sum of two subalgebras left and right this phenomenon is basically the same as observed recently by avan and bellon on the case of principal chiral models in addition to these rather familiar symmeties a new type of symmetries referred to as coordinate transformation type are also introduced generators of the above dynamical flows are all included therein these two types of symmetries altogether form a big lie algebra which lead to more satisfactory understanding of symmetry properties of integrable systems of guage fields

asymptotic geometry of the hitchin metric we study the asymptotics of the natural l2 metric on the hitchin moduli space with group g mathrm su 2 our main result which addresses a detailed conjectural picture made by gaiotto et al adv math 234 239 403 2013 is that on the regular part of the hitchin system this metric is well approximated by the semiflat metric from gaiotto et al 2013 we prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion and hence that the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay new work by dumas neitzke and later fredrickson shows that the convergence is actually exponential

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

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

generalised monopole equations on kahler surfaces in this article we establish a hitchin kobayashi type correspondence for generalised seiberg witten monopole equations on kahler surfaces we show that the stability criterion we obtain for the existence of solutions coincides with that of the usual seiberg witten monopole equations this enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors in this article we establish a hitchin kobayashi type correspondence for generalised seiberg witten monopole equations on kahler surfaces we show that the stability criterion we obtain for the existence of solutions coincides with that of the usual seiberg witten monopole equations this enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors

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

a class of renyi information estimators for multidimensional densities a class of estimators of the renyi and tsallis entropies of an unknown distribution f in r m is presented these estimators are based on the k th nearest neighbor distances computed from a sample of n i i d vectors with distribution f we show that entropies of any order q including shannon s entropy can be estimated consistently with minimal assumptions on f moreover we show that it is straightforward to extend the nearest neighbor method to estimate the statistical distance between two distributions using one i i d sample from each

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

conditional renyi entropies there is no generally accepted definition of conditional renyi entropy the unconditional renyi entropy depends on a parameter which for the case of min entropy takes the value even for this particular case there are several proposals for the definition of conditional entropy this paper describes three general definitions of conditional renyi entropy that were found or suggested in the literature their properties are studied and their values as a function of are compared the particular case of min entropy is widely used in cryptography as a security parameter this case is studied in some detail

new hyperkahler metrics and new supermultiplets we present new constructions of hyperkahler metrics along with the three new classes ofn 4 supermultiplets that they derive from further we provide a general setting for understanding the constructions and give a detailed description of the multiplets inn 2 andn 4 superspace

nahm s equations and hyperkahler geometry the geometry of certain moduli spaces of solutions to nahm s equations is studied and a family of gravitational instationsj is shown to arise as a deformation of theh atiyah hitchin maniford

reduction of hkt structures kt geometry is the geometry of a hermitian connection whose torsion is a 3 form hkt geometry is the geometry of a hyper hermitian connection whose torsion is a 3 form we identify nontrivial conditions for a reduction theory for these types of geometry

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

on the homology of su n instantons in this paper we study the homology of the moduli spaces of instantons associated to principal su n bundles over the four sphere this is accomplished by exploiting an iterated loop space structure implicit in the disjoint union of all moduli spaces associated to a fixed su n with arbitrary instanton number and relating these spaces to the known homology structure of the four fold loop space on bsu n moduli spaces of instantons self dual connections with respect to a conformal class of metrics associated to principal g bundles pk g p over s4 have proven to be basic objects in modern geometry here g is any simple compact lie group and k is the integer that classifies the bundle p and is referred to as the instanton number we denote these moduli spaces by at g there is a natural inclusion 0 1 ik g z g qk bg induced by forgetting the self duality condition where we have identified the moduli space of based gauge equivalence classes of all connections on p with the kth component of the four fold loop space q 4bg 5 in a fundamental paper atiyah and jones 5 studied the inclusion 0 1 for g su 2 when s4 has its standard conformally flat metric and posed several fundamental questions in a series of papers taubes cf 24 26 proved several basic existence theorems stability theorems in terms of k and provided a basis framework to describe how the topology of ak changes as k increases taubes work is much more general than we describe here in that he not only studied general lie groups g but also replaced 54 by an arbitrary compact closed riemannian four manifold with arbitrary conformal class of metric in 7 it was observed that over the four sphere but with g sp n the disjoint union of k over all k form a homotopy c4 space and that iterated loop space techniques may be profitably used to study h xk for individual k certain computational results may be obtained immediately from these received by the editors october 17 1989 1980 mathematics subject classification 1985 revision primary 53c55 58e05 secondary 53c57 55p35

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

curvature and characteristic numbers of hyper kahler manifolds it is perhaps surprising to see this norm expressed solely in terms of the volume andpontryagin classes in fact the context for this formula is the work of rozansky andwitten who showed in 10 how one could associate to each compact hyperk ahlermanifold a three manifold invariant in more abstract terms what they show is that

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

conification of kahler and hyper kahler manifolds given a kahler manifold m endowed with a hamiltonian killing vector field z we construct a conical kahler manifold hat m such that m is recovered as a kahler quotient of hat m similarly given a hyper kahler manifold m g j1 j2 j3 endowed with a killing vector field z hamiltonian with respect to the kahler form of j1 and satisfying mathcal l zj 2 2j 3 we construct a hyper kahler cone hat m such that m is a certain hyper kahler quotient of hat m in this way we recover a theorem by haydys our work is motivated by the problem of relating the supergravity c map to the rigid c map we show that any hyper kahler manifold in the image of the c map admits a killing vector field with the above properties therefore it gives rise to a hyper kahler cone which in turn defines a quaternionic kahler manifold our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c map

irreducible representations of the poincare parasuperalgebra we explicitly describe all the irreducible unitary representations of the poincare parasuperalgebra i e the parasupersymmetric extension of the lie algebra of the poincare group this parasuperalgebra includes as a particular case the usual poincare superalgebra and can serve as the group theoretical foundation of parasupersymmetric quantum field theory

on learning with integral operators a large number of learning algorithms for example spectral clustering kernel principal components analysis and many manifold methods are based on estimating eigenvalues and eigenfunctions of operators defined by a similarity function or a kernel given empirical data thus for the analysis of algorithms it is an important problem to be able to assess the quality of such approximations the contribution of our paper is two fold 1 we use a technique based on a concentration inequality for hilbert spaces to provide new much simplified proofs for a number of results in spectral approximation 2 using these methods we provide several new results for estimating spectral properties of the graph laplacian operator extending and strengthening results from von luxburg et al 2008

consistent nonparametric tests of independence three simple and explicit procedures for testing the independence of two multi dimensional random variables are described two of the associated test statistics l1 log likelihood are defined when the empirical distribution of the variables is restricted to finite partitions a third test statistic is defined as a kernel based independence measure two kinds of tests are provided distribution free strong consistent tests are derived on the basis of large deviation bounds on the test statistics these tests make almost surely no type i or type ii error after a random sample size asymptotically level tests are obtained from the limiting distribution of the test statistics for the latter tests the type i error converges to a fixed non zero value and the type ii error drops to zero for increasing sample size all tests reject the null hypothesis of independence if the test statistics become large the performance of the tests is evaluated experimentally on benchmark data