moment maps and riemannian symmetric pairs we study hamiltonian actions of a compact lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold the fixed point set of the involution on the manifold is a lagrangian submanifold we investigate its image under the moment map and conclude that the intersection with the weyl chamber is an easily described subpolytope of the kirwan polytope of special interest is the integral kahler case where much stronger results hold in particular we obtain convexity theorems for closures of orbits of the noncompact dual group in the sense of the theory of symmetric pairs in the abelian case these results were obtained earlier by duistermaat we derive explicit inequalities for polytopes associated with real flag varieties

a schur horn theorem for symplectic eigenvalues abstract let x and y be positive n vectors we show that there exists a 2 n 2 n positive definite real matrix whose symplectic spectrum is y and the symplectic spectrum of whose diagonal is x if and only if x is weakly supermajorised by y

a variable selection heuristic for k means clustering one of the most vexing problems in cluster analysis is the selection and or weighting of variables in order to include those that truly define cluster structure while eliminating those that might mask such structure this paper presents a variable selection heuristic for nonhierarchical k means cluster analysis based on the adjusted rand index for measuring cluster recovery the heuristic was subjected to monte carlo testing across more than 2200 datasets with known cluster structure the results indicate the heuristic is extremely effective at eliminating masking variables a cluster analysis of real world financial services data revealed that using the variable selection heuristic prior to the k means algorithm resulted in greater cluster stability

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

on the first eigenvalue of invariant kahler metrics given a simply connected compact generalized flag manifold m together with its invariant kahler einstein metric bar g we investigate the functional given by the first eigenvalue of the hodge laplacian on c infty m restricted to the space of invariant kahler metrics we give sufficient and necessary conditions so that the metric bar g is a critical point for this functional moreover we prove that when m is a full flag manifold the metric bar g is critical if and only if m mathrm su 3 t 2 and in this case bar g is a maximum

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

a new approach to cluster analysis the clustering function based method the purpose of the paper is to present a new statistical approach to hierarchical cluster analysis with n objects measured on p variables motivated by the model of multivariate analysis of variance and the method of maximum likelihood a clustering problem is formulated as a least squares optimization problem simultaneously solving for both an n vector of unknown group membership of objects and a linear clustering function this formulation is shown to be linked to linear regression analysis and fisher linear discriminant analysis and includes principal component regression for tackling multicollinearity or rank deficiency polynomial or b splines regression for handling non linearity and various variable selection methods to eliminate irrelevant variables from data analysis algorithmic issues are investigated by using sign eigenanalysis

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

bohr sommerfeld orbits in the moduli space of flat connections and the verlinde dimension formula we show how the moduli space of flatsu 2 connections on a two manifold can be quantized in the real polarization of 15 using the methods of 6 the dimension of the quantization given by the number of integral fibres of the polarization matches the verlinde formula which is known to give the dimension of the quantization of this space in a kahler polarization

random walks on symmetric spaces and inequalities for matrix spectra abstract using harmonic analysis on symmetric spaces we reduce the singular spectral problem for products of matrices to the recently solved spectral problem for sums of hermitian matrices this proves r c thompson s conjecture matrix spectral inequalities johns hopkins university press baltimore md 1988

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

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

ideal triangles in euclidean buildings and branching to levi subgroups abstract let g denote a connected reductive group defined and split over z and let m g denote a levi subgroup in this paper we study varieties of geodesic triangles with fixed vector valued side lengths in the bruhat tits buildings associated to g along with varieties of ideal triangles associated to the pair m g the ideal triangles have a fixed side containing a fixed base vertex and a fixed infinite vertex such that other infinite side containing has fixed ideal length and the remaining finite side has fixed length we establish an isomorphism between varieties in the second family and certain varieties in the first family the pair and the triple satisfy a certain relation we apply these results to the study of the hecke ring of g and the restriction homomorphism r g r m between representation rings we deduce some new saturation theorems for constant term coefficients and for the structure constants of the restriction homomorphism

negative norm error control for second kind convolution volterra equations summary we consider a piecewise constant finite element approximation to the convolution volterra equation problem of the second kind find u such that u f phi u in a time interval 0 t an a posteriori estimate of the error measured in the w 1 p 0 t norm is developed and used to provide a time step selection criterion for an adaptive solution algorithm numerical examples are given for problems in which phi is of a form typical in viscoelasticity theory

hermite predictor corrector scheme for regular volterra integral equations and for some integro differential equations of turbulent diffusion abstract in this paper we present a hermite predictor corrector scheme for the nonlinear regular volterra equations of the second kind and some integro differential equations of turbulent diffusion the method developed here closely parallels the corresponding theory for ordinary differential equations

convex analysis on cartan subspaces in 1937 von neumann 31 gave a famous characterization of unitarily invariant matrix norms that is norms f on cp q satisfying f uxv f x for all unitary matrices u and v and matrices x in cp q his result states that such norms are those functions of the form g where the map x cp q 7 x r has components the singular values 1 x 2 x p x of x assuming p q and g is a norm on rp invariant under sign changes and permutations of components furthermore he showed the respective dual norms satisfy

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

discrete schemes for treating hereditary problems of viscoelasticity and applications abstract mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are presented and their discretisation using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described theoretical error estimates in appropriate sobolev space settings are given both as they arise as a result of using a gronwall inequality and also from employing a more sensitive comparison theorem which for the quasistatic problem and under physically reasonable assumptions on the relaxation function yields much sharper constants in the estimates the range of applicability of the mathematical models and hence the numerical schemes and error estimates are discussed in the context of various materials primarily polymeric materials and extensions of the techniques to the modelling of manufacturing processes such as thermoforming are presented

a hilbert mumford criterion for polystability in kaehler geometry consider a hamiltonian action of a compact lie group k on a kaehler manifold x with moment map x assume that the action of k extends to a holomorphic action of the complexification g of k we characterize which g orbits in x intersect 1 0 in terms of the maximal weights lim t e its x s where s we do not impose any a priori restriction on the stabilizer of x under some mild restrictions on the action k x we view the maximal weights as defining a collection of maps for each x x x k g where k g is the boundary at infinity of the symmetric space k g we prove that g x 1 0 0 if 1 x is everywhere nonnegative 2 any boundary point y such that x y 0 can be connected with a geodesic in k g to another boundary point y satisfying x y 0 we also prove that the maximal weight functions are g equivariant for any g g and any y k g we have g x y x y g

the transient solution of time dependent m m 1 queues the transient behavior of time dependent m m 1 queues is studied the boundary probability function pi sub 0 t which is the probability that the queue is empty at time t is shown with analyticity arguments to satisfy a volterra type integral equation the boundary integral equation is derived and a numerical algorithm is used to solve the integral equation and to find the expected queue size from pi sub 0 t the approach can be applied to many other types of time dependent queues examples are given

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

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

discontinuous galerkin method with a posteriori l_p 0 t_i error estimate for second kind volterra problems we consider a discontinuous galerkin finite element method applied in time to a model volterra equation of the second kind a residual based computable l p 0 t i galerkin error estimate is derived for 1 le p le infty this estimate does not explicitly contain the time step and therefore the time step control must be based on a heuristic criterion the estimate can then be used to demonstrate the integrity or otherwise of the finite element solution after performing some numerical experiments we conclude that this approach is at least competetive with classical discretizations since it is computationally simple to implement but has the added advantage of reliable error feedback

complex analytic quotients of algebraic g varieties it is shown that any compact semistable quotient of a normal variety by a complex reductive lie group g in the sense of heinzner and snow is a good quotient this reduces the investigation and classification of such complex analytic quotients to the corresponding questions in the algebraic category as a consequence of our main result we show that every compact space in nemirovski s class fancyscript q g has a realisation as a good quotient and that every complete algebraic variety in fancyscript q g is unirational with finitely generated cox ring and at worst rational singularities in particular every compact space in class fancyscript q t where t is an algebraic torus is a toric variety

satake furstenberg compactifications the moment map and λ 1 let g be a complex semisimple lie group k a maximal compact subgroup and tau an irreducible representation of k on v denote by m the unique closed orbit of g in bbb p v and by cal o its image via the moment map for any measure gamma on m we construct a map psi gamma from the satake compactification of g k associated to v to the lie algebra of k if gamma is the k invariant measure then psi gamma is a homeomorphism of the satake compactification onto the convex envelope of cal o for a large class of measures the image of psi gamma is the convex envelope as an application we get sharp upper bounds for the first eigenvalue of the laplacian on functions for an arbitrary k ahler metric on a hermitian symmetric space

stokes matrices poisson lie groups and frobenius manifolds the purpose of this paper is to point out and then draw some consequences of the fact that the poisson lie group g dual to g gln c may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin g will be fully described in sect 2 the key feature of this point of view is that there is a holomorphic map g g from the dual of the lie algebra to the group g for each choice of diagonal matrix a0 with distinct eigenvalues the irregular type this map is essentially the riemann hilbert map or de rham morphism for such connections we will call it the monodromy map it is generically a local analytic isomorphism the main result is

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

compact kahler quotients of algebraic varieties and geometric invariant theory abstract given an action of a complex reductive lie group g on a normal variety x we show that every analytically zariski open subset of x admitting an analytic hilbert quotient with projective quotient space is given as the set of semistable points with respect to some g linearised weil divisor on x applying this result to hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to git semistability in the sense of mumford and hausen it follows that the number of compact momentum map quotients of a given algebraic hamiltonian g variety is finite as further corollary we derive a projectivity criterion for varieties with compact kahler quotient

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g in this paper we study the geometric quantization of the symplectic quotient m g guillemin and sternberg invent math 67 515 538 1982 have shown under suitable regularity assumptions that there is a natural invertible map between the quantum hilbert space over m g and the g invariant subspace of the quantum hilbert space over m

lie poisson structure on some poisson lie groups poisson lie groups appeared in the work of drinfel d see e g drl dr2 as classical objects corresponding to quantum groups going in the other direction we may say that a poisson lie group is a group of symmetries of a phase space that are allowed to twist in a certain sense the symplectic or poisson structure the poisson structure on the group controls this twisting in a precise way quantizing both the phase space and the symmetries one may obtain a quantum group acting on a quantum phase space in recent work lu and ratiu lr used so called standard poisson structures on a compact semisimple lie group k and on its poisson dual k in order to give a new proof of the nonlinear convexity theorem of kostant ko their method is analogous to the famous symplectic proof of the linear convexity theorem given by atiyah a and guillemin and stemnberg gs the nonlinear convexity theorem like the linear one follows from a very general result on convexity of the image of the momentum map a gs however in lr it is applied not to a coadjoint orbit in t but to a symplectic leaf in k the main result of this paper is that the standard poisson structure on the poisson dual k to a compact semisimple poisson lie group k is actually isomorphic to the linear one on t this theorem seems to be related to some facts in the theory of quantum groups namely for generic q the universal enveloping algebra u t and its quantum deformation uq t are isomorphic as algebras though not as coalgebras of course since the quantum version is not cocommutative in particular there is a bijective correspondence between their representations a direct connection between our work and its quantum analogues though is still to be found the present work supplies a positive answer to question 5 1 in lr and strongly depends on that paper to simplify reading we keep the notations of lr wherever possible on one hand but give all necessary definitions on the other our work is also related to du in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in 5 the paper is organized as follows in 2 we define poisson lie groups discuss

cartan decomposition of the moment map we investigate a class of actions of real lie groups on complex spaces using moment map techniques we establish the existence of a quotient and a version of luna s slice theorem as well as a version of the hilbert mumford criterion a global slice theorem is proved for proper actions we give new proofs of results of mostow on decompositions of groups and homogeneous spaces

applications and numerical analysis of partial differential volterra equations a brief survey this article contains a concise survey of the numerical analysis of volterra equations and leads up to some recent results on a posteriori error estimation for finite element approximations

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

torus actions on weakly pseudoconvex spaces we show that the univalent local actions of the complexification of a compact connected lie group k on a weakly pseudoconvex space where k is acting holomorphically have a universal orbit convex weakly pseudoconvex complexification we also show that if k is a torus then every holomorphic action of k on a weakly pseudoconvex space extends to a univalent local action of k c

stability of measures on kahler manifolds abstract let m be a kahler manifold and let k be a compact group that acts on m in a hamiltonian fashion we study the action of k c on probability measures on m first of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability semi stability and polystability next we apply this setting to the action of k c on measures we get various stability criteria for measures on kahler manifolds the same circle of ideas gives a very general surjectivity result for a map originally studied by hersch and bourguignon li yau

discrimillant analysis with discrete and continuous variables a paradigmatic methodologic approach does not exist for the problem of discriminant analysis with both discrete and continuous explanatory variables procedures that have been employed in this situation include fisher s linear discriminant function the quadratic discriminant function the linear discriminant furlction with higher order terms discriminant functions with logistic regression estimates of the coefficients and the location model average optimal error rates for these procedures are reported for the case of one normal variable and two binary variables these approaches are also compared for the prediction of two year survival following recovery from myocardial infarction employing resubstitution jackknife and independent sample estimates of the error rates recommendations for applications are presented

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

kahlerian potentials and convexity properties of the moment map since kostant proved his convexity theorems for torus actions on ag varieties ko there have been numerous contributions to this subject in the more general symplectic and k ahlerian setting for example let k be a compact lie group acting in a hamiltonian fashion on a connected compact symplectic manifold x then the intersection x of the image of the moment map x liek with a positive weyl chamber t is convex thus x k x where x is a natural convex section this was proved by atiyah and guillemin sternberg a g s in the abelian case i e where k t is a compact torus by mumford m for x projective algebraic with an integral k ahlerian structure and k not necessarily abelian and in its nal form in the compact symplectic case by kirwan k a precise description of x x t in the compact k ahlerian framework can be given in terms of the image of the components of the set of t xed points in x in the projective case where the k ahlerian structure comes from a projective embedding there are explicit connections to the representations theory of g kc which are due to brion and to mumford ness b n the purpose of the present paper is to prove convexity properties of in the setting of non compact k ahlerian spaces note that by removing an appropriate k invariant subset from x one can easily construct non convex images from convex ones on the other hand hilgert neeb plank proved h n p that convexity holds for a proper moment map x liek another convexity result has been obtained by sjamaar for x a ne and the restriction of the moment map of a representation s2 here the representation theoretical approach of brion is used now let x be an irreducible complex space endowed with a holomorphic action of a complex reductive group g assume that a maximal

harmonic maps to buildings and singular perturbation theory the notion of a uni versal building associated with a point in the hitchin base is introduced the universal building is a building equipped with a harmonic map from the universal cover of the given riemann surface that is initial among harmonic maps which induce the given cameral cover of the riemann surface the notion of a versal building is obtained by relaxing the uniqueness condition in the definition of an initial object in the rank one case the universal building is the leaf space of the quadratic differential defining the point in the hitchin base the main conjectures of this paper are 1 a versal building always exists 2 the asymptotics of the riemann hilbert correspondence and the non abelian hodge correspondence are controlled by the harmonic map associated to any versal building 3 spectral networks arise as inverse images of singularities of the versal building and 4 versal buildings encode the data of a 3d calabi yau category whose space of stability conditions has a connected component that contains the hitchin base the main theorem establishes the existence of the universal building conjecture 3 as well as the riemann hilbert part of conjecture 2 in the case of the rank two example introduced in the seminal work of berk nevins roberts on higher order stokes phenomena it is also shown that the asymptotics of the riemann hilbert correspondence are always controlled by a harmonic map to a certain building which is constructed as the asymptotic cone of a symmetric space

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

a combination theorem for anosov subgroups we prove an analogue of klein combination theorem for anosov subgroups by using a local to global principle for morse quasigeodesics

asymptotic metrics for su n monopoles with maximal symmetry breaking we compute the asymptotic metrics for moduli spaces of su n monopoles with maximal symmetry breaking these metrics are exponentially close to the exact monopole metric1 as soon as for each simple root the individual monopoles corresponding to that root are well separated we also show that the estimates can be differentiated term by term in natural coordinates which is a new result even for su 2 monopoles

clustering objects on subsets of attributes with discussion read before the royal statistical society at a meeting organized by the research section on wednesday may 5th 2004 professor j t kent in the chair summary a new procedure is proposed for clustering attribute value data when used in conjunction with conventional distance based clustering algorithms this procedure encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously the relevant attribute subsets for each individual cluster can be different and partially or completely overlap with those of other clusters enhancements for increasing sensitivity for detecting especially low cardinality groups clustering on a small subset of variables are discussed applications in different domains including gene expression arrays are presented