dunkl operator and quantization of ℤ2 singularity let x be a symplectic orbifold which is locally like the quotient of a z2 action on r let a x be a deformation quantization of x constructed via the standard fedosov method with characteristic class being in this paper we construct a universal deformation of the algebra a x parametrized by codimension 2 components of the associated inertia orbifold x this partially confirms a conjecture of dolgushev and etingof see 4 in the case of z2 orbifolds to do so we generalize the interpretation of moyal star product as a composition of symbols of pseudodifferential operators in the case where partial derivatives are replaced with dunkl operators the star products we obtain can be seen as globalizations of symplectic reflection algebras 6

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

high probability analysis of the condition number of sparse polynomial systems let f f1 fn be a random polynomial system with fixed n tuple of supports our main result is an upper bound on the probability that the condition number of f in a region u is larger than 1 e the bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the newton polytopes and underlying variances are all identical we also consider polynomials with real coefficients and give bounds for the expected number of real roots and restricted condition number using a kahler geometric framework throughout we also express the expected number of roots of f inside a region u as the integral over u of a certain mixed volume form thus recovering the classical mixed volume when u c n

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

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

examples of non kahler hamiltonian torus actions an important question with a rich history is the extent to which the symplectic category is larger than the kahler category many interesting examples of non kahler symplectic manifolds have been constructed t m g however sufficiently large symmetries can force a symplectic manifolds to be kahler d kn in this paper we solve several outstanding problems by constructing the first symplectic manifold with large non trivial symmetries which does not admit an invariant kahler structure the proof that it is not kahler is based on the atiyah guillemin sternberg convexity theorem at gs using the ideas of this paper c woodward shows that even the symplectic analogue of spherical varieties need not be kahler w

the hochschild cohomology ring of a global quotient orbifold abstract we study the cup product on the hochschild cohomology of the stack quotient x g of a smooth quasi projective variety x by a finite group g more specifically we construct a g equivariant sheaf of graded algebras on x whose g invariant global sections recover the associated graded algebra of the hochschild cohomology of x g under a natural filtration this sheaf is an algebra over the polyvector fields t x p o l y on x and is generated as a t x p o l y algebra by the sum of the determinants det n x g of the normal bundles of the fixed loci in x we employ our understanding of hochschild cohomology to conclude that the analog of kontsevich s formality theorem for the cup product does not hold for deligne mumford stacks in general we discuss in the case of a symplectic group action on a symplectic variety x relationships with orbifold cohomology and ruan s cohomological conjectures in describing the hochschild cohomology in the symplectic situation we employ compatible trivializations of the determinants det n x g which requires for the cup product a nontrivial normalization missing in previous literature

convex bodies of states and maps we give a general solution to the question of when the convex hulls of orbits of quantum states on a finite dimensional hilbert space under unitary actions of a compact group have a non empty interior in the surrounding space of all density operators the same approach can be applied to study convex combinations of quantum channels the importance of both problems stems from the fact that usually only sets with non vanishing volumes in the embedding spaces of all states or channels are of practical importance for the group of local transformations on a bipartite system we characterize maximally entangled states by the properties of a convex hull of orbits through them we also compare two partial characteristics of convex bodies in terms of the largest balls and maximum volume ellipsoids contained in them and show that in general they do not coincide separable states mixed unitary channels and k entangled states are also considered as examples of our techniques

on poisson actions of compact lie groups on symplectic manifolds let g p be a compact simple poisson lie group equipped with a poisson structure p and m omega be a symplectic manifold assume that m carries a poisson action of g p and there is an equivariant moment map in the sense of lu and weinstein which maps

g twisted braces and orbifold landau ginzburg models given an algebra with a group g action we construct brace structures for its g twisted hochschild cochains an an application we construct g frobenius algebras for orbifold landau ginzburg b models and present explicit orbifold cup product formula for all invertible polynomials

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

factorization reduction and embedding in integrable cellular automata factorized dynamics in soliton cellular automata with quantum group symmetry is identified with a motion of particles and anti particles exhibiting pair creation and annihilation an embedding scheme is presented showing that the d 1 n automaton contains as certain subsectors the box ball systems and all the other automata associated with the crystal bases of non exceptional affine lie algebras the results extend the earlier ones to higher representations by a certain reduction and to a wider class of boundary conditions

a weighted selection algorithm for certain tree structured linear programs we solve a simple class of linear programming problems with tree structured constraint matrices our procedure separates the problem into a sequence of continuous knapsack problems each of which requires linear time to solve for total solution time no worse than proportional to the number of nonzero entries in the original constraint matrix the model arises in portfolio selection advertising and inventory and production planning

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

openness and convexity for momentum maps the purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps it is shown that they are of a topological nature more specifically we show that convexity follows if the map is open onto its image and has the so called local convexity data property these conditions are satisfied in all the classical convexity theorems and hence they can in principle be obtained as corollaries of a more general theorem that has only these two hypotheses we also prove a generalization of the so called local to global principle that only requires the map to be closed and to have a normal topological space as domain instead of using a properness condition this allows us to generalize the flaschka ratiu convexity theorem to noncompact manifolds

eigenvalue distributions of reduced density matrices given a random quantum state of multiple distinguishable or indistinguishable particles we provide an effective method rooted in symplectic geometry to compute the joint probability distribution of the eigenvalues of its one body reduced density matrices as a corollary by taking the distribution s support which is a convex moment polytope we recover a complete solution to the one body quantum marginal problem we obtain the probability distribution by reducing to the corresponding distribution of diagonal entries i e to the quantitative version of a classical marginal problem which is then determined algorithmically this reduction applies more generally to symplectic geometry relating invariant measures for the coadjoint action of a compact lie group to their projections onto a cartan subalgebra and can also be quantized to provide an efficient algorithm for computing bounded height kronecker and plethysm coefficients

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

frankel s theorem in the symplectic category we prove that if an n 1 dimensional torus acts symplectically on a 2n dimensional symplectic manifold then the action has a fixed point if and only if the action is hamiltonian one may regard it as a symplectic version of frankel s theorem which says that a kahler circle action has a fixed point if and only if it is hamiltonian the case of n 2 is the well known theorem by mcduff

cohomology of regular embeddings we give in this paper an explicit description of the rational cohomology ring of a complete symmetric variety or regular compactification of an algebraic symmetric variety the motivation for this computation comes from the desire to describe an explicit schubert calculus for these varieties which have been considered since the work of chasles and schubert in several special cases the technique we use is the result of our attempts to understand better the work of jurkiewicz and danilov in the case of torus embeddings the combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology the key point is that the reisner stanley algebra of the torus embedding coincides with its equivariant cohomology ring we consider a class of spaces regular embeddings that contains both complete symmetric varieties and smooth torus embeddings by suitably generalizing the notion of reisner stanley algebra we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

relative equilibria and bifurcations in the generalized van der waals 4d oscillator abstract a uniparametric 4 dof family of perturbed hamiltonian oscillators in 1 1 1 1 resonance is studied as a generalization for several models for perturbed keplerian systems normalization by lie transforms only first order is considered here as well as geometric reduction related to the invariants associated to the symmetries is used based on the previous work of the authors a description is given for the lower dimensional relative equilibria in such normalized systems for which we introduce in this context the new concept of moment polytope in addition bifurcations of relative equilibria corresponding to 3d tori are studied in some particular cases where we focus on hamiltonian hopf bifurcations and bifurcations in the 3d van der waals and zeeman systems

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

hypergeometric polynomials are optimal with any integer convex polytope p subset mathbb r n we associate a multivariate hypergeometric polynomial whose set of exponents is mathbb z n cap p this polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of horn s type we prove that under certain nondegeneracy conditions any such polynomial is optimal in the sense of 7 i e that the topology of its amoeba 13 is as complex as it could possibly be using this we derive optimal properties of several classical families of multivariate hypergeometric polynomials

new polytope decompositions and euler maclaurin formulas for simple integral polytopes abstract we give new weighted decompositions for simple polytopes generalizing previous results of lawrence varchenko and brianchon gram we start with witten s non abelian localization principle in equivariant cohomology for the norm square of the moment map in the context of toric varieties to obtain a decomposition for delzant polytopes then by a purely combinatorial argument we show that this formula holds for any simple polytope as an application we study euler maclaurin formulas

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

noncommutative poisson structures on orbifolds author s halbout gilles tang xiang abstract in this paper we compute the gerstenhaber bracket on the hoch schild cohomology of c infty m rtimes g for a finite group g acting on a compact manifold m using this computation we obtain geometric descriptions for all noncommutative poisson structures on c infty m rtimes g when m is a symplectic manifold we also discuss examples of deformation quantizations of these noncommutative poisson structures

convexity properties of gradient maps abstract we consider the action of a real reductive group g on a kahler manifold z which is the restriction of a holomorphic action of a complex reductive group h we assume that the action of a maximal compact subgroup u of h is hamiltonian and that g is compatible with a cartan decomposition of h we have an associated gradient map p z p where g k p is the cartan decomposition of g for a g stable subset y of z we consider convexity properties of the intersection of p y with a closed weyl chamber in a maximal abelian subspace a of p our main result is a convexity theorem for real semi algebraic subsets y of z p v where v is a unitary representation of u

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

analysis of a particle antiparticle description of a soliton cellular automaton we present a derivation of a formula that gives dynamics of an integrable cellular automation associated with crystal bases this automaton is related to type d affine lie algebra and contains usual box ball systems as a special case the dynamics is described by means of such objects as carriers particles and antiparticles we derive it from an analysis of a recently obtained formula of the combinatorial r an intertwiner between tensor products of crystals that was found in a study of geometric crystals

a new cohomology theory of orbifold based on the orbifold string theory model in physics we construct a new cohomology ring for any almost complex orbifold the key theorem is the associativity of this new ring some examples are computed

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

the gerstenhaber bracket as a schouten bracket for polynomial rings extended by finite groups we apply new techniques to compute gerstenhaber brackets on the hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group in characteristic 0 we show that the gerstenhaber brackets can always be expressed in terms of schouten brackets on polyvector fields we obtain as consequences some conditions under which brackets are always 0 and show that the hochschild cohomology is a graded gerstenhaber algebra under the codimension grading strengthening known results

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

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

group actions on algebras and the graded lie structure of hochschild cohomology hochschild cohomology governs deformations of algebras and its graded lie structure plays a vital role we study this structure for the hochschild cohomology of the skew group algebra formed by a nite group acting on an al gebra by automorphisms we examine the gerstenhaber bracket with a view toward deformations and developing bracket formulas we then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory deformations in this context include graded hecke alge bras and symplectic reection algebras we give some general results describing when brackets are zero for polynomial skew group algebras which allow us in particular to nd noncommutative poisson structures for abelian groups we express the bracket using inner products of group characters lastly we inter pret results for graded hecke algebras

hochschild and cyclic homology of the crossed product of algebraic irrational rotational algebra by finite subgroups of sl 2 z abstract let sl 2 z be a finite subgroup acting on the irrational rotational algebra a via the restriction of the canonical action of sl 2 z consider the crossed product algebra a alg obtained by the restriction of the action on the algebraic irrational rotational algebra in this paper we prove many results on the homology group of the crossed product algebra a alg we also analyse the case of the smooth crossed product algebra a and calculate some of its homology groups

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

algorithms for scheduling a single machine to minimize the weighted number of late jobs this paper considers the problem of scheduling n jobs each having a processing time a due date and a weight on a single machine to minimize the weighted number of late jobs an o n log n algorithm is given for solving the linear programming problem obtained by relaxing the integrality constraints in a zero one programming formulation of the problem this linear programming lower bound is used in a reduction algorithm that eliminates jobs from the problem also a branch and bound algorithm that uses the linear programming lower bound is proposed computational results with branch and bound algorithms that use this and other lower bounds and with a dynamic programming algorithm for problems with up to 1000 jobs are given

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

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

finite groups acting linearly hochschild cohomology and the cup product abstract when a finite group acts linearly on a complex vector space the natural semi direct product of the group and the polynomial ring over the space forms a skew group algebra this algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics its hochschild cohomology predicts various hecke algebras and deformations of the orbifold in this article we investigate the ring structure of the hochschild cohomology of the skew group algebra we show that the cup product coincides with a natural smash product transferring the cohomology of a group action into a group action on cohomology we express the algebraic structure of hochschild cohomology in terms of a partial order on the group modulo the kernel of the action this partial order arises after assigning to each group element the codimension of its fixed point space we describe the algebraic structure for coxeter groups where this partial order is given by the reflection length function a similar combinatorial description holds for an infinite family of complex reflection groups

technical note the multiperiod knapsack problem in the multiperiod knapsack problem the decision maker faces a horizon of m periods associated with each period are a number of types of items each with a value and weight subject to the requirement that the cumulative capacity of the knapsack in each period i cannot be exceeded by items chosen in periods 1 i the decision maker chooses the most valuable knapsack possible a branch and bound algorithm exploits the special structure of the multiperiod knapsack problem by calculating bounds by the direct solution of linear programs with m constraints in 0 m operations computational experience is reported on problems ranging in size up to 200 constraints and 1 000 general integer variables

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

polar and coisotropic actions on kahler manifolds the main result of the paper is that a polar action on a compact irreducible homogeneous kahler manifold is coisotropic this is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

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

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

the am 1 automata related to crystals of symmetric tensors a soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra uq am 1 is introduced it is a crystal theoretic formulation of the generalized box ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous scattering matrices of two solitons coincide with the combinatorial r matrices of uq am 1 1 a piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete kadomtsev petviashivili equation a class of n soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter

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

the convexity package for hamiltonian actions on conformal symplectic manifolds consider a hamiltonian action of a compact connected lie group on a conformal symplectic manifold we prove a convexity theorem for the moment map under the assumption that the action is of lee type which establishes an analog of kirwan s convexity theorem in conformal symplectic geometry

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

moment polytopes for symplectic manifolds with monodromy abstract a natural way of generalising hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map for a class of such almost toric 4 manifolds which admits a hamiltonian s 1 action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of atiyah guillemin sternberg as an application we derive a duistermaat heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system