brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

okounkov bodies and seshadri constants abstract okounkov bodies which are closed convex sets defined for big line bundles have rich information on the line bundles on the other hand seshadri constants are invariants which measure the positivity of line bundles in this paper we prove that okounkov bodies give lower bounds of seshadri constants

growth of multiplicities of graded families of ideals let r m be a noetherian local ring of dimension d 0 let i in n n be a graded family of m primary ideals in r we examine how far off from a polynomial can the length function lr r in be asymptotically more specifically we show that there exists a constant 0 such that for all n 0 lr r in 1 lr r in nd 1

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

okounkov bodies and restricted volumes along very general curves abstract given a big divisor d on a normal complex projective variety x we show that the restricted volume of d along a very general complete intersection curve c x can be read off from the okounkov body of d with respect to an admissible flag containing c from this we deduce that if two big divisors d 1 and d 2 on x have the same okounkov body with respect to every admissible flag then d 1 and d 2 are numerically equivalent

volume functions of linear series the volume of a cartier divisor is an asymptotic invariant which measures the rate of growth of sections of powers of the divisor it extends to a continuous homogeneous and log concave function on the whole neron severi space thus giving rise to a basic invariant of the underlying projective variety analogously one can also define the volume function of a possibly non complete multigraded linear series in this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties in the multigraded setting inspired by the work of lazarsfeld and musta ann inst fourier grenoble 56 6 1701 1734 2006 on okounkov bodies we show that any continuous homogeneous and log concave function appears as the volume function of a multigraded linear series by contrast we show that there exists countably many functions which arise as the volume functions of projective varieties we end the paper with an example where the volume function of a projective variety is given by a transcendental formula emphasizing the complicated nature of the volume in the classical case

local positivity in terms of newton okounkov bodies abstract in recent years the study of newton okounkov bodies on normal varieties has become a central subject in the asymptotic theory of linear series after its introduction by lazarsfeld musta and kaveh khovanskii one reason for this is that they encode all numerical equivalence information of divisor classes by work of jow at the same time they can be seen as local positivity invariants and kuronya lozovanu have studied them in depth from this point of view we determine what information is encoded by the set of all newton okounkov bodies of a big divisor with respect to flags centered at a fixed point of a surface by showing that it determines and is determined by the numerical equivalence class of the divisor up to negative components in the zariski decomposition that do not go through the fixed point

cavity analysis on the robustness of random networks against targeted attacks influences of degree degree correlations we developed a scheme for evaluating the size of the largest connected subnetwork giant component in random networks and the percolation threshold when sites nodes and or bonds edges are removed from the networks based on the cavity method of statistical mechanics of disordered systems we apply our scheme particularly to random networks of bimodal degree distribution two peak networks which have been proposed in earlier studies as robust networks against random failures of site and or targeted random degree dependent attacks on sites our analysis indicates that the correlations among degrees affect a network s robustness against targeted attacks on sites or bonds nontrivially depending on details of network configurations

hankel hyperdeterminants and selberg integrals we investigate the simplest class of hyperdeterminants defined by cayley in the case of hankel hypermatrices tensors of the form ai1i2 ik f i1 i2 ik it is found that many classical properties of hankel determinants can be generalized and a connection with selberg type integrals is established in particular selberg s original formula amounts to the evaluation of all hankel hyperdeterminants built from the moments of the jacobi polynomials many higher dimensional analogues of classical hankel determinants are evaluated in closed form the toeplitz case is also briefly discussed in physical terms both cases are related to the partition functions of one dimensional coulomb systems with logarithmic potential

newton okounkov bodies on projective bundles over curves in this article we study newton okounkov bodies on projective vector bundles over curves inspired by wolfe s estimates used to compute the volume function on these varieties we compute all newton okounkov bodies with respect to linear flags moreover we characterize semi stable vector bundles over curves via newton okounkov bodies

newton okounkov bodies semigroups of integral points graded algebras and intersection theory generalizing the notion of newton polytope we dene the newtonokounkov body respectively for semigroups of integral points graded algebras and linear series on varieties we prove that any semigroup in the lattice z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone applying this we obtain several results we show that for a large class of graded algebras the hilbert functions have polynomial growth and their growth coecients satisfy a brunn minkowski type inequality we prove analogues of the fujita approximation theorem for semigroups of integral points and graded algebras which imply a generalization of this theorem for arbitrary linear series applications to intersection theory include a far reaching generalization of the kushnirenko theorem from newton polytope theory and a new version of the hodge inequality we also give elementary proofs of the alexandrov fenchel inequality in convex geometry and its analogue in algebraic geometry

log concavity of multiplicities with application to characters ofu the log concavity of the reduction multiplicities for the classical groups of typean bn cnis proved moreover the skew schur functionss are shown to be log concave coefficient by coefficient the results are applied to the calculation of the characters of the infinite dimensional classical groups the log concavity of density of the push forward of the liouville measure on coadjoint orbits under moment map is proved

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

okounkov bodies on projectivizations of rank two toric vector bundles abstract the global okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety in the case of a rank two toric vector bundle e on a smooth projective toric variety we use its klyachko filtrations to give an explicit description of the global okounkov body of p e in particular we show that this is a rational polyhedral cone and that p e is a mori dream space

convex bodies appearing as okounkov bodies of divisors abstract based on the work of okounkov okounkov 1996 15 2003 16 lazarsfeld and musta 2009 13 and kaveh and khovanskii preprint 10 have independently associated a convex body called the okounkov body to a big divisor on a smooth projective variety with respect to a complete flag in this paper we consider the following question what can be said about the set of convex bodies that appear as okounkov bodies we show first that the set of convex bodies appearing as okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable we then give a complete characterisation of the set of convex bodies that arise as okounkov bodies of r divisors on smooth projective surfaces such okounkov bodies are always polygons satisfying certain combinatorial criteria finally we construct two examples of non polyhedral okounkov bodies in the first one the variety we deal with is fano and the line bundle is ample in the second one we find a mori dream space variety such that under small perturbations of the flag the okounkov body remains non polyhedral

integrable systems toric degenerations and okounkov bodies let x be a smooth projective variety of dimension n over mathbb c equipped with a very ample hermitian line bundle mathcal l in the first part of the paper we show that if there exists a toric degeneration of x satisfying some natural hypotheses which are satisfied in many settings then there exists a surjective continuous map from x to the special fiber x 0 which is a symplectomorphism on an open dense subset u from this we are then able to construct a completely integrable system on x in the sense of symplectic geometry more precisely we construct a collection of real valued functions h 1 ldots h n on x which are continuous on all of x smooth on an open dense subset u of x and pairwise poisson commute on u moreover our integrable system in fact generates a hamiltonian torus action on u in the second part we show that the toric degenerations arising in the theory of newton okounkov bodies satisfy all the hypotheses of the first part of the paper in this case the image of the moment map mu h 1 ldots h n x rightarrow mathbb r n is precisely the newton okounkov body delta delta r v associated to the homogeneous coordinate ring r of x and an appropriate choice of a valuation v on r our main technical tools come from algebraic geometry differential kahler geometry and analysis specifically we use the gradient hamiltonian vector field and a subtle generalization of the famous ojasiewicz gradient inequality for real valued analytic functions since our construction is valid for a large class of projective varieties x this manuscript provides a rich source of new examples of integrable systems we discuss concrete examples including elliptic curves flag varieties of arbitrary connected complex reductive groups spherical varieties and weight varieties

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

local okounkov bodies and limits in prime characteristic this article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic these sequences which we call p families of ideals are ubiquitous in prime characteristic commutative algebra e g they occur naturally in the theories of tight closure hilbert kunz multiplicity and f signature we associate to each p family of ideals an object in euclidean space that is analogous to the newton okounkov body of a graded family of ideals which we call a p body generalizing the methods used to establish volume formulas for the hilbert kunz multiplicity and f signature of semigroup rings we relate the volume of a p body to a certain asymptotic invariant determined by the corresponding p family of ideals we apply these methods to obtain new existence results for limits in positive characteristic an analogue of the brunn minkowski theorem for hilbert kunz multiplicity and a uniformity result concerning the positivity of a p family

asymptotic base loci via okounkov bodies abstract an okounkov body is a convex subset of euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag in this paper we recover the asymptotic base loci from the okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving seshadri constants consequently we obtain the nefness and ampleness criteria of divisors in terms of the okounkov bodies furthermore we compute the divisorial zariski decomposition by the okounkov bodies and find upper and lower bounds for moving seshadri constants given by the size of simplexes contained in the okounkov bodies