gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

higher order schemes with cip method and adaptive soroban grid towards mesh free scheme a new class of body fitted grid system that can keep the third order accuracy in time and space is proposed with the help of the cip constrained interpolation profile cubic interpolated propagation method the grid system consists of the straight lines and grid points moving along these lines like abacus soroban in japanese the length of each line and the number of grid points in each line can be different the cip scheme is suitable to this mesh system and the calculation of large cfl 10 at locally refined mesh is easily performed mesh generation and searching of upstream departure point are very simple and almost mesh free treatment is possible adaptive grid movement and local mesh refinement are demonstrated

mirror symmetry for closed open and unoriented gromov witten invariants abstract in the first part of this paper we obtain mirror formulas for twisted genus 0 two point gromov witten gw invariants of projective spaces and for the genus 0 two point gw invariants of fano and calabi yau complete intersections this extends previous results for projective hypersurfaces following the same approach but we also completely describe the structure coefficients in both cases and obtain relations between these coefficients that are vital to the applications to mirror symmetry in the rest of this paper in the second and third parts of this paper we confirm walcher s mirror symmetry conjectures for the annulus and klein bottle gw invariants of calabi yau complete intersection threefolds these applications are the main results of this paper in a separate paper the genus 0 two point formulas are used to obtain mirror formulas for the genus 1 gw invariants of all calabi yau complete intersections

localization in equivariant intersection theory and the bott residue formula we prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections and schubert varieties 1 introduction the purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections cf bfq and schubert varieties let t be a split torus acting on a scheme x the 7 equivariant chow groups of x are a module over rt sym t where f is the character group of t the localization theorem states that up to torsion the equivariant chow groups of the fixed locus xt are isomorphic to those of x such a theorem is a hallmark of any equivariant theory the earliest version for equivariant cohomology is due to borel bo subsequently it theory versions were proved by segal se in topological a theory quart qu for actions of a cyclic group and thomason th2 for algebraic theory th2 for equivariant chow groups the localization isomorphism is given by the equivariant push forward induced by the inclusion of xt to x an interesting aspect of this theory is that the push forward is naturally defined on the level of cycles even in the singular case the closest topological analogue of this is equivariant borel moore homology see e g3 for a definition and a similar proof establishes localization in that theory for smooth spaces the inverse to the equivariant push forward can be written explicitly it was realized independently by several authors i n a b b v

interpolating ensembles of random unitary matrices we investigate transitions between ensembles of unitary random matrices modeling changes of statistical properties of quantum chaotic systems which are periodically time dependent the transitions simulating change of integrability are modeled with the help of numerically generated one parameter families of ensembles interpolating between an ensemble of diagonal unitary matrices and a circular unitary or circular orthogonal ensemble in an analogous manner we described transitions between circular orthogonal and unitary ensembles corresponding to the time reversal symmetry breaking perturbation of a chaotic system in all cases we present results concerning statistics of the quasi energy levels and eigenvectors textcopyright 1996 the american physical society

adelic chern forms and applications let x be a variety over a field of characteristic 0 given a vector bundle e on x we construct chern forms c v t x ax here a x is the sheaf of beilinson adeles and v is an adelic connection when x is smooth wt x a x h r x the algebraic de rham cohomology and c q v are the usual chern classes we include three applications of the construction 1 existence of adelic secondary chern simons characteristic classes on any smooth x and any vector bundle e 2 proof of the bott residue formula for a vector field action and 3 proof of a gauss bonnet formula on the level of differential forms namely in the de rham residue complex 0 introduction let x be a scheme of finite type over a field k according to beilinson be given any quasi coherent x module m and an integer q9 there is a flasque ox module a dcm called the sheaf of adeles this is a gen eralization of the classical adeles of number theory cf example 2 3 moreover there are homomorphisms d aj ed m a m which make ared m into a complex and m ared m is quasi isomorphism now let qx be the algebra of kahler differential forms on x in hy we proved that the sheaf

bott s formula and enumerative geometry we outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of bott as an example gromov witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed the results are consistent with predictions made from mirror symmetry compu tations we also compute degrees of some loci in the linear system of plane curves of degrees less than 10 like those corresponding to sums of powers of linear forms and curves carrying inscribed polygons mathematical institute university of oslo p 0 box 1053 n 0316 oslo norway e mail address ellingsr0math uio no mathematical institute university of bergen alleg 55 n 5007 bergen norway e mail address strommeti i uib no this content downloaded from 157 55 39 29 on tue 12 apr 2016 10 23 14 utc all use subject to http about jstor org terms

exactly conservative semi lagrangian scheme for multi dimensional hyperbolic equations with directional splitting technique abstract a new numerical method that guarantees exact mass conservation is proposed to solve multidimensional hyperbolic equations in semi lagrangian form the method is based on the constrained interpolation profile cip scheme and keeps the many good characteristics of the original cip scheme the cip strategy is applied to the integral form of variables although the advection and nonadvection terms are separately treated mass conservation is kept in the form of a spatial profile inside a grid cell therefore it retains various advantages of the semi lagrangian solution with exact conservation which has been beyond the capability of conventional semi lagrangian schemes

on the structure of the small quantum cohomology rings of projective hypersurfaces we give an explicit procedure which computes for degree d 3 the correlation functions of topological sigma model a model on a projective fano hypersurface x as homogeneous polynomials of degree d in the correlation functions of degree 1 number of lines we extend this formalism to the case of calabi yau hypersurfaces and explain how the polynomial property is preserved our key tool is the construction of universal recursive formulas which express the structure constants of the quantum cohomology ring of x as weighted homogeneous polynomial functions of the constants of the fano hypersurface with the same degree and dimension one more we propose some conjectures about the existence and the form of the recursive laws for the structure constants of rational curves of arbitrary degree our recursive formulas should yield the coefficients of the hypergeometric series used in the mirror calculation assuming the validity of the conjectures we find the recursive laws for rational curves of degree four

varieties with minimal secant degree and linear systems of maximal dimension on surfaces abstract in this paper we prove using a refinement of terracini s lemma a sharp lower bound for the degree of higher secant varieties to a given projective variety which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space moreover we study varieties for which the bound is attained proving some general properties related to tangential projections e g these varieties are rational in particular we completely classify surfaces and curves for which the bound is attained it turns out that these surfaces enjoy some maximality properties for their embedding dimension in terms of their degree or sectional genus this is related to classical beautiful results of castelnuovo and enriques that we revise here in terms of adjunction theory

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

mirror manifolds in higher dimension we describe mirror manifolds in dimensions different from the familiar case of complex threefolds we isolate certain simplifying features present only in dimension three and supply alternative methods that do not rely on these special characteristics and hence can be generalized to other dimensions although the moduli spaces for calabi yaud folds are not special kahler manifolds whend 3 they still have a restricted geometry and we indicate the new geometrical structures which arise we formulate and apply procedures which allow for the construction of mirror maps and the calculation of order by order instanton corrections to yukawa couplings mathematically these corrections are expected to correspond to calculating chern classes of various parameter spaces hilbert schemes for rational curves on calabi yau manifolds our mirror aided calculations agree with those chern class calculations in the limited number of cases for which the latter can be carried out with current mathematical tools finally we make explicit some striking relations between instanton corrections for various yukawa couplings derived from the associativity of the operator product algebra

the cubic interpolated pseudo particle cip method application to nonlinear and multi dimensional hyperbolic equations abstract the basic of cip as a method to solve hyperbolic equations is re examined from a different viewpoint and the scheme is modified into an explicit finite difeerence form the method gives a stable and less diffusive result for square wave propagation compared with fct and a better result for propagation of a sine wave with a discontinuit the scheme is extended to nonlinear and multi dimensional problems

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

the surface gradient method for the treatment of source terms in the shallow water equations a novel scheme has been developed for data reconstruction within a godunov type method for solving the shallow water equations with source terms in contrast to conventional data reconstruction methods based on conservative variables the water surface level is chosen as the basis for data reconstruction this provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a riemann solver the main advantages are 1 a simple centered discretization is used for the source terms 2 the scheme is no more complicated than the conventional method for the homogeneous terms 3 small perturbations in the water surface elevation can be accurately predicted and 4 the method is generally suitable for both steady and unsteady shallow water problems the accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems excellent agreement has been obtained between the numerical predictions and analytical solutions the results indicate that the new scheme is accurate simple efficient and robust

shallow water model on cubed sphere by multi moment finite volume method a global numerical model for shallow water flows on the cubed sphere grid is proposed in this paper the model is constructed by using the constrained interpolation profile multi moment finite volume method cip mm fvm two kinds of moments i e the point value pv and the volume integrated average via are defined and independently updated in the present model by different numerical formulations the lax friedrichs upwind splitting is used to update the pv moment in terms of a derivative riemann problem and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the via moment the cubed sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry highly localized reconstruction in cip mm fvm is well suited for the cubed sphere grid especially in dealing with the discontinuity in the coordinates between different patches the mass conservation is completely achieved over the whole globe the numerical model has been verified by williamson s standard test set for shallow water equation model on sphere the results reveal that the present model is competitive to most existing ones

the space of triangles vanishing theorems and combinatorics abstract we consider compactifications of pn 3 ij the space of triples of distinct points in projective space one such space is a singular variety of configurations of points and lines another is the smooth compactification of fulton and macpherson and a third is the triangle space of schubert and semple we compute the sections of line bundles on these spaces and show that they are equal as gl n representations to the generalized schur modules associated to bad generalized young diagrams with three rows borel weil theorem on the one hand this yields weyl type character and dimension formulas for the schur modules on the other a combinatorial picture of the space of sections cohomology vanishing theorems play a key role in our analysis

chern classes of tautological sheaves on hilbert schemes of points on surfaces we give an algorithmic description of the action of the chern classes of tautological bundles on the cohomology of hilbert schemes of points on a smooth surface within the framework of nakajima s oscillator algebra this leads to an identification of the cohomology ring of hilb n a2 with a ring of explicitly given differential operators on a fock space we end with the computation of the top segre classes of tautological bundles associated to line bundles on hilb n up to n 7 extending computations of severi lebarz tikhomirov and troshina and give a conjecture for the generating series

multi dimensional conservative semi lagrangian method of characteristics cip for the shallow water equations a new characteristic approach that guarantees conservative property is proposed and is applied to the shallow water equations cip csl constrained interpolation profile conservative semi lagrangian interpolation is applied to the cip method of characteristics in order to enhance the mass conservation of the numerical result although the characteristic formulation is originally derived from non conservative form present scheme achieves complete mass conservation by solving mass conservation simultaneously and reflecting conserving mass in interpolation profile present method has less height error compared to the cip method of characteristics by several orders of magnitude by the enhanced conservation property present scheme is applicable to nonlinear problem such as shock furthermore application to two dimensions including the coriolis term is straightforward with directional splitting technique

opening mirror symmetry on the quintic aided by mirror symmetry we determine the number of holomorphic disks ending on the real lagrangian in the quintic threefold we hypothesize that the tension of the domainwall between the two vacua on the brane which is the generating function for the open gromov witten invariants satisfies a certain extension of the picard fuchs differential equation governing periods of the mirror quintic we verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space we further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the a model and verifying ooguri vafa integrality this is the first exact result on open string mirror symmetry for a compact calabi yau manifold