equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

stanley symmetric functions and quiver varieties abstract we prove that the coefficients obtained when stanley symmetric functions are expanded in the basis of schur functions are special cases of the coefficients in the formula for quiver varieties given by the author and w fulton 1999 invent math 135 665 687 and we discuss the relations of this with a conjectured littlewood richardson rule for these quiver coefficients in addition we generalize a factorization formula for schubert polynomials of a product of two permutations

chern class formulas for quiver varieties our goal in this paper is to prove a formula for the general degeneracy locus xr associated to an oriented quiver of type an if we are given a sequence of vector bundles and vector bundle maps e0 1 e1 2 e2 y y eny1 n en on an algebraic variety x and a collection r rij 0 i j n of non negative integers there is a degeneracy locus xr xr e xr e de ned by xr fx 2 x j rank ei x ej x rij 8 i jg 1 1

divisors of secant planes to curves abstract inside the symmetric product of a very general curve we consider the codimension one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension we compute the classes of such divisors and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve by letting the moduli of the curve vary we study more generally the classes of the related divisors inside the moduli space of stable pointed curves

on quantum cohomology rings of partial flag varieties 0 introduction the main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces sln c p where p is a parabolic subgroup the quantum cohomology ring of a smooth projective variety or more generally of a symplectic manifoldx has been introduced by string theorists see va and w1 roughly speaking it is a deformation of the usual cohomology ring with parameter space given byh x the multiplicative structure of quantum cohomology encodes the enumerative geometry of rational curves on x in the past few years the highly nontrivial task of giving a rigorous mathematical treatment of quantum cohomology has been accomplished both in the realm of algebraic and symplectic geometry in various degrees of generality this can be found in beh behm km lit1 lit2 mcs and rt as well as in the surveys fp and t if one restricts the parameter space to h 1 1 x one gets the small quantum cohomology ring this ring in the case of partial flag varieties is the object of the present paper in order to state our main results we first describe briefly the classical side of the story we interpret the homogeneous space f sln c p as the complex projective variety parametrizing flags of quotients of c of given ranks say nk n1 by a classical result of ehresmann e the integral cohomology of f can be described geometrically as the free abelian group generated by the schubert classes these are the poincare duals of fundamental classes of certain subvarieties w f one for each element of the subset s s n1 nk of the symmetric group sn consisting of permutations w with descents in n1 nk a description of the multiplicative structure is provided by yet another classical theorem due to borel bor which gives a presentation forh f z specifically let 1 1 1 n1 2 1 2 n2 n1 k 1 1 k 1 n nk be n independent variables define an to be the block diagonal matrix diag d1 d2 dk 1 where

f regularity of large schubert varieties let g denote a connected reductive algebraic group over an algebraically closed field k and let x denote a projective g g equivariant embedding of g the large schubert varieties in x are the closures of the double cosets bgb where b denotes a borel subgroup of g and g g we prove that these varieties are globally f regular in positive characteristic resp of globally f regular type in characteristic 0 as a consequence the large schubert varieties are normal and cohen macaulay 1 introduction the class of globally f regular varieties was introduced by smith in 21 these are projective algebraic varieties in positive character istics such that all the ideals in their homogeneous coordinate rings are tightly closed the globally f regular varieties and their analogues in characteristic 0 the varieties of globally f regular type have remarkable properties e g they are normal and cohen macaulay and the higher cohomology groups of all nef invertible sheaves are trivial examples of globally f regular varieties include the projective toric varieties prop 6 4 in 21 in this note we obtain the global f regularity of a wider class of varieties with algebraic group action the g g equivariant projective embeddings of any connected reductive group g and the closures in any such embedding of the double cosets bgb where b denotes a borel subgroup of g and g g is arbitrary by the bruhat decomposition these large schubert varieties are parametrized by the weyl group of g examples are the closures of parabolic subgroups we also show that the large schubert varieties are of globally f regular type in characteristic 0 for this we exploit the close relation between global f regularity and frobe nius splitting established in 21 and the frobenius splitting properties of the large schubert varieties proved in 5 and 19 another key ingredient is the global f regularity of the flag varieties and their schubert varieties 12 note that unlike for schubert varieties no desingularization of large schubert varieties is known in general this makes our arguments somewhat indirect as a consequence of our result the large schubert varieties in any equivariant embedding of g are normal and cohen macaulay this was first proved in the case of the canonical compactification of a semisimple adjoint group by frobenius

presenting the cohomology of a schubert variety we extend the short presentation due to borel 53 of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its schubert varieties our result is stated in a root system uniform manner by introducing the essential set of a coxeter group element generalizing and giving a new characterization of fulton 92 s definition for permutations further refinements are obtained in type a

plane curves in boxes and equal sums of two powers let f z x1 x2 x3 be an absolutely irreducible form of degree d producing a plane curve in p the central aim of this paper is to analyze the density of rational points on such curves which are contained in boxes with unequal sides we shall see below how such considerations may be used to obtain new paucity results for equal sums of two powers suppose that p p1 p2 p3 for fixed real numbers 1 p1 p2 p3 say then we define

schubert polynomials and classes of hessenberg varieties abstract regular semisimple hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory numerical analysis representation theory algebraic geometry and combinatorics we give a giambelli formula expressing the classes of regular semisimple hessenberg varieties in terms of chern classes in fact we show that the cohomology class of each regular semisimple hessenberg variety is the specialization of a certain double schubert polynomial giving a natural geometric interpretation to such specializations we also decompose such classes in terms of the schubert basis for the cohomology ring of the flag variety the coefficients obtained are nonnegative and we give closed combinatorial formulas for the coefficients in many cases we introduce a closely related family of schemes called regular nilpotent hessenberg schemes and use our results to determine when such schemes are reduced

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

subword complexes in coxeter groups abstract let be a coxeter system an ordered list of elements in and an element in determine a subword complex as introduced in knutson and miller ann of math 2 2003 to appear subword complexes are demonstrated here to be homeomorphic to balls or spheres and their hilbert series are shown to reflect combinatorial properties of reduced expressions in coxeter groups two formulae for double grothendieck polynomials one of which appeared in fomin and kirillov proceedings of the sixth conference in formal power series and algebraic combinatorics dimacs 1994 pp 183 190 are recovered in the context of simplicial topology for subword complexes some open questions related to subword complexes are presented

double schubert polynomials for the classical groups abstract for each infinite series of the classical lie groups of type b c or d we construct a family of polynomials parametrized by the elements of the corresponding weyl group of infinite rank these polynomials represent the schubert classes in the equivariant cohomology of the appropriate flag variety they satisfy a stability property and are a natural extension of the single schubert polynomials of billey and haiman which represent non equivariant schubert classes they are also positive in a certain sense and when indexed by maximal grassmannian elements or by the longest element in a finite weyl group these polynomials can be expressed in terms of the factorial analogues of schur s q or p functions defined earlier by ivanov

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

on the representation of integers by binary forms let f be a binary form with integer coefficients non zero discriminant and degree d with d at least 3 let r f z denote the number of integers of absolute value at most z which are represented by f we prove that there is a positive number c f such that r f z is asymptotic to c f z frac 2 d

schubert complexes and degeneracy loci the classical thom porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of schur polynomials a proof of this formula was given by pragacz by expressing this alternating sum as the euler characteristic of a schur complex which gives an explanation for the signs fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of schubert polynomials building on the schubert functors of kraand pragacz we introduce schubert complexes and show that fulton s alternating sum can be realized as the euler characteristic of this complex thereby providing a conceptual proof for why an alternating sum appears

totally nonnegative cells and matrix poisson varieties we describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices that is the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by a postnikov in order to do this we relate the totally nonnegative cells to torus orbits of symplectic leaves of the poisson varieties of complex matrices in particular we describe the minors that vanish on a torus orbit of symplectic leaves we prove that such families of minors are exactly the admissible families and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices

positivity of quiver coefficients through thom polynomials abstract we use the thom polynomial theory developed by feher and rimanyi to prove the component formula for quiver varieties conjectured by knutson miller and shimozono this formula expresses the cohomology class of a quiver variety as a sum of products of schubert polynomials indexed by minimal lace diagrams and implies that the quiver coefficients of buch and fulton are non negative we also apply our methods to give a new proof of the component formula from the grobner degeneration of quiver varieties and to give generating moves for the kms factorizations that form the index set in k theoretic versions of the component formula

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

combinatorics of fulton s essential set 1 introduction an n by n permutation matrix can be represented by an nby n array of squares with one dot in each row and column and all other squares empty the diagram of a permutation matrix defined in 1800 by rothe is obtained by shading every row from the dot and eastwards and shading every column from the dot and southwards the essential set is the set of southeast corners of the connected components of the diagram the essential set together with a rank function was introduced by fulton 7 in a pioneering paper from 1992 to study the irreducible loci in spaces of matrices although we are not going to pursue this geometric issue much but rather concentrate on the essential set per se we need a short recollection of the background fulton studies varieties given by ideals of minors subject to certain rank conditions in a generic matrix one concern is given a prescribed rank function r i j to determine if there exist matrices such that the rank of the upper left i by j submatrices is r i j for every position i j he observes that if such matrices exist then there is in particular some permutation matrix with this property so it is enough to consider permutation matrices more generally he is interested in prescribing the ranks only for a few of the submatrices such that all the other ranks follow from these in other words to find a subset of the upper left submatrices of a permutation matrix such that the permutation is uniquely determined by their ranks fulton shows that the essential set of a permutation matrix is such a set that is the permutation is determined by its essential set and the ranks of the corresponding upper left submatrices in section 2 we present more detailed definitions and the relevant results of fulton our first result in section 3 is an elementary algorithm that retrieves a permutation from its ranked essential set the algorithm is flexible in that the input set may be bigger and include any other squares as well as be smaller as long as it contains a certain core of the essential set by this flexibility the algorithm ought to be useful as a practical tool in general the core is much smaller than the essential set so this is a significant strengthening fulton used the essential sets mainly for vexillary 2143 avoiding permutations he gave a characterization of the ranked essential set for vexillary permutations which may be expressed by three simple conditions and he pointed out that it might be useful to have an analogous characterization of the ranked sets of squares that arise as ranked essential sets of all permutations as an application of the algorithm we are able to state such a characterization which keeps

rigorous analysis of a circular patch antenna excited by a microstrip transmission line boundary conditions are enforced on a portion of the microstrip feed line as well as the patch antenna the integral equation for the unknown currents on the antenna and feed is solved by applying the galerkin method of moments in the fourier transform domain the validity of the solution is tested by comparison of computed results with experimental data the theoretical treatment proves to be applicable to the most common feeding arrangements namely the direct edge feed and proximity coupling excitation in the latter case two layer substrates having distinct dielectric constants are studied the purpose of the study is to deduce for a given overall substrate thickness the smallest line ground plane separation for which a match of the radiator to the feed line is still possible the advantages of such a configuration are discussed

explicit uniform estimation of rational points ii hypersurface coverings by using the slope method we obtain an explicit uniform estimation for the density of rational points in an arithmetic projective variety with given degree and dimension embedded in a given arithmetic projective space

segre classes and kempf laksov formula in algebraic cobordism in this paper we extend to algebraic cobordism the classical damon kempf laksov formula which expresses the chow ring schubert classes of grassmann bundles as schur determinants in chern classes the basic building block of our closed formula which is written as a sum of determinants is represented by a generalisation of segre classes that we introduce and describe

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

on the density of rational and integral points on algebraic varieties let x n be a projective geometrically integral variety over of dimension r and degree d 4 suppose that there are only finitely many r 1 planes over on x the main result of this paper is a proof of the fact that the number n x b of rational points on xwhich have height at most b satisfies for any 0 the implied constant depends at most on d n and

rational points of bounded height on projective surfaces we give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces

the enumerative geometry of del pezzo surfaces via degenerations this paper investigates low codimension degenerations of del pezzo surfaces as an application we determine certain characteristic numbers of del pezzo surfaces finally we analyze the relation between the enumerative geometry of del pezzo surfaces and the gromov witten invariants of the hilbert scheme of conics in p n

explicit uniform estimation of rational points i estimation of heights by using the slope method in arakelov geometry we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic hilbert samuel function

analysis of conformal microstrip antennas with the discrete mode matching method an extension of the discrete mode matching dmm method is used to analyze microstrip antennas conformed to cylinders the structure needs to be discretized only along the azimuthal and the axial directions by sampling the fields and current densities in the interfaces between dielectric layers the solution in the radial direction is obtained analytically with the dyadic green s function which is calculated using the full wave equivalent circuit fwec in the spectral domain the structure is truncated in the axial direction by employing absorbing boundary conditions abcs the mathematical formulation is given in this paper followed by its application to compute the main properties of conformal microstrip antennas such as resonant frequency radiation patterns and input impedance validation with the commercial software hfss has been done and good agreement between the results has been observed

k theoretic chern class formulas for vexillary degeneracy loci abstract using raising operators and geometric arguments we establish formulas for the k theory classes of degeneracy loci in classical types we also find new determinantal and pfaffian expressions for classical cases considered by giambelli the loci where a generic matrix drops rank and where a generic symmetric or skew symmetric matrix drops rank in an appendix we construct a k theoretic euler class for even rank vector bundles with quadratic form refining the chow theoretic class introduced by edidin and graham we also establish a relation between top chern classes of maximal isotropic subbundles which is used in proving the type d degeneracy locus formulas

a scheme related to the brauer loop model abstract we introduce the brauer loop scheme e m m n c m m 0 where is a certain degeneration of the ordinary matrix product its components of top dimension n 2 2 correspond to involutions s n having one or no fixed points in the case n even this scheme contains the upper upper scheme from a knutson some schemes related to the commuting variety j algebraic geom in press math ag 0306275 as a union of n 2 of its components one of those is a degeneration of the commuting variety of pairs of commuting matrices the brauer loop model is an integrable stochastic process studied in j de gier b nienhuis brauer loops and the commuting variety j stat mech 2005 p01006 math ag 0410392 based on earlier related work in m j martins b nienhuis r rietman an intersecting loop model as a solvable super spin chain phys rev lett 81 1998 504 507 cond mat 9709051 and some of the entries of its perron frobenius eigenvector were observed conjecturally to equal the degrees of the components of the upper upper scheme our proof of this equality follows the program outlined in p di francesco p zinn justin inhomogeneous model of crossing loops and multidegrees of some algebraic varieties math ph 0412031 in that paper the entries of the perron frobenius eigenvector were generalized from numbers to polynomials which allowed them to be calculated inductively using divided difference operators we relate these polynomials to the multi degrees of the components of the brauer loop scheme defined using an evident torus action on e as a consequence we obtain a formula for the degree of the commuting variety previously calculated up to 4 4 matrices

when is a schubert variety gorenstein abstract a normal variety is gorenstein if it is cohen macaulay and its canonical sheaf is a line bundle this property is a weakening of smoothness which implies that a variety behaves like a smooth one for various algebro geometric purposes we give a combinatorial characterization of which schubert varieties are gorenstein using in part a new generalization of pattern avoidance we also give an explicit description as a line bundle of the canonical sheaf of a gorenstein schubert variety

a directed graph structure of alternating sign matrices abstract we introduce a new directed graph structure into the set of alternating sign matrices this includes bruhat graph bruhat order of the symmetric groups as a subgraph subposet drake gerrish skandera 2004 2006 6 7 gave characterizations of bruhat order in terms of total nonnegativity tnn and subtraction free laurent sfl expressions for permutation monomials with our directed graph we extend their idea in two ways first from permutations to alternating sign matrices second q analogs which we name q tnn and q sfl properties as a by product we obtain a new kind of permutation statistic the signed bigrassmannian statistics using dodgson s condensation on determinants

analysis and measurements of conformal patch array antennas on multilayer circular cylinder in this paper we present a method of moments mom program to analyze different configurations of arrays of cylindrical rectangular patches the patches can be located inside or on the surface of a multilayer circular cylindrical structure and can be arbitrarily rotated the antenna structure is rigorously taken into account by using proper green s functions and the array is analyzed by using an element by element approach the elements of the mom matrix are calculated in the spectral domain and special attention is given to their numerical treatment when analyzing cylindrical antennas with large radii a laboratory model is built to test different configurations of patches and to validate calculated results the agreement between measurements and calculations is very good

a grobner basis for kazhdan lusztig ideals it kazhdan lusztig ideals a family of generalized determinantal ideals investigated in woo yong 08 provide an explicit choice of coordinates and equations encoding a neighborhood of a torus fixed point of a schubert variety on a type a flag variety our main result is a gr o bner basis for these ideals this provides a single geometric setting to transparently explain the naturality of pipe dreams on the it rothe diagram of a permutation and their appearance in begin itemize item combinatorial formulas fomin kirillov 94 for schubert and grothendieck polynomialsof lascoux sch u tzenberger 82 item the equivariant k theory specialization formula of buch rim a nyi 04 and item a positive combinatorial formula for multiplicities of schubert varieties in good cases including those for which the associated kazhdan lusztig ideal is homogeneous under the standard grading end itemize our results generalize with alternate proofs knutson miller 05 s gr o bner basis theorem for schubert determinantal ideals and their geometric interpretation of the monomial positivity of schubert polynomials we also complement recent work of knutson 08 09 on degenerations of kazhdan lusztig varieties in general lie type as well as work of goldin 01 on equivariant localization and of lakshmibai weyman 90 rosenthal zelevinsky 01 and krattenthaler 01 on grassmannian multiplicity formulas

some degenerations of kazhdan lusztig ideals and multiplicities of schubert varieties abstract we study hilbert samuel multiplicity for points of schubert varieties in the complete flag variety by grobner degenerations of the kazhdan lusztig ideal in the covexillary case we give a manifestly positive combinatorial rule for multiplicity by establishing with a grobner basis a reduced limit whose stanley reisner simplicial complex is homeomorphic to a shellable ball or sphere we show that multiplicity counts the number of facets of this complex we also obtain a formula for the hilbert series of the local ring in particular our work gives a multiplicity rule for grassmannian schubert varieties providing alternative statements and proofs to formulae of lakshmibai and weyman 1990 26 rosenthal and zelevinsky 2001 37 krattenthaler 2001 22 kodiyalam and raghavan 2003 21 kreiman and lakshmibai 2004 24 ikeda and naruse 2009 13 and woo and yong 2009 40 we suggest extensions of our methodology to the general case

additive representation in short intervals ii sums of two like powers we establish that for almost all natural numbers n there is a sum of two positive integral cubes lying in the interval n n 7 18 varepsilon n here the exponent 7 18 lies half way between the trivial exponent 4 9 stemming from the greedy algorithm and the exponent 1 3 constrained by the number of integers not exceeding x that can be represented as the sum of two positive integral cubes we also provide analogous conclusions for sums of two positive integral k th powers when k ge 4

the nash blow up of a cominuscule schubert variety abstract we compute the nash blow up of a cominuscule schubert variety in particular we show that the nash blow up is algebraically isomorphic to another schubert variety of the same lie type as a consequence we give a new characterization of the smooth locus and for grassmannian schubert varieties determine when the nash blow up is a resolution of singularities we also study the induced torus action on the nash blow up and give a bijection between its torus fixed points and peterson translates on the schubert variety