quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

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

asymptotically optimal quantizers for detection of i i d the asymptotic probability of error for quantization in maximum likelihood tests is analyzed the authors assume quantizers with large numbers of levels generated from a companding function a theorem that relates the companding function to the asymptotic probability of error is proved the companding function is then optimized

dimension formula for the affine deligne lusztig variety x mu b x μ b the study of certain union x mu b of affine deligne lusztig varieties in the affine flag varieties arose from the study of shimura varieties with iwahori level structure in this paper we give an explicit dimension formula for x mu b associated to sufficiently large dominant coweight mu

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

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

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

quantum cohomology and the periodic toda lattice we describe a relation between the periodic one dimensional toda lattice and the quantum cohomology of the periodic flag manifold an infinite dimensional kahler manifold this generalizes a result of givental and kim relating the open toda lattice and the quantum cohomology of the finite dimensional flag manifold we derive a simple and explicit differential operator formula for the necessary quantum products which applies both to the finite dimensional and to the infinite dimensional situations

upper bounds to the asymptotic performance of block quantizers upper bounds to the asymptotic performance of block quantizers with difference distortion measures are derived in many eases these upper bounds approach known lower bounds as the block length of the quantizer approaches infinity a condition for the optimal point density function of the output levels is derived it is shown to particularize to a known result of gersho the behavior of the bounds for large block lengths is investigated

quantum and affine schubert calculus and macdonald polynomials abstract we definitively establish that the theory of symmetric macdonald polynomials aligns with quantum and affine schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong bruhat order poset on the type a affine weyl group we construct two one parameter families of functions that respectively transition positively with hall littlewood and macdonald s p functions and specialize to the representatives for schubert classes of homology and cohomology of the affine grassmannian our approach leads us to conjecture that all elements in a defining set of 3 point genus 0 gromov witten invariants for flag manifolds can be formulated as strong covers

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

quantum cohomology of flag manifolds abstract we study two aspects of quantum schubert calculus a presentation of the small quantum cohomology ring of partial flag manifolds and a quantum giambelli formula our proof gives a relationship between universal schubert polynomials as defined by fulton and quantum schubert polynomials as defined by fomin gelfand and postnikov and later extended by ciocan fontanine intersection theory on hyperquot schemes is an essential element of the proof

quantum cohomology of partial flag manifolds we give elementary geometric proofs of the structure theorems for the small quantum cohomology of partial flag varieties sl n p including the quantum pieri and quantum giambelli formulas and the presentation

k schur expansions of catalan functions abstract we make a broad conjecture about the k schur positivity of catalan functions symmetric functions which generalize the parabolic hall littlewood polynomials we resolve the conjecture with positive combinatorial formulas in cases which address the k schur expansion of 1 hall littlewood polynomials proving the q 0 case of the strengthened macdonald positivity conjecture from 24 2 the product of a schur function and a k schur function when the indexing partitions concatenate to a partition describing a class of gromov witten invariants for the quantum cohomology of complete flag varieties 3 k split polynomials solving a substantial special case of a problem of broer and shimozono weyman on parabolic hall littlewood polynomials 37 in addition we prove the conjecture that the k schur functions defined via k split polynomials 25 agree with those defined in terms of strong tableaux 21

affine approach to quantum schubert calculus this paper presents a formula for products of schubert classes in the quantum cohomology ring of the grassmannian we introduce a generalization of schur symmetric polynomials for shapes that are naturally embedded in a torus then we show that the coefficients in the expansion of these toric schur polynomials in terms of the regular schur polynomials are exactly the 3 point gromov witten invariants which are the structure constants of the quantum cohomology ring this construction implies three symmetries of the gromov witten invariants of the grassmannian with respect to the groups s3 z nz 2 and z 2z the last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q our construction gives a solution to a problem posed by fulton and woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two schubert classes the curious duality switches the smallest such power of q with the highest power we also discuss the affine nil temperley lieb algebra that gives a model for the quantum cohomology

ordered set partitions generalized coinvariant algebras and the delta conjecture abstract the symmetric group s n acts on the polynomial ring q x n q x 1 x n by variable permutation the invariant ideal i n is the ideal generated by all s n invariant polynomials with vanishing constant term the quotient r n q x n i n is called the coinvariant algebra the coinvariant algebra r n has received a great deal of study in algebraic and geometric combinatorics we introduce a generalization i n k q x n of the ideal i n indexed by two positive integers k n the corresponding quotient r n k q x n i n k carries a graded action of s n and specializes to r n when k n we generalize many of the nice properties of r n to r n k in particular we describe the hilbert series of r n k give extensions of the artin and garsia stanton monomial bases of r n to r n k determine the reduced grobner basis for i n k with respect to the lexicographic monomial order and describe the graded frobenius series of r n k just as the combinatorics of r n are controlled by permutations in s n we will show that the combinatorics of r n k are controlled by ordered set partitions of 1 2 n with k blocks the delta conjecture of haglund remmel and wilson is a generalization of the shuffle conjecture in the theory of diagonal coinvariants we will show that the graded frobenius series of r n k is up to a minor twist the t 0 specialization of the combinatorial side of the delta conjecture it remains an open problem to give a bigraded s n module v n k whose frobenius image is even conjecturally equal to any of the expressions in the delta conjecture our module r n k solves this problem in the specialization t 0

q deformation of witt burnside rings in this paper we construct a q deformation of the witt burnside ring of a profinite group over a commutative ring where q ranges over the set of integers when q 1 it coincides with the witt burnside ring introduced by dress and siebeneicher adv math 70 87 132 1988 to achieve our goal we first show that there exists a q deformation of the necklace ring of a profinite group over a commutative ring as in the classical case i e the case q 1 q deformed witt burnside rings and necklace rings always come equipped with inductions and restrictions we also study their properties as a byproduct we prove a conjecture due to lenart j algebra 199 703 732 1998 finally we classify mathbb w g q up to strict natural isomorphism in case where g is an abelian profinite group

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

quantization error in hexagonal sensory configurations the authors develop mathematical tools for estimating quantization error in hexagonal sensory configurations these include analytic expressions for the average error and the error distribution of a function of an arbitrary number of independently quantized variables these two quantities are essential for assessing the reliability of a given algorithm they can also be used to compare the relative sensitivity of a particular algorithm to quantization error for hexagonal and other spatial samplings e g square and can have an impact on sensor design furthermore it is shown that the ratio of hexagonal error to square error is bounded between 0 90 and 1 05

a reconstruction theorem in quantum cohomology and quantum k theory a reconstruction theorem for genus 0 gravitational quantum cohomology and quantum k theory is proved a new linear equivalence in the picard group of the moduli space of genus 0 stable maps relating the pull backs of line bundles from the target via different markings is used for the reconstruction result examples of calculations in quantum cohomology and quantum k theory are given

quantum cohomology of g p and homology of affine grassmannian let g be a simple and simply connected complex algebraic group p g a parabolic subgroup we prove an unpublished result of d peterson which states that the quantum cohomology qh g p of a flag variety is up to localization a quotient of the homology h gr g of the affine grassmannian gr g of g as a consequence all three point genus zero gromov witten invariants of g p are identified with homology schubert structure constants of h gr g establishing the equivalence of the quantum and homology affine schubert calculi

quantum schubert polynomials where in is the ideal generated by symmetric polynomials in x1 xn without constant term another geometric description of the cohomology ring of the flag manifold is based on the decomposition of fln into schubert cells these are even dimensional cells indexed by the elements w of the symmetric group sn the corresponding cohomology classes oa called schubert classes form an additive basis in h fln 2 to relate the two descriptions one would like to determine which elements of 2 xl xn in correspond to the schubert classes under the isomorphism 1 1 this was first done in 2 see also 8 for a general case of an arbitrary complex semisimple lie group later lascoux and schiitzenberger 22 came up with a combinatorial version of this theory for the type a by introducing remarkable polynomial representatives of the schubert classes oa called schubert polynomials and denoted gw recently motivated by ideas that came from the string theory 31 30 mathematicians defined for any kahler algebraic manifold x the small quantum cohomology ring qh x 2 which is a certain deformation of the classical cohomology ring see e g 28 19 14 and references therein the additive structure of qh x 2 is essentially the same as that of ordinary cohomology in particular qh fln z is canonically isomorphic as an abelian group to the tensor product h fln 2 0 z ql qn 1 where the qi are formal variables deformation parameters the multiplicative structure of the quantum cohomology is however

positivity in equivariant quantum schubert calculus a conjecture of d peterson proved by w graham states that the structure constants of the t equivariant cohomology of a homogeneous space g p satisfy a certain positivity property in this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology 1 introduction it is well known that the integral cohomology of the homogeneous space x g p for g a connected semisimple complex lie group and p a parabolic subgroup satisfies a positivity property its structure constants are nonnegative integers equal to the number of intersection points of three schubert varieties in general position whose codimensions add up to the dimension of x recently graham gr has proved a conjecture of peterson p asserting that h x the t equivariant cohomology of x where t c r is a maximal torus in g enjoys a more general positivity property

multidimensional digitization of data followed by a mapping corresp in many applications it is necessary to digitize data knowing only that later on some random function of the digitized data will be of interest this problem is investigated when the data digitizers are allowed to be multidimensional i e they map a k dimensional data vector into one of a set of k dimensional output vectors it is shown that very complex distortion measures arise naturally results are given for the error measure defined as the squared value of the difference between the function of the digitized data and the function of the undigitized data

quantum sheaf cohomology and duality of flag manifolds we study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds realized as the ope ring of a 2 twisted two dimensional theories with 0 2 supersymmetry quantum sheaf cohomology generalizes the notion of quantum cohomology complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models glsms and for nonabelian glsms describing grassmannians in this paper we continue to explore the quantum sheaf cohomology of nonabelian theories we first propose a method to compute the generating relations for 0 2 glsms with 2 2 locus we apply this method to derive the quantum sheaf cohomology of products of grassmannians and flag manifolds the dual deformation associated with the biholomorphic duality gives rise to an explicit ir duality of two a 2 twisted 0 2 gauge theories

distributed functional scalar quantization simplified distributed functional scalar quantization dfsq theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired we address two limitations of previous works prohibitively expensive decoder design and a restriction to source distributions with bounded support we show that a much simpler decoder has equivalent asymptotic performance to the conditional expectation estimator studied previously thus reducing decoder design complexity the simpler decoder features decoupled communication and computation blocks moreover we extend the dfsq framework with the simpler decoder to source distributions with unbounded support finally through simulation results we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis and we give new insight on the rate of convergence

necklace rings and logarithmic functions in this paper we develop the theory of the necklace ring and the logarithmic function regarding the necklace ring we introduce the necklace ring functor nr from the category of special rings into the category of special rings and then study the associated adams operators as far as the logarithmic function is concerned we generalize the results in bryant s paper free lie algebras and formal power series j algebra 253 1 2002 167 188 to the case of graded lie super algebras with a group action by applying the euler poincare principle

distributed scalar quantization for computing high resolution analysis and extensions communication of quantized information is frequently followed by a computation we consider situations of distributed functional scalar quantization distributed scalar quantization of possibly correlated sources followed by centralized computation of a function under smoothness conditions on the sources and function companding scalar quantizer designs are developed to minimize mean squared error mse of the computed function as the quantizer resolution is allowed to grow striking improvements over quantizers designed without consideration of the function are possible and are larger in the entropy constrained setting than in the fixed rate setting as extensions to the basic analysis we characterize a large class of functions for which regular quantization suffices consider certain functions for which asymptotic optimality is achieved without arbitrarily fine quantization and allow limited collaboration between source encoders in the entropy constrained setting a single bit per sample communicated between encoders can have an arbitrarily large effect on functional distortion in contrast such communication has very little effect in the fixed rate setting

corrigendum to analogue of the burnside ring of profinite groups and free lie algebras adv math 190 2005 1 46

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

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

witt burnside ring and burnside ring over a special λ ring abstract given an arbitrary group g we construct a covariant functor f g from the category of special rings to that of commutative rings with unity when g is a strongly complete profinite group it will be used to reconstruct the functor n r g which was first introduced in y t oh r analogue of the burnside ring of profinite groups and free lie algebras adv math 190 2005 1 46 to investigate the structure of witt burnside rings over special rings more precisely given a special ring r we show that n r g r is functorially isomorphic to f g r j g r for some ideal j g r of f g r the connection to the functor f g due to j j graham generalized witt vectors adv math 99 1993 248 263 will also be demonstrated

generalized burnside grothendieck ring functor and aperiodic ring functor associated with profinite groups abstract for every profinite group g we construct two covariant functors g and ap g which are equivalent to the functor w g introduced in a dress c siebeneicher the burnside ring of profinite groups and the witt vectors construction adv math 70 1988 87 132 we call g the generalized burnside grothendieck ring functor and ap g the aperiodic ring functor associated with g in case g is abelian we also construct another functor ap g from the category of commutative rings with identity to itself as a generalization of the functor ap introduced in k varadarajan k wehrhahn aperiodic rings necklace rings and witt vectors adv math 81 1990 1 29 finally it is shown that there exist q analogues of these functors i e w g g ap g and ap g in case g is the profinite completion of the multiplicative infinite cyclic group c