a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

computing motivic zeta functions on log smooth models we give an explicit formula for the motivic zeta function in terms of a log smooth model it generalizes the classical formulas for snc models but it gives rise to much fewer candidate poles in general this formula plays an essential role in recent work on motivic zeta functions of degenerating calabi yau varieties by the second named author and his collaborators as a further illustration we explain how the formula for newton non degenerate polynomials can be viewed as a special case of our results

duals of colored quantum universal enveloping algebras and colored universal t matrices we extend the notion of dually conjugate hopf super algebras to the colored hopf super algebras hc that we recently introduced we show that if the standard hopf super algebras hq that are the building blocks of hc have hopf duals hq then the latter may be used to construct coloured hopf duals hc endowed with colored algebra and antipode maps but with a standard coalgebraic structure next we review the case where the hq s are quantum universal enveloping algebras of lie super algebras uq g so that the corresponding hq s are quantum super groups gq we extend the fronsdal and galindo universal t matrix formalism to the colored pairs uc g gc by defining colored universal t matrices we then show that together with the colored universal r matrices previously introduced the latter provide an algebraic formulation of the colored rtt relations proposed by basu mallick this establishes a link between the colored extensions of drinfeld jimbo and faddeev reshetikhin takhtajan pictures of quant

iterated vanishing cycles convolution and a motivic analogue of a conjecture of steenbrink we prove a motivic analogue of steenbrink s conjecture 25 conjecture 2 2 on the hodge spectrum proved by m saito in 21 to achieve this we construct and compute motivic iterated vanishing cycles associated with two functions we are also led to introduce a more general version of the convolution operator appearing in the motivic thom sebastiani formula throughout the article we use the framework of relative equivariant grothendieck rings of varieties endowed with an algebraic torus action

vanishing of principal value integrals on surfaces principal value integrals are associated to multi valued rational differential forms with normal crossings support on a non singular algebraic variety we prove their vanishing on rational surfaces in the context of a conjecture of denef jacobs as an appli cation we obtain a strong vanishing result for candidate poles of p adic and motivic igusa zeta functions

towards the singular locus of the space of arcs we give a theory on the space of arcs x infty of a singular variety x based on the finiteness property of the stable points of x infty we also show how computations can be made in this theory as a consequence we obtain a process of determining invariants of x from its space of arcs x infty and from its spaces of wedges arcs in the space of arcs more precisely we give two concepts of regularity for the stable points of x infty and we show that we have regularity in codimension 1 and for all stable points of the space of arcs of a curve as another application we show concrete examples obtained from the spaces of arcs of noetherian 1 dimensional local rings which are analytically ramified

estimation of the negative binomial parameter κ by maximum quasi likelihood we investigate estimation of the parameter k of the negative binomial distribution for small samples using a method of moments estimate mme and a maximum quasi likelihood estimate mqle previous work is reviewed the importance of indirect estimation of k through its reciprocal a and of allowance for negative estimates of k or a are discussed samples of size 50 are simulated 10 000 times for each of several parameter combinations to examine the properties of the estimates samples of sizes 10 20 30 and 50 are simulated 1 000 times to investigate the effect of sample size both estimators perform reasonably well except when the mean is small and the sample size does not exceed 20 three examples are given one of a designed experiment for which the mqle is especially suited confidence limits are derived for the mqle further work along these lines is required for adequate assessment of the usual maximum likelihood estimate

on up q gl 2 and a p q virasoro algebra the quantum algebra up q gl 2 with two independent deformation parameters p q is studied and in particular its universal r matrix is constructed using reshetikhin s method a contraction procedure then leads to the p q deformed heisenberg algebra up q h 1 and its universal r matrix using a sugawara construction employing an infinite number of copies of these heisenberg modes a p q deformed virasoro algebra is obtained the closure property of the p q virasoro algebra necessitates two parameters alpha beta for the generators lm alpha beta while the parameter alpha may be taken as an integer the parameter beta is continuous on a complex path and imparts an integral equation structure to the p q deformed virasoro algebra

motivic height zeta functions we consider a motivic analogue of the height zeta function for integral points of equivariant partial compactifications of affine spaces we establish its rationality and determine its largest pole a major problem of diophantine geometry is to understand the distribution of rational or integral points of algebraic varieties defined over number fields for example a well studied question put forward by manin in 1 is that of an asymptotic expansion for the number of rational integral points of bounded height a basic tool is the height zeta function which is a dirichlet series around 2000 e peyre suggested to consider the analogous problem over function fields which has then an even more geometric flavor since it translates as a problem of enumerative geometry namely counting algebraic curves of given degree and establishing properties of the corresponding generating series in view of the developments of motivic integration by kontsevich denef loeser 11 etc it is natural to look at a more general generating series which not only counts the number of such algebraic curves but takes into account the space they constitute in a suitable hilbert scheme a natural coefficient ring for the generating series is the grothendieck ring of varieties kvark if k is a field this ring is generated as a group by the isomorphism classes of k schemes of finite type the addition being subject to obvious cut and paste relations and the product is induced by the product of k varieties one interest of this generalization is that it also makes sense in a purely geometric context where no counting is available such a situation has been first studied in a paper by kapranov in 15 where the analogy with the mentioned diophantine problem is not pointed out later bourqui made some progress on the motivic analogue of manin s problem see 4 as well as his survey report 5 manuscript received april 14 2014 revised march 3 2015 research of the first author supported by the institut universitaire de france by the national science foundation under agreement no dms 0635607 as well as by the positive project of agence nationale de la recherche under agreement anr 2010 blan 0119 01 research of the second author supported in part by the european research council under the european community s seventh framework programme fp7 2007 2013 erc grant agreement nr 246903 nmnag also partially supported by valcomo anr 13 bs01 0006 american journal of mathematics 138 2016 1 59 c 2016 by johns hopkins university press

parabosons virasoro type algebras and their deformations two parameter p q deformation of a parabosonic algebra underlying the two particle calogero model is considered the fock space representation bargmann fock representation coherent states spectrum generating algebra and a hidden supersymmetry are discussed the paraboson algebra induces a centreless virasoro type algebra with a doubling of the space of generators a two parameter deformation of this virasoro type algebra is studied

poincare series of a toric variety abstract for an affine toric variety x we compute the poincare series of the multi index filtration defined by a finite number of monomial divisorial valuations on the ring o x 0 we give an alternative description of the poincare series as an integral with respect to the euler characteristic over the projectivization of the space of germs o x 0 in particular we study divisorial valuations on the ring o c d 0 that arise by considering toric constellations we give an explicit formula for the poincare series and a nice geometric description this generalizes an expression of the poincare series for curves and rational surface singularities

free fermions and the alexander conway polynomial we show how the conway alexander polynomial arises from theq deformation of z2 graded sl n n algebras in the simplestsl 1 1 case we then establish connection between classical knot theory and its modern versions based on quantum groups we first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the burau representation of the braid group the burau matrix is then transformed into theu q sl 1 1 r matrix by going to the exterior power algebra using a det str identity this allows us to recover the state model of k2 89 as well we also show how theu q sl 1 1 algebra describes free fermions propagating on the knot diagram we rewrite the conway alexander polynomial as a berezin integral and thus as an apparently new determinant

extended moment series and the parameters of the negative binomial distribution recent studies indicate that for finite sample sizes moment estimators may be superior to maximum likelihood estimators in some regions of parameter space in this paper a statistic based on the central moment of the sample is expanded in a taylor series using 24 derivatives and many more terms than previous expansions a summary algorithm is required to find meaningful approximants using the higher order coefficients a example is presented and a comparison between theoretical assessment and simulation results is made

vanishing cycles for formal schemes let k be a non archimedean field and let x be a formal scheme locally finitely presented over the ring of integers k see 1 in this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre x of x which is a k analytic space to the category of etale sheaves on the closed fibre xs of x which is a scheme over the residue field of k we prove that if x is the formal completion x of a scheme x finitely presented over k along the closed fibre then the vanishing cycles sheaves of x are canonically isomorphic to those of x as defined in sga7 exp xiii in particular the vanishing cycles sheaves of x depend only on x and any morphism x induces a homomorphism from the pullback of the vanishing cycles sheaves of x under s ys xs to those of y furthermore we prove that for each x one can find a nontrivial ideal of k such that if two morphisms x coincide modulo this ideal then the homomorphisms between the vanishing cycles sheaves induced by and coincide these facts were conjectured by p deligne in 1 we associate with a formal scheme x locally finitely presented over k a k analytic space x in the sense of ber1 and ber2 in 2 we find that the morphism y x which is induced by an etale morphism of formal schemes y x possesses a certain property morphisms of k analytic spaces with this property are called quasi etale and they give rise to a quasi etale site xqet of a k analytic space x there is a canonical morphism of sites xqet xet where xet is the etale site introduced in ber2 we show that the inverse image functor identifies the category of etale sheaves x et with a full subcategory of

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

covariant differential calculus on the quantum superplane and generalization of the wess zumino formalism it is shown that the construction of covariant differential calculus on quantum planes developed by wess and zumino 1990 for arbitrary quadratic algebras can generalize to the quantum superplane if one accounts for the differences due to the graduation of the superplane

a trace formula for varieties over a discretely valued field we study the motivic serre invariant of a smoothly bounded alge braic or rigid variety x over a complete discretely valued field k with perfect residue field k if k has characteristic zero we extend the definition to arbi trary k varieties using bittner s presentation of the grothendieck ring and a process of neron smoothening of pairs of varieties the motivic serre invariant can be considered as a measure for the set of unramified points on x under certain tameness conditions it admits a cohomological interpretation by means of a trace formula in the curve case we use t saito s geometric criterion for cohomological tameness to obtain more detailed results we discuss some applications to weil chatelet groups chow motives and the structure of the grothendieck ring of varieties msc 2000 11s15 14g05 14g22

the motivic zeta function and its smallest poles abstract let f be a regular function on a nonsingular complex algebraic variety of dimension d we prove a formula for the motivic zeta function of f in terms of an embedded resolution this formula is over the grothendieck ring itself and specializes to the formula of denef and loeser over a certain localization we also show that the space of n jets satisfying f 0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension d n 2 finally we look at the consequences for the poles of the motivic zeta function

motivic serre invariants and weil restriction we study the interactions between weil restriction for formal schemes and rigid varieties greenberg schemes and motivic serre invariants and their behavior with respect to finite extensions of the base ring r which is a complete discrete valuation ring with perfect residue field

motivic zeta functions of degenerating calabi yau varieties we study motivic zeta functions of degenerating families of calabi yau varieties our main result says that they satisfy an analog of igusa s monodromy conjecture if the family has a so called galois equivariant kulikov model we provide several classes of examples where this condition is verified we also establish a close relation between the zeta function and the skeleton that appeared in kontsevich and soibelman s non archimedean interpretation of the syz conjecture in mirror symmetry

differential arcs and regular types in differential fields we introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point using differential arcs we show that if k 1 n is a differentially closed field of characteristic zero with n commuting derivations and p s k is a regular type over k then either p is locally modular or there is a definable subgroup g k of the additive group having a regular generic type that is nonorthogonal to p

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

dimensions of jet schemes of log singularities we characterize kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes our main result is theorem 2 4

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

on the representations of glp q 2 glp q 1 mod 1 and non commutative spaces explicit realizations of the quantum groups glp q 2 and glp q 1 mod 1 corresponding to unimodular values of the deformation parameters p and q are given in terms of the canonically conjugate x p operators using the heisenberg weyl commutation relations matrix representations are also discussed some observations are made on similar realizations of the non commutative coordinate spaces on which the quantum groups act as endomorphisms

equivariant motivic integration and proof of the integral identity conjecture for regular functions we develop denef loeser s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions in comparison with hartmann s work the equivariant grothendieck ring defined in this article is more elementary and it yields the application to the conjecture

on the representations of glq n using the heisenberg weyl relations using the representations of the heisenberg weyl relations the authors develop a systematic scheme for constructing finite and infinite dimensional representations of the elements of the quantum groups glq n where the deformation parameter q is a primitive root of unity explicit results for the examples glq 2 glq 3 and glq 4 are discussed

motivic integration on smooth rigid varieties and invariants of degenerations we develop a theory of motivic integration for smooth rigid varieties as an application we obtain a motivic analogue for rigid varieties of serre s invariant for p adic varieties our construction provides new geometric birational invariants of degenerations of algebraic varieties for degenerations of calabi yau varieties our results take a stronger form

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

on the two parameter quantum supergroups and quantum superplanes starting from the assumption that the generators of quantum supergroup are not supercommutatives with the coordinates of quantum superplane instead of usual supercommutativity in the manin approach we construct in a natural way the two parameter quantum supergroup the virtue of this formulation is that it leads to many probable quantum superplanes associated with a given quantum supergroup some interesting examples are presented as a by product of this formulation a correspondence between the one and two parameter deformed differential structures on the quantum superplane is presented

a trace formula for rigid varieties and motivic weil generating series for formal schemes we establish a trace formula for rigid varieties x over a complete discretely valued field which relates the set of unramified points on x to the galois action on its etale cohomology next we show that the analytic milnor fiber of a morphism f at a point x completely determines the formal germ of f at x we develop a theory of motivic integration for formal schemes of pseudo finite type over a complete discrete valuation ring r and we introduce the weil generating series of a regular formal r scheme mathfrak x of pseudo finite type via the construction of a gelfand leray form on its generic fiber when mathfrak x is the formal completion of a morphism f from a smooth irreducible variety to the affine line then its weil generating series coincides modulo normalization with the motivic zeta function of f when mathfrak x is the formal completion of f at a closed point x of the special fiber f 1 0 we obtain the local motivic zeta function of f at x

motivic zeta functions of abelian varieties and the monodromy conjecture we prove for abelian varieties a global form of denef and loeser s motivic monodromy conjecture in arbitrary characteristic more precisely we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field its motivic zeta function has a unique pole at chai s base change conductor c a of a and that the order of this pole equals one plus the potential toric rank of a moreover we show that for every embedding of ql in c the value exp 2 ic a is an l adic tame monodromy eigenvalue of a the main tool in the paper is edixhoven s filtration on the special fiber of the neron model of a which measures the behavior of the neron model under tame base change

negative binomial process count and mixture modeling the seemingly disjoint problems of count and mixture modeling are united under the negative binomial nb process a gamma process is employed to model the rate measure of a poisson process whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an nb process for count modeling a draw from the nb process consists of a poisson distributed finite number of distinct atoms each of which is associated with a logarithmic distributed number of data samples we reveal relationships between various count and mixture modeling distributions and construct a poisson logarithmic bivariate distribution that connects the nb and chinese restaurant table distributions fundamental properties of the models are developed and we derive efficient bayesian inference it is shown that with augmentation and normalization the nb process and gamma nb process can be reduced to the dirichlet process and hierarchical dirichlet process respectively these relationships highlight theoretical structural and computational advantages of the nb process a variety of nb processes including the beta geometric beta nb marked beta nb marked gamma nb and zero inflated nb processes with distinct sharing mechanisms are also constructed these models are applied to topic modeling with connections made to existing algorithms under poisson factor analysis example results show the importance of inferring both the nb dispersion and probability parameters

definable sets motives and p adic integrals 0 1 let x be a scheme reduced and separated of finite type over z for p a prime number one may consider the set x zp of its zp rational points for every n in n there is a natural map n x zp x z p assigning to a zp rational point its class modulo p the image yn p of x zp by n is exactly the set of z p rational points which can be lifted to zp rational points denote by nn p the cardinality of the finite set yn p by a result of the first author 7 the poincare series

proofs of the integral identity conjecture over algebraically closed fields recently it has become well known that the conjectural integral identity is of crucial importance in the motivic donaldson thomas invariants theory for noncommutative calabi yau threefolds the purpose of this article is to consider different versions of the identity for regular functions and formal functions and to give them the positive answer for the algebraically closed ground fields technically the result on the motivic milnor fiber by hrushovski loeser using hrushovski kazhdan s motivic integration and nicaise s computations on motivic integrals on special formal schemes are main tools

the thom sebastiani theorem for the euler characteristic of cyclic l algebras abstract let l be a cyclic l algebra of dimension 3 with finite dimensional cohomology only in dimension one and two by transfer theorem there exists a cyclic l algebra structure on the cohomology h l the inner product plus the higher products of the cyclic l algebra defines a superpotential function f on h 1 l we associate with an analytic milnor fiber for the formal function f and define the euler characteristic of l is to be the euler characteristic of the etale cohomology of the analytic milnor fiber in this paper we prove a thom sebastiani type formula for the euler characteristic of cyclic l algebras as applications we prove the joyce song formulas about the behrend function identities for semi schur objects in the derived category of coherent sheaves over calabi yau threefolds a motivic thom sebastiani type formula and a conjectural motivic joyce song formula for the motivic milnor fiber of cyclic l algebras are also discussed

on grothendieck rings and algebraically constructible functions we investigate grothendieck rings appearing in real geometry notably for arc symmetric sets and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by mccrory and parusinski we study in particular the duality and link operators including their behaviour with respect to motivic milnor fibres with signs

topology of nonarchimedean analytic spaces and relations to complex algebraic geometry this note surveys basic topological properties of nonarchimedean analytic spaces in the sense of berkovich including the recent tameness results of hrushovski and loeser we also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry

geometric criteria for tame ramification we prove an a campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field k as a first application we relate the orders of the tame monodromy eigenvalues on the l adic cohomology of a k curve to the geometry of a relatively minimal sncd model and we show that the semi stable reduction theorem and saito s criterion for cohomological tameness are immediate consequences of this result as a second application we compute the error term in the trace formula for smooth and proper k varieties we see that the validity of the trace formula would imply a partial generalization of saito s criterion to arbitrary dimension

formal loops iii additive functions and the radon transform to any algebraic variety x and closed 2 form on x we associate the symplectic action functional t which is a function on the formal loop space lx introduced by the authors earlier the correspondence t can be seen as a version of the radon transform we give a characterization of the functions of the form t in terms of factorizability infinitesimal analog of additivity in holomorphic pairs of pants as well as in terms of vertex operator algebras these results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on x whose lien is the sheaf of closed 2 forms and the determinantal gerbe of the tangent bundle of lx whose lien is the sheaf of invertible functions on lx on the level of liens this relation associates to a closed 2 form the invertible function expt

finite dimensional representations of the quantum group glp q 2 using the exponential map from up q gl 2 using the fronsdal galindo formula for the exponential mapping from the quantum algebra up q gl 2 to the quantum group glp q 2 we show how the 2j 1 dimensional representations of glp q 2 can be obtained by exponentiating the well known 2j 1 dimensional representations of up q gl 2 for j 1 3 2 j 1 2 corresponds to the defining two dimensional t matrix the earlier results on the finite dimensional representations of glq 2 and slq 2 or suq 2 are obtained when p q representations of uq q 2 q in c mod r and uq 2 q in r mod 0 are also considered the structure of the clebsch gordan matrix for up q gl 2 is studied the same clebsch gordan coefficients are applicable in the reduction of the direct product representations of the quantum group glp q 2

poles of the topological zeta function associated to an ideal in dimension two to an ideal in mathbb c x y one can associate a topological zeta function this is an extension of the topological zeta function associated to one polynomial but in this case we use a principalization of the ideal instead of an embedded resolution of the curve in this paper we will study two questions about the poles of this zeta function first we will give a criterion to determine whether or not a candidate pole is a pole it turns out that we can know this immediately by looking at the intersection diagram of the principalization together with the numerical data of the exceptional curves afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two the same results are valid for related zeta functions as for instance the motivic zeta function

multistage estimation compared with fixed sample size estimation of the negative binomial parameter k the negative binomial distribution often fits biological data well since the mean is of primary interest the parameterization in terms of the mean and a parameter k has become almost standard given a precise estimate of k sequential methods of estimating the mean are available however precise estimation of k has been elusive in this paper estimators of k by the method of moments and by maximum likelihood estimation based on samples of size 50 and 100 are investigated by using monte carlo methods although the maximum likelihood estimators tend to have smaller mean square errors than the method of moments estimators neither can be considered good a multistage approach to the estimation of k is presented and monte carlo results indicate that this procedure gives a mean square error less than that of comparable fixed sample size estimators two truncation rules for the multistage procedure are also given and compared

a p q deformed virasoro algebra a conformal dimension delta dependent p q deformed virasoro p q virasoro algebra with two independent deformation parameters p q is constructed the comultiplication rule for the generating functional for the delta 0 1 case is established and found to be depending on p and q individually the central charge term for the p q virasoro algebra is described a p q deformed nonlinear equation p q kdv corresponding to p q virasoro algebra for delta 0 1 case is obtained

supersymmetry and the formal loop space abstract for any algebraic super manifold m we define the super ind scheme lm of formal loops and study the transgression map radon transform on differential forms in this context applying this to the super manifold m s x the spectrum of the de rham complex of a manifold x we obtain in particular that the transgression map for x is a quasi isomorphism between the 2 3 truncated de rham complex of x and the additive part of the 1 2 truncated de rham complex of l x the proof uses the super manifold ss x and the action of the lie super algebra sl 1 2 on this manifold this quasi isomorphism result provides a crucial step in the classification of sheaves of chiral differential operators in terms of geometry of the formal loop space

non commutative periods and mirror symmetry in higher dimensions we study an analog for higher dimensional calabi yau manifolds of the standard predictions of mirror symmetry we introduce periods associated with non commutative deformations of calabi yau manifolds these periods define a map on the moduli space of such deformations which is a local isomorphism using these non commutative periods we introduce invariants of variations of semi infinite generalized hodge structures living over the moduli space it is shown that the generating function of such invariants satisfies the system of wdvv equations exactly as in the case of gromov witten invariants we prove that the total collection of rational gromov witten invariants of complete intersection calabi yau manifold can be identified with the collection of invariants of variations of generalized semi infinite hodge structures attached to the mirror variety the basic technical tool utilized is the deformation theory

two parameter deformation of grassmann matrix group and supergroup the two parameter quantum deformations of 2 2 grassmann matrices gr 2 and supermatrices gr 1 1 are presented gr 2 whose matrix elements are all grassmannian variables is called the superdual of the general linear group gl 2 and gr 1 1 whose diagonal matrix elements are grassmannian variables is called the superdual of the supergroup gl 1 1 whose nondiagonal elements are grassmannian noncentral dual superdeterminant for grassmann supermatrices belonging to grp q 1 1 is constructed as with the 2 2 quantum matrices the relations satisfied by the matrix elements of the grassmann matrices and supermatrices are expressed in terms of an r matrix the properties of the nth power of a grassmann supermatrix are given as an appendix