the classification of differential structures on quantum 2 spheres exterior algebras of differential forms on quantum 2 spheress qc 2 q 1 1 0 c 0 c 0 forq 1 are classified in the definition of exterior algebras we assume the invariance w r t the action of the quantumsu 2 group and dimensionality conditions which imply that we deal with two dimensional manifolds the exterior algebras exist only forc 0 and are unique in that case the corresponding generalized directional derivatives are provided

riemannian geometry of noncommutative surfaces a riemannian geometry of noncommutative n dimensional surfaces is developed as a first step toward the construction of a consistent noncommutative gravitational theory historically as well riemannian geometry was recognized to be the underlying structure of einstein s theory of general relativity and led to further developments of the latter the notions of metric and connections on such noncommutative surfaces are introduced and it is shown that the connections are metric compatible giving rise to the corresponding riemann curvature the latter also satisfies the noncommutative analog of the first and second bianchi identities as examples noncommutative analogs of the sphere torus and hyperboloid are studied in detail the problem of covariance under appropriately defined general coordinate transformations is also discussed and commented on as compared to other treatments

the green rings of the 2 rank taft algebra and its two relatives twisted abstract in the paper the representation rings or the green rings for a family of hopf algebras of tame type the 2 rank taft algebra at q 1 and its two relatives twisted by 2 cocycles are explicitly described via a representation theoretic analysis it turns out that the green rings can serve to detect effectively the twist equivalent hopf algebras here

traces in monoidal categories the main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category this construction is a common generalization of the trace for endomorphisms of dualizable objects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

covariant connections on bicovariant differential calculus abstract given a bicovariant differential calculus e d such that the braiding map is diagonalisable in a certain sense the bimodule of two tensors admits a direct sum decomposition into symmetric and anti symmetric tensors this is used to prove the existence of a bicovariant torsionless connection on e following heckenberger and schmudgen we study invariant metrics and the compatibility of covariant connections with such metrics a sufficient condition for the existence and uniqueness of bicovariant levi civita connections is derived this condition is shown to hold for cocycle deformations of classical lie groups

on solitary wave solutions of a class of nonlinear partial differential equations based on the function 1 coshn αx βt abstract the method of simplest equation is applied for obtaining exact solitary traveling wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives the used simplest equation is f 2 n 2 f 2 f 2 n 2 n the developed methodology is illustrated on examples of classes of nonlinear partial differential equations that contain i only monomials of odd grade with respect to participating derivatives ii only monomials of even grade with respect to participating derivatives iii monomials of odd and monomials of even grades with respect to participating derivatives the obtained solitary wave solution for the case i contains as particular cases the solitary wave solutions of korteweg devries equation and of a version of the modified korteweg devries equation one of the obtained solitary wave solutions for the case ii is a solitary wave solution of the classic version of the boussinesq type equation

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

de rham complex for quantized irreducible flag manifolds it is shown that quantized irreducible flag manifolds possess a canonical q analogue of the de rham complex generalizing the well known situation for the standard podle quantum sphere this analogue is obtained as the universal differential calculus of a distinguished first order differential calculus the corresponding differential d can be written as a sum of differentials and the universal differential calculus corresponding to the first order differential calculi d and are given in terms of generators and relations relations to well known quantized exterior algebras are established the dimensions of the homogeneous components are shown to be the same as in the classical case the existence of a volume form is proven

modified method of simplest equation and its application to nonlinear pdes abstract we search for traveling wave solutions of the class of pdes p 1 n 1 a p q p q t p r 2 n 2 b r q q t r s 1 n 3 c s q s q x s u 2 n 4 d u q q x u f q 0 where a p q b r q c s q d u q and f q are polynomials of q the basis of the investigation is a modification of the method of simplest equation the equations of bernoulli riccati and the extended tanh function equation are used as simplest equations the obtained general results are illustrated by obtaining exact solutions of versions of the generalized kuramoto sivashinsky equation reaction diffusion equation with density dependent diffusion and the reaction telegraph equation

fibres de schwarzenberger et fibres logarithmiques generalises on propose une generalisation des fibres logarithmiques et des fibres de schwarzenberger sur mathbb p n mathbb p n mathbb c en rang plus grand que n les premiers sont associes a des ensembles finis de points de mathbb p n vee et les seconds a des courbes de degre plus grand que n sur mathbb p n vee sur le plan projectif nous montrons que deux fibres logarithmiques generalises sont isomorphes si et seulement s ils sont associes au meme ensemble de points ou bien si les deux ensembles de points sont sur une meme courbe de degre egal au rang des fibres

on semisimple hopf algebras of dimension pq2 let p and q be distinct prime numbers we prove a result on the existence of nontrivial group like elements in a certain class of semisimple hopf algebras of dimension pqr we conclude the classification of semisimple hopf algebras a of dimension pq2 over an algebraically closed field k of characteristic zero such that both a and a are of frobenius type we also complete the classification of semisimple hopf algebras of dimension pq2 100

noncommutative instantons on the 4 sphere from quantum groups we describe an approach to the noncommutative instantons on the 4 sphere based on quantum group theory we quantize the hopf bundle 7 4 making use of the concept of quantum coisotropic subgroups the analysis of the semiclassical poisson lie structure of u 4 shows that the diagonal su 2 must be conjugated to be properly quantized the quantum coisotropic subgroup we obtain is the standard su q 2 it determines a new deformation of the 4 sphere 4 q as the algebra of coinvariants in q 7 we show that the quantum vector bundle associated to the fundamental corepresentation of su q 2 is finitely generated and projective and we compute the explicit projector we give the unitary representations of 4 q we define two 0 summable fredholm modules and we compute the chern connes pairing between the projector and their characters it comes out that even the zero class in cyclic homology is non trivial

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

quantum differentials on cross product hopf algebras abstract we construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product hopf algebras namely double cross products a a h h double cross coproducts biproducts and bicrossproducts on the assumption that the factors have strongly bicovariant calculi a h or a braided version b we use super versions of each of the constructions moreover the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable in the case of the drinfeld double d a h a op h where a is dually paired to h we show that its canonical actions on a h are differentiable examples include a canonical c q g l 2 c 2 for the quantum group of affine transformations of the quantum plane and c poin c 1 1 for the bicrossproduct poincare quantum group in 2 dimensions we also show that c q g l 2 itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

moduli of quantum riemannian geometries on 4 points we classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously s majid commun math phys 225 131 2002 the full moduli space is found for 3 points and a restricted moduli space for 4 points generalized levi civita connections and their curvatures are found for a variety of models including models of a discrete torus the topological part of the moduli space is found for 9 points based on the known atlas of regular graphs we also remark on aspects of quantum gravity in this approach

quantum group gauge theory on quantum spaces we construct quantum group valued canonical connections on quantum homogeneous spaces including aq deformed dirac monopole on the quantum sphere of podles with quantum differential structure coming from the 3d calculus of woronowicz onsu q 2 the construction is presented within the setting of a general theory of quantum principal bundles with quantum group hopf algebra fibre associated quantum vector bundles and connection one forms both the base space spacetime and the total space are non commutative algebras quantum spaces

quantization by cochain twists and nonassociative differentials we show that several standard associative quantizations in mathematical physics can be expressed as cochain module algebra twists in the spirit of moyal products at least to o 3 but to achieve this we twist not by a 2 cocycle but by a 2 cochain this implies a hidden nonassociativity not visible in the algebra itself but present in its deeper noncommutative differential geometry a phenomenon first seen in our previous work on semiclassicalization of differential structures the quantizations are induced by a classical group covariance and include enveloping algebras u g as quantizations of g a fedosov type quantization of the sphere s2 under a lorentz group covariance the mackey quantization of homogeneous spaces and the standard quantum groups cq g we also consider the differential quantization of rn for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the dirac operator

yetter drinfeld modules under cocycle twists abstract we give an explicit formula for the correspondence between simple yetter drinfeld modules for certain finite dimensional pointed hopf algebras h and those for cocycle twists h of h this implies an equivalence between modules for their drinfeld doubles to illustrate our results we consider the restricted two parameter quantum groups u r s sl n under conditions on the parameters guaranteeing that u r s sl n is a drinfeld double of its borel subalgebra we determine explicit correspondences between u r s sl n modules for different values of r and s and provide examples where no such correspondence can exist our examples were obtained via the computer algebra system singular plural

quantum bundle description of quantum projective spaces we realise heckenberger and kolb s canonical calculus on quantum projective n 1 space c q c p n 1 as the restriction of a distinguished quotient of the standard bicovariant calculus for the quantum special unitary group c q su n we introduce a calculus on the quantum sphere c q s 2n 1 in the same way with respect to these choices of calculi we present c q c p n 1 as the base space of two different quantum principal bundles one with total space c q su n and the other with total space c q s 2n 1 we go on to give c q c p n 1 the structure of a quantum framed manifold more specifically we describe the module of one forms of heckenberger and kolb s calculus as an associated vector bundle to the principal bundle with total space c q su n finally we construct strong connections for both bundles

the serre spectral sequence of a noncommutative fibration for de rham cohomology for differential calculi on noncommutative algebras we construct a twisted de rham cohomology using flat connections on modules this has propertie s similar in some respects to sheaf cohomology on topological spaces we also discuss generalised mapping properties of these theories and relations of these properties to corings using t his we give conditions for the serre spectral sequence to hold for a noncommutative fibration th is might be better read as giving the definition of a fibration in noncommutative differential geo metry we also study the multiplicative structure of such spectral sequences finally we show that some noncommutative homogeneous spaces satisfy the conditions to be such a fibration and in th e process clarify the differential structure on these homogeneous spaces we also give two explicit examples of differential fibrations these are built on the quantum hopf fibration with two differe nt differential structures

vollstandige fast vollstandige und mengen theoretisch vollstandige durchschnitte in steinschen mannigfaltigkeiten

extended simplest equation method for nonlinear differential equations abstract the modified simplest equation method to look for the exact solutions of the nonlinear differential equations is presented our approach is applied to search for the exact solutions of the sharma tasso olver and the burgers huxley equations the new exact solutions of these equations are obtained

kawamata viehweg vanishing on rational surfaces in positive characteristic we prove that the kawamata viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to w2 k of certain log pairs on smooth rational surfaces as a corollary the kawamata viehweg vanishing theorem holds on log del pezzo surfaces in positive characteristic

riemannian geometry of quantum groups and finite groups with nonuniversal differentials we construct noncommutative riemannian manifold structures on dual quasitriangular hopf algebras such as q su 2 with its standard bicovariant differential calculus using the quantum frame bundle approach introduced previously the metric is provided by the braided killing form on the braided lie algebra on the tangent space and the n bein by the maurer cartan form we also apply the theory to finite sets and in particular to finite group function algebras g with differential calculi and killing forms determined by a conjugacy class the case of the permutation group s 3 is worked out in full detail and a unique torsion free and cotorsion free or levi civita connection is obtained with noncommutative ricci curvature essentially proportional to the metric an einstein space we also construct dirac operators in the metric background including on finite groups such as s 3 in the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form

derivative complex bgg correspondence and numerical inequalities for compact kahler manifolds given an irregular compact kahler manifold x one can form the derivative complex of x which governs the deformation theory of the groups h i x as varies over pic x together with its variants it plays a central role in a body of work involving generic vanishing theorems the purpose of this paper is to present two new applications of this complex first we show that it fits neatly into the setting of the so called bernstein gel fand gel fand bgg correspondence between modules over an exterior algebra and linear complexes over a symmetric algebra cf bgg efs eis co what comes out here is that some natural cohomology modules associated to x have a surprisingly simple algebraic structure and that conversely one can read off from these modules some basic geometric invariants associated to the albanese mapping of x secondly under an additional hypothesis the derivative complex determines a vector bundle on the projectivized tangent space to pic x at the origin we show that this bundle which encodes the infinitesimal behavior of the hilbert scheme of paracanonical divisors along the canonical linear series can be used to generate inequalities among numerical invariants of x

noncommutative riemannian and spin geometry of the standard q sphere we study the quantum sphere open image in new window as a quantum riemannian manifold in the quantum frame bundle approach we exhibit its 2 dimensional cotangent bundle as a direct sum 0 1 1 0 in a double complex we find the natural metric volume form hodge operator laplace and maxwell operators and projective module structure we show that the q monopole as spin connection induces a natural levi civita type connection and find its ricci curvature and q dirac operator open image in new window we find the possibility of an antisymmetric volume form quantum correction to the ricci curvature and lichnerowicz type formulae for open image in new window we also remark on the geometric q borel weil bott construction

noether lefschetz space curves and mathematical instantons there are three areas of study of geometric entities living in the three dimensional projective space over the complex numbers which have proved to be strictly connected the classification of space curves surfaces and vector bundles despite all the production in these areas several key unsolved problems still challenge us after the work of many classical and modern authors see h1 h2 for a bibliography the problem of classifying space curves has been divided into two main parts 1 the so called halphen problem find all the triples of integers n g s such that there exists a smooth irreducible curve c ip 3 of degree n and genus g lying on a surface of degree s but not on any surface of lower degree 2 study the properties of the hilbert scheme hn g of smooth irreducible curves in ip namely its irreducible components dimension singular points let g n s be the maximum genus of a curve of degree n not contained in a surface of degree less than s then part 1 above falls in turn into three ranges for n and s

noncommutative manifolds the instanton algebra and isospectral deformations we give new examples of noncommutative manifolds that are less standard than the nc torus or moyal deformations of n they arise naturally from basic considerations of noncommutative differential topology and have non trivial global features

twisting of quantum differentials and the planck scale hopf algebra we show that the crossed modules and bicovariant differential calculi on two hopf algebras related by a cocycle twist are in 1 1 correspondence in particular for quantum groups which are cocycle deformation quantisations of classical groups the calculi are obtained as deformation quantisations of the classical ones as an application we classify all bicovariant differential calculi on the planck scale hopf algebra this is a quantum group which has an limit as the functions on a classical but non abelian group and a limit as flat space quantum mechanics we further study the noncommutative differential geometry and fourier theory for this hopf algebra as a toy model for planck scale physics the fourier theory implements a t duality like self duality the noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone

solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives the method of simplest equation is applied to nonlinear pdes the pdes contain monomials of odd and even order with respect to derivatives solitary wave solutions are studied numerous examples of the methodology are presented we apply the method of simplest equation for obtaining exact solitary traveling wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives we consider first the general case of presence of monomials of the both odd and even grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade the methodology is illustrated by numerous examples

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

principal fibrations from noncommutative spheres we construct noncommutative principal fibrations s 7 s 4 which are deformations of the classical su 2 hopf fibration over the four sphere we realize the noncommutative vector bundles associated to the irreducible representations of su 2 as modules of coequivariant maps and construct corresponding projections the index of dirac operators with coefficients in the associated bundles is computed with the connes moscovici local index formula the algebra inclusion is an example of a not trivial quantum principal bundle

knot homology via derived categories of coherent sheaves ii mathfrak sl _ m case using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum mathfrak sl m knot polynomial our knot homology naturally satisfies the categorified moy relations and is conjecturally isomorphic to khovanov rozansky homology our construction is motivated by the geometric satake correspondence and is related to manolescu s by homological mirror symmetry

backlund transformations and the atiyah ward ansatz for non commutative anti self dual yang mills equations we present backlund transformations for the non commutative anti self dual yang mills equation where the gauge group is g gl 2 and use it to generate a series of exact solutions from a simple seed solution the solutions generated by this approach are represented in terms of quasi determinants we also explain the origins of all the ingredients of the backlund transformations within the framework of non commutative twistor theory in particular we show that the generated solutions belong to a non commutative version of the atiyah ward ansatz

noncommutative principal bundles through twist deformation we construct noncommutative principal bundles deforming principal bundles with a drinfeld twist 2 cocycle if the twist is associated with the structure group then we have a deformation of the fibers if the twist is associated with the automorphism group of the principal bundle then we obtain noncommutative deformations of the base space as well combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space more in general the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new hopf galois extensions as twists of hopf galois extensions a sheaf approach is also considered and examples presented

modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations we discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities as simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of weierstrass and jacobi we show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions this system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation the methodology is illustrated by obtaining solitary wave solutions for the generalized korteweg devries equation and by obtaining solutions of the higher order korteweg devries equation

exterior differentials of higher order and their covariant generalization we investigate a particular realization of generalized q differential calculus of exterior forms on a smooth manifold based on the assumption that dn 0 while dk 0 for k n it implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order special attention is paid to the cases n 3 and n 4 a covariant basis of the algebra of such q grade forms is introduced and the analogs of torsion and curvature of higher order are considered we also study a zn graded exterior calculus on a generalized clifford algebra

braided cyclic cocycles and nonassociative geometry we use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a drinfeld type cochain twist these are the so called quasialgebras and include the octonions as braided commutative but nonassociative coordinate rings as well as quasialgebra versions cq g of the standard q deformation quantum groups we introduce the notion of ribbon algebras in the category which are algebras equipped with a suitable generalized automorphism and obtain the required generalization of cyclic cohomology we show that this braided cyclic cohomology is invariant under a cochain twist we also extend to our generalization the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra the paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry as well as the algebraic noncommutative torus as an associative example

gromov witten donaldson thomas correspondence for toric 3 folds we prove the equivariant gromov witten theory of a nonsingular toric 3 fold x with primary insertions is equivalent to the equivariant donaldson thomas theory of x as a corollary the topological vertex calculations by agangic klemm marino and vafa of the gromov witten theory of local calabi yau toric 3 folds are proven to be correct in the full 3 leg setting

twistor theory at fifty from contour integrals to twistor strings we review aspects of twistor theory its aims and achievements spanning the last five decades in the twistor approach space time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold the twistor space after giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics anti self duality equations on yang mills or conformal curvature can be encoded into twistor cohomology these twistor correspondences yield explicit examples of yang mills and gravitational instantons which we review they also underlie the twistor approach to integrability the solitonic systems arise as symmetry reductions of anti self dual asd yang mills equations and einstein weyl dispersionless systems are reductions of asd conformal equations we then review the holomorphic string theories in twistor and ambitwistor spaces and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes finally we discuss the newtonian limit of twistor theory and its possible role in penrose s proposal for a role of gravity in quantum collapse of a wave function

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

application of the method of simplest equation for obtaining exact traveling wave solutions for the extended korteweg de vries equation and generalized camassa holm equation a version of the method of the simplest equation called modified method of simplest equation is applied to the extended korteweg de vries equation and to generalized camassa holm equation exact traveling wave solutions of these two nonlinear partial differential equations are obtained the equations of bernoulli riccati and the extended tanh equation are used as simplest equations some of the obtained solutions correspond to surface water waves

projective module description of the q monopole the dirac q monopole connection is used to compute projector matrices of quantum hopf line bundles for arbitrary winding number the chern connes pairing of cyclic cohomology and k theory is computed for the winding number 1 the non triviality of this pairing is used to conclude that the quantum principal hopf fibration is non cleft among general results we provide a left right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode we also provide for arbitrary strong connections on algebraic quantum principal bundles hopf galois extensions their associated covariant derivatives on projective modules

quantisation of twistor theory by cocycle twist we present the main ingredients of twistor theory leading up to and including the penrose ward transform in a coordinate algebra form which we can then quantise by means of a functorial cocycle twist the quantum algebras for the conformal group twistor space mathbb cp 3 compactified minkowski space mathbb cm and the twistor correspondence space are obtained along with their canonical quantum differential calculi both in a local form and in a global algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties we outline how the penrose ward transform then quantises as an example we show that the pull back of the tautological bundle on mathbb cm pulls back to the basic instanton on s 4 subset mathbb cm and that this observation quantises to obtain the connes landi instanton on deformed s4 as the pull back of the tautological bundle on our deformed mathbb cm we likewise quantise the fibration mathbb cp 3 rightarrow s 4 and use it to construct the bundle on deformed mathbb cp 3 that maps over under the transform to the deformed instanton

noncommutative kahler structures on quantum homogeneous spaces abstract building on the theory of noncommutative complex structures the notion of a noncommutative kahler structure is introduced in the quantum homogeneous space case many of the fundamental results of classical kahler geometry are shown to follow from the existence of such a structure allowing for the definition of noncommutative lefschetz hodge kahler dirac and laplace operators quantum projective space endowed with its heckenberger kolb calculus is taken as the motivating example the general theory is then used to show that the calculus has cohomology groups of at least classical dimension

the dirac operator on compact quantum groups for the q deformation gq 0 q 1 of any simply connected simple compact lie group g we construct an equivariant spectral triple which is an isospectral deformation of that defined by the dirac operator d on g our quantum dirac operator dq is a unitary twist of d considered as an element of ug cl g the commutator of dq with a regular function on gq consists of two parts one is a twist of a classical commutator and so is automatically bounded the second is expressed in terms of the commutator of the associator with an extension of d we show that in the case of the drinfeld associator the latter commutator is also bounded