gravity coupled with matter and the foundation of non commutative geometry we first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds its unitary representations correspond to riemannian metrics and spin structure whileds is the dirac propagatords x x d 1 whered is the dirac operator we extend these simple relations to the non commutative case using tomita s involutionj we then write a spectral action the trace of a function of the length element which when applied to the non commutative geometry of the standard model will be shown cc to give the sm lagrangian coupled to gravity the internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of sm with all correct quantum numbers in this slightly non commutative case the group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group

hamiltonian gravity and noncommutative geometry a version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry the adm version of the gravitational action is expressed in terms of these variables it is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action

discrete spectral triples and their symmetries we classify zero dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure we investigate also whether such spectral triples admit a symmetry arising from the hopf algebra structure of the finite algebra we discuss examples of commutative algebras and group algebras

noncommutative geometry topology and the standard model vacuum as a ramification of a motivational discussion for previous joint work in which equations of motion for the finite spectral action of the standard model were derived we provide a new analysis of the results of the calculations therein switching from the perspective of spectral triple to that of fredholm module and thus from the analogy with riemannian geometry to the premetrical structure of the noncommutative geometry using a suggested noncommutative version of morse theory together with algebraic k theory to analyze the vacuum solutions the first two summands of the algebra for the finite triple of the standard model arise up to morita equivalence we also demonstrate a new vacuum solution whose features are compatible with the physical mass matrix

quantum spin coverings and statistics slq 2 at primitive odd roots of unity ql 1 is studied as a quantum cover of the complex rotation group so 3 in terms of the associated hopf algebras of quantum polynomial functions we work out the irreducible corepresentations the decomposition of their tensor products and a coquasitriangular structure with the associated braiding or statistics as an example the case l 3 is discussed in detail

quantum symmetry groups of noncommutative spheres we show that the noncommutative spheres of connes and landi are quantum homogeneous spaces for certain compact quantum groups we give a general construction of homogeneous spaces which support noncommutative spin geometries

localizing gauge theories from noncommutative geometry abstract we recall the emergence within noncommutative geometry of a generalized gauge theory from a real spectral triple for a unital c algebra a this includes a gauge group determined by the unitaries in a and gauge fields arising from a so called perturbation semigroup which is associated to a our main new result is the construction of an upper semi continuous c bundle on a hausdorff base space x whose space of continuous sections is isomorphic to a the gauge group acts by vertical automorphisms on this c bundle and can under some conditions be identified with the space of continuous sections of a group bundle on x moreover in some cases the gauge group is found to coincide with the group of inner automorphisms of a we discuss two classes of examples of our construction yang mills theory and toric noncommutative manifolds and we show that they actually give rise to continuous c bundles moreover in these examples the gauge groups coincide with the inner automorphism groups and can be realized as spaces of sections of group bundles that we explicitly determine

limits and degenerations of unitary conformal field theories in the present paper degeneration phenomena in conformal field theories are studied for this purpose a notion of convergent sequences of cfts is introduced properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of cfts which as familiar from large volume limits of non linear sigma models can be regarded as commutative degenerations of the corresponding quantum geometries as an application the large level limit of the a series of unitary virasoro minimal models is investigated in detail in particular its geometric interpretation is determined

dirac operators and the calculation of the connes metric on arbitrary infinite graphs as an outgrowth of our investigation of non regular spaces within the context of quantum gravity and non commutative geometry we develop a graph hilbert space framework on arbitrary infinite graphs and use it to study spectral properties of graph laplacians and graph dirac operators we define a spectral triplet sharing most of the properties of what connes calls a spectral triple with the help of this scheme we derive an explicit expression for the connes distance function on general directed or undirected graphs we derive a series of a priori estimates and calculate it for a variety of examples of graphs as a possibly interesting side we show that the natural setting for approaching such problems may be the framework of non linear programming or optimization we compare our results arrived at within our particular framework with those of other authors and show that the seeming differences depend on the use of different graph geometries and or dirac operators

noncommutative manifolds from graph and k graph c algebras in pren we constructed smooth 1 summable semifinite spectral triples for graph algebras with a faithful trace and in prs we constructed k summable semifinite spectral triples for k graph algebras in this paper we identify classes of graphs and k graphs which satisfy a version of connes conditions for noncommutative manifolds

type iii representations and modular spectral triples for the noncommutative torus abstract it is well known that for any irrational rotation number the noncommutative torus a must have representations such that the generated von neumann algebra a is of type iii therefore it could be of interest to exhibit and investigate such kind of representations together with the associated spectral triples whose twist of the dirac operator and the corresponding derivation arises from the tomita modular operator in the present paper we show that this program can be carried out at least when is a liouville number satisfying a faster approximation property by rationals in this case we exhibit several type i i and ii i 0 1 factor representations and modular spectral triples the method developed in the present paper can be generalised to ccr algebras based on a locally compact abelian group equipped with a symplectic form

spectral triples and manifolds with boundary abstract we investigate manifolds with boundary in noncommutative geometry spectral triples associated to a symmetric differential operator and a local boundary condition are constructed we show that there is no tadpole for classical dirac operators with a chiral boundary condition on spin manifolds

traces of compact operators and the noncommutative residue abstract we extend the noncommutative residue of m wodzicki on compactly supported classical pseudo differential operators of order d and generalise a connes trace theorem which states that the residue can be calculated using a singular trace on compact operators contrary to the role of the noncommutative residue for the classical pseudo differential operators a corollary is that the pseudo differential operators of order d do not have a unique trace pseudo differential operators can be non measurable in connes sense other corollaries are given clarifying the role of dixmier traces in noncommutative geometry including the definitive statement of connes original theorem

noncommutative geometry and reality we introduce the notion of real structure in our spectral geometry this notion is motivated by atiyah s kr theory and by tomita s involution j it allows us to remove two unpleasant features of the connes lott description of the standard model namely the use of bivector potentials and the asymmetry in the poincare duality and in the unimodularity condition

reproducing kernel method for solving singularly perturbed differential difference equations with boundary layer behavior in hilbert space abstract we consider singularly perturbed differential difference equation which contains negative shift in the differentiated term with boundary layer behavior rkhsm reproducing kernel hilbert space method without gram schmidt orthogonalization process is considered in the present paper we decompose the domain of the problem into two subintervals one of them has not the boundary layer and the other one has the side of the interval in which the boundary layer exists is important if the boundary layer of this problem exists on the left side of interval the rkhsm will provide a proper approximation of solution otherwise for the implement of rkhsm we need to change the variable of the singularly perturbed problem to shift the boundary layer region

noncommutative instantons from twisted conformal symmetries we construct a five parameter family of gauge nonequivalent su 2 instantons on a noncommutative four sphere s theta 4 and of topological charge equal to 1 these instantons are critical points of a gauge functional and satisfy self duality equations with respect to a hodge star operator on forms on s theta 4 they are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere a completeness argument for this family is obtained by means of index theorems the dimension of the tangent space to the moduli space is computed as the index of a twisted dirac operator and turns out to be equal to five a number that survives deformation

on a classification of irreducible almost commutative geometries a second helping we complete the classification of almost commutative geometries from a particle physics point of view given by iochum schucker and stephan j math phys to be published four missing krajewski diagrams will be presented after a short introduction into irreducible nondegenerate spectral triples

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

the graded product of real spectral triples forming the product of two geometric spaces is one of the most basic operations in geometry but in the spectral triple formulation of non commutative geometry the standard prescription for taking the product of two real spectral triples is problematic among other drawbacks it is non commutative non associative does not transform properly under unitaries and often fails to define a proper spectral triple in this paper we explain that these various problems result from using the ungraded tensor product by switching to the graded tensor product we obtain a new prescription where all of the earlier problems are neatly resolved in particular the new product is commutative associative transforms correctly under unitaries and always forms a well defined spectral triple

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

poincare duality for cuntz pimsner algebras abstract we present a new approach to poincare duality for cuntz pimsner algebras we provide sufficient conditions under which poincare self duality for the coefficient algebra of a hilbert bimodule lifts to poincare self duality for the associated cuntz pimsner algebra with these conditions in hand we can constructively produce fundamental classes in k theory for a wide range of examples we can also produce k homology fundamental classes for the important examples of cuntz krieger algebras following kaminker putnam and crossed products of manifolds by isometries and their non commutative analogues

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes

spectral distance on the circle a building block of non commutative geometry is the observation that most of the geometric information of a compact riemannian spin manifold m is encoded within its dirac operator d especially via connes distance formula one is able to extract from the spectral properties of d the geodesic distance on m in this paper we investigate the distance d encoded within a covariant dirac operator on a trivial u n fiber bundle over the circle with arbitrary connection it turns out that the connected components of d are tori whose dimension is given by the holonomy of the connection for n 2 we explicitly compute d on all the connected components for n 2 we restrict to a given fiber and find that the distance is given by the trace of the module of a matrix the latest is defined by the holonomy and the coordinate of the points under consideration this paper extends to arbitrary n and arbitrary connection the results obtained in a previous work for u 2 bundle with constant connection it confirms interesting properties of the spectral distance with respect to another distance naturally associated to connection namely the horizontal or carnot caratheodory distance dh especially in case the connection has irrational components the connected components for d are the closure of the connected components of dh within the euclidean topology on the torus