We exhibit large classes of examples of noncommutative finitedimensional 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 R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θ-deformations, which we generalize in any dimension and various contexts, and study in some details. Here, and

We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative Z2-graded algebras tending in a suitable limit to a dense subalgebra of the Z2-graded algebra of H1-functions on the (2|2)-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the superdeRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the "fuzzy level".

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms.

L.P.T.H.E.-ORSAY 95/78 ESI-preprint 285 1Laboratoire associ'e au Centre National de la Recherche Scientifique- URA D0063 1 Abstract We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework. 1 Introduction In [23], J.L. Koszul described a powerful algebraic version of differential geometry in terms of a commutative associative algebra C, C-modules and connections ("derivation laws") on these modules. For the applications to differential geometry, C is the algebra of smooth functions on a manifold and the C-modules are modules of smooth sections of smooth vector bundles over the manifold. The fact that classical differential geometry admits such an algebraic formulation is at the very origin of the idea of noncommutative differential geometry. Historically, the motivation of noncommutative geometry was the development of quantum theory [12]. In noncommutative geometry, one replaces the commutative associative algebra C by an associative algebra A which is not assumed to be commutative. However this replacement raises several problems which will be discussed in this lecture.

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Abstract. Let X be a locally compact space, and let A and B be C0(X)-algebras. We define the notion of an asymptotic C0(X)-morphism from A to B and construct representable E-theory groups RE(X; A, B). These are the universal groups on the category of separable C0(X)-algebras that are C0(X)-stable, C0(X)-homotopy-invariant, and half-exact. If A is RKK(X)-nuclear, these groups are naturally isomorphic to Kasparov’s representable KKtheory groups RKK(X; A, B). Applications and examples are also discussed. 1.

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Abstract. This is an extended version of a communication made at the international conference Noncommutative Geometry and Physics held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary ber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)-vector bundle, and its di erential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs elds and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes. LPT-Orsay/07-57

In recent years there has been much progress in developing theories of noncommutative geometry and exploring their applications in physics. Many viewpoints were adopted and different mathematical approaches were followed by different researchers. Connes ’ theory [10] (see also [20]) formulated within the framework of C ∗-algebras is the most successful approach to noncommutative differential