فاطمه محمد مصطفى العظمى

        K-theory is the branch of algebraic topology concerned with the study of vector bundles by algebraic means. The first notion of K-theory were developed by Grothendieck in his work on Riemann- Roch theorem in algebraic geometry. K-theory as a part of algebraic topology was begun by Atiyah and Hirzebruch [1961]

       In the mid 80’s, Prof. Alain Connes developed non-commutative geometry, in an effort to relate the theory of algebra of operators on a Hilbert space to differential geometry and algebraic topology. Non-commutative geometry, under the influence of quantum physics replaces sets of points by classes of functions.

      One of the corner stone of non-commutative geometry is the famous theorem of Gelfand and Naimark in 1943. In the commutative case it states: Every commutative C*- algebra is isomorphic to C₀(X) (the algebra of all continuous complex valued functions on a locally compact Hausdorff space X vanishing at infinity).

      Connes’s main idea was to first reformulate the constructions of algebraic topology and differential geometry in terms of commutative C*- algebra, and then try to generalize these constructions to arbitrary C*- algebra. And this leads to the subject of K-theory for C*-algebra. The work of Cuntz, Higson, Rosenberg and Schochet shows that algebraic K-theory can be characterized for certain C*-algebra by simple set of axioms analogous to the steenrod axioms of cohomology.

2- Bara - براء عبد ربه

Chern Character and Noncommutative Geometry

      In algebraic topology and differential geometry, the (classical) Chern classes are a particular type of characteristic classes associated to complex vector bundles and they are topological invariants. Chern classes are named for Prof. Shiing-Shen Chern, who first gave a general definition of them in the 40’s. The original approach to the Chern classes was via algebraic topology, but Prof. Chern expressed the Chern classes as polynomials in the coefficients of the curvature form. If X is a compact smooth manifold and E is a smooth complex vector bundle over X, then he showed how to use the notion of curvature of the bundle E to define the Chern classes. These classes can be put together to get an invariant called the Chern character Ch(E), which has some useful properties. The Chern character (which we will refer to it as classical Chern character), extends to a ring homomorphism from the topological K-theory (which is defined using equivalence classes of complex vector bundles) to the de Rham cohomology of the vector bundle. It should be noted that K-theory is the branch of algebraic topology concerned with the study of vector bundles by algebraic means. The notion of K-theory as a part of algebraic topology was first developed by Atiyah and Hirzebruch [1961].

      In the mid 80’s, Prof. Alain Connes developed noncommutative geometry. One of the corner stone of non-commutative geometry is the famous theorem of Gelfand and Naimark [1943]. In the commutative case it states: There is a 1-1 correspondence between locally compact Hausdorff topological spaces and commutative C*-algebras. Connes’s main idea was to first reformulate the constructions of algebraic topology and differential geometry in terms of commutative C*- algebra, and then try to generalize these constructions to arbitrary C*- algebra. And this leads to the subject of K-theory for C*-algebras. Both K-theories coincides in the case of commutative unital C*-algebra by the Gelfand-Naimark theorem.

       As one defines the connection and curvatures on complex vector bundles, which leads to the Chern classes and Chern character.     In this project; we study the classical Chern classes which are defined as polynomials in the coefficients of the curvature form. The Chern character is expressed as a ring homomorphism from the topological K-theory to the de Rham cohomology. To define the algebraic K-theory for unital C*-algebras, we study different equivalence relations defiend on projections on C*-algebras, this yields an abelian semi group whose Grothendieck construction yields the K-theory for unital C*-algebras. To define the noncommutative Chern character, from algebraic K-theory to cyclic homology theory, the concept of differential graded algebra and differential calculus is introduced. Then a connection on a finitely generated projective module E over unital algebra A is defiend in a certain way, and this leads to the definition of Chern classes and Chern character. Finally we study the cyclic homology.