The Atiyah-Singer index theorem gives a 'geometric-topological' formula for the index of an elliptic differential operator, like the Dirac operator or the signature operator. The index is analytically defined (via solutions of differential equations), but the answer is of purely topological nature. The best way to discuss the Atiyah-Singer index theorem, to obtain its many applications and to establish powewrful generalizations uses K-theory and K-homology, and this not only of spaces but of C*-algebras. Indeed, with the appropriate definition of K-homology using non-commutative C*-algebras the elliptic differential operators define on the nose elements in K-homology, the (analytic) index is the image under a certain homomorphism out of K-homology, and the index theorem is the identification of this homomorphism with a topologically defined second homomorphism where the image can be calculated; and this is a general picture which applies to many "higher index" situations exactly in the same way. This has numerous applications, e.g. to the construction of homotopy invariants of smooth manifolds, or to obstruct and distinguish Riemannian metrics of positive scalar curvature. In the lectures, we will - introduce analytic K-homology - develop the abstract settings of higher index theory - indicate the routes of proof of index theorems of Atiyah-Singer and its generalizations - derive applications, in particular obstructions to the existences of metrics of positive scalar curvature on spin manifolds The targeted audience are master students and PhD students which have a basic knowledge of C*-algebras and their K-theory, as well as in differential topology.
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