260155 VO Spinors: classical and quantum. Elements of Noncommutative Riemannian Geometry (2013W)

These lectures concentrate on the latest layer of Noncommutative Geometry (NCG): Riemannian and Spin. It is encoded in terms of spectral triple and its main ingredient, the Dirac operator. The canonical spectral triple on a Riemannian and spin manifold will be described starting with basic notions of multilinear algebra and differential geometry. Its basic properties, and then certain additional requirements that permit to reconstruct the underlying geometry will be presented. They are essential for further fascinating generalizations to noncommutative spaces by A. Connes.

Some previous levels of NCG will be briefly mentioned, that regard the (differential) topology and calculus, like the equivalence between (locally compact) topological spaces and C*-algebras, and between vector bundles and finite projective modules, projectors and K theory, the Hochschild and cyclic cohomology, noncommutative integral, and others. In the last part the concept of symmetries (isometries, diffeomorphisms) will be presented and generalized to Hopf algebras and quantum groups, and to equivariant spectral triples. The product of spectral triples and noncommutative principal bundles will be also discussed. Among the NCG examples we plan to describe the noncommutative torus, quantum spheres and - if time permits - the almost commutative geometry.

Due to the necessary selection among the wealth of available material some well established topics (e.g. the index theory) will not be discussed, and just few indispensable facts from well known theory of the (elliptic) Laplace operator will be used. Such a choice hopefully will lead us fast to some of the active and interesting fields of current research.

The presentation style will be oriented towards both mathematicians and mathematical physicists. The only prerequisites are basics of multilinear algebra, differential geometry and Hilbert space operators.