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

Main goal of the lectures is to introduce the students in mathematics and mathematical physicists
to the latest layer - Riemannian and Spin - of Noncommutative Geometry. This is a
relatively new field of mathematics, whose sources from physics are quantum mechanics, gauge
theory and general relativity. It is encoded in terms of spectral triple and its main ingredient,
the Dirac operator on Hilbert space of spinor fields.
The canonical spectral triple on a Riemannian 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.
In the second 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, behind the Standard model of elementary
particles.
A suitable selection among the wealth of available material hopefully will lead the students
to some of the active and interesting fields of current research.