## Introduction to noncommutative geometry

### Prof. Ludwik Dabrowski

Number of cycles: 2

Second semester

These lectures concentrate on the latest layer of Noncommutative Geometry (NCG): Riemannian and Spin. It is encoded in terms of the Dirac-type operator, and more precisely in terms of `spectral triple'. 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 additional requirements that completely characterize this operator will be presented. They are essential for a further fascinating generalization to noncommutative spaces by A. Connes. In the course of the account various previous levels of NCG will be mentioned, and among them (differential) topology and calculus; including 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, pseudodifferential calculus and local index formula.

In the second part, which can possibly be organized in form of a `running seminar', the concept of symmetries will be described in terms of Hopf algebras and quantum groups and applied to (equivariant) spectral triples. Moreover notions of noncommutative principal bundles and connections with a suitable differential calculus will be discussed.

By concentrating the material in relatively few lectures some well established topics (e.g. the index theory) will not be discussed. Just an indispensable minimum from the well known theory of the (elliptic) Laplace operator will be used. Such a selection among the wealth of available material hopefully will lead us fast to some of the active and interesting fields of current research.

### Plan:

- Introduction.
- Exterior and Clifford algebras. Spin groups. Spinors.
- Spin structures.
- Dirac operator.
- Some analytic properties. Spectral triple.
- Other (seven) characteristic features:
- dimension (finite summability)
- regularity (smoothness)
- finiteness & projectivity
- reality
- first order
- orientation
- Poincaré duality

- Statement of the `reconstruction theorem' of A. Connes.
- Group actions (isometries, diffeomorphisms)
- Quantum symmetries: Hopf algebras and equivariant spectral triples.
- Other notions: principal bundles, connections, pseudodifferential calculus, geodesic flows...
- NCG Examples: noncommutative tori, spheres, finite dimensional spectral triple.