Introduction to Noncommutative Riemaniann Spin Geometry

Prof. Ludwik Dabrowski

These lectures concentrate on the latest layer of Noncommutative Geometry: Riemannian and Spin. It is encoded in terms of a 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 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 some previous levels of NCG will be mentioned regarding the (differential) topology and calculus (C*-algebras, Hochschild and cyclic cohomology, noncommutative integral, pseudodifferential calculus and local index formula).

By concentrating the material in relatively few lectures some well established topics (e.g. the index theory) will be necessarily omitted. 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.

  1. Introduction.
  2. Exterior and Clifford algebras. Spin groups. Spinors.
  3. Spin structures.
  4. Dirac operator.
  5. Some analytic properties. Spectral triple.
  6. Other (seven) characteristic features:
    • dimension (finite summability)
    • regularity (smoothness)
    • finiteness & projectivity
    • reality
    • first order
    • orientation
    • Poincare duality
  7. Statement of the 'reconstruction theorem' of A. Connes.
  8. Other notions in N.C. vein: pseudodifferential calculus, geodesic flows.
  9. Examples (time permitting): quantum tori, spheres...