  These lectures notes are an intoduction for physicists to several ideas and
applications of noncommutative geometry. The necessary mathematical tools are
presented in a way which we feel should be accessible to physicists. We
illustrate applications to Yang-Mills, fermionic and gravity models, notably we
describe the spectral action recently introduced by Chamseddine and Connes. We
also present an introduction to recent work on noncommutative lattices. The
latter have been used to construct topologically nontrivial quantum mechanical
and field theory models, in particular alternative models of lattice gauge
theory.
  Here is the list of sections:
  1. Introduction. 2. Noncommutative Spaces and Algebras of Functions. 3.
Noncommutative Lattices. 4. Modules as Bundles. 5. The Spectral Calculus. 6.
Noncommutative Differential Forms. 7. Connections on Modules. 8. Field Theories
on Modules. 9. Gravity Models. 10. Quantum Mechanical Models on Noncommutative
Lattices.
  Appendices: Basic Notions of Topology. The Gel'fand-Naimark-Segal
Construction. Hilbert Modules. Strong Morita Equivalence. Partially Ordered
Sets. Pseudodifferential Operators
