- 97-62 Giovanni Landi
- An introduction to noncommutative spaces and their geometries
(442K, LaTeX, 181 + iv pages, 26 figs in the source file)
Feb 11, 97
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Abstract. These lectures notes are an introduction to several ideas
and applications of noncommutative geometry. We feel that
the necessary mathematical tools are presented in an accessible way.
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
Appendices: Basic Notions of Topology. The Gel'fand-Naimark-Segal
Construction. Hilbert Modules. Strong Morita Equivalence. Partially
Ordered Sets. Pseudodifferential Operators