#
ELEMENTS OF GEOMETRY (CLASSICAL AND QUANTUM)

## by L. Dabrowski, SISSA

This introductive course (20 hours ca.) is meant as a
`piecewise continuous' journey through few selected aspects
of geometry of a `classical' space *X* in terms of
(commutative) algebra of functions *A* =
*F(X)*. In particular, Riemannian structure will be
encoded by means of Dirac operator. Possible
generalizations to noncommutative algebra *A* will be briefly
indicated.

Proposed topics:

- Clifford algebras, spinorial and orthogonal groups
- Spinors, charge conjugation
- Spin structures and spinor fields
- Dirac operator
- Connes `reconstruction' theorem
- quantum
- groups (Hopf algebras)
- topology (C*-algebras)
- vector bundles (projective modules)
- principal bundles (Hopf-Galois extensions)
- differential calculus, connections and Dirac operator

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