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:

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

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