Part I: Higgs bundles and the Non Abelian Hodge correspondence

  1. Riemann surfaces and holomorphic bundles
  2. Two classical theorems
  3. Higher rank holomorphic bundles
  4. Harmonic mappings
  5. Non abelian Hodge correspondence
  6. The Hitchin fibration and its fibers
  7. The Hitchin section Theorem
  8. Symmetric Spaces
  9. Non positively curved manifolds
  10. The heat flow method and harmonic mappings
  11. Existence Theorem for the solutions of Hitchin Self Duality equations

Part II: Anosov representations and positivity

  1. Background on hyperbolic surfaces
  2. Contracting maps and Anosov bundles
  3. Anosov representations: definition and limit maps
  4. Orbit equivalence, reparametrisation and periods
  5. The geodesic flow of an Anosov representation, with applications