(Tentative) Program for Semester I

The following topics will be (hopefully!) discussed in the first semester. Some of them will take more than one lecture, though I will try to keep the break between lectures as logical as possible.

1. Analytic differential equations (introduction).
2. Geometry: Complex phase portraits and Holomorphic foliations.
3. Algebra: Formal series. Derivations, authomorphisms. Exponentiation and formal embedding. 
4. Formal normal form of a vector field at a singular point. Hyperbolic and elementary singularities.
5. Holomorphic (convergent) transformations. Poincare and Siegel domains. Holomoprhic invariant manifolds.
6. Finitely generated groups of conformal germs. Rigidity phenomenon.
7. Local geometric analysis of isolated singularities. Multiplicity and order. Desingularization (blow-up).
8. Desingularization theorem for planar holomorphic vector fields.
9. Linear systems: General facts.
10. Local theory of linear systems. Fuchsian singular points.
11. Global theory of linear systems: Holomorphic vector bundles and meromorphic connexions on these bundles.
12. Riemann–Hilbert problem.