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