Analytic ODEs in real and complex domain: similarities and differences.
- Background on holomorphic functions. Weierstrass compactness principle.
- (Ordinary) Differential Equations and their solutions.
- Contracting mapping principle (recall).
- Picard integral operator and its contractivity.
- Existence/uniqueness theorem.
- Example: Matrix exponent and its computation.
Holomorphic vector fields and their trajectories. Equivalence of vector fields.Flow box theorem and rectification theorem for nonsingular vector fields.
Attached is Section 1. It will be available on these pages for a limited time and is password-protected from printing 
… I must obey the requirements of the Publisher.
Disclaimer. In full compliance with the strike rules (were it still be underway), this meeting is defined as a research/orientation seminar on a novel teaching technology. 
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.
