# Sergei Yakovenko's blog: on Math and Teaching

## Monday, October 22, 2007

### (Tentative) Program for Semester I

Filed under: Analytic ODE course — Sergei Yakovenko @ 11:15
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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.

## Sunday, October 21, 2007

### Experimenting with fashionable gadgets: Blogged Course

Filed under: Analytic ODE course — Sergei Yakovenko @ 11:03
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This blog is essentially created as a companion to the course Analytic and Geometric Theory of Ordinary Differential Equations, to be served at the Weizmann Institute in 2007/8 academic year. Here are a few questions that are usually asked at the beginning.

1. The course is covered by the textbook “Lectures on Analytic Differential Equations” by Ilyashenko and Yakovenko (to arrive at the WIS library shortly). In the meantime the draft version (full of errors and typos) is available.
2. The course will end up with an exam. The exam is take-home, about 10 problems for a couple of weeks.
3. Pre-requisites: basic proficiency in the language of geometry (manifolds, vector fields, differential forms) and functions of (one) complex variable won’t harm, though I will try to make this course as self-consistent as possible. A primitive First Aid is available on functions of several variables and Riemann surfaces. An extensive treatment can be found in the W. Ebeling’s book (highly recommended).
4. There will be no home assignments, however, it is highly advised to look at the Problems and Exercises sections of the textbook, at least for self-control. Questions (and answers) are most welcome on the pages of this blog.

See you on Thursdays at Zyskind Building, Lecture Room 1, between 9:00 and 11:00.

P.S.This hosting (WordPress) allows for easy insertion of LaTeX code, which makes mathematical discussions here especially pleasant: e.g., one of the main heroes of the course will be the equation

$\dot x=P(x,y),\ \dot y=Q(x,y),\ P,Q\in\mathbb C[x,y]$.