# Sergei Yakovenko's blog: on Math and Teaching

## Wednesday, October 31, 2007

### “Auxiliary Lesson” שעור עזר) #2) November 1, 2007

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 10:07

The meeting will again be devoted to explanation of background material for the course to be taught after the strike. The supplementary material covered on this meeting includes:

1. Autonomous differential equations and (holomorphic) vector fields. Flow of vector field.
2. Algebraic description of vector fields as derivations of the algebra oh holomorphic functions.  One-parametric groups of automorphisms.
3. Rectification of vector fields near nonsingular points. Singularities.
4. Localization and germs.
5. Holomorphic foliations: definition and examples.
6. Holonomy of foliations.
7. Inverse problem: construction of a foliation with the specified holonomy. Suspension.

## Monday, October 22, 2007

### “Auxiliary Lesson” #1 שעור עזר (Oct. 25, 2007)

Analytic ODEs in real and complex domain: similarities and differences.

1. Background on holomorphic functions. Weierstrass compactness principle.
2. (Ordinary) Differential Equations and their solutions.
3. Contracting mapping principle (recall).
4. Picard integral operator and its contractivity.
5. Existence/uniqueness theorem.
6. Example: Matrix exponent and its computation.
7. Holomorphic vector fields and their trajectories. Equivalence of vector fields.
8. 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.

### (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

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

### Blogging rules.

For those not familiar with the blogging subculture (are there such people?): you are welcome to leave your comments/questions/remarks next to the relevant posts. Please introduce yourself when commenting. My wet dream is mathematical discussions between the students attending the course on these pages. Don’t be afraid to express yourself and teach others. I promise not to abuse my rights as a moderator here.

Any language is accepted, though I strongly urge to write in the l.c.d. (= Simple English).

## Thursday, October 18, 2007

### Good textbook on holomorphic functions in several complex variables

Filed under: books — Sergei Yakovenko @ 7:55

Functions of Several Complex Variables and Their Singularities
Wolfgang Ebeling, Leibniz Universität Hannover, Germany

“Poor man’s Arnold-Gussein-Zade-Varchenko“. Easy to read, contains quick introduction into different areas (integral representations, monodromy, analytic geometry etc.). Highly recommended for a geometrically/analytically minded reader.

Link to the AMS Bookstore page for this book.

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