The minicourse is planned to provide the audience with the basic notions of real and complex analytic vector fields. We begin by describing different notions of normal forms for elementary singularities of vector fields and desingularization (blow-up) for non-elementary planar singularities. The minicourse will culminate with the proof of the finiteness theorem for limit cycles of polynomial vector fields having only nondegenerate singularities.

The material is covered in the recent textbook: Ilyashenko, Yakovenko, Lectures on Analytic Differential Equations (AMS 2007), Chapter I and Chapter IV (sect. 24).

The draft version of this textbook is available online at http://www.wisdom.weizmann.ac.il/~yakov/thebook1.pdf

Several first separate sections of the final version can be found on the blog http://yakovenko.wordpress.com/ (more specifically, at http://yakovenko.wordpress.com/category/lecture/ )