Model theory, algebra and differential equations

In this thesis, we applied ideas and techniques from model theory, to study the structure of the sets of solutions XII - XV I , in a differentially closed field, of the Painlevé equations. First we show that the generic XII - XV I , that is those with parameters in general positions, are strongly minimal and geometrically trivial. Then, we prove that the generic XII , XIV and XV are strictly disintegrated and that the generic XIII and XV I are ω-categorical.
These results, already known for XI , are the culmination of the work started
by P. Painlevé (over 100 years ago), the Japanese school and many others on
transcendence and the Painlevé equations. We also look at the non generic
second Painlevé equations and show that all the strongly minimal ones are
geometrically trivial.