Geometry of Painlevé equations
Résumé
The goal of the course is to provide a description of the foliation associated to the Painlevé VI equation. Painlevé equations form 6 families of differential equations, the first 5 arising from degenerescence of the 6th one, with 4 parameters. They were dis- covered by Painlevé and his students as non linear second order ODE having the Painlevé property: solutions are well-behaved with respect to analytic continuation. These differential equations, or their solutions, are now used in many areas of mathematics and physics. Our goal is to focus on the 6th family and explain how this family arise as isomonodromy equation, and then provide a description of its phase portrait in the 3-space, its semi-compactification, its monodromy, and how this has been used to classify algebraic solutions.