XXVI International Fall Workshop on Geometry and Physics

Braga, 4-7 September 2017

Painlevé equations are ubiquitous in both Mathematics and Physics and there are still many unanswered questions about them. For this talk we shall focus on the computation of the Galois-Malgrange pseudo-group. Roughly speaking the Galois-Malgrange pseudo-group is "large" (maximal) provided that the equation has a sufficiently complicated dynamical behavior and this principle has enabled Cantat and Loray to compute these groups in the case of P6. Their method however cannot be applied to equations such as P1 and P2 since the dynamics associated with these equations are trivial: however the Galois-Malgrange pseudo-group can still be maximal as it also has an algebraic nature, and this is actually confirmed in the case of P1 by a result of Casale which relies on algebraic techniques.

The purpose of this talk is to describe a general approach to this questions that relies on making sense of a certain "dynamics of infinity" which is non-trivial for all Painlevé equations and explains how this dynamics captures the algebraic character of the Galois-Malgrange group and enables its computation. If time permits, we might also indicate how these ideas also yield new asymptotic estimates for the solutions of P1 and P2 in particular.

Click here to access the slides.