Joel Nagloo
(University of Leeds)
Model Theory and
the Painlevé Equations

The Painlevé equations are nonlinear 2nd order ODE and come in six families P1-P6, where P1 consists of the single equation y'' = 6y2+t, and P2-P6 come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20th century but have arisen in a variety of important physical applications including for example random matrix theory and general relativity.

In this talk I will explain how one can use model theory to study the structure of the sets of solutions of these equations. Indeed, after recalling the basics of the theory of differentially closed fields of characteristic zero, I will explain how the trichotomy theorem, a powerful result proved by Hrushovski and Sokolovic in the early 90's, plays a fundamental role in this work.