We begin by discussing the Painlevé VI equation, a nonlinear second order ordinary differential equation discovered by R. Fuchs (1906), which has several beautiful properties and applications. We describe how the classification of its algebraic solutions, completed by Lisovyy-Tykhyy (2008), connects to mapping class group dynamics of the four punctured sphere on certain moduli spaces. Time permitting, we present an analogous classification in mapping class group dynamics for higher genus surfaces (which turns out to be much simpler), and discuss open conjectures.