I'll describe some concrete questions of linear algebra arising from the dynamics of mapping class group actions on character varieties. Variants of these questions go back to work of Painlevé, Fuchs, Gambier, and Schlesinger at the beginning of the 20th century, but their (partial) answers depend on modern techniques from algebraic and arithmetic geometry. I'll describe how this work resolves (arithmetic) conjectures of Esnault-Kerz, Budur-Wang, Kisin, and Whang, and makes progress towards a (topological) conjecture of Putman-Wieland. Everything is joint with Aaron Landesman.