Title: Painlevé VI: the search for canonical representations

Abstract: This talk is intended to be a colloquium style talk that is widely accessible to mathematicians in all fields. In 1902, Painlevé classified second order differential equations whose only movable singularities are poles, thereby obtaining the six Painlevé equations. Algebraic solutions to Painlevé's sixth equation correspond to canonical triples of 2 by 2 complex matrices. One can alternatively view these canonical triples of matrices either as canonical representations of fundamental groups of surfaces or as local systems on certain moduli spaces of curves. In this talk, based on joint work with Josh Lam and Daniel Litt, we will survey what is known about these canonical representations.