Abstract : First we explain how recent results on the asymptotic solutions of Hitchin's equations on a curve allow one to prove Simpson's Geometric P = W conjecture in the Painlevé 6 case. In the second part of the talk we outline the main ideas of our ongoing work on the extension of this result to the Garnier case with 5 parabolic points.

Abstract: We study connection problems of q-Painlevé equations. This problem is divided Into three parts:

1. Study q-connection problems on hypergeometric equations

2. Study q-connection problems on linearized equations of the Painlevé equations

3. Consider q-connection problems on the Painlevé equations

Abstract : In this talk, I will discuss the quantisation of the Painlevé monodromy manifolds as a special class of quantum del Pezzo surfaces. In particular I will introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.

Abstract : The weight of a differential equation is defined through the Newton diagram of the equation. It gives the weighted projective space, that is the natural compactification of the phase space of a differential equation. In this talk, I show how to construct a weighted projective space, analysis of the Painlevé equation using the geometry of a weighted projective space, and related topics.