Apparaît dans la collection : 2025 - T2 - WS1 - Higher rank geometric structures, Higgs bundles and physics

In this talk, I will present how to obtain explicit formulas for the Hamiltonians and Lax matrices arising in isomonodromic deformations of generic rank 2 connections. Then, I will present how to proceed in the reverse way, i.e. how to build formal wave matrices solutions to a Lax system from a classical spectral curve using Topological Recursion. Finally I will discuss on the Painlevé 1 example, how one can upgrade formal power solutions to analytic solution known as tritronquées solutions of Painlevé 1 and how to define exact WKB on the formal wave matrix to formulate a corresponding Riemann-Hilbert problem via the characterization of the Stokes matrices.