Topological recursion, isomonodromic systems and WKB

Among other applications, topological recursion is believed to provide a way to compute the coefficients of the WKB expansion of formal solutions of some linear differential systems depending on a small parameter $\hbar$. If the characterization of such systems is a hard problem, isomonodromic systems are conjectured to provide examples of such systems. In this talk, I will report on a work in progress aiming at building such a $\hbar$-deformed system from any isomonodromic system by using the underlying Poisson structure, giving partial results in the $sl_2$ case. Based on a joint work with Olivier Marchal