Quantum curves and q-deformed Painlevé equations

In this talk I will first review some aspects of Painlevé equations and their connection to four dimensional gauge theory; then I will generalise this construction to q-difference Painlevé equations and topological string theory. I will show that their tau-functions are Fredholm determinant of operators associated to quantum mirror curves on a corresponding geometry. As a consequence, the zeroes of these tau-functions compute the exact spectrum of the associated quantum integrable systems. I will focus on the particular example of q-Painlevé III_3 which is related to topological string on local P1xP1 and to relativistic Toda system."