The talk starts with a naive question: What is the mirror symmetric dual to the Catalan numbers? Answering this question turns out to be opening a door to an entirely new territory, where the Eynard-Orantin theory interacts, in an essential way, with the theory of Hitchin integrable systems and Hitchin fibrations. I will explain the idea of quantum curves, using an elementary and elegant mathematical example. Then I will present the general principle that Hitchin fibrations are designed to work for the quantization of spectral curves using the Eynard-Orantin theory.

UC Davis