Algebraic geometry of quantum curves and classical differential equations

Quantum curves, a notion originated in physics, are conjectured to capture the information of quantum invariants (such as certain Gromov-Witten invariants, Seiberg-Witten invariants, and knot polynomials) in a compact manner. Several rigorous examples have been constructed by mathematicians, including the one for GW invariants of the projective line. A systematic construction of quantum curves has been established (jointly with Olivia Dumitrescu of Hannover) for the spectral curve of a possibly singular SL(2)-Higgs field on a projective algebraic curve, by generalizing the topological recursion of Eynard and Orantin. Classical equations (the Airy, Hermite, and Gauss hyper-geometric equations) are the simplest examples of quantum curves. The talk is aimed at surveying some of the recent mathematical developments on quantum curves.