Appears in collection : Workshop on Quantum Geometry

Under certain conditions, it is possible to compute the spectrum of a polynomial differential operator via its Birkhoff normal form. In this talk, I will explain a geometric approach for obtaining the Birkhoff normal form of a quantized Hamiltonian using the variation of Hodge structure for a formal deformation of a complex Lagrangian fibration. This is joint work in progress with Maxim Kontsevich.