Normal forms, integrable systems, and quantum periods

A theorem of Sjöstrand says that, under certain conditions, one can compute the asymptotic expansions of the eigenvalues of a Schrödinger operator with polynomial potential using the quantum analogue of the Birkhoff normal form. In my talk, I will give a brief introduction to both the classical and quantum Birkhoff normal forms, then explain how each may be interpreted geometrically, in terms of an equivalence on the germs of a complex integrable system or (for the quantum version) its quantization. In the latter case, this gives rise to "quantum periods", which may be understood either as formal deformations of variations of Hodge structures or as functionals on Hochschild homology. This is joint work in progress with Maxim Kontsevich.