Quantized integrable systems, normal forms, and variation of Hodge structures

Abstract: A classical theorem due to Birkhoff states that on a complex symplectic manifold a function near its Morse critical point can be transformed by a formal symplectomorphism into a normal form given by a power series in the pairwise sums of squares of the coordinates. Using a quantum analog of this normal form, one can compute the eigenvalues of the Schrödinger operator, given certain conditions. In my talk, I will explain how to obtain the Birkhoff normal form of a quantum Hamiltonian geometrically, relating it to the quantization of integrable systems and to formal deformations of variations of Hodge structures. This is joint work in progress with Maxim Kontsevich.

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar