This paper concerns spectral invariants of the Laplacian on a compact
Riemannian manifold (M,g) known as wave invariants. If U(t) denotes the wave
group of (M,g), then the trace Tr U(t) is singular when t = 0 or when ti is the
length of a closed geodesic. It has a special type of singularity expansion at
each length and the coefficients are known as the wave invariants. Our main
purpose is to calculate the wave invariants explicitly in terms of curvature,
Jacobi fields etc. when the closed geodesic is non-degenerate elliptic. We do
this by putting the Laplacian into quantum Birkhoff normal form at the closed
geodesic. Such a normal form was previously introduced by V. Guillemin. We give
a new algorithm for calculating it, and for expressing wave invariants in terms
of normal form invariants.