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This talk will describe some results on the inverse spectral problem on a compact Riemannian manifold (possibly with boundary) which are based on V.Guillemin's strategy of normal forms. It consists of three steps: First, put the wave group into a normal form around each closed geodesic. Second, determine the normal form from the spectrum of the Laplacian. Third, determine the metric from the normal form. We will try to explain all three steps and to illustrate with simple examples such as surfaces of revolution.

Normal form of the wave group and inverse spectral theory