Geometry of reflecting rays and inverse spectral problems

Part 1 Preliminaries from differential topology and microlocal analysis: jets and transversality theorems generalized bicharacteristics wave front sets of distributions. Part 2 Reflecting rays: billiard ball map periodic rays for several convex bodies Poincare map scattering rays examples. Part 3 Generic properties of reflecting rays: generic properties and smooth embeddings elementary generic properties absence of tangent segments non-degeneracy of reflecting rays. Part 4 Bumpy metrics: Poincare map for closed geodesics local perturbations of smooth surfaces non-degeneracy and transversality global perturbations of smooth surfaces. Part 5 Poisson relation for manifolds with boundary: Poisson relation for convex domains Poisson relation for arbitrary domains. Part 6 Poisson summation formula for manifolds with boundary: global parametrix for mixed problem Poisson summation formula. Part 7 Inverse spectral results for generic bounded domains: planar domains interpolating Hamiltonians approximations of closed geodesics by periodic reflecting rays Poisson relation for generic strictly convex domains. Part 8 Poisson relation for the scattering kernel: representation of the scattering kernel Poisson relation for the scattering kernel. Part 9 Singularities of the scattering kernel for generic domains. Part 10 Scattering invariants for several strictly convex domains: hyperbolicity of scattering trajectories existence of scattering rays and asymptotic of their sojourn times asymptotic of the coefficients of the main singularity.