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.