1 Mather Sets for Twist Maps and Geodesics on Tori.- 2 Connecting Orbits in Scalar Reaction Diffusion Equations.- 3 Qualitative Theory of Nonlinear Resonance by Averaging and Dynamical Systems Methods.- 4 An Algorithmic Approach for Solving Singularly Perturbed Initial Value Problems.- 5 Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points.