Background from geometry and algebra: manifolds, Riemannian metric, geodesics, exponential and inverse exponential map, Karcher/Frechet means; functional spaces of interest: set of probability density functions (pdfs), set of warping functions, group, group actions, quotient spaces.Phase variability and the registration problem; L2 metric and pinching problem; dynamic programming algorithm; penalized L2 metric.Invariance, square-root velocity function (SRVF); Fisher-Rao Riemannian metric; change of variables; registration using SRVFs; exact solution and approximation; quotient space metric, amplitude and phase distances.