This is an expository talk on various methodologies to integrate and build geometries as they relate to analytic tasks.
The simplest paradigm is the construction of good metrics on a space given a collection of functions. This requires building a metric on the functions as well as on the space (duality).
Various geometric organization methods are going to be described as well as applications to empirical Riemannian geometry using Deep Neural Nets leading to a discrete version of isometric embeddings.