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.

Abstract: The notion of K-stability has been first defined by differential geometers to characterise the existence of canonical metrics on a polarised variety. In the case of Fano varieties, where K-stability corresponds to the existence of a Kahler-Einstein metric, the purely algebraic study of K-stability now has become an independent subject in higher dimensional algebraic geometry, which attracts lots of recent interests. In our lecture, we will explain abundant recent progress that people have made by using deep machinery such as the minimal model program. We will also discuss some remaining challenging questions.