The goal of this project is to create a mathematical theory of how to recognize and how to align geometric shapes, and developing algorithms to implement it. This is a major research challenge with far reaching implications. As we interact with our environment we constantly compare the shapes of objects. When we scan the outside of an object, with our eyes or with a laser scanner, we are analyzing the shape of a 2-dimensional surface. The theory of conformal, harmonic and isometric mappings, which has played a central role in geometry and analysis over the last century, are now finding applications in shape matching. However, the computational and algorithmic aspects of the classical theory is still in infancy. Developing a computable discrete theory for conformal, harmonic and isometric mappings and fast algorithms are major challenging mathematical problems. This project explores fundamental issues in this exciting area, which is on the cusp of seeing major innovations.