Workshop on Topology: Identifying Order in Complex Systems

The problem of comparing shapes turns up in different guises in numerous fields. I will discuss a new metric on the space of smooth Riemannian 2-spheres that is well suited for comparing geometric similarity. The metric is based on a distortion energy defined on the space of diffeomorphisms between a pair of genus-zero surfaces. I’ll also discuss a related idea based on hyperbolic orbifold structures on a surface. I will also present results of experiments on applying these ideas to biological data. This involves comparing the similarity of collections of teeth, bones, proteins and brain cortices. This is joint work with Patrice Koehl. The mathematical theory can be found in Hass J and Koehl P. A metric for genus-zero surfaces, (http://arxiv.org/abs/1507.00798) Some biological applications are in Koehl P and Hass J. Landmark-free geometric methods in biological shape analysis, J. R. Soc. Interface 12: 2015.