Geometric and Topological Shape Analysis: Investigating and summarising the shape of data

Abstract

Tools from geometry and topology can be applied in a wide variety of settings. In particular, they are adept at exploring and summarising the shape of data. By considering the shape of objects, we can answer questions related to object reconstruction and classification problems. These problems often face difficulty distinguishing signal from noise, which can be overcome using ideas from persistent homology and computational geometry. We first look at an application of geometry and topology to learning the abstract structure of embedded stratified spaces, and then consider an object classification problem relating to human mesenchymal stem cells.