S^7 : Statistics and Stochastics for Sheaves, Shapes, Stratified and Sub-Riemannian Structures

Abstract

The mathematical description of shapes as elements of non-linear geometric spaces has been central to the field of geometric statistics and has spawned a variety of new links to other fields of mathematics: Geometric data requires models for stochastic processes in geometric spaces, for example, for data in structural and evolutionary biology, in biomedical imaging, and in morphology such as plant sciences, and non-holonomic constraints lead to sub-Riemannian geometric structures. The prevalence of shape data has created a need for statistical inference tools to draw conclusions from observed shapes in complex, non-linear settings, and new numerical techniques now allow for studying geometries experimentally in novel ways, for example, using neural networks to allow simulation of diffusion bridges in sub-Riemannian spaces. The ability to model local information with sheaves gives the ability to analyze shapes with neural networks.