In topological data analysis, the "manifold hypothesis" is that a dataset lying in Euclidean space in fact lies on, or near, some submanifold of that space. The submanifold is understood to provide information about the non-linear relationships between different variables.

The goal of statistical inference is to use some set of sample data to understand aspects of an entire population and express a mathematically justifiable level of confidence in the statements made. In this talk I will introduce geometric inference, which aims to understand the geometry of a submanifold of Euclidean space using a finite sample of points drawn from the submanifold.

I will introduce some of the philosophy and tools of inference which are probably not familiar to most practicing geometers and survey some results in this field. I will have a particular focus on the reach, where I proved new bounds with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar, which Berenfeld and others recently showed were optimal.