Large Scale Geometry

Large-scale geometry, also called coarse geometry, is often non-formally described as the study of geometric objects “seen from afar”. In particular, local properties, also called “small-scale” properties, are not perceived by this notion of geometry.

In general, the objects of interest in this area are metric spaces. But we can abstract the key features of metric spaces that allow us to talk about bounded sets and thus obtain what we call coarse spaces. In this context, the morphisms of interest are the so-called coarse maps; which give rise to coarse equivalences/embeddings.

In this course, I plan to cover the basics of coarse spaces with a special emphasis on the metric case. Course topics will include asymptotic dimension, property A, expanding graphs, uniform Roe algebras, the rigidity problem for uniform Roe algebras and their relation with the algebras of quasi-local operators. As time permits, more topics will be added.

References:
1) Class notes.
2) Lectures in Coarse Geometry, John Roe
3) Large Scale Geometry, Piotr Nowak e Guoliang Yu