In analysis we tend to focus on the "small scale" structure of a space. For example, both derivatives and continuity only depend on a very small neighborhood around a point. Coarse geometry on the other hand focuses on the "large scale" structure of a space. Coarse spaces generalize metric spaces in a way that provides an appropriate framework to study large-scale geometry. Coarse geometry is used to study: higher index theory, elliptical operators, the coarse Baum-Connes conjecture and as a consiquence the Novikov conjecture.
In this talk we will discuss what a coarse structure is, both in terms of metric spaces and in full generality. Then we will look at a few examples. Next, We will introduce uniform Roe algebras and examine their relationship to coarse structures along with recent advances in solving the rigidity problem. Then, time permitting, we will look at uniform Roe modules.