## Graduate Student Organization Seminar

**Date:** November 13, 2019

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 628

**Speaker:** Diego Martinez

**Title:** *Coarse Geometry and Inverse Semigroups*

**Abstract:** In this talk we will study two seemingly disconnected notions: coarse geometry and inverse semigroups. Geometry often studies certain objects (such as sets or manifolds) equipped with a distance function. For instance, one classical problem would be to classify every compact manifold up to diffeomorphism. Coarse geometry shifts the point of view, and defines two sets to be coarse equivalent if they look the same from far away. In this way, for instance, a point and a sphere are indistinguishable from each other. Coarse geometry then studies properties that remain invariant under this weak equivalence relation, that is, properties of the space that only appear at infinity. On the other hand, an inverse semigroup is a natural generalization of the notion of group, and is closely related to the idea of groupoid. Starting with one of these objects we will introduce how to construct a metric space, in the same fashion as the Cayley graph construction in the context of groups. We will then study its coarse structure, in particular its property A and its amenability. Time permitting, we will also relate these properties to analogue properties in some operator algebras.