What is a geometry? The Greeks had an answer, but then the 19th century showed that they didn't have all of them. In this talk we explore a more modern answer to this question---a (G, X)-structure---describe the geometries that arise in 2 and 3 dimensions from this perspective, and show how projective geometry can encompass all of these. In particular, there are paths of projective geometries which exhibit transitions from one geometry to another: we will attempt to explain these geometric transitions. Speaker(s): Feng Zhu (UM)