In this first of two talks, we study the collection of maximal orders in the ring of 2X2 matrices (over a local field) by constructing a graph whose vertices are the maximal orders, with edges determined by a notion of distance between maximal orders. We show that this graph is actually a tree. At the end of the talk, we introduce a class of nonmaximal orders called split orders and give a geometric characterization in terms of the tree.

A large part of this first talk is dedicated to defining the objects of study which will include basic definitions such as localization and p-adic numbers for those first-year students who come.