Maximal orders on algebraic surfaces

Abstract: 

Maximal orders over discrete valuation rings were classically studied by many authors. In the more recent times, Michael Artin started the local study of maximal orders over two dimensional rings. Following his vision, Daniel Chan, Colin Ingalls and I have embarked on the study of maximal orders on projective surfaces. The classification of these orders up to the so called ramification data is now complete. We will review these developments of the last decade. Most of the talk will be a leisurely walk through basic ideas in the subject.