Algebraic Geometry Seminar: How should maximal orders move?

I will discuss joint work in progress with Rajesh Kulkarni on the moduli of maximal orders on surfaces. In contrast to the "classical" case of Azumaya algebras, ramified maximal orders have several potentially interesting moduli spaces. I will discuss three different scheme structures on the same set of points: a naive structure, a structure arising from a non-commutative version of Koll\'ar's condition on moduli of stable surfaces, and a structure that comes from hidden Azumaya algebras on stacky models of the underlying surface. Only (?) the third admits a natural compactification carrying a virtual fundamental class, giving rise to potentially new numerical invariants of division algebras over function fields of surfaces.