(i) Noncommutative algebras and (commutative) algebraic geometry

I will give an introduction to the emerging role which noncommutative algebras play in commutative algebraic geometry.  I will review the Kapranov-Vasserot interpretation of the McKay correspondence in detail, and survey some of the many extensions of this idea which have been made.

(ii) Quivers for flops

Work of Bridgeland, as interpreted by van den Bergh, describes the effect of a flop on the derived category of coherent sheaves, as well as describing a noncommutative algebra supported on the (singular) "blown down flop" whose derived category of modules is isomorphic to the derived category of coherent sheaves on either smooth variety involved in the flop.  One expects that this algebra could be described by means of a quiver with relations, but this does not appear to have been done except in the simplest cases.  We will describe some joint work in progress with Aspinwall to determine quivers for more general flops.