In Part I we introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining appropriate conventional resolutions using noncommutative coordinate rings, specifically matrix-valued functions, and we verify this conjecture in a number of examples. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, nonconventional resolutions that are hidden if only commutative functions are considered. Geometrically, these non-conventional resolutions result from shrinking exceptional loci to stack-like (non-Azumaya) points.

In Part II we consider superpotential algebras that are square, that is, the quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares (examples are provided by brane tilings in physics). Such an embedding reveals much of the algebras representation theory through a device we introduce called an impression. Using impressions, we classify all simple representations of maximal k-dimension of all homogeneous square superpotential algebras. Moreover, we show that the localization of each algebra is a noncommutative crepant resolution with a 3 dimensional normal Gorenstein center, and hence a local Calabi-Yau algebra. Another special property of these algebras, equipped with an impression, is that crystal melting (a type of stability change) and quiver mutation may be regarded as a single operation.

A particular class of square superpotential algebras, the Y ^p,q algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and we make the proposal that each \stack-like" maximal ideal sitting over the singular locus is the exceptional locus of a blowup of the center shrunk to zero size.