The classical McKay correspondence gives the close connection between the minimal resolution of rational double points and the Dynkin diagram of type ADE. Also, the relationships between these objects and some non-commutative algebras had been discovered in the last few decades. For example, it is known that the minimal resolution of a rational double point is derived equivalent to a certain non-commutative algebra which is Morita equivalent to the preprojective algebra of the extended Dynkin quiver of type ADE. Furthermore, the stable category of Z-graded maximal Cohen-Macaulay modules over a rational double point is equivalent to the bounded derived category of finitely generated modules over the path algebra of the Dynkin quiver of type ADE.
In my talk, I will explain about some analogues of the above correspondences for three dimensional Gorenstein toric singularities. The main ingredients are geometric and non-commutative crepant resolutions of this singularity arising from dimer models.