PBW deformations and quiver GIT for noncommutative resolutions

In this thesis we first investigate PBW deformations of Koszul, Calabi-Yau algebras, and we then study moduli spaces of representations of algebras defined by tilting bundles. These classes of algebras are generalisations of the skew group algebras appearing as noncommutative resolutions in the McKay correspondence, and the results we prove are motivated by corresponding results for skew group algebras. Koszul, Calabi-Yau algebras are Morita equivalent to path algebras with relations defined by superpotentials, and we classify which PBW deformations of these algebras still have relations defined by a superpotential and show that these deformations also retain the Calabi-Yau property. As an application of these results we show that symplectic reflection algebras are Calabi-Yau and can be interpreted as path algebras with relations defined by a superpotential. We then investigate when a variety with a tilting bundle can be produced as a moduli space of representations of an algebra defined by the tilting bundle. We find a set of conditions ensuring a variety and tilting bundle can be reconstructed in such a manner, and we show that these conditions hold in a large class of examples, which includes situations arising in the minimal model Program where the variety may be singular. As an example we show that the minimal resolution of a rational surface singularity can be produced as a moduli space for a noncommutative algebra such that the tautological bundle is a tilting bundle defining the noncommutative algebra