# Singularity categories via the derived quotient

Given a reasonable commutative ring R and a noncommutative partial resolution A of R, the singularity category of A relative to R is controlled by a connective dg algebra, the derived exceptional locus, which can be obtained as a derived quotient of A. In fact, one can identify the derived exceptional locus as the connective cover of an endomorphism dg algebra of an object in the singularity category of R. When R is a complete local hypersurface, the derived exceptional locus in fact recovers the dg singularity category of R, which - by a result of Hua and Keller - recovers the isomorphism type of R itself. I’ll talk about the above before giving an application: the classification of singular threefold flops via their derived contraction algebras. Derived methods must come into play here since, in the singular setting, the usual contraction algebra does not classify, in contrast to a recent theorem of Jasso, Keller, and Muro in the smooth setting.