Uniqueness of Minimal Acyclic Complexes over Monomial Algebras

In this talk we discuss conditions for uniqueness among minimal acyclic complexes of finitely generated free modules over a commutative local ring.  Although uniqueness exists over Gorenstein rings, the question has been asked whether two minimal acyclic complexes in general can be isomorphic to the left and non-isomorphic to the right.  We answer the question for certain cases, including periodic complexes, sesqui-acyclic complexes, rings with $\m^3=0$, and monomial algebras.  Additionally, we look at the classification of, and the effects of a Conca generator on monomial algebras.