On Uses for and Constructions of Differential Graded Algebras

This thesis is comprised of two projects (in reverse chronological order) with the unifying theme of differential graded (dg) algebras. The first is a collaborative effort with Hugh Geller and Desiree Martin, in which we completely classify the trees and cycles whose edge ideals are minimally resolved by dg algebras. To do this, we pull from discrete Morse theory and homological algebra to give descriptions of the dg algebra structures at play. We also use a "pruning" technique due to Boocher to give a combinatorial obstruction to the existence of dg algebra structure on minimal free resolutions of edge ideals of graphs containing certain forbidden induced subgraphs. The second project is an individual work, where I use dg algebra structure to construct the minimal free resolution of the module of derivations $\Der_{R \mid \Bbbk}$ on an $n \times (n+1)$ determinantal ring $R$. I follow a method of Iyengar, which requires the construction of explicit dg algebra and dg module structures. The result of this construction is a relative bar resolution, and the structures and maps involved show that this resolution is minimal, i.e., $\Der_{R \mid \Bbbk}$ is a Golod $R$-module.