Realizing Algebra Structures on Free Resolutions of Grade 3 Perfect Ideals

Perfect ideals $I$ of grade $3$ in a local ring $(R,\mathfrak{m},\mathbb{k})$ can be classified based on multiplicative structures on ${\text{Tor}}_{\bullet }^R(R\mathrm{/}I,\mathbb{k})$. The classification is incomplete in the sense that it remains open which of the possible algebra structures actually occur; this realizability question was formally posed by Avramov in 2012. Of five classes of algebra structures, the realizability question has been answered for one class. In this work, we answer the realizability question for two more classes and contribute towards an answer for a third.