Betti numbers of Koszul algebras and codimension two matrix factorizations

This thesis consists of two projects on the structure of free resolutions in commutative algebra. After developing some necessary background, we prove a structure theorem in Chapter 3 for the defining ideals of Koszul almost complete intersections and, in the process, give an affirmative answer for all such rings to a question of Avramov, Conca, and Iyengar about the Betti numbers of Koszul algebras. In Chapter 4, we study the codimension two matrix factorizations of Eisenbud and Peeva. Each matrix factorization compactly encodes the data of a free resolution of its corresponding matrix factorization module. By showing that each matrix factorization also encodes a canonical system of higher homotopies on this free resolution, we are able to construct a functor from codimension two matrix factorizations to the singularity category of the corresponding complete intersection. This represents the first step towards reconciling higher codimension matrix factorizations with known generalizations of a theorem of Buchweitz and Orlov in the hypersurface case