Finite atomic lattices and their relationship to resolutions of monomial ideals

This thesis studies monomial ideals and their resolutions by using combinatorial methods. In the study of cellular resolutions of monomial ideals it is often useful to consider the LCM lattice of the given monomial ideal. It has been shown that all finite atomic lattices can realized as the LCM lattice of some monomial ideal, and that the parameter space of these lattices, L (n), is itself a finite atomic lattice. This thesis focuses on exploring this notion that finite atomic lattices are abstract monomial ideals and aims to use the structure of L( n) as a tool to provide new insights into concepts such as deformation of exponents. The main results of this thesis fall into three categories: structural results about L(n), results relating to deformation of exponents, and results relating these constructions to those found in recent work by Floystad.;I also include two appendices describing computer packages written to aid in my research. One is an implementation in Haskell which uses reverse search to enumerate L(n), and the other is a package for Macaulay2 which introduces posets as a new data type.