There are several issues in relating to cosmology that I would like to pursue in connection with this project. One concerns the physical basis for the inflationary behavior of the universe that has motivated the reintroduction of a cosmological constant into the Einstein Field Equation. Another project, somewhat related to the first, is the construction of discrete models of geometry of the universe that can be coarse-grained to yield recognizably Relativistic space-times. This is an application of the new approach to topology that I have been developing for the past few years. One simple toy model that I have been investigating yields an exponentially inflating three-dimensional space-time, and the inflation can be traced to rather basic features of the rules for constructing the discrete geometry. I would like to investigate how both the supposed exponential inflation immediately after the Big Bang and the more gentle inflation now being observed might be incorporated into such a model. I hope that this might illustrate how a fundamentally discrete geometry might shed light on cosmological structure.