Invited Lectures ; Hubert L. Bray ; http://www.math.duke.edu/~bray/Presentations/2013_SCGAS.pdf ; More than 95% of the present day curvature of the universe is not a result of regular baryonic matter represented by the periodic table of elements. About 73% is well described by a geometrically natural cosmological constant, also referred to as dark energy, which results in a very small amount of curvature uniformly spread throughout the universe. We will explore the possibility that the remaining 23%, commonly referred to as dark mater, could also be explained very naturally from a geometric point of view. We propose geometric axioms which result in the Einstein-Klein-Gordon equations and model dark matter with the resulting scalar field. We will present simulations and models of the resulting ``wave dark matter'' theory (aka scalar field dark matter and boson stars) in dark matter dominated systems including spiral galaxies, elliptical galaxies, and dwarf spheroidal galaxies. We will compare these predictions to the observed data and photos of actual galaxies.