Luo, Xianghui, 1983-2011-09-012011-09-012011-06https://hdl.handle.net/1794/11543ix, 111 p.This dissertation contains the results obtained from a study of two subjects in mathematical general relativity. The first part of this dissertation is about the existence of Killing symmetries in spacetimes containing a compact Cauchy horizon. We prove the existence of a nontrivial Killing symmetry in a large class of analytic cosmological spacetimes with a compact Cauchy horizon for any spacetime dimension. In doing so, we also remove the restrictive analyticity condition and obtain a generalization to the smooth case. The second part of the dissertation presents our results on the global stability problem for a class of cosmological models. We investigate the power law inflating cosmological models in the presence of electromagnetic fields. A stability result for such cosmological spacetimes is proved. This dissertation includes unpublished co-authored material.en-USTheoretical physicsMathematicsApplied mathematicsCauchy horizonCosmologyGeneral relativityGlobal stabilityHyperbolic partial differential equationsMathematical relativityThesis