Precompactness of the Ricci flow and a maximum principle on combinatorial Yamabe flow

In the first part we look at sequences of solutions to the Ricci flow. Richard Hamilton has proven a compactness theorem for sequences of solutions to the Ricci flow with bounded curvature and injectivity radius. We use metric space geometry, especially the compactness theorem of Gromov and Fukaya's study of local geometry of collapsing limits, to study the case when the injectivity radius is not bounded. We then prove a precompactness theorem and study the limit space. In the second part we define a combinatorial version of the Yamabe How. We then explore the notion of discrete parabolic equations and show that the combinatorial Yamabe flow is almost parabolic for a certain class of simplicial complexes.