Some Problems of Uniqueness for the Ricci Flow

Many of the fundamental structure-preserving properties of the Ricci flow and other geometric PDE are underwritten by some principle of uniqueness, either forward or backward in time. We will discuss some of these principles in connection with a recent project related to the backward propagation of warped-product structures, as well as some of the analytic issues which arise from the gauge degeneracy of the equation. Time-permitting, we will discuss some applications of these results to the analysis of the singular behavior of the flow.