Abstract: 

One of the most basic questions one can ask of any evolutionary differential equation is that of uniqueness: whether two solutions which agree at some time must agree at all times. This question can be quite subtle, however, especially when the domain under consideration is noncompact. For the heat equation on Euclidean space, the answer is conditional. A famous example of Tychonoff demonstrates that there are smooth solutions to the heat equation on Euclidean space which vanish identically at t=0 but are nontrivial at all other times. However, uniqueness can be assured assuming, for example, that the solution is either bounded below or does not grow too quickly at infinity in either direction.

The question of uniqueness has added significance in the study of geometric heat flows such as the Ricci flow and mean curvature flow, where (in various guises) it governs the propagation of symmetry and other geometric structure along the flow. On noncompact spaces, however, solutions to these (nonlinear) equations may be influenced by their geometry at infinity in ways that the usual analogy with the linear heat equation fails to predict, and many basic questions regarding uniqueness and preservation of structure for such solutions remain open. In this talk, I will give an overview of some uniqueness results for these equations, highlighting some of the challenges in the geometric setting, and will discuss what is currently known and unknown about the uniqueness and propagation of structure along general complete solutions of potentially unbounded curvature.