Time-invariant surfaces in evolution equations

A time-invariant level surface is a (codimension one)

spatial surface on which, for every fixed time, the solution of an

evolution equation equals a constant (depending on the time). A

relevant and motivating case is that of the heat equation. The

occurrence of one or more time-invariant surfaces forces the solution

to have a certain degree of symmetry. In my talk, I shall present a

set of results on this theme and sketch the main ideas involved, that

intertwine a wide variety of old and new analytical and geometrical

techniques.