Urysohn's metric space U is a complete, separable metric space that (a) contains an isometric copy of each finite metric space and (b) is isometrically homogeneous for its finite subspaces. It is easy to see that U is uniquely determined by these properties, and that it contains an isometric copy of each separable metric space.

Let Aut(U) be the group of isometries of U onto itself. Given a separable metric space M, Aut(U) acts naturally on the set of isometric embeddings of M into U (by composition). Condition (b) above is equivalent to saying that when M is finite, this action is transitive. It was shown in the 1950s that transitivity also holds when M is a compact metric space, and some examples of noncompact M were known for which this action is not transitive.

In these seminars we will discuss how model theoretic ideas can be used to obtain new results about these actions.