Motivated by recent work of Mok and others on local holomorphic
isometric embeddings, we consider local conformal maps from an irreducible
Hermitian symmetric space of compact type into the products of such
manifolds. We allow the conformal factors to be arbitrary real numbers and
derive a necessary and sufficient condition for the global rigidity to
hold. This is a joint work with Yuan Yuan.

We show that there is a countable structure M such that for any set X, X computes a copy o M if and only if X is not hyperarithmetic. This gives a strong generalisation of the Slaman-Wehner theorem to the hyperdegrees. On the other hand, the generalization of the Slaman-Wehner theorem to the degrees of constructibility is false.
This is related to work of Kalimullin and his co-authors on structures whose degree spectrum has full measure. We show, for example, that there are only countably many such structures.
We also touch on the possible form of structures M as above. For example, they can be linear orderings, but not have uncountably categorical theories. Many open questions remain. Among them: can we similarly capture the nonarithmetic degrees?
Joint work with Antonio Montalban and Ted Slaman.