In 1992 Lascar proved that the group of field automorphisms of the complex numbers which fix pointwise the algebraic closure of the rationals is simple, assuming the continuum hypothesis. His proof used strongly the topological features of the group of automorphisms of a countable structure, as a Polish group.
In 1997 Lascar gave a different proof of the above, without assuming the continuum hypothesis. In a recent preprint with T. Blossier, Z. Chatzidakis and C. Hardouin, we have adapted a proof of Lascar to show that certain groups of automorphisms of various theories of fields with operators are simple. It particularly it applies to uncountable (universal) differentially closed fields as well as to certain existentially closed fields equipped with an automorphism.
