Title: Abelian reductions and internality to the constants.

Abstract: I will report on joint work with Remi Jaoui on the structure of finite rank types in differentially closed fields. Given a type p that is internal and weakly orthogonal to the constants, we introduce and study its “abelian reduction”, namely an image q of p such that the binding group of q is the abelian part in the Chevalley decomposition of the binding group of p. I will discuss a number of applications of this new tool, including the following result: If a type over constant parameters is nonorthogonal to the constants then its second Morley power is not weakly orthogonal to the constants.