Abstract: This is a continuation of our discussion of the connectedness of $Symp(CP^2)$ an other symplectomorphism groups.

Abstract: Differential-algebraic geometry is the expansion of algebraic geometry where in addition to polynomial equations one considers (partial) algebraic differential equations. Similarly, using automorphisms instead of derivations, we have difference-algebraic geometry. In this talk I will give an introduction to these subjects and to describe recent work with Tom Scanlon in which we develop a new foundation that unifies and generalises them. Our approach is informed by the theory of iterative Hasse-Schmidt derivations and is based upon an abstract notion of "prolongation" for an algebraic variety.