In his 1979 paper, Algebraic aspects of nonlinear differential equations, Yu. Mann describes and critiques three possible languages for their study: Classical, geometric, and differential algebraic, and points out the strengths and challenges of each.

My aim in these talks is to sketch aspects of the theory of linear differential algebraic groups over ordinary as well as partial differential fields that illustrate both the power and computational efficiency of differential algebraic geometry as well as the challenges we face if we wish to carry out what we might call the Manin program.

We work in the spirit of C. Chevalier and A. Borel. So, we begin with some basic notions of the Kolchin topology on affine space over a differentially closed differential field. The Ritt-Kolchin correspondence between radical differential polynomial ideals and Kolchin closed sets, as well as the Ritt basis theorem imply that a linear differential algebraic group can be described as the stabilizer of a line in a Chevalley-type construction. An associated finite set of differential polynomial semi-invariants of the regular representation completely determine the group. (Techniques for computing the semi-invariants need to be developed.)

The morphisms in differential algebraic geometry are differential rational -- instances of the Baecklund transformations whose absence in the early Ritt theory was a central concern of Manin's in his 1979 paper. The power of these transformations for us is best illustrated by the crossed homomorphism from the group to its Lie algebra known as the logarithmic derivative map. This map is the tool used to define the gauge action of the group on its Lie algebra, which is central to the classification of the semisimple differential algebraic groups.
Time permitting, we will give some examples of intriguing differential algebraic group actions on differential algebraic varieties. The examples bring us into algebraic analysis, and exhibit the challenges we face in our attempts to initiate the Manin program.