Let (K,<,d) be an ordered diffenrential field, and v the natural valuation. We assume that d is compatible with v, i.e. that v is a differential valuation in the sense of M. Rosenlicht. Denote by k the residue field and by (G,Ψ) the induced asymptotic couple; i.e. G = v(K) is the value group endowed with the map Ψ(v(a)) := v(a′: =a).

The purpose of this workshop is to study a differential Kaplansky theory in this setting. We want to achieve progress on the following problem: Find necessary and su±cient conditions on (K; <, d) so that: (i) the data (G, Ψ) allows to define a derivation d on the field of generalized series k((G)); (ii) the induced asymptotic couple is precisely (G,Ψ); (iii) there is an order preserving di®erential embedding of (K,<,d) in (k((G)),<,d); (iv) the embedding may be chosen to be truncation closed; i.e. the image of the embedding is closed under the operation of taking initial segments of series. Partial progress has been achieved on this topic, for example regarding item (i), we have described the construction of "well-defined" derivations on k((G)). Regarding item (iii), J.M.Aroca and J. Del Blanco have considered the case of archimedean value group. Other approaches to this problem are described in the works of M. Aschenbrenner - L. v. D. Dries on H - fields, and the works of J. v. D. Hoeven on Transseries.

J. Del Blanco Marana, Lou van den Drie