Further thoughts on transseries and valued differential fields
This concerns work with Aschenbrenner and van der Hoeven
related to but not included in our book, open problems, and recent results
by my students Camacho, Gehret, and Hakobyan.
In particular, I will discuss a natural notion of
dimension for definable sets in the differential field $\mathbb{T}$, with dimension $0$ of special
interest. I will also describe the group of strong automorphisms of $\mathbb{T}$.
Camacho shows the preservation of truncation closedness
in fields of transseries under natural ``differential''
operations like taking the Liouville closure. Gehret aims to do for
the model theory of the differential field $\mathbb{T}_{\log}$ of
logarithmic transseries what we did for $\mathbb{T}$. Hakobyan has generalized Scanlon's thesis on
differential-henselian valued fields with many constants
by removing the assumption of having many constants.