The work on Workpackage 1 was influenced by refutation of the main conjectures by P. Simon. The main results on this Workpackage were obtained by Dobrowolski in collaboration with F.-V. Kuhlamann on generalized IFS(iterated function system)-attractors (which was a continuation of an earlier collaboration). Some natural examples of IFS-attractors (called also fractals) occur very naturally in algebra, via certain valued fields, many of them being also important objects in model theory. The goal of the work with Kuhlmann was to find a generalization of the notion of an IFS-attractor allowing more algebraic and model-theoretic examples, but being in close analogy with the original definition.
Several such possible generalizations were introduced, and for each of them topological consequences were proven, and classes of algebraic examples were found.
On Workpackage 2, in a joint work, Dobrowolski, B. Kim, A. Kolesnikov, and J. Lee studied a notion of localized Lascar-Galois group - this is a natural analogue of the notion of Lascar group - a very important invariant of a first order theory. In the work by Dobrowolski, Kim, Kolesnikov, and Lee, the questions about some desirable properties of these groups (related also to homology groups) hold were addressed. A positive answer was given under some additional assumptions, and it was proven that none of these assumptions can be removed by constructing several counterexamples to stronger statements.
Another work within this Workpackage was done by Dobrowolski in collaboration with B. Kim and N. Ramsey on Kim-independence in NSOP1 theories. Kim-independence is a variant of forking-independence - an independence relation enjoying very nice properties in so-called simple theories, thus closely resembling linear independence in linear spaces and algebraic independence in fields. In fact, simple theories are exactly those in which forking independence behaves nicely, and NSOP1 theories as those in which Kim-independence behaves nicely - the latter was established in an earlier work by I. Kaplan and N. Ramsey. However, there was a limitation in their setting - they considered only independence over a model. In the work by Dobrowolski, Kim, and Ramsey this limitation was removed, at a cost of a very mild assumption of forking existence, which is conjectured to be true in all NSOP1 theories.
On Workpackage 3, several algebraic consequences of tameness assumptions on groups were proven. Dobrowolski in collaboration with J. Goodrick proved that left-ordered groups of inp-rank equal to one (where inp rank is a cardinal-valued rank giving a hierarchy on the class of NTP2 theories) are abelian, generalizing a previous result of P. Simon on bi-ordered groups. Secondly, it was proven by Dobrowolski and F. Wagner that omega-categorical groups of finite inp-rank are finite-by-abelian-by-finite, thus very close to being commutative. Finally, Dobrowolski, D. Hoffmann, and J. Lee, generalized a well-known theorem characterizing elementary equivalence of pseudo algebraically closed (PAC) fields in terms of isomorphisms of their absolute Galois groups to the setting of pseudo algebraically closed structures, an abstract generalization of the class of PAC (perfect) fields.
Dobrowolski has established the project website
http://www.math.uni.wroc.pl/~dobrowol/mc.html(odnośnik otworzy się w nowym oknie)where the outcomes of the project are presented. In particular, information on where to find the papers resulting from the project in open access can be found is provided there. The results were also disseminated widely through seven invited conference talks, and through several seminar talks in Leeds, Wroclaw, Jerusalem, Lyon, Paris, and Manchester.
