Project description
Definable and pseudofinite groups
The EU-funded DEFGROUP project is exploring logic (model theory) and algebra (group theory), by studying groups known as 'first-order definable' in structures that satisfy certain model-theoretic restrictions. The project is working to show that groups definable in tame fields are closely related to the rational points of an algebraic group. The project is also working to prove that the soluble radical of any pseudofinite group with a simple theory is soluble. The project also aims to demonstrate that any infinite group definable in a pseudofinite field (or in any pseudofinite structure) is pseudofinite.
Objective
The proposed research links logic (model theory) and algebra (group theory). It explores groups which are first-order definable in structures that satisfy certain model-theoretic restrictions. The structures, here called `tame’, are stable, simple, NIP, or NTP2, concepts from Shelah's `generalised stability theory'. WP 1 concerns groups definable in tame fields, possibly equipped with extra operators. The aim is to show that a definable group in such fields must be closely related to the rational points of an algebraic group and to investigate the structure fixed pointwise by a generic automorphism in a generic differential difference field. WP 2 revolves around pseudofinite groups (infinite groups satisfying every sentence true of all finite groups), and gives a model theoretic perspective on finite group theory. One goal is to prove that the soluble radical of any pseudofinite group with a simple theory is soluble. Another is to solve the following question, possibly with a pseudofinite counterexample: given a `tame’ group G and a soluble subgroup H of G, is there a definable soluble subgroup of G containing H? The final WP2 objective concerns primitive pseudofinite permutation groups and in particular the question whether elementary extensions preserve primitivity. The two Workpackages are well-linked: for example, any infinite group definable in a pseudofinite field (or in any pseudofinite structure) is pseudofinite. The Fellow, Hempel, will receive training through research in Leeds, from the supervisor Macpherson and from the wider model theory group. She will benefit from the background of Macpherson on pseudofinite groups and permutation groups, and the broader experience of other Leeds model theorists. She will transfer to Leeds specific expertise on conditions such as NTP2, and on tame groups. She will receive training and opportunities complementary to her previous experience, on topics such as outreach, project management, and PhD student support.
Fields of science
Programme(s)
Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
LS2 9JT Leeds
United Kingdom