TOPMODTHE · Topological Model Theory
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2006-04-01 → 2007-03-31
- EU contribution
- €40,000
- Participants
- 1
- Scheme
- ERG
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Results in brief
Final Activity Report Summary - TOPMODTHE (Topological Model Theory)
The project was divided into three parts: (a) investigation of definable and type definable groups in structures with various minimality conditions; (b) studying the structure of countable models with minimality conditions; (c) abstract approach to first order topological structures with a special attention to the trichotomy results. During the European Reintegration Grant Roman Wencel worked mainly on (a) and (c). In part (a) R. Wencel worked on generalisation of methods which appeared in the proof of Pillay's group conjecture. He developed forking theory in the context of weakly o-minimal non-valuational expansions of real closed fields. He also worked on development of an analogue of homology theory in the context of weekly o-minimal non-valuational expansions of real closed fields. R. Wencel's research in part (b) concerned weak orthogonality of types in weekly o-minimal structures with the strong cell decomposition property. He showed that non-orthogonal types in theories of weakly o-minimal non-valuational structures with the strong cell decomposition property behave as in the o-minimal setting. As far as (c) is concerned, he found a weak version of Peterzil-Starchenko trichotomy theorem in case of weakly o-minimal structures satisfying the strong cell decomposition property. He did not succeed to prove an analogue of trichotomy theorem in the abstract framework of first order topological structures. He proved a version of Pillay's topologisation theorem for groups definable in first order topological structures with dimension function satisfying certain reasonable axioms.
Data: CORDIS, © European Union
Project objective
The proposed research principally belongs to model theory, but it is also linked to algebra and combinatorial geometry. The aim of the project is to study first order topological structures, both from specific and abstract point of view. The class of first order topological structures generalises classes of models satisfying various minimality conditions, a subject of my initial MC Fellowships at the University of Leeds.This project is a natural continuation of my research concerning minimality conditions in model theory. Firstly, I am planning to continue the study of definable and type-definable groups in structures with various minimality conditions, mainly to generalise the recent positive solution of Pillay's group conjecture. Second objective of the project is investigation of countable models with small theories, satisfying various minimality conditions (like weak o-minimality, C-minimality, P-minimality) and to make attempts towards proving Vaught's conjecture for weakly o-minimal theories (and possibly other classes of theories with some minimality conditions). Finally, I am going to develop an abstract (axiomatic) approach to first order topological structures.This will be mainly inspired by the work of Hrushovski and Zilber on Zariski-type structures. Realisation of this project will be an excellent opportunity to combine the experience I gained at the University of Leeds (minimality conditions) with the expertise of the Wroclaw model theory team (definable and type-definable groups, Vaught's conjecture). It will contribute to transferring methods from stability theory to unstable contexts, and to the classification of models of first order theories without the independence property.
Original text from CORDIS.
Participants
- WROCLAW UNIVERSITY · WROCLAWCoordinatorCity levelPoland
Links
Data: CORDIS, © European Union