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.
(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.