WILDMOD: Model Theory of wild mathematical structures, new perspectives via geometries and positive logic.

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This project will investigate wild algebraic structures through two overlapping perspectives in model theory: a geometric approach and a positive logic approach. Model theory is a branch of mathematical logic that deals with definable sets (a generalisation of the notion of algebraic variety). It identifies deep notions of tameness which draw dividing lines among mathematical structures (e.g. groups, fields, graphs...). Tame structures (for instance stable structures) are the terrain of applications and connections between model theory and other areas (e.g. algebraic and diophantine geometry). This project will push the boundaries of tameness to structures previously considered wild but ripe for model-theoretic treatment. In Workpackage 1, we describe a geometric approach for wild structures which includes real-closed fields and algebraically closed valued fields expanded by generic subgroups andgeneric homomorphisms, as well as the ring of algebraic integers. In Workpackage 2 we describe how recent rekindlement in positive model theory shall be used to work with the classes of generic difference pseudo-finite/real-closed fields, filling a gap in the model theory of difference fields.D'Elbée - the Fellow - will receive training in the model theory group in Leeds and during a 3-month secondment in Munster. The transfer of knowledge will be two-way; d'Elbée will benefit from broad expertise in Leeds (his supervisor Macpherson, also Mantova, Eleftheriou and Brooke-Taylor) and in Munster (his supervisor Hils, also Tent and Jahnke), and will transfer to the Leeds and Munster teams his knowledge of wild expansions of fields and of positive logic (through study groups, seminars and discussions). The project incorporates outreach activities for all ages at MathCity Leeds, public talks for a general audience, training in PhD supervision, a website for the research project, and a final 3-day workshop on the research themes of his project.