Model theory is traditionally done with “classical first-order logic”, the logic which allows unlimited use of the operators AND, OR, NOT, with the EXISTS and FOR ALL quantifiers. There is also a category-theoretic approach to logic, which is more general but less deep. More recently, positive logic has emerged as a useful generalisation of classical logic, taking some ideas from category theory. For a theory in positive logic, one can specific how much you are allowed to use the NOT operator: either without restriction (to get the classical case) or not at all, or much less. Positive logic is more appropriate to use directly for some applications in algebra, such as for modules, and for theories which do not admit quantifier elimination. Some constructions in classical logic, such as “hyperimaginaries” can be understood much more readily in positive logic.
Theories in classical logic can be classified according to their combinatorial complexity, via a number of dividing lines which are mostly due to Shelah. A map of this classification appears at http://forkinganddividing.com. We know how to extend some of these dividing lines to positive logic, including stability and simplicity, but not all.
This PhD project will aim to extend other dividing lines such as o-minimality, superstability, or NIP, and will explore new applications of the results obtained.
Students should have some knowledge of mathematical logic and preferably also model theory or categorical logic, and are advised to contact Dr Kirby directly to discuss their application.