Model theory: classification theory in positive logic (KIRBYJ_U21SF)

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. More recently, positive logic has emerged as a useful generalisation. 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 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.  

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. This PhD project will aim to extend other dividing lines such as NIP, and will explore new applications of the results obtained.  

Students should have some knowledge of mathematical logic and preferably also model theory, and are advised to contact Dr Kirby directly to discuss their application. 

It may be possible to undertake this PhD project on a part time basis but applicants should discuss with Dr Kirby in the first instance 

References:

1. Gabriel Conant: http://forkinganddividing.com. 

2. Levon Haykazyan, Spaces of Types in Positive Model Theory, J. Symbolic Logic (2019). 

3. Levon Haykazyan, Jonathan Kirby, Existentially Closed Exponential Fields, to appear in the Israel Journal of Mathematics, arXiv:1812.08271 (2018). 

4. Ehud Hrushovski, Definability patterns and their symmetries, arXiv:1911.01129 

Funding Notes:

This PhD project is offered on a self-funding basis.  It is open to applicants with funding or those applying to funding sources.  Details of tuition fees can be found at https://www.uea.ac.uk/about/university-information/finance-and-procurement/finance-information-for-students/tuition-fees 

A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research.  Applicants should contact the primary supervisor for further information about the fee associated with the project.