PhD Studentship: Ranks and Types in Positive Logic

Overview

This PhD project at the University of East Anglia, supervised by Dr Jonathan Kirby, studies ranks and types in positive logic. Model theory is traditionally done with classical first-order logic, which allows unlimited use of AND, OR, NOT and the EXISTS and FOR ALL quantifiers. Positive logic is a more recent generalisation in which the use of NOT and FOR ALL can be restricted to varying degrees, and it suits some applications in algebra, such as modules, and theories without quantifier elimination.

Theories in classical logic are classified by combinatorial complexity through dividing lines, many due to Shelah, and many of these relate to ordinal-valued ranks such as Morley rank, U-rank and dp-rank, which describe the complexity of a formula or type by how it relates to other formulas within a theory. Some of the dividing lines have been extended to positive logic, but ranks have not yet been studied there. The project will develop suitable analogues of Morley rank, U-rank and dp-rank in positive theories and explore new applications of the results.

Requirements

The minimum entry requirement is a first class BSc or a 2:1 Masters, or equivalent, in mathematics. Students should know some mathematical logic and preferably model theory, and are advised to contact the supervisor directly to discuss their application.

What you get

The project is offered on a self-funded basis, open to applicants who are self-funded or are securing external funding. A bench fee is payable in addition to the tuition fee, and details are available from the supervisor. UEA alumni may be eligible for a tuition fee discount.

Key dates

Study can be full or part time, starting on 1 February 2027. The closing date is 30 November 2026.