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Model theory: definable sets as accessible functors (KIRBYJ2U20SF)

  • Full or part time
  • Application Deadline
    Sunday, May 31, 2020
  • Self-Funded PhD Students Only
    Self-Funded PhD Students Only

Project Description

Model theory is a branch of mathematical logic which considers the category of all models of a given theory, for example all groups, or all algebraically-closed fields. One feature of the models is that they can always be built up from small models by a process called amalgamation. This is due to the downward Lowenheim-Skolem theorem.

Another approach to models due to Fraisse is to start with a category of small models (usually finite) and consider what models you can build out of them, instead of starting with a theory. Accessible categories are a different approach to the same ideas which are more general, but recent work of Lieberman, Boney, Rosicky and others has shown that the two subjects have much to offer each other.

Given a model M, for example a field, a mathematical formula such as
“there exists an x such that x^2 = y”
defines the subset of M consisting of all the y’s for which the formula is true. This relates mathematical formulas to so-called definable sets. We can view this assignment as a functor from the category of models M to the category of Sets (in fact, an accessible functor). This PhD project will study definable sets from this new perspective, as accessible functors, and look at generalisations to positive and continuous logic, which are increasing central in model theory and its applications.

Students should have some knowledge of model theory and/or category theory and are advised to contact Dr Kirby directly to discuss their application.

For more information on the project’s supervisor, please visit: https://people.uea.ac.uk/jonathan_kirby
Type of programme: PhD
Start date of project: October 2020
Mode of study: full time or part time
Studentship length: 3 years.(3 year studentships have a (non-funded) 1 year ‘registration only’ period).
Location: UEA
Entry requirements:
a) acceptable first degree in: Mathematics
b) 1st class BSc or 2:1 Masters or equivalent.

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 View Website.

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.

References

i) Sebastien Vasey, Accessible categories, set theory, and model theory: an invitation, arXiv:1904.11307
ii) Ehud Hrushovski, Definability patterns and their symmetries, arXiv:1911.01129
iii) Michael Lieberman, Jirí Rosický, Classification theory for accessible categories, J. Symbolic Logic (2016)

Related Subjects

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