Model theory and categories (KIRBYU16SF)
Model theory is a branch of mathematical logic which considers the category of all models of a given theory. 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 idea 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.
Categories also arise in other ways in model theory, for example apart from the category of models of a theory, the category of definable sets also plays a large role. Category theory is now the language of much of modern mathematics, but model theory has been relatively slow to adopt it. In this project we will recast some of the main ideas of model theory in category-theoretic terms and then use the new language to push some model-theoretic ideas to new examples, not easily within reach of standard methods. Applicants should have some knowledge of category theory and at least one of mathematical logic, model theory and algebraic geometry. The PhD project can be tailored to suit the application, so please contact Dr Kirby directly to discuss your application.
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 http://bit.ly/1Jf7KCr
A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. The amount charged annually will vary considerably depending on the nature of the project and applicants should contact the primary supervisor for further information about the fee associated with the project.
i) Jonathan Kirby, On quasiminimal excellent classes, J. Symbolic Logic 75 (2010), no. 2, 551–564.
ii) Anand Pillay, Geometric Stability Theory, OUP, 1996
iii) Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kesala, and Jonathan Kirby, Quasiminimal structures and excellence, Bull. Lond. Math. Soc. 46 (2014), no. 1, 155–163.
iv) Michael Lieberman, Jirí Rosický, Classification theory for accessible categories, arXiv:1404.2528