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A category-theoretic approach to classification theory (KIRBYJU19SF)

  • Full or part time
  • Application Deadline
    Friday, May 31, 2019
  • 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 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.

Classification theory studies when the models of a theory can be classified, and when they are too complicated to admit any classification. One of the main paradigms of the subject is that, in a classifiable theory, the way a model is built up from small models is highly structured. Recent work is extending these ideas from the original setting to a more general category-theoretic setting. This project will continue that work, extending techniques on 3-amalgamation to accessible categories.

Students are advised to contact Dr Kirby directly to discuss their application.

Type of Programme: PhD
Start date of project: October 2019
Mode of study : full time
Acceptable first degree: 1st class degree or equivalent in Mathematics or Biological Sciences.
Application deadline: 31 May 2019. Applications are processed as soon as they are received and the project may be filled before the closing date, so early application is encouraged.

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. 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.

References

i) Jonathan Kirby, On quasiminimal excellent classes, J. Symbolic Logic (2010)
ii) Saharon Shelah, Classification Theory, 2nd ed, North Holland, (1990)
iii) Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kesala, and Jonathan Kirby, Quasiminimal structures and excellence, Bull. Lond. Math. Soc. (2014)
iv) Michael Lieberman, Jirí Rosický, Classification theory for accessible categories, J. Symbolic Logic (2016)
v) Saharon Shelah, Sebastien Vasey, Categoricity and multidimensional diagrams, arXiv:1805.06291, (2018)

Related Subjects

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