University of Leeds Featured PhD Programmes
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King’s College London Featured PhD Programmes

Model theory and quasiminimality for analytic functions (KIRBYJ1U20SF)

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

Project Description

This PhD project is based on the recent exciting developments in the application of model theory, a branch of mathematical logic, to analytic functions such as exponentiation. These include Wilkie’s proof that the real exponential function has a tame (so-called o-minimal) geometry, and the programme started by Zilber studying the complex exponential function by algebraic / model-theoretic means. There are also exciting relations to number theory, particularly transcendence theory (for example proofs of functional transcendence theorems by Kirby, Kowalski, Pila and others), and to Diophantine geometry, for example the formulation of the Zilber-Pink conjecture, and recent progress on it.

On the model-theoretic side, there have been new developments in abstract stability theory developing the tools such as quasiminimality, used particularly for studying the complex exponential. Dr Kirby is at the forefront of several of these developments.

This project will take some functions arising in complex or p-adic analysis, such as correspondences between elliptic curves or the Iwasawa logarithm, and attempt to show that their logical theory is quasiminimal: that every definable set is either countable or the complement of a countable set. Methods likely to be useful arise from model theory, topology, algebraic geometry, and real, complex and p-adic analysis. Candidates should have knowledge of some of these areas and are advised to contact Dr Kirby 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 / 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) standard minimum entry requirement is 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) Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kesala, and Jonathan Kirby, Quasiminimal structures and excellence, Bull. Lond. Math. Soc. (2014).
ii) Martin Bays and Jonathan Kirby, Pseudo-exponential maps, variants, and quasiminimality, Algebra and Number Theory (2018).
iii) Jonathan Kirby, Angus Macintyre, and Alf Onshuus, The algebraic numbers definable in various exponential fields, J. Inst. Math. Jussieu (2012).
iv) Jonathan Kirby and Boris Zilber, Exponentially closed fields and the conjecture on intersections with tori, Annals of Pure and Applied Logic (2014).
v) David Marker, A remark on Zilber’s pseudoexponentiation, J. Symbolic Logic (2006).

Related Subjects

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