PhD Studentship: Model Theory of Complex Powers

Project Supervisor - Dr Jonathan Kirby

The field of complex numbers is in many ways the most elegant and important example of a mathematical structure in model theory. It has been the prototype for many developments in model theory, starting with strong minimality, as well as providing some of the most important applications where model-theoretic tools are applied to algebraic geometry. 

After the field operations of addition and multiplication, raising to a fixed power is the most basic and universal operation in mathematics. However, the model theory of the complex field with fixed (irrational) powers is only just starting to be understood, following on from seminal work of Zilber which showed that when we restrict to a sufficiently generic power we get a superstable first-order theory. More recent work of Gallinaro and Kirby shows that the complex field with all complex powers is quasiminimal – the definable subsets are countable or co-countable – so in particular the real field is not definable. 

This project will investigate both the algebra of these powered fields and the model theory, aiming to give a description of all the definable sets, and of their geometry. This geometry is intermediate between the algebraic geometry of algebraic varieties (which deals with polynomial equations only) and analytic geometry (which allows more general analytic functions). 

Applicants should have some knowledge of mathematical logic or Galois theory or algebraic geometry. Knowledge of model theory would be an advantage but is not essential. They are advised to contact Dr Kirby directly to discuss their application. 

Entry Requirements

Acceptable first degree - Mathematics.

1st class BSc or 2:1 Masters or equivalent.

Start Date: 1 October 2026 

Additional Funding Information

This PhD project is in a competition for a Faculty of Science funded studentship. Funding is available to UK applicants and comprises ‘home’ tuition fees and an annual stipend for 3 years.

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