The 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 and currently has an EPSRC-funded project in this area.
In this project, you 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.
When this can be proved, systems of equations involving this function will have solutions which can be understood geometrically, in a similar way to algebraic geometry, which applies to polynomial functions.
Methods likely to be useful arise from model theory, topology, algebraic geometry, and real, complex and p-adic analysis.
Candidates should have knowledge of at least one or two of these areas and are advised to contact Dr Kirby to discuss their application.
Funding Notes :
Successful candidates who meet UKRI’s eligibility criteria will be awarded an EPSRC funded studentship covering fees, stipend (£15,285 pa, 2020-21), and research funding for years.
International applicants (EU / non-EU) are eligible for fully-funded EPSRC studentships. The eligibility requirements are detailed in the UKRI Training Grant Guidance :
Applicants to this project will also be considered for a 3 year UEA funded studentship covering stipend (£15,285 pa, 2020-21), tuition fees (Home only) and research costs.
International applicants (EU / non-EU) are eligible for UEA funded studentships but they are required to fund the difference between Home and International tuition fees (see