Model theory and quasiminimality for analytic functions

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The model theory of analytic functions has been a major research area within mathematical logic for the last 20 or so years, growing in importance as more connections are made with other branches of mathematics including number theory. However, some fundamental questions are still wide open. Chief amongst these is the question of which functions are "tame" enough to be studied with model-theoretic methods. The central example is the complex exponential function, which is ubiquitous in pure and applied mathematics, as it incorporates both the real exponential function which models exponential growth and decay, and the sine function upon which all periodic and wave functions are based. That is the focus of my current EPSRC grant.This project will look at other functions for which it should be easier to prove tameness (specifically quasiminimality), bringing it within reach of a PhD project. The student will start by learning some methods which have already been used for other functions, but it is expected that some new ideas will be needed (depending on the functions chosen), and these new ideas should also feed back and be useful in a wider context. So the student should quickly be brought into the wider research community, which is very important for career progress.Major Aims: 1) For one or more suitable chosen functions, give axioms describing all relevant functional equations. 2) Use these to explain the necessary restrictions of systems of equations having solutions.3) Prove that any system which satisfies these restrictions does indeed have solutions, and deduce quasiminimality.