Model theory and quasiminimality for analytic functions

解析函数的模型理论在过去20年左右的时间里一直是数理逻辑的一个主要研究领域，随着与数论等其他数学分支的联系越来越多，它的重要性也越来越大。然而，一些基本问题仍有待解决。其中最主要的问题是，哪些函数足够“驯服”，可以用模型理论方法来研究。中心的例子是复指数函数，它在纯数学和应用数学中无处不在，因为它既包含了模拟指数增长和衰减的真实指数函数，也包含了所有周期函数和波函数所基于的正弦函数。这是我目前EPSRC资助的重点。这个项目将研究其他更容易证明驯服性（特别是准最小性）的函数，使其在博士项目的范围内。学生将从学习一些已经用于其他函数的方法开始，但预计将需要一些新的想法（取决于所选择的函数），这些新想法也应该反馈并在更广泛的背景下有用。因此，学生应该迅速进入更广泛的研究社区，这对职业发展非常重要。主要目标：1)对于一个或多个合适的选择函数，给出描述所有相关函数方程的公理。2)用这些来解释有解的方程组的必要限制。3)证明任何满足这些限制的系统确实有解，并推导出拟极小性。

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.