Model theory and quasiminimality for analytic functions

解析函数的模型理论和拟极小性

• 批准号: 2602989
• 金额: --
• 依托单位: University of East Anglia
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2602989
• 关键词: Model; theory; quasiminimality; analytic; functions

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.

在过去的20年左右的时间里，分析功能的模型理论一直是数学逻辑中的主要研究领域，随着与数学理论在内的其他数学分支之间建立更多的联系，重要性越来越重要。但是，某些基本问题仍然很开。其中最主要的是哪些功能“驯服”足以通过模型理论方法进行研究。中心示例是复杂的指数函数，它在纯和应用数学中无处不在，因为它既包含了模拟指数生长和衰减的真实指数函数，又结合了所有周期性和波浪函数所基于的正弦函数。这是我当前的EPSRC授予的重点。此项目将研究其他功能，这些功能应该更容易证明可驯服（特别是准临时性），从而使其在博士学位项目的范围内。学生将开始学习一些已经用于其他功能的方法，但是可以预期需要一些新想法（取决于所选的功能），这些新想法也应该反馈并在更广泛的背景下有用。因此，应该快速将学生带入更广泛的研究社区，这对职业进步非常重要。Major的目的：1）对于一个或多个合适的选定功能，可以使公理描述所有相关的功能方程。 2）使用这些来解释具有解决方案的方程式系统的必要限制。3）证明，满足这些限制的任何系统确实具有解决方案，并推断出准时性。