Model Theory and Differential and Difference Equations

在本项目中，研究者将利用数理逻辑的一个分支模型论，对一些经典的微分方程和差分方程进行分类研究。模型论是数理逻辑的一个分支，研究数学结构及其可定义集合（即，“生活”在这些结构中的基本集合）。另一方面，微分方程和差分方程是描述事物如何随时间变化的方程，是现代科学和工程用来模拟物理现实的基本工具。该项目的基本思想是将微分或差分方程的解集视为适当结构中的可定义集，并使用模型理论的工具来理解其属性。该项目的重点方程-Painlevé微分和差分方程和Schwarzian微分方程-出现在许多重要的物理应用中，如统计力学和广义相对论，以及数论中的重要问题。该项目建立在模型理论家最近的成功，使用几何稳定性理论的技术来分类这些微分方程的特殊情况下，根据在特征零的微分封闭领域中的微分定理。调查员将延长他的工作对通用Painlevé（微分）方程的非通用的，以及Schwarzian方程。我们的目标是充分描述的结构的解决方案，并使用这样的描述，以更好地理解几何平凡集的理论。这项工作预计将有应用到数论和功能超越理论的问题。该项目还将传输一些用于研究经典微分方程的思想和技术的差异设置。主要的挑战将是发展一个概念的不可约（在意义上的经典功能）的差分方程，并表明，差分Painlevé方程是不可约的。除了模型理论，这项工作还将采用代数和几何技术，并从微分/差分方程的分析研究。

In this project, the investigator will use model theory, a branch of mathematical logic, to classify and study some classical differential and difference equations. Model theory is a branch of mathematical logic that studies mathematical structures and their definable sets (i.e., the basic sets that "live" in those structures). On the other hand, differential and difference equations are equations that describe how things change with time and are the fundamental tools that modern science and engineering use to model physical reality. The basic idea of the project will be to view the set of solutions of a differential or difference equation as a definable set in an appropriate structure, and use the tools of model theory to understand its properties. The equations that the project focuses on -- the Painlevé differential and difference equations and the Schwarzian differential equations -- appear in many important physical applications such as statistical mechanics and general relativity, as well as in important problems in number theory. The project builds on recent successes by model theorists to use techniques from geometric stability theory to classify special cases of these differential equations according to the trichotomy theorem in differentially closed fields of characteristic zero. The investigator will extend his work on the generic Painlevé (differential) equations to non-generic ones as well as to the Schwarzian equations. The objective is to both fully describe the structure of the sets of solutions and to use such descriptions to better our understanding of geometrically trivial sets in the theory. This work is expected to have applications to problems in number theory and functional transcendence theory. The project will also transport some of the ideas and techniques used to study classical differential equations to the difference setting. The main challenge will be to develop a notion of irreducibility (in the sense of classical functions) for difference equation and show that the difference Painlevé equations are irreducible. In addition to model theory, the work will also employ techniques from algebra and geometry and from the analytic study of differential/difference equations.